# LPV H-infinity Control for the Longitudinal Dynamics of a Flexible Air-Breathing Hypersonic Vehicle

код для вставкиСкачатьABSTRACT HUGHES, HUNTER DOUGLAS. LPV H∞ Control for the Longitudinal Dynamics of a Flexible Air-Breathing Hypersonic Vehicle. (Under the direction of Dr. Fen Wu.) This dissertation establishes the method needed to synthesize and simulate an H∞ Linear Parameter-Varying (LPV) controller for a flexible air-breathing hypersonic vehicle model. A study was conducted to gain the understanding of the elastic effects on the open loop system. It was determined that three modes of vibration would be suitable for the hypersonic vehicle model. It was also discovered from the open loop study that there is strong coupling in the hypersonic vehicle states, especially between the angle of attack, pitch rate, pitch attitude, and the flexible modes of the vehicle. This dissertation outlines the procedure for synthesizing a full state feedback H∞ LPV controller for the hypersonic vehicle. The full state feedback study looked at both velocity and altitude tracking for the flexible vehicle. A parametric study was conducted on each of these controllers to see the effects of changing the number of gridding points in the parameter space and changing the parameter variation rate limits in the system on the robust performance of the controller. As a result of the parametric study, a 7 × 7 grid ranging from Mach 7 to Mach 9 in velocity and from 70,000 feet to 90,000 feet in altitude, and a parameter variation rate limit of [.5 200]T was used for both the velocity tracking and altitude tracking cases. The resulting H∞ robust performances were γ = 2.2224 for the velocity tracking case and γ = 1.7582 for the altitude tracking case. A linear analysis was then conducted on five different selected trim points from the H∞ LPV controller. This was conducted for the velocity tracking and altitude tracking cases. The results of linear analysis show that there is a slight difference in the response of the H∞ LPV controller and the fixed point H∞ controller. For the tracking task, the H∞ controller responds more quickly, and has a lower H∞ performance value. Next, the H∞ LPV controller was simulated using the nonlinear flexible hypersonic model for both the velocity tracking and altitude tracking cases. Both of these cases were subject to a ramp input and a multi-step input both with and without perturbation in the model. The results of the simulation show that the tracking state follows the command signal successfully though the perturbed system does show some higher frequency characteristics in the non-tracking states. It was discovered that there is an issue with integral windup when switching takes place in the controller, so an algorithm was implemented to reset the integration of the error on the tracking state when the switch takes place. It was also seen that there was a decline in altitude when tracking velocity, and a large change in velocity that occurred during altitude tracking. These results lead to the decision to include a unity gain regulation state on velocity for the altitude tracking and the altitude for the velocity tracking during the output feedback control synthesis. The procedure for synthesizing an output feedback H∞ LPV controller for the hypersonic vehicle is also discussed in this dissertation. The output feedback design looked at velocity tracking and altitude tracking with rigid body motion variables for both the flexible and rigid body hypersonic vehicle models. As with the full state feedback controller, a parametric study was conducted on each of these controllers to determine the number of gridding points in the parameter space and the parameter variation rate limits in the system. The parametric study reveals a 7×7 grid ranging from Mach 7 to Mach 9 in velocity and from 70,000 feet to 90,000 feet in altitude, and a parameter variation rate limit of [.1 200]T is preferable for both the velocity tracking and altitude tracking cases with both the flexible and rigid body assumptions. The resulting H∞ robust performances were γ = 113.2146 for the flexible body velocity tracking case, γ = 83.6931 for the rigid body velocity tracking case, γ = 107.2043 for the flexible body altitude tracking case, and γ = 97.7403 for the rigid body altitude tracking case. A linear analysis was then conducted on five different selected trim points from the H∞ LPV controller. The results of this analysis show that there is a larger difference in the response of the H∞ LPV controller and the H∞ controller. For the tracking task, the H∞ controller responds more quickly, and has a lower H∞ performance value. Next, the H∞ LPV controller was applied to the flexible nonlinear plant model. The rigid body controllers were applied to the flexible plant model to see if the flexible nature of the vehicle could be treated as a perturbation to the system. Additionally, there were simulations run both with and without sensor noise and parametric uncertainty. The results of simulation show that the rigid body controller is able to successfully apply to the flexible body model for the velocity tracking case, but is unable to stabilize the altitude tracking case. It was also seen that the system is able to track the command signal while minimizing the variations seen in the altitude for the velocity tracking case and in the velocity during the altitude tracking case. Additionally, there was no obvious effect of perturbations in the system on the tracking state or secondary regulation state. There were high frequency responses associated with the other perturbed states. c Copyright 2010 by Hunter Douglas Hughes All Rights Reserved LPV H∞ Control for the Longitudinal Dynamics of a Flexible Air-Breathing Hypersonic Vehicle by Hunter Douglas Hughes A dissertation submitted to the Graduate Faculty of North Carolina State University in partial fulfillment of the requirements for the Degree of Doctor of Philosophy Mechanical Engineering Raleigh, North Carolina 2010 APPROVED BY: Dr. Paul Ro Dr. Larry Silverberg Dr. Gregg Buckner Dr. Fen Wu Chair of Advisory Committee UMI Number: 3442646 All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is dependent upon the quality of the copy submitted. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion. UMI 3442646 Copyright 2011 by ProQuest LLC. All rights reserved. This edition of the work is protected against unauthorized copying under Title 17, United States Code. ProQuest LLC 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, MI 48106-1346 DEDICATION I would like to dedicate this work to my parents for their ongoing love and support. This work would not have been possible without their emotional and financial support. I would also like to dedicate this work to my beautiful fiancée Audrey who has helped to take care of me and keep me sane throughout my course of study. ii BIOGRAPHY The author was born in Chattanooga, Tennessee. He attended Tennessee Technological University and received a Bachelor of Science degree in mechanical engineering. Upon graduation from Tennessee Technological University, he moved to Raleigh, North Carolina where he attended North Carolina State University. In 2006, the author received a Master of Science degree in mechanical engineering. Be duly motivated, the author continued to work at North Carolina State University to complete this dissertation in preparation for the doctoral degree requirements in mechanical engineering. iii ACKNOWLEDGEMENTS I would like to thank my committee for their guidance and help with my research. Without their instruction, none of this would have been possible. I would also like to thank my advisor, Dr. Fen Wu, for his commitment and dedication to my research. His patience, knowledge, and experience has been a crucial part of my research. I would also like to take this opportunity to thank Michael Bolender and David Doman from the Air Force Research Lab for providing the code and model for the hypersonic vehicle. I also would like to take the time to thank the mechanical and aerospace engineering department at North Carolina State University as well as the state of North Carolina and its taxpayers for the financial support that I have recieved. Also I would like to give a special thanks to the Leon family for the fellowship that I received from their endowment. Also, I would like to thank the North Carolina Space Grant for their funding of my research. Last but certainly not least, I would like to thank Mark Osborne, Scott Hays, and Xeujing Cai from Dr. Wu’s research group for their help and collaboration efforts in regards to this research. iv TABLE OF CONTENTS List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi Chapter 1 Introduction . . . . . . 1.1 Hypersonic Vehicle Controls . 1.2 H∞ LPV Control Techniques 1.3 Dissertation Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 2 Hypersonic Vehicle Model . . . . . . . . . 2.1 Flexible Aircraft Model . . . . . . . . . . . . . . . 2.2 Equations of Motion . . . . . . . . . . . . . . . . . 2.2.1 Hypersonic Vehicle Free Body Diagram and 2.2.2 Hypersonic Vehicle Aerodynamics . . . . . 2.2.3 Actuator Dynamics . . . . . . . . . . . . . . 2.3 Model Linearization . . . . . . . . . . . . . . . . . 2.4 Open Loop Analysis . . . . . . . . . . . . . . . . . 2.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 . 1 . 10 . 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Force Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 19 23 24 27 35 36 39 44 47 47 48 57 63 63 64 72 80 80 82 83 87 87 99 111 Chapter 3 Full State Feedback Control for Hypersonic Vehicle 3.1 Control Synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Velocity Tracking . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Altitude Tracking . . . . . . . . . . . . . . . . . . . . . 3.1.3 Summary of Control Synthesis . . . . . . . . . . . . . . 3.2 Linear Control Analysis . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Velocity Tracking . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Altitude Tracking . . . . . . . . . . . . . . . . . . . . . 3.3 Nonlinear HSV Analysis and LPV Control Implementation . . 3.3.1 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Robustness Analysis . . . . . . . . . . . . . . . . . . . . 3.3.3 LPV Control Switching Algorithm . . . . . . . . . . . . 3.4 Nonlinear Simulation Results . . . . . . . . . . . . . . . . . . . 3.4.1 Velocity Tracking . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Altitude Tracking . . . . . . . . . . . . . . . . . . . . . 3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 4 Output Feedback Control 4.1 Control Synthesis . . . . . . . . . 4.1.1 Velocity Tracking . . . . . 4.1.2 Altitude Tracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 . 114 . 115 . 126 for Hypersonic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v Vehicle . . . . . . . . . . . . . . . . . . . . . . 4.2 4.3 4.4 4.5 4.1.3 Summary of Control Synthesis . . Linear Control Analysis . . . . . . . . . . 4.2.1 Velocity Tracking . . . . . . . . . . 4.2.2 Altitude Tracking . . . . . . . . . Nonlinear HSV Analysis and LPV Control 4.3.1 Setup . . . . . . . . . . . . . . . . 4.3.2 Robustness Analysis . . . . . . . . 4.3.3 LPV Control Switching Algorithm Nonlinear Simulation Results . . . . . . . 4.4.1 Unstable Linear Control . . . . . . 4.4.2 Stable Velocity Tracking . . . . . . 4.4.3 Stable Altitude Tracking . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 136 136 144 152 152 154 156 158 159 167 187 199 Chapter 5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 5.1 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 5.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 Appendix A Controller Reference Number Table For Full State Feedback . . . . . . . 213 vi LIST OF TABLES Table Table Table Table Table 2.1 2.2 2.3 2.4 2.5 LH2 fuel . . . . . . . . . . . . . . . . . . . Actuator Saturation Limits . . . . . . . . . Natural Frequencies for Hypersonic Vehicle Flexible Open Loop Eigenvalues . . . . . . Rigid Open Loop Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 37 39 41 41 Table Table Table Table Table Table Table Table Table 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 γ performance for different parameter variation rates . . . . . . . . . . . γ performance for different number of griding points . . . . . . . . . . . . Closed Loop Eigenvalues for Selected Velocity Tracking Trim Conditions γ performance for different parameter variation rates . . . . . . . . . . . γ performance for different number of griding points . . . . . . . . . . . . Closed Loop Eigenvalues for Selected Altitude Tracking Trim Conditions H∞ γ Performance Values . . . . . . . . . . . . . . . . . . . . . . . . . . H∞ γ Performance Values . . . . . . . . . . . . . . . . . . . . . . . . . . H∞ γ Performance Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 54 56 60 60 62 64 65 73 Table 4.1 γ performance for different parameter variation rates . . . . . . . . . . . Table 4.2 γ performance for different number of gridding points . . . . . . . . . . . Table 4.3 Closed Loop Eigenvalues for Selected Flexible Body Velocity Tracking Trim Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Table 4.4 Closed Loop Eigenvalues for Selected Rigid Body Velocity Tracking Trim Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Table 4.5 γ performance for different parameter variation rates . . . . . . . . . . . Table 4.6 γ performance for different number of gridding points . . . . . . . . . . . Table 4.7 Closed Loop Eigenvalues for Selected Flexible Body Altitude Tracking Trim Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Table 4.8 Closed Loop Eigenvalues for Selected Rigid Body Altitude Tracking Trim Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Table 4.9 H∞ γ Performance Values . . . . . . . . . . . . . . . . . . . . . . . . . . Table 4.10 H∞ γ Performance Values . . . . . . . . . . . . . . . . . . . . . . . . . . Table 4.11 H∞ γ Performance Values . . . . . . . . . . . . . . . . . . . . . . . . . . Table 4.12 Sensor noise variance and seed values . . . . . . . . . . . . . . . . . . . . . 121 . 122 . 124 . 125 . 130 . 131 . 132 . . . . . 133 135 137 145 155 Table A.1 Controller Reference Numbers . . . . . . . . . . . . . . . . . . . . . . . . . 214 vii LIST OF FIGURES Figure Figure Figure Figure 1.1 1.2 1.3 1.4 Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 Figure 3.1 Figure 3.2 Figure 3.3 Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12 3.13 3.14 3.15 3.16 3.17 3.18 3.19 3.20 Figure 3.21 Figure 3.22 Figure 3.23 Hypersonic Vehicle Isometric View . . . Hypersonic Vehicle Side View . . . . . Block Diagram of Uncontrolled System Block Diagram of Controlled System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 . 2 . 11 . 13 Hypersonic Vehicle Free Body Diagram . . . Hypersonic Vehicle Geometry [8, 7] . . . . . . Scramjet Cross Section [8, 7] . . . . . . . . . Bode plot of δe to Vt for the open loop plant Bode plot of φ to Vt for the open loop plant . Bode plot of Ad to Vt for the open loop plant Bode plot of δe to h for the open loop plant . Bode plot of φ to h for the open loop plant . Bode plot of Ad to h for the open loop plant Hypersonic Vehicle Mode Shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Parameterized Space with Linearized Hypersonic Vehicle Grid . . . . . . Open Loop Interconnected System For Velocity Tracking . . . . . . . . Parameterized Space with Linearized Controller Grid for Different Cases (Note Larger Blocks Are Inclusive of Smaller Blocks) . . . . . . . . . . . Open Loop Interconnected System For Altitude Tracking . . . . . . . . Velocity Tracking Step Response . . . . . . . . . . . . . . . . . . . . . . Velocity Tracking Step Response . . . . . . . . . . . . . . . . . . . . . . Velocity Tracking Step Response . . . . . . . . . . . . . . . . . . . . . . Velocity Tracking Step Response . . . . . . . . . . . . . . . . . . . . . . Velocity Tracking Step Response . . . . . . . . . . . . . . . . . . . . . . Velocity Tracking Step Response . . . . . . . . . . . . . . . . . . . . . . Altitude Tracking Step Response . . . . . . . . . . . . . . . . . . . . . . Altitude Tracking Step Response . . . . . . . . . . . . . . . . . . . . . . Altitude Tracking Step Response . . . . . . . . . . . . . . . . . . . . . . Altitude Tracking Step Response . . . . . . . . . . . . . . . . . . . . . . Altitude Tracking Step Response . . . . . . . . . . . . . . . . . . . . . . Altitude Tracking Step Response . . . . . . . . . . . . . . . . . . . . . . Block Diagram of Closed Loop System for the Velocity Tracking Case . Block Diagram of Closed Loop System for the Altitude Tracking Case . Controllable Region for Linearized Controller in 2D Parameter Space . . Switching Threshold for Linearized Controller in 2D Parameter Space Along the Mach Number Axis . . . . . . . . . . . . . . . . . . . . . . . . Velocity Tracking Ramp Response . . . . . . . . . . . . . . . . . . . . . Velocity Tracking Ramp Response . . . . . . . . . . . . . . . . . . . . . Velocity Tracking Ramp Response . . . . . . . . . . . . . . . . . . . . . viii 24 28 31 42 42 42 43 43 43 44 . 48 . 49 . . . . . . . . . . . . . . . . . 53 57 67 68 69 70 71 72 75 76 77 78 79 80 81 82 86 . . . . 86 90 91 92 Figure Figure Figure Figure Figure Figure Figure Figure Figure 3.24 3.25 3.26 3.27 3.28 3.29 3.30 3.31 3.32 Velocity Tracking Step Response . Velocity Tracking Step Response . Velocity Tracking Step Response . Altitude Tracking Ramp Response Altitude Tracking Ramp Response Altitude Tracking Ramp Response Altitude Tracking Step Response . Altitude Tracking Step Response . Altitude Tracking Step Response . Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 4.12 4.13 4.14 4.15 4.16 4.17 4.18 4.19 4.20 4.21 4.22 4.23 4.24 4.25 4.26 4.27 4.28 4.29 4.30 4.31 4.32 4.33 4.34 4.35 Open Loop Interconnected System For Velocity Tracking Output Feedback117 Open Loop Interconnected System For Altitude Tracking Output Feedback127 Velocity Tracking Step Response . . . . . . . . . . . . . . . . . . . . . . . 139 Velocity Tracking Step Response . . . . . . . . . . . . . . . . . . . . . . . 140 Velocity Tracking Step Response . . . . . . . . . . . . . . . . . . . . . . . 141 Velocity Tracking Step Response . . . . . . . . . . . . . . . . . . . . . . . 142 Velocity Tracking Step Response . . . . . . . . . . . . . . . . . . . . . . . 143 Velocity Tracking Step Response . . . . . . . . . . . . . . . . . . . . . . . 144 Altitude Tracking Step Response . . . . . . . . . . . . . . . . . . . . . . . 147 Altitude Tracking Step Response . . . . . . . . . . . . . . . . . . . . . . . 148 Altitude Tracking Step Response . . . . . . . . . . . . . . . . . . . . . . . 149 Altitude Tracking Step Response . . . . . . . . . . . . . . . . . . . . . . . 150 Altitude Tracking Step Response . . . . . . . . . . . . . . . . . . . . . . . 151 Altitude Tracking Step Response . . . . . . . . . . . . . . . . . . . . . . . 152 Block Diagram of Closed Loop System for the Velocity Tracking Case . . 153 Block Diagram of Closed Loop System for the Altitude Tracking Case . . 154 Velocity Tracking Ramp Response Unstable . . . . . . . . . . . . . . . . . 161 Velocity Tracking Ramp Response Unstable . . . . . . . . . . . . . . . . . 162 Velocity Tracking Ramp Response . . . . . . . . . . . . . . . . . . . . . . 163 Altitude Tracking Ramp Response Unstable . . . . . . . . . . . . . . . . . 165 Altitude Tracking Ramp Response Unstable . . . . . . . . . . . . . . . . . 166 Altitude Tracking Ramp Response Unstable . . . . . . . . . . . . . . . . . 167 Velocity Tracking Ramp Response . . . . . . . . . . . . . . . . . . . . . . 171 Velocity Tracking Ramp Response . . . . . . . . . . . . . . . . . . . . . . 172 Velocity Tracking Ramp Response . . . . . . . . . . . . . . . . . . . . . . 173 Velocity Tracking Ramp Response . . . . . . . . . . . . . . . . . . . . . . 174 Velocity Tracking Ramp Response . . . . . . . . . . . . . . . . . . . . . . 175 Velocity Tracking Ramp Response . . . . . . . . . . . . . . . . . . . . . . 176 Velocity Tracking Step Response . . . . . . . . . . . . . . . . . . . . . . . 181 Velocity Tracking Step Response . . . . . . . . . . . . . . . . . . . . . . . 182 Velocity Tracking Step Response . . . . . . . . . . . . . . . . . . . . . . . 183 Velocity Tracking Step Response . . . . . . . . . . . . . . . . . . . . . . . 184 Velocity Tracking Step Response . . . . . . . . . . . . . . . . . . . . . . . 185 Velocity Tracking Step Response . . . . . . . . . . . . . . . . . . . . . . . 186 Altitude Tracking Ramp Response . . . . . . . . . . . . . . . . . . . . . . 190 ix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 97 98 102 103 104 108 109 110 Figure Figure Figure Figure Figure 4.36 4.37 4.38 4.39 4.40 Altitude Altitude Altitude Altitude Altitude Tracking Tracking Tracking Tracking Tracking Ramp Response Ramp Response Step Response . Step Response . Step Response . x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 192 196 197 198 NOMENCLATURE α The angle of attack for the hypersonic vehicle δc Deflection angle of the canard δe Deflection angle of the elevator η(t) Time modal coordinate of vibration γ H∞ performance index m̂ Mass of the hypersonic vehicle ω Natural frequency λ Upper bound vector of minimization νk Upper bound of parameter variation rate φ Fuel equivalence ratio φ(x) Spatial modal coordinate of vibration ρ LPV Scheduling parameter vector θ The pitch angle of the hypersonic vehicle λ Lower bound vector of minimization νk Lower bound of parameter variation rate ω ~ The angular velocity vector of the hypersonic vehicle ~a The acceleration vector of the hypersonic vehicle F~ The force vector of the hypersonic vehicle F~b The force vector of the hypersonic vehicle in the body frame p~ The momentum vector of the hypersonic vehicle ~ V The velocity vector of the hypersonic vehicle A System matrix of the plant Acl System matrix for the closed loop system xi Ad Diffuser area ratio Aeq Minimization matrix Ak System matrix for the controller B Input matrix of the plant Bcl Input matrix for the closed loop system beq Minimization vector Bk Input matrix for the controller C Output matrix of the plant c Coefficient of damping for the hypersonic vehicle Ccl Output matrix for the closed loop system Ck Output matrix for the controller D Feedthrough matrix of the plant D The drag of the hypersonic vehicle d Disturbance vector Dcl Feedthrough matrix for the closed loop system Dk Feedthrough matrix for the controller E Modulus of elasticity for the hypersonic vehicle e Output vector f (x) Function that returns a scalar g The gravitational acceleration I Identity matrix I Moment of inertia for the hypersonic vehicle L The lift of the hypersonic vehicle M The moment of the hypersonic vehicle xii m NR The mass of the hypersonic vehicle h The bases of the null spaces of B2T NS h i The bases of the null spaces of C2 D21 T D12 i Nn (t) The generalized modal force acting on the hypersonic vehicle a s a function of time P Matrix associated with Lyapunov Function P The roll rate of the hypersonic vehicle p(x, t) Distributed force acting on the hypersonic vehicle Pj (t) Concentrated force acting on the hypersonic vehicle Q The pitch rate of the hypersonic vehicle R The yaw rate of the hypersonic vehicle T The thrust of the hypersonic vehicle u, v, w The velocity of the hypersonic vehicle in the body frame V Lyapunov Function Vt The true airspeed of the hypersonic vehicle w(x, t) Beam displacement as a function of time and distance along the x-axis of the beam x Minimization vector x State variable vector of the plant xcl State vector for the closed loop system xk State variable matrix for the controller P Parameter set R Optimization variables for H∞ problem S Optimization variables for H∞ problem u The control input vector y The measurement output vector xiii Chapter 1 Introduction 1.1 Hypersonic Vehicle Controls Research into air-breathing hypersonic vehicles started in the 1960’s and continued through the 1990’s with the National Aerospace Plane [8, 7]. Hypersonic vehicles provide several possibilities that current technology cannot achieve. They are being considered as a means of achieving both low Earth orbit, and affordable, reliable outer space access [8, 7, 20, 28, 60]. Hypersonic vehicles have also been proposed as a means of delivering a quick response to global threats [28, 60]. It has even been suggested that hypersonic vehicles could be used in commercial and military applications to reduce flight times since hypersonic vehicles have the ability to carry larger payloads, due to the requirement of fuel and oxygen for rockets as opposed to just fuel for the hypersonic vehicle, than the equivalent rocket powered systems [47, 21]. More recently, NASA has successfully designed and flown the X-43A. The X-43A is a 12 foot long hypersonic vehicle designed by NASA which incorporates an integrated scramjet engine. The X-43A has had successful flight tests. The top recorded speed of the X-43A was Mach 9.6, which was achieved in November of 2004 over the Pacific ocean west of California [57, 28, 27]. Hypersonic vehicles require a highly integrated design approach which causes many significant design challenges with respect to the controls engineer. For hypersonic flight, an airframe with a highly integrated scramjet is required for optimum performance [60, 8]. Deriving models and control systems for hypersonic aircraft can be very difficult, and attempts to provide more integrated approaches to modeling and controlling flexible aircraft have been underway for some time [41]. In figure 1.1, one can see the basic design and layout of a hypersonic vehicle. Notice the scramjet on the bottom of the vehicle. This particular configuration shows a canard on the front of the vehicle. Figure 1.2 shows how the scramjet and integrated airframe work together. The bow shock off the nose of the hypersonic vehicle acts as a compression stage 1 Figure 1.1: Figure 1.2: Hypersonic Vehicle Isometric View Hypersonic Vehicle Side View 2 to the scramjet. This is essential to generating the proper pressure and flow rate needed to maintain the combustion required to produce the thrust needed for hypersonic flight. The offset of the scramjet causes a strong coupling between the thrust, lift, drag, and pitching moment of the vehicle [60, 27]. The vehicle must also be considered flexible since it is long and slender with a relatively light weight [28]. This flexibility will have an effect on the propulsion of the hypersonic vehicle. As the vehicle flexes, the bow shock position will change. Additionally, the overall drag and lift for the vehicle will change. Therefore, the flexible nature of the vehicle must be considered in the construction of the plant model [60, 28, 47]. These challenges present problems in the potential controls design. The modeling of hypersonic vehicles has been an ongoing research topic. One of the earlier studies in this area was performed by Shaughnessy et al. [50]. This preliminary study was a look into developing a model for a hypersonic vehicle with the winged-cone configuration. The design objective at the time was to develop a single-stage-to-orbit (SSTO) hypersonic vehicle that would maximize the propulsion efficiency while minimizing the aerodynamic heating and structural load. This study has since been improved upon by other researchers in the field. The winged-cone configuration has been abandoned in favor of other vehicle configurations. Since the work by Shaughnessy et al., Schmidt and his coworkers have contributed to the advancement of hypersonic vehicle dynamics [49]. Their study investigates the interaction between the airframe, engine, and structural dynamics with respect to the pitch attitude of the vehicle. It is suggested that the use of a control effort to change the diffuser area ratio of the hypersonic vehicle may be necessary to have stable combustion. More importantly, this particular study recognizes the strong couplings that are present between the airframe, the scramjet engine, and the elastic modes of the vehicle. This study also realized that there are issues with using a scramjet engine as well. There is a potential for a flame out (where either the combustion occurs past the combustion chamber, or the flame is completely extinguished). The scramjet is most directly affected by the pitch control inputs to the airframe. Schmidt and his coworkers also discovered that the actuation of the diffuser area ratio must be of the same bandwidth or more than the fuel flow rate in the combustion chamber. More recently, Chavez and Schmidt have made an effort to model the air-breathing hypersonic vehicle [12]. This work is the foundation of the model used in this dissertation. The conclusion of their study is that hypersonic vehicles are dependent upon both aerodynamic and propulsive effects. Chavez and Schmidt reported that since the hypersonic vehicle is elastic, the deformation of the vehicle’s forebody and the vehicle’s pitch response will affect the inlet conditions of the propulsion system. This will cause disturbances in the engine if this is not 3 properly modeled. Their method simplifies the hypersonic modeling problem by using 2 dimensional Newtonian impact theory to characterize the aerodynamic pressure distribution. More recent studies have shown that 2 dimensional Newtonian impact theory does not accurately capture the location of the shock wave for all flight conditions [8, 7]. Therefore, this method will not be directly implemented in this dissertation. Bilimoria and Schmidt have also worked on the flexible hypersonic vehicle model [4]. In their study, they attempt to describe the hypersonic vehicle dynamics using the Lagrangian approach. This modeling effort was an attempt at a complete usable set of kinematic equations for the hypersonic vehicle. Rigid body motion, elastic deformation, fluid flow, rotating machinery, wind, and the curvature of the Earth were all considered in this study. This model was compared to the results of a SSTO. Though this dissertation will not be looking at a SSTO configuration, the results of their study confirmed that there was a strong coupling of the aerodynamic forces and moments with the elastic deformation of vehicle. Similarly, Mirmirani et al. have managed to incorporate many of the coupled dynamics and physics of the hypersonic vehicle using a computational fluid dynamics (CFD) approach [42]. Their approach uses a high fidelity CFD based model in conjunction with multi-physics software to model the dynamics of the hypersonic vehicle. The model is based off the design of the X-43 vehicle and represents an air-breathing hypersonic vehicle model with an integrated airframe and propulsion system. Their model relies on CFD analysis in hopes of comparing these results with wind tunnel and flight test data for verification. Their study also investigates the couplings between the different aerodynamic states and the resulting aerodynamic forces. It is shown that there is a strong coupling between the pitching moment of the vehicle and the propulsion of the scramjet. Essentially, there is a nose up pitching moment for the vehicle when the scramjet is off, and a nose down pitching moment when the scramjet is on. This is due to the lift generated on the aft section of the vehicle being increased by the exhaust of the vehicle. Additionally, there is a coupling between the angle of the vehicle and the thrust. For negative angles of attack, the thrust is larger than that for positive angles of attack. This is due mainly to the location of the shock wave. For negative angles of attack, the shock wave is deeper into the throat of the scramjet, whereas for positive angles of attack the shock wave presents an amount of spillage since the shock wave is not completely captured by the scramjet. This results in a greater amount of thrust for negative angles of attack as opposed to positive angles of attack, but the optimum is achieved for a zero angle of attack case. Their study gives much insight to the hypersonic vehicle, and considers many of the physical effects that take place in the system. 4 There are currently many different control techniques that have been proposed for the hypersonic vehicle. NASA suggested simply using classical controls and simple gain scheduling for the control of their hypersonic vehicle [16]. More recently, a controller design for the hypersonic vehicle was suggested by Parker [47]. This control design incorporates the high fidelity plant model introduced by Bolender and Doman [8, 7], and obtains the force and moment as functions by using a curve fitting technique. Weak couplings and slow dynamics were neglected in this approach. An inner loop feedback linearization with an outer loop LQR controller with integral augmentation was applied to the derived system. Though this system was able to show some robust capabilities with respect to small parameter variations on the length, mass and moment of inertia of the vehicle, the system is not optimized for parameter variation tolerance or disturbance rejection. It has shown improvement in the field of hypersonic control theory, but still has some limitations. Specifically, this method simplifies the equations of motion for the hypersonic vehicle such that it does not include the altitude of the vehicle. The dynamics for altitude are typically slower than the other dynamics for aircraft, but given the speed of the hypersonic vehicle, this may not be a valid assumption. This method also requires the measurement of the Lee derivatives to be taken for the purpose of feedback linearization. This may not be a feasible thing to measure for a hypersonic vehicle. Also, LQR is not a prevailing approach for robustness. Though it is possible to tune an LQR controller to reduce the sensitivity function giving improved tolerance to parameter variation, it is generally accepted that the H∞ approach is superior for disturbance rejection and handling uncertainty in the system. Nevertheless, it is difficult to apply H∞ control techniques to the nonlinear hypersonic vehicle, so a new method of control is needed to achieve hypersonic vehicle robustness and disturbance rejection capabilities. Another proposed control technique for the hypersonic vehicle uses a linear approximation for a plant around a trim condition and an LQR controller [28]. A linearized plant model is used to obtain a linear controller, then the linear controller is applied to the nonlinear plant. This type of controller was implemented using tracking for the velocity, angle of attack, and altitude of the hypersonic vehicle simultaneously. This type of controller does work well on an idealized situation, but it is only applied to a single linearized equilibrium position, and therefore has a limited operating range. Again, this LQR approach can be tuned to provide some robustness for modeling uncertainty, but an H∞ approach would be more suitable to optimize the system for modeling uncertainty and disturbance rejection. Additionally, if a wider operating range is desired, then this control technique is not suitable. It is suggested that a gain scheduling technique will be required with this methodology to ensure control through a large operating region. Their study also assumed full state feedback is possible. It may not always be feasible to measure all of the system states, so it would be nice to have an output feedback controller 5 developed for the hypersonic vehicle. The work done by Xu et al. attempted to use an adaptive sliding mode controller with an observer for output feedback [62]. This method is useful since the sliding mode controller stabilizes the system and gives robust capabilities to the system. It is however limited in that sliding mode control requires large control forces to be generated as well as full state feedback. The adaptive observer added into this system took care of some of these issues, however the work done in this study still left some issues that should be addressed. This controller did get a good performance, but it did not take into account the flexibility of the aircraft. There is also a problem with chattering that is associated with system response when using sliding mode control. This is an undesired effect, and could potentially make this method unsuitable for application to the hypersonic vehicle. Earlier, the importance of taking this flexibility into account was discussed. Though the model used in the study conducted by Xu et al. did achieve good performance with limited control authority in the presence of parameter uncertainty, the model used for control synthesis and simulation was in fact very simplistic. It is important to have a valid model when designing controllers for a hypersonic vehicle as the dynamics are very complicated and coupled. The model used in their study did not take into account actuator dynamics, the flexibility of the vehicle, or a model for the propulsion system. It also neglected the saturation limits of the control efforts. This controller may indeed be a good design, but the model used was overly simplified. This dissertation intends to use a more realistic model for the dynamics of the hypersonic vehicle to design a robust control algorithm. The work done by Mooij incorporates a model reference adaptive control (MRAC) algorithm [43]. Using only the outputs, this system has to track a reference model that can be a simple approximation of the actual system. The reference model is a linearized reference model for a single velocity. This method could prove to be a valid method for controlling hypersonic vehicles, but there are also some limitations. First this method uses a large number of design parameters, or weighting functions, that must be tuned to influence the performance. This is not favorable to a designer as it may be difficult to tune the weighting functions to find the appropriate values to obtain the necessary performance. Additionally, the choice of the wrong weight function can result in the instability of the system due to the discontinuities from the linearization. Experience is needed in order to make the appropriate design decisions using this method. Another drawback to this particular method is that the model used in Mooij’s study does not include the flexible effects of the hypersonic vehicle. Additionally, large control efforts may be needed with this method of control, and it may not be possible for the system to provide the appropriate level of control effort. This method also is based upon linearization at a single operating point. Consequently, this does not make it a suitable controller for a large 6 range of operation for the hypersonic vehicle. The work done by Buschek and Calise, as well as Gregory et al., investigates the application of H∞ and µ-synthesis control algorithms to the hypersonic vehicle [10, 11, 26]. The study conducted by Gregory et al. looked into the feasibility of controlling a single-stage-to-orbit (SSTO) air-breathing hypersonic vehicle. It intended to use velocity and altitude tracking with angle of attack regulation for the hypersonic vehicle. There were two controllers designed in their study. The first controller was the H∞ controller which had no structured uncertainty block. The second controller was a µ-synthesis controller with a structured uncertainty block. Their study showed that the µ-synthesis controller had a better performance than the H∞ controller when actuator uncertainty was introduced into the system [26]. Similar studies by Buschek and Carlise show that µ-synthesis has good robust performance properties using a structured uncertainty block [10, 11]. Using a canonical realization and a homotopy algorithm, Buschek and Carlise managed to design a fixed order µ-synthesis controller which are then incorporated with structured uncertainty blocks for robust stability and performance [11]. The real issue with these methods is the modeling. First, the vehicle model and controller are only good for a single linearized trim condition. This of course severely limits the operating range of the hypersonic vehicle. Secondly, the propulsion system is not well modeled. All of these studies treat the variation of the propulsion system as part of the structured uncertainty block. In all of these cases, the propulsion system is assumed to be unaffected by the angle of attack of the vehicle. This is not a good assumption to make for the hypersonic vehicle because the propulsion system is strongly dependent upon the angle of attack. These studies also assume that the vehicle body is rigid. The flexible effects are modeled as uncertainty in the system. This can be a problem if there is not an accurate model for the propulsion system because the flexible effects of the vehicle also have interaction with the thrust that the scramjet can produce. The control algorithms used in these studies do work well under the assumed conditions on the system, but the limitations of modeling do not make these studies a good metric for actual implementation. This dissertation will look at integrating a high fidelity model for the hypersonic vehicle with a set of controllers designed about a set of linearized trim conditions for the vehicle such that a robust control can be achieved for a large range of operation. The work by T. Gibson et al. looks into the use of adaptive control for a hypersonic vehicle in the presence of modeling uncertainty [25]. Their study investigates a hypersonic vehicle model with aerodynamic uncertainty, parametric uncertainty in the location of the center of gravity, actuator saturation and failure, and time delays in the system. Their study does a good job of evaluating different uncertainties that can exist in the system, but the control algorithm is 7 designed for a linearized model of the system at a single point. This means that the controller, though robust to the uncertainties modeled in the system, is not validated for its robustness margins. This method also models the flexibility of the hypersonic vehicle as two cantilever beams with constant cross sectional area. This model for the flexibility has been replaced by the assumed modes methodology later on [33]. The study presented in this dissertation will look to include a wider range of operation and a more accurate hypersonic vehicle model than what is presented in the work done by Gibson et al. B. Fidan et al. have done work on the longitudinal motion control of a hypersonic vehicle based on time-varying models [19]. Their study uses linear time-varying models (which is a type of LPV model) with adaptive and non-adaptive control systems. The advantage of doing this is that it allows for fast parameter variations and a large range of motion as is the case with the LPV control system. Fidan et al. use the model developed by Mirmirani et al. for their approach [42]. This method has a lot of potential, but there are a few shortcomings in this particular study. First, the study uses full state feedback, which may not be a viable option for a hypersonic vehicle as there may be some states that cannot be measured. Secondly, it is a mathematical derivation for the control algorithm which shows great promise, but there is no simulated data to show the results of this study. This study also neglects to mention any disturbances or measurement noise that may be present in the system. The work done by Sigthorsson et al. investigates tracking control for an overactuated hypersonic vehicle with steady state constraints [51]. This study uses the flexible hypersonic vehicle model developed by Bolender and Doman [8, 7]. The goal of their study is to track the velocity and the altitude of the hypersonic vehicle. To achieve this, they design a controller that ensures asymptotic tracking of altitude and velocity while using the redundancy in the inputs to optimize the performance in steady-state. By linearizing the nonlinear model, they are able to synthesize a linear regulator which consists in letting the output of a system track a reference trajectory or rejecting a disturbance generated by an autonomous linear time invariant (LTI) system. Their study uses full state feedback and LQR control for stabilizing control, and optimization methods for steady state control for both the constrained and unconstrained cases. Though there is the potential for some robust capabilities with this method, it does not insure that the system will be robust. Currently, there have been no tests to show the systems robustness to disturbances and uncertainty in the system. This study also uses the linearized plant model for all but one of the cases presented. These results may not contain the level of accuracy needed to control the nonlinear system if the range of operation becomes too large. Further investigation into this control method is needed. 8 Different from linear control techniques, another nonlinear control method has been presented by Wilcox et al. This method involves robust nonlinear control of a hypersonic aircraft in the presence of aerothermal effects [59]. This method uses temperature dependent parameter-varying state space models. It includes an uncertain parameter varying state matrix and uncertain non-square parameter varying input matrix and nonlinear additive bounded disturbance. From there, a Lyapunov based continuous robust output feedback controller is developed that has global exponential tracking of a reference model. This model makes the modulus of elasticity a function of temperature. This controller is supposed to be robust to sensor noise, exogenous perturbations, parametric uncertainty, and plant nonlinearities, but currently there is no simulation results to back up the theory. The mathematical model is provided, but is not verified using nonlinear simulation. The work done by Jankovsky et al. involves applying output feedback control to the hypersonic vehicle model as well as investigating the need for proper sensor placement [33]. This study also uses the high fidelity model developed by Bolender and Doman [8, 7]. In their study, there are two proposed output feedback controllers. The first output feedback controller uses an observer to reconstruct the full state information of the system. The other output feedback controller uses a robust output feedback to ensure stabilization without an observer in the system. The model used in this approach also takes into account the sensor models and placement. The observer based controller did not feasibly control the system due to the limitations of the linearized observer. The second controller employs pre-compensation of the unstable zero-dynamics, dynamic extension, and a robust servomechanism design based on time-scale methods. The results for the second output feedback controller show that there are some favorable results, but the gains for the system are very high. Also, the reference trajectories are tracked, but the tracking is very slow. Their study has suggested that gain scheduling or adaptive control may be able to yield an improved performance for the output feedback problem. Linear Parameter-Varying control techniques have been applied to the flexible hypersonic vehicle model in the past. This technique involved using a multi loop controller where the inner controller was an LPV controller used to augment active structural damping in the aeroelastic modes while the outer loop was a traditional rigid body aircraft controller [36]. The inner controller for this algorithm uses the angle of attack and the desired angle of attack as the input, while the output is the elevator. This multi-loop control technique is successful in using LPV to control the hypersonic vehicle, but the research does not go further to investigate the use of H∞ and robust control techniques. It also uses an out of date hypersonic vehicle model. Newer models for hypersonic vehicles have shown the need for a canard in order to reduce the 9 pitching moment caused by having a scramjet that is below the neutral axis of the vehicle [8]. This dissertation will design and implement an LPV H∞ controller for the longitudinal dynamics of the flexible air-breathing hypersonic vehicle as modeled by Bolender and Doman [8, 7]. The proposed study will consider both full state feedback, and output feedback. This is not meant to be an exhaustive study, but rather a preliminary study into the feasibility of using an LPV H∞ controller as applied to the longitudinal dynamics of a flexible hypersonic vehicle. A comparison will be made between the output feedback case, and the full state feedback case. Additionally, this study will investigate the feasibility of using the flexibility of the aircraft as a disturbance to the system by synthesizing a controller using rigid body assumptions and simulating the results using a flexible body controller. Disturbance and parametric uncertainty will be investigated in the system to show the robust capabilities of the proposed control technique. Precaution will be taken to make sure that the model is valid, though some assumptions will be made to simplify the problem. Actuator dynamics will also be considered in the process of modeling the system in order to yield more realistic results. It should be noted that more extensive study will need to be done before implementation of such a controller on an actual hypersonic system is achievable. 1.2 H∞ LPV Control Techniques The background for building this type of controller is founded in the H∞ and Linear Parameter Varying (LPV) control theory [30, 31, 1]. Therefore, it will be important to understand the basic concepts of H∞ control for the purposes of understanding this report. The H∞ control problem can be solved using linear matrix inequality (LMI) techniques [23]. Using the system seen in figure 1.3, the Linear Time Invariant (LTI) system equations of G is ẋ(t) = Ax(t) + Bd(t), (1.1) e(t) = Cx(t) + Dd(t), (1.2) where the state x ∈ Rn , the disturbance d ∈ Rnd and controlled output e ∈ Rne . Assuming that the matrix A is asymptotically stable by letting the eigenvalue of A be less than zero, it is possible to determine the H∞ performance from d to e. The H∞ norm of the system G 1.1-1.2 is defined as kek2 , d,kdk2 6=0 kdk2 kGk∞ = max where the signal e’s 2-norm is kek22 = R∞ 0 eT (t)e(t)dt and similar for d. 10 (1.3) Figure 1.3: Block Diagram of Uncontrolled System Using a quadratic Lyapunov function V (x) = xT P x, P = P T > 0, (1.4) to analyze the stability and system performance, the following equation will be yielded V̇ + Since dV dt = ∂V ∂x ẋ, 1 T e e − γdT d < 0. γ (1.5) then the above equation becomes ẋT P x + xT P ẋ + 1 T e e − γdT d < 0, γ which can be reduced to, kek2 < γkdk2 . (1.6) Therefore, the H∞ norm of the system is bounded from above by γ. From this equation, it can be seen that the smaller γ is, the greater the energy amplification of the system becomes. This will be important as one of the stated goals is to decrease power consumption. To minimize the γ, apply the following linear matrix inequality by finding a 11 positive definite matrix P > 0 such that T A P + PA PB −γI BT P C D CT DT < 0. −γI (1.7) This is known as the bounded real Lemma and is given by an LMI [37, 63]. An LMI is a special type of convex problem which can be solved using efficient interior-point optimization techniques [9]. For the H∞ control problem, assume that the open loop plant is in the following form ẋ(t) = Ax(t) + B1 d(t) + B2 u(t), (1.8) e(t) = C1 x(t) + D11 d(t) + D12 u(t), (1.9) y(t) = C2 x(t) + D21 d(t) + D22 u(t), (1.10) where the control input u ∈ Rnu and the measurement output y ∈ Rny . It is assumed that the triple (A, B2 , C2 ) is stabilizable and detectable. Now that the open loop plant has been defined, it is necessary to determine the closed loop plant in order to solve the H∞ optimization problem. This closed loop system can be seen in figure 1.4. It is desirable to design an output feedback controller with the following set of equations. ẋk (t) = Ak xk (t) + Bk y(t), (1.11) u(t) = Ck xk (t) + Dk y(t), (1.12) where xk ∈ Rnk is the controller states. In order to find the optimized controller, it will be necessary to optimize the closed loop H∞ performance using a dynamic controller. This optimized γ will be referred to as γopt . This problem will now need to be formulated as a Linear Matrix Inequality (LMI) optimization problem as well. For a dynamic output feedback control in the form of equations 1.11-1.12, the closed loop system becomes, ẋcl (t) = Acl xcl (t) + Bcl d(t), (1.13) e(t) = Ccl xcl (t) + Dcl d(t), (1.14) 12 Figure 1.4: h where xcl = xT xTk iT Block Diagram of Controlled System and the state space matrices of the closed loop system are, " Acl = B k C2 " Bcl = A + B 2 Dk C2 B 2 Ck B1 + B2 Dk D12 Ak # Bk D12 # , , (1.15) (1.16) i h Ccl = C1 + D12 Dk C2 D12 Ck , (1.17) Dcl = D11 + D12 Dk D12 . (1.18) From this set of equations, an optimal controller can be found using H∞ analysis condition 1.7. Specifically as an LMI optimization problem, the following three conditions must also be met to solve the control synthesis problem [23]: find positive definite matrices R, S > 0 such 13 that " #T AR + RAT NR 0 C1 R 0 I B1T #T AT S + SA " NS 0 B1T S 0 I C1 RC1T B1 " NR −γI D11 0 T D11 −γI SB1 C1T " NS T −γI D11 0 D11 −γI " R I h where NR and NS are the bases of the null spaces of B2T defined as h i 0 # I 0 S T D12 i (1.19) < 0, (1.20) ≥ 0, (1.21) # I I < 0, # h i and C2 D21 which are T , NR = ker B2T D12 h i NS = ker C2 D21 . (1.22) (1.23) The control synthesis condition 1.19-1.21 is again in the form of LMIs and can be solved using a commercial software LMIlab [24]. This concludes the H∞ synthesis of the LTI system. The next section will discuss how to take these results for LTI systems and extend them into LPV formulations. Now that the H∞ synthesis conditions have been defined, it is important to look at how this parameterization affects the control problem, and the synthesis conditions. The LPV system is a class of linear systems with its state space matrices depending on a time-varying vector ρ(t) ∈ Rs , ẋ(t) = A(ρ(t))x(t) + B1 (ρ(t))d(t) + B2 (ρ(t))u(t), (1.24) e(t) = C1 (ρ(t))x(t) + D11 (ρ(t))d(t) + D12 (ρ(t))u(t), (1.25) y(t) = C2 (ρ(t))x(t) + D21 (ρ(t))d(t) + D22 (ρ(t))u(t), (1.26) It is assumed that the scheduling parameter ρ evolves continuously over time and its range is limited to a compact set ρ ∈ P. In addition, its time derivative is often assumed to be bounded and satisfy ν k < ρ̇k < ν̄k , k = 1, 2, · · · , s. Moreover, assume that (A1) The matrix function triple (A(ρ), B2 (ρ), C2 (ρ)) is parameter-dependent stabilizable and detectable, h i h i T (ρ) have full row rank for all ρ ∈ P, (A2) The matrices C2 (ρ) D21 (ρ) and B2T (ρ) C12 14 (A3) D11 (ρ) = 0 and D22 (ρ) = 0. Similar to the open loop description, the LPV synthesis conditions also change with parameterization. For full state feedback, y = x, we consider the static state feedback control law in the form of u = F (ρ)x (1.27) The synthesis condition of LPV state feedback control problem is given by ( " #T I NR (ρ) 0 0 A(ρ)R(ρ) + R(ρ)AT (ρ) P ∂R − si=1 {ν i , ν i } ∂ρ i R(ρ)C1T (ρ) −γI C1 (ρ)R(ρ) B1T (ρ) ) T (ρ) D11 # NR (ρ) 0 <0 × 0 I B1 (ρ) D11 (ρ) −γI " (1.28) for any ρ ∈ P. Consequently, the resulting LPV state feedback control gain will be T T F (ρ) = −(D12 (ρ)D12 (ρ))−1 γB2T (ρ)R−1 (ρ) + D12 (ρ)C1 (ρ) . (1.29) For the LPV output feedback control problem, the following controller formulation is needed. ẋk (t) = Ak (ρ(t), ρ̇(t))xk (t) + Bk (ρ(t))y(t), u(t) = Ck (ρ(t))xk (t) + Dk (ρ(t))y(t). (1.30) (1.31) Using a parameter-dependent quadratic Lyapunov function V (x) = xTcl P (ρ)xcl , the solution of H∞ synthesis problem of an LPV output feedback controller [3, 61] is to find a pair of 15 continuously differentiable matrix functions R(ρ), S(ρ) > 0 which satisfy ( " #T I NR (ρ) 0 0 A(ρ)R(ρ) + R(ρ)AT (ρ) P ∂R − si=1 {ν i , ν i } ∂ρ i ) R(ρ)C1T (ρ) −γI C1 (ρ)R(ρ) B1T (ρ) B1 (ρ) D11 (ρ) −γI T (ρ) D11 # NR (ρ) 0 × < 0, 0 I ) ( AT (ρ)S(ρ) + S(ρ)A(ρ) #T " S(ρ)B1 (ρ) C1T (ρ) Ps ∂S + {ν ν } , NS (ρ) 0 i i ∂ρi i=1 T T 0 I B1 (ρ)S(ρ) −γI D11 (ρ) C1 (ρ) D11 (ρ) −γI " # NS (ρ) 0 < 0, × 0 I " # R(ρ) I ≥ 0, I S(ρ) " (1.32) (1.33) (1.34) h i T (ρ) , for all ρ ∈ P. Similar to the LTI case,NR (ρ) = ker B2T (ρ) D12 h i NS (ρ) = ker C2 (ρ) D21 (ρ) . The resulting controller gains become [2], s̄ X ∂S R A + S (A + B2 F + LC2 ) R + ρ̇i ∂ρi ( Ak := − T (1.35) i=1 + γ −1 S (B1 + LD21 ) B1T + γ −1 C1T (C1 + D12 F ) R } (I − RS) −T Bk := SL (1.36) Ck := F R (I − RS)−T (1.37) Dk := 0 (1.38) Where L and F in are defined as, T T −1 D21 D21 D21 (1.39) −1 T −1 T −1 T T B2 R F = D12 D12 D12 C1 + γD12 D12 D12 (1.40) L = − B1 + γS −1 C2T T −1 D21 D21 D21 16 Now that the LPV solution techniques have been established, it is possible to find a solution to a given problem. It should be noted here that the synthesis conditions and closed loop dynamics of the plant are LPV. This means that the LMI’s and closed loop equations involve an infinite number constraints. To simplify this problem, the parameters will be discretized. This means that a finite number of points for each parameter in the system will be chosen to represent the system as a whole. It is important to realize that each of these discretized parameter points will represent a linear model of the system. These griding points will be chosen in a way such that the discrete points are close enough that the range for which each linearized region is valid overlaps with another linearized griding point. In addition, R(ρ) and S(ρ) must also be parameterized using a finite number of basis functions [3]. This will take the form of, R(ρ) = Nf X fi (ρ)Ri , (1.41) gj (ρ)Sj , (1.42) i=1 Ng S(ρ) = X j=1 where fi (ρ), i = 1, 2, · · · , Nf and gj (ρ), j = 1, 2, · · · , Ng are user specified basis functions. It should be noted here that for the special cases where R or S is constant, the controller gains will not depend on ρ̇, but on ρ only. Once this parameterization of the system has taken place, the synthesis problem can now be solved. 1.3 Dissertation Outline This dissertation intends to explore the solution technique used to apply an H∞ LPV controller to a hypersonic vehicle. It will investigate the methods needed to synthesize robust controllers for the full state feedback and output feedback cases. It will also analyze the performance of the hypersonic vehicle based upon its ability to track a reference signal. Chapter 2 will discuss the hypersonic vehicle model. It show the methods used to model the flexibility of the hypersonic vehicle used in this study as well as some of the assumptions made. It will look at the free body diagram of the hypersonic vehicle and derive the basic equations of motion for the longitudinal dynamics of the flexible hypersonic vehicle. Additionally, it will give a brief review of the aerodynamics used for the hypersonic vehicle in this model, and discuss the significance of the resulting equations. Chapter 2 will also investigate the open loop dynamics of the hypersonic vehicle to give the reader a better understanding of the open loop performance of the system. Actuator models will be discussed, and the results of the modeling 17 will be discussed in detail. Chapter 3 will discuss the full state feedback analysis of the hypersonic vehicle. This chapter will investigate the velocity tracking case and the altitude tracking case. For each of these cases, comparisons will be made between the performance of the H∞ LPV controller at a single operating point, and a linear H∞ controller. Conclusions will be drawn with regard to the effects of optimal control for velocity and altitude tracking. These conclusions will justify the use of the combined velocity tracking with loose altitude regulation case. The system will also be simulated with some perturbation injected into the system to determine the controller’s robust capabilities. Additionally, the control synthesis process will be analyzed to see what effects changing the number of discrete trim points and the parameter variation rate have on the robust performance. The results of this study will be analyzed and discussed in detail. Chapter 4 will cover the output feedback control synthesis, simulation, and analysis for the H∞ LPV controller for the hypersonic vehicle. In this chapter, the velocity tracking with loose altitude regulation and altitude tracking with loose velocity regulation cases will be discussed in detail. Simulation of the hypersonic vehicle will be provided for both a rigid body case and a flexible body case. Conclusions will be drawn as to whether or not making a rigid body assumption when synthesizing a controller is legitimate or not. Also, chapter 4 will discuss the disturbance rejection capabilities. Simulations will be run with perturbation in the system to show the robust capabilities of the controller. These results will be compared with simulations without perturbation in the system. Additionally, the control synthesis process will be analyzed to see what effects changing the number of discrete trim points and the parameter variation rate have on the robust performance. The results of this study will be analyzed and discussed in detail. Chapter 5 will give a summary of the results of the dissertation. It will take all of the simulations and results and make conclusions based on a broader understanding of the robust control methods used in this study. An attempt will be made to show the successes and the shortcomings of this study. A discussion of future work will also be included in this chapter so that this line of research may be continued. 18 Chapter 2 Hypersonic Vehicle Model The main focus of this chapter is to present the aerodynamic model of a hypersonic vehicle as presented by Blender and Doman [8, 7] to obtain a nonlinear plant model from which its LPV model will be derived and an H∞ LPV controller will be synthesized. A minimal amount of effort will be spent discussing the aerodynamics of the system, so it is recommended that the readers refer to the references [8, 7] for further information. 2.1 Flexible Aircraft Model It is important to first understand the importance of including the flexibility of the vehicle into the equations of motion. The vibration of the hypersonic vehicle has an effect on the angle of the bow shock. As a result, this has an effect on the pressures downstream of the shock wave as well. These changes in pressure have an effect on the performance of the scramjet. Often times due to the angle of the shock wave, the scramjet will be operating outside of its optimal design region. This of course affects the thrust and moment of the hypersonic vehicle. It is therefore important to take the vibrational effects of the vehicle into consideration when modeling its equations of motion. The particular hypersonic vehicle model used in this study incorporates a flexible body model. A modal solution to the Euler-Bernoulli beam equation will yield the modal forces which can then be used in the equations of motion for the hypersonic vehicle. The Euler-Bernoulli equation is a fourth order differential equation in both time and space. The Euler-Bernoulli equation is shown in equation 2.1 [52, 64, 32, 5, 48]. EI ∂ 4 w(x, t) ∂ 2 w(x, t) + m̂ =0 ∂x4 ∂t2 19 (2.1) In order to solve this partial differential equation, one must assume a solution in the form of w(x, t) = φ(x)η(t) (2.2) Using the separation of variables, Equation 2.1 can be separated into two ordinary differential equations as can be seen in Equations 2.3 and 2.4. EI It is assumed that β 4 = ω 2 m̂ EI , d4 φ(x) − ω 2 m̂φ(x) = 0 dx4 (2.3) d2 η(t) + ω 2 η(t) = 0 dx2 (2.4) then equation 2.2 becomes, d4 φ(x) − β 4 φ(x) = 0 dx4 (2.5) φ(x) = A sin(βx) + B cos(βx) + C sinh(βx) + D cosh(βx) (2.6) The solution to Equation 2.5 is, Since the hypersonic vehicle is assumed to behave as a free-free beam, the following boundary conditions will be applied, φ00 (0) = φ000 (0) = φ00 (L) = φ000 (L) = 0 (2.7) Applying the boundary conditions yields the following transcendental equation. cos(βL) cosh(βL) − 1 = 0 (2.8) There are an infinite number of solutions βn that satisfy equation 2.8. Using modal analysis yields: φn (x) = [(− sin(βn L) + sinh(βn L)) · (cos(βn x) + cosh(βn x)) +(cos(βn L) − cosh(βn L)) · (sin(βn x) + sinh(βn x))] (2.9) This value of φn (x) represents the mode shape for a free-free beam with a constant cross section and uniform material. The hypersonic vehicle used in this study, however, does not have a constant cross sectional area. Therefore, it will be necessary to use an alternative method to solve for the mode shapes to account for the change in the mass and the moment of inertia of 20 the vehicle along the x axis. There are a few different ways of doing this. One method is to use finite element analysis. Finite element analysis is very useful for getting accurate descriptions of the mode shapes for difficult geometries. This method results in high-dimensioned systems, and therefore is not a favorable option for a system with a relatively simplistic geometry [34]. The method used in this study is the assumed modes method. This method will use a Ritz approximation to model the continuous beam and transform it into a discrete system in which both moment of inertia and the mass of the system can change with respect to the spatial coordinate. In this particular study, the solution from the Euler-Bernoulli equation will be used as a comparison function for the assumed modes method. The result will be a discretized representation of the mode shape for the beam. The first step in applying the assumed modes method will be to consider the displacement described as a sum of the modes of vibration of the system which can be seen in equation 2.10. w(x, t) = ∞ X φ̄n (x)ηn (t) (2.10) n=1 It is important to note at this point that φ̄n represents the assumed mode shape and not the Euler-Bernoulli mode shape which be used as a comparison function for the assumed mode approach. From the solution to the Euler-Bernoulli equation, equation 2.9 and the following equation are obtained. 00 φn (x) = [βn2 (cos(βn L) − cosh(βn L)) · (−sin(βn x) + sinh(βn x)) −βn2 (sin(βn L) − sinh(βn L)) · (−cos(βn x) + cosh(βn x))] (2.11) From here, it is necessary to discretize and append the mode shapes to include the rigid body modes. Since this study will consider three flexible modes of vibration and there are two rigid body modes, there will be a total of five modes considered for the analysis. The hypersonic vehicle length is 100 feet. This length was divided into 1001 discrete points evenly spaced .1 feet from each other. The resulting rigid body modes are, 1 . . φ1 = . 1 (2.12) φ2 = xT − xcg (2.13) 21 where x is a vector of the discretized points along the body axial, and xcg is the location of the center of gravity. The center of gravity of the vehicle can be determined by, 1 · m(x) · xT · ∆x Mt xcg = (2.14) Using the kinetic and potential energy of the system yields, N T (t) = N 1 XX 1 [M ]ij η̇i (t)η̇j (t) = η̇ T M η̇ 2 2 (2.15) i=1 j=1 N V (t) = N 1 XX 1 [K]ij ηi (t)ηj (t) = η T M η 2 2 (2.16) i=1 j=1 where the matrices [M ]ij and [K]ij are the (i, j)th element of the symmetric mass matrix M and the symmetric stiffness matrix K which both depend on the mode shapes and the mass and stiffness distribution for the system. These matrices are calculated by, Z [M ]ij = φTi (x) · m(x) · φTj (x)dx 0 Z [K]ij = L L 00 00 φi T (x) · EI(x) · φj T (x)dx 0 where m(x) represents the mass of the hypersonic vehicle with respect to length, and EI(x) represents change in the moment of inertia and the modulus of elasticity with respect to the length of the hypersonic vehicle. Now that these expressions have been formulated, it is possible to derive vibrational equations of the system. Using Lagrange’s equation, d dt ∂T ∂ η˙n − ∂T ∂V + = Nn ∂ηn ∂ηn (2.17) and substituting in equations 2.16 and 2.15 yields, N X [M ]nj η̈j (t) + j=1 N X [K]nj ηj (t) = Nn (2.18) j=1 which can be written in matrix form as, M η̈(t) + Kη(t) = N 22 (2.19) Now solve for the Eigenvalues and Eigenvectors for equation 2.19 using, 2 ωn I − M −1 K Vn = 0 (2.20) where the Eigenvalues of equation 2.20 are ωn2 , and the Eigenvectors are Vn . Now that the Eigenvalues and Eigenvectors have been calculated, it is possible to come up with an expression for the mode shapes. The new mode shapes, φ̄n , can be expressed as, φ̄n = VnT · φTn (2.21) −1 2 (2.22) which can be mass normalized by, φ̄m n = φ̄n M Similarly, the modal equations for the time coordinate, ηn , can be expressed as, η¨n + ωn2 ηn = Nn (t) (2.23) Perturbation analysis shows that for light damping, mode shapes do not change. Therefore a legitimate way to add damping into the system is to artificially add it into the modal equations [53, 17]. For this study, the damping ratio ζ was chosen to be 0.02. This is much less than 0.1 which is a general heuristic for what is considered to be a lightly damped system [53, 17]. This causes equation 2.23 to become, η¨n + 2ζωn η˙n + ωn2 ηn = Nn (t) (2.24) Using a distributed and concentrated force, the following equation expresses the generalized modal force, Z Nn (t) = L φ̄m n (x)p(x, t)dx + 0 l X φ̄m n (xj )Pj (t) j=1 where l is the number of concentrated forces applied to the beam. 2.2 Equations of Motion This section will discuss the derivation of the equations of motion for the hypersonic vehicle. This is not meant to be a complete derivation of the vehicle dynamics as this topic is beyond the scope of the research presented in this dissertation. Instead, this derivation is meant to give the reader a basic understanding of the equations of motion for the hypersonic vehicle. For a full derivation of the vehicle dynamics the reader should refer to Bolender and Doman’s work 23 Figure 2.1: Hypersonic Vehicle Free Body Diagram [8, 7]. 2.2.1 Hypersonic Vehicle Free Body Diagram and Force Equations There are multiple ways of deriving the equations of motion for the hypersonic vehicle. Waszak and Schmidt used the Lagrangian approach to solve for the longitudinal dynamics of a flexible hypersonic vehicle [58]. Bolender and Doman use the same approach in their paper as well [8]. For the purpose of this simplicity, this derivation will build up the equations of motion based upon the momentum equation [29]. It is assumed that only the longitudinal motion of the vehicle will be considered. Additionally, it is assumed that there is no side slip, no lateral motion, and no roll for the hypersonic vehicle. Also, the flat Earth assumption is applied (i.e. the curvature of the Earth is neglected). It will be assumed that the vehicle has a constant mass, and that the thrust from the scramjet is axial in the body frame (the moment caused by the scramjet being below the centroid of the vehicle will be incorporated into the calculation of the moment of the vehicle, so this assumption is purely for the derivation of the equations of motion but will have no effect on the dynamics in the simulation or control synthesis). The resulting ~. free body diagram can be seen in figure 2.1. The momentum of the vehicle gives,~ p = mV Taking the time derivative of the momentum of the hypersonic vehicle yields: ~ dV d~ p =m = m~a F~ = dt dt 24 (2.25) Taking equation 2.25 and putting it into the body frame gives the following: F~b = du where, ~ dV dt = dt dv dt dw dt d~ p dt +ω ~ × p~ = m b ~ dV dt ! ~ + m~ ω×V (2.26) b u̇ = v̇ . Simplifying equation 2.26 by substituting dV̄ dt yields, ẇ ê1 ê2 ê3 Qw − Rv u̇ u̇ F~b = m v̇ + m P Q R = m v̇ + m Ru − P w u v w P v − Qu ẇ ẇ (2.27) From the assumptions made for the problem, v̇ = v = P = R = 0. These assumptions reduce equation 2.27 to be as follows. u̇ Qw F~b = m 0 + m 0 ẇ −Qu (2.28) Solving equation 2.28 for u̇ and ẇ respectively gives the following equations u̇ = Fx − Qw m (2.29) ẇ = Fz + Qu m (2.30) From the free body diagram in figure 2.1, summing the forces in the body frame x and z directions give the following equations, Fx = T − mg sin θ − D cos α + L sin α (2.31) Fz = mg cos θ − L cos α − D sin α (2.32) Substituting equations 2.31 and 2.32 into equations 2.29 and 2.30 respectively yields the following equations. T − D cos α + L sin α − g sin θ − Qw m −L cos α − D sin α ẇ = + g cos θ + Qu m u̇ = 25 (2.33) (2.34) By definition, the following relationship can be seen between the true airspeed velocity, Vt , and its components. p Vt = u2 + w2 (2.35) The expression of Vt can also be rewritten as, u = Vt cos α and w = Vt sin α. Taking the derivative of equation 2.35 gives, uu̇ + wẇ V̇t = √ u2 + w2 (2.36) Now by substituting u and w as well as equations 2.33 and 2.34 into equation 2.36, the following equation can be obtained. V̇t = T cos α − D − g sin (θ − α) m (2.37) Equation 2.37 is the first equation of motion that will be used. From the free body diagram, the following expression can also be obtained. sin α = w Vt (2.38) Taking the time derivative of equation 2.38 gives, α̇ cos α = Vt ẇ − wV̇t Vt2 (2.39) By substituting equations 2.34 and 2.37 into equation 2.39 and simplifying, the following equation is obtained. α̇ = −L − T sin α g + cos (θ − α) + Q mVt Vt (2.40) Equation 2.40 is the second equation of motion that will be used. The following set of equations represent the equations of motion for the rigid body, where equations 2.43, 2.45, and 2.44 are derived directly from the free body diagram. T cos α − D − g · sin (θ − α) m (2.41) −L − T sin α g + cos (θ − α) + Q mVt Vt (2.42) M Iyy (2.43) V̇t = α̇ = Q̇ = ḣ = Vt sin (θ − α) (2.44) θ̇ = Q (2.45) This set of equations of motion describe the rigid body dynamics for the hypersonic vehicle in 26 terms of lift, thrust, drag, and moment. It should be noted that lift, thrust, drag, and moment are functions of the system states. For a detailed explanation on how to calculate these values, the reader is referred to the work of Bolender and Doman [8, 7] and subsection 2.2.2. It has already been established that for this study, an assumed mode analysis will be used to model the flexible effects of the hypersonic vehicle. The real question is exactly how many modes should be considered for this particular system. To answer this, first the reader should refer to the study done by Williams et al. and Bolender et al. [60, 8]. The conclusions drawn from these studies show that as the number of modes is increased, the convergence of the assumed modes method for estimating the first natural frequency will be improved. Since the first natural frequency is the lowest natural frequency for the system, it will have the greatest amount of influence on the displacement of the vehicle body from the rigid body location. It will be necessary to use a model that can acurately converge on the first natural frequency of the system without adding too many flexible states to the system as adding flexible states will make the controller synthesis more complicated. Bolender et al. used only three modes of vibration in their study, and so three modes were chosen in this study as well since Williams et al. has shown that there is a good convergence of the first natural frequency of the vehicle with only three modes of vibration, especially when an accurate comparison function is used as opposed to a poorly conditioned admissable function for the assumed modes method [60, 8]. Simulation and open loop analysis will later verify that three modes of vibration will accurately capture the flexible effects of the aircraft. Assuming that there will only be three modes of vibration, the following three equations will also be used in the development of the flexible body plant dynamics. 2 η̈1 = −2ζωn1 η̇1 − ωn1 η1 + N1 (2.46) 2 η̈2 = −2ζωn2 η̇2 − ωn2 η2 + N2 (2.47) 2 η̈3 = −2ζωn3 η̇3 − ωn3 η3 + N3 (2.48) Equations 2.41-2.45 and 2.46-2.48 represent the total nonlinear equations of motion for the flexible hypersonic vehicle. Later in this chapter, a look at the open loop dynamics will be considered to justify using only the first three modes of vibration. 2.2.2 Hypersonic Vehicle Aerodynamics This subsection will go through a brief review of the aerodynamics for the hypersonic vehicle as defined in the works by Bolender and Doman [8, 7]. In order to calculate the lift, thrust, drag, and moment on the hypersonic vehicle, it will be necessary to calculate the pressures on the vehicle body, the forces and interactions of the scramjet, and the pressures on the control 27 Ln Lf La 1,u xB 1,l M zB α 2 s Figure 2.2: Hypersonic Vehicle Geometry [8, 7] surfaces. Figure 2.2 shows the geometry of the hypersonic vehicle used to determine the forces and moments acting on the vehicle during flight. First consider the forces on the forebody and upper surface of the hypersonic vehicle. There are two different ways to calculate the pressure on the forebody of the hypersonic vehicle. If the angle of attack is greater than the −τ1,l , then oblique shock theory is used. Using this method, the shock angle with respect to the horizontal is a function of angle of attack and τ1,l . In order to solve for the shock angle, the following polynomial must be solved for sin2 (θs ) [8, 7], sin6 θs + b · sin4 θs + c · sin2 θs + d = 0 (2.49) where, 2 +2 M∞ − γ · sin2 δ 2 M∞ 2 +1 2M∞ (γ + 1)2 γ − 1 c= + + sin2 δ 4 2 M∞ 4 M∞ b=− d=− cos2 δ 4 M∞ It is noted that oblique shock angle is given by the second root to the polynomial in equation 2.49. The first root corresponds to the strong shock, and the second root corresponds to the weak shock solution. With the oblique shock angle determined, it is possible to calculate the corresponding pressure, temperature, and Mach number behind the shock wave. This can be 28 done with the following equations [8, 7], 2 · sin2 θ − 1 p1 7M∞ s = p∞ 6 (2.50) 2 · sin2 θ − 1 M 2 · sin2 θ + 5 7M∞ T1 s s ∞ = 2 · sin2 θ T∞ 36M∞ s (2.51) M12 sin2 (θs − δ) = 2 · sin2 θ + 5 M∞ s 2 2 7M∞ · sin θs − 1 (2.52) The above set of equations will calculate the angle of the shock wave, and the corresponding temperature, pressure, and Mach number behind the shock wave if the previous stated condition for using oblique shock theory is true. If the angle of attack is less than −τ1,l , then PrandtlMeyer expansion fan theory should be used. This means that first the angle ν1 will need to be calculated using the following equation [8, 7], r ν1 = γ+1 tan−1 γ−1 r p γ+1 2 − 1) − tan−1 M 2 − 1 (M∞ ∞ γ−1 (2.53) Using the angle ν1 and the resulting angle ν2 = ν1 + δ, where δ is the angle of the expansion ramp in radians. To find the Mach number after the expansion, the following equation must be solved numerically [8, 7], r f (M∞ ) = γ+1 tan−1 γ−1 r p γ+1 2 − 1) − tan−1 M 2 − 1 − ν (M∞ 2 ∞ γ−1 (2.54) Now the pressure and temperature can be calculated from the following equations [8, 7]. 1+ h γ−1 2 i 2 M∞ γ γ−1 p1 h i = γ−1 p∞ 2 1+ M 2 h i 2 M∞ T1 h i = γ−1 T∞ 2 1+ M 1+ (2.55) 1 γ−1 2 2 (2.56) 1 It should be noted that the solution to equation 2.54 may not exist. This will impose limitations on the angle of attack and on control surface deflection for the hypersonic vehicle operation. 29 Now that the values for the pressure downstream have been calculated, it is possible to calculate the resulting forces on the forebody of the vehicle using the following equations [8, 7], Fx,f = −pf Lf tan(τ1,l ) (2.57) Fz,f = −pf Lf (2.58) where Lf is the length of the forebody, and pf is the static pressure on the forebody. The pitching moment can be calculated by, Mf = zf Fx,f − xf Fz,f (2.59) The aerodynamic center of the lower forebody is at the midpoint of the surface, (xf , zf ), because the pressure distribution is uniform over the surface. This is also true for the upper surface of the aircraft. The force and moment equations for the upper surface of the forebody is given by, Fx,u = −pu Lu tan(τ1,u ) (2.60) Fz,u = pu Lu (2.61) Mu = zu Fx,u − xu Fz,u (2.62) where (xu , zu ) is the center of the upper surface panel relative to the center of mass. Now that the forces and moments for the lower and upper forebody of the hypersonic vehicle have been defined, it is necessary to come up with a model for the scramjet engine. The model for the scramjet comes from the work done by Bolender and Doman [8, 7]. In figure 2.3, the scramjet has been divided into stages. At the first stage, the conditions are essentially given by the Mach number and angle of attack at which the vehicle is flying at. It is assumed in this model that the flow through diffuser is isentropic. There are two control efforts inside of the scramjet. These are the diffuser area ratio, Ad , and the fuel equivalence ratio, Φ. The fuel equivalence ratio controls the change in temperature of the section of the scramjet between cross section two and cross section 3 by controlling the fuel flow rate into the combustion chamber. This is favorable in doing the calculations for combustion which are necessary for calculating thrust since the air flow rate is known. Additionally, by controlling the diffuser area ratio, it is possible to control the Mach number and static pressure of the air entering the the combustion chamber. It should be noted at this point that the scramjet is sensitive to changes in fuel flow rate, diffuser area ratio, and inlet pressure and Mach number. There is a region of operability, but under the wrong conditions it is possible to choke the flame in the combustion chamber by providing too much fuel or too much air. 30 A1 M 1 , p1 Ae A3 A2 Figure 2.3: Scramjet Cross Section [8, 7] To calculate the thrust of the vehicle, the parameters at each stage of the combustion must be calculated. Using the continuity equation on the diffuser yields the following equation which is used to determine the Mach number at the diffuser exit/combuster inlet given the diffuser inlet Mach. 1+ h γ−1 2 i M22 M22 γ+1 γ−1 = A2d 1+ h γ−1 2 i M12 γ+1 γ−1 M12 (2.63) The pressure and temperature at the combustion chamber inlet (A2 ) can be determined by the Prandtl-Meyer theory in equations 2.55 and 2.56. The combuster itself is modeled as having no friction with heat addition. The Mach number at the combustion chamber exit can be calculated by, M22 1 + 12 (γ − 1)M22 M32 1 + 21 (γ − 1)M32 M22 ∆Tc = + 2 · 2 2 2 2 2 T2 γM3 + 1 γM2 + 1 γM2 + 1 (2.64) and the pressure and temperature at the exit are calculated by, p3 = p2 1 + γM22 1 + γM32 2 M32 1 + γM22 T3 = 2 · 2 M2 1 + γM32 (2.65) (2.66) The change in temperature in equation 2.64 is due to the burning of fuel in the scramjet. There are some scaling problems when simply looking at ∆Tc , so the fuel equivalence ratio, φ will be used. The fuel equivalence ratio is dependent upon the fuel to air ratio (f = ṁf /ṁa ), and the stoichiometric fuel to air ratio, fst . The fuel equivalence ratio is simply the ratio of these two values (φ = f /fst ). Looking at the enthalpy flux for the combustion chamber, ηc ṁf Hf = ṁa (ht3 − ht2 ) + ṁf ht3 31 (2.67) Table 2.1: LH2 fuel Thermodynamic Properties fst Hf ,BTU/lbm cp ,BT U/lbm◦ R .00291 51,500 .24 The stoichiometric fuel to air ratio and the fuel lower heating value are going to depend on the type of fuel used for the combustion. In this study, it is assumed that the fuel used is LH2. The properties for this fuel are found in table 2.1. Also, the combustion efficiency, ηc for this study is going to be assumed to be 0.9. Since the total enthalpy is, ht = cp Tt (2.68) Substituting back into equation 2.67 yields the following equation: cp Tt2 + Hf ηc fst φ Tt3 = Tt2 cp Tt2 + fst φcp Tt2 (2.69) Now that the ratio for total combustion chamber temperature has been calculated, it is possible to obtain the total change in temperature, ∆Tc = Tt3 − Tt2 . Using the Mach, pressure, and temperature calculated from equations 2.64-2.66 it is possible to calculate the exit Mach number from the scramjet with the following equation. 1+ h γ−1 2 i Me2 Me2 γ+1 γ−1 = A2n 1+ h γ−1 2 i M32 γ+1 γ−1 M32 (2.70) The temperature and pressure at the exit of the scramjet can be calculated by applying equations 2.55 and 2.56. Now that the scramjet properties have been determined, it is possible to calculate the thrust generated by the scramjet by using the momentum theorem from fluid mechanics to a control volume that encloses the scramjet engine. This yields the following equation, T = ṁa (Ve − V∞ ) + (pe − p∞ ) Ae − (p1 − p∞ ) Ai (2.71) where the mass flow rate of air into the engine, ṁa , is given by, r ṁa = p∞ M∞ γ cos (τ12 ) hi sin (θs ) RT∞ sin (θs − τ12 − α) 32 (2.72) If the shock wave angle is zero, then equation 2.72 becomes, r ṁa = p∞ M∞ γ hi RT∞ (2.73) In order to calculate the thrust of the engine, it is important to properly calculate the forces and moments at the engine inlet. Therefore, it is necessary to look at the inlet turning force. The inlet turning force is the force imparted on the aircraft by turning the incoming airflow which is parallel to the forebody after the oblique shock wave to be parallel to the engine centerline as it goes into the scramjet. This results in the following equations for the force and moment imparted on the vehicle. Fx,inlet = γM12 p1 [1 − cos (τ1,l + α)] Fz,inlet = γM12 p1 sin (τ1,l + α) Ae 1 b Ad An Ae 1 b Ad An Minlet = zinlet Fx,inlet − xinlet Fz,inlet (2.74) (2.75) (2.76) Now that the forces and moments have been determined for the forebody of the hypersonic vehicle, and the thrust for the scramjet with the resulting moments and forces have been described, the next step is to calculate the forces and moments on the aftbody of the vehicle. The exhaust from the scramjet uses the aft of the vehicle as a nozzle. The upper part of the nozzle is formed by the vehicle body itself, while the lower part of the nozzle is formed by the shear layer coming off of the back of the vehicle. The resulting pressure distribution can be expressed as, pa = 1+ sa La p e pe p∞ −1 (2.77) where sa is the length along the aftbody panel. It should be noted that equation 2.77 is an approximation of the aftbody pressure, and that in order to accurately capture the pressure, the location of the shear layer must be known. This is computationally expensive, and the study conducted by Bolender and Doman has shown that this approximation is sufficient for estimating the pressure on the aft of the vehicle [8, 7]. With this pressure estimation, the following force and moment equations can be derived, Fx,a = p∞ La pe p∞ ln (pe /p∞ ) tan (τ2 + τ1,u ) (pe /p∞ ) − 1 Fz,a = −p∞ La pe p∞ 33 ln (pe /p∞ ) (pe /p∞ ) − 1 (2.78) (2.79) Ma = za Fx,a − xa Fz,a (2.80) where the pitching moment acts on the mean value of the pressure distribution RL p̄a = 0 a pa (x)dx/La , and (xa , za ) are the coordinates of the mid-point of the aft body relative to the center of mass of the hypersonic vehicle. The last thing that needs to be modeled in the aerodynamic study is the control surfaces (i.e. the canard and elevator). These will be modeled as flat plates, and the resulting forces and moments will be calculated. One side of the plate will be calculated by using oblique shock theory, the other side of the plate will use Prandtl-Meyer expansion fan theory. The reader should note that the location of these control surfaces changes as the vehicle flexes during flight. The calculation for the forces and moments for the canard are the same as those used for the elevator. The results of the oblique shock theory and the Prandtl-Meyer theory give the pressures on the top and bottom surface of the plate. These pressures are then used to calculate the forces and moments using the following equations, Fx,e = − (pe,l − pe,u ) sin (δe ) Se (2.81) Fz,e = − (pe,l − pe,u ) cos (δe ) Se (2.82) Me = ze Fx,e − xe Fz,e (2.83) These equations will be the same for the canard with the subscript being changed from an e to a c, so they will not be repeated here. Now all of the individual components of the hypersonic vehicle have been taken into account, so it is possible to calculate the resulting lift, drag, and moment for the vehicle. The thrust has already been shown in equation 2.71. The final moment equation is expressed by, M = Mf + Mu + Ma + Me + Mc + Minlet + zT T (2.84) where zT is the distance from the centerline of the scramjet to the center of gravity along the z axis in the body frame. Similarly, the forces in the x and z directions can be summed up in the body frame. The lift and drag on the vehicle can then be expressed as, L = Fx sin(α) − Fz cos(α) (2.85) D = −Fx cos(α) − Fz sin(α) (2.86) 34 where, Fx = Fx,f + Fx,e + Fx,c + Fx,a + Fx,u + Fx,inlet (2.87) Fz = Fz,f + Fz,e + Fz,c + Fz,a + Fz,u + Fz,inlet + Fz,b (2.88) Fz,b = pn · Ln (2.89) and, where, Where pn is the pressure on the bottom of the nacelle, and Ln is the length of the nacelle. This concludes the review of the hypersonic vehicle aerodynamics. The next section will explore the model linearization. 2.2.3 Actuator Dynamics Though it is not a part of the open loop system as shown in the previous section, it is important to discuss the actuator dynamics of the system. When designing a control system for a hypersonic vehicle, it is important to take the actuator dynamics into account. There are bandwidth limitations as well as saturation limits imposed by the actuators in the system. If the controller depends upon the actuators providing control effort that is outside of the operating range for the actuator or if the needed response of the actuator is faster than the actuator can achieve, then stability in the system maybe lost. Therefore, it is important to include some sort of actuator dynamics in the system for this reason. For the hypersonic vehicle in this study, there is no information on the actual actuators being used since the model is a hypothetical design. Others have attempted to come up with simple second order differential equations for the actuator dynamics [25, 47]. This model does capture a description that could very well describe the actuator dynamics, but for the purposes of this study it was decided that a more general approach would be suitable. By taking the method suggested by Groves et al. and Sigthorsson et al., the actuators are modeled as low pass filters [28, 51]. This incorporates a bandwidth limitation on the response of the actuator output without making assumptions to the internal workings of the actuators themselves. Since the actuators are hypothetical, it would be difficult to characterize them accurately. It has been assumed for this study that this method for modeling the actuator dynamics is sufficient. It will also be necessary to apply saturation limits to the actuators themselves, but this will be taken care of in the simulation. For the particular hypersonic vehicle used in this study, there will be a total of four control inputs to the system. These four control inputs are defined in equation 2.96. They are δe , the 35 elevator angle, δc , the canard angle, φ, the fuel equivalence ratio, and Ad , the diffuser area ratio. For a realistic model of the hypersonic vehicle, it will be necessary to describe the dynamics of these actuators in the control synthesis. It is not possible for an actuator to give a nearly infinite amount of control effort in a differentially small amount of time, therefore actuator dynamics must be described. The dynamics are as follows, −20 0 0 0 0 −20 0 0 Ad = 0 −10 0 0 0 0 0 −10 (2.90) 20 0 0 0 0 20 0 0 Bd = 0 0 10 0 0 0 0 10 (2.91) ẋd = Ad · xd + Bd · ud (2.92) where, Equations 2.90-2.92 describe the model used for the actuators in this study. This model of the actuators does not represent a real set of actuator dynamics, but rather is chosen to set certain frequency response limitations on the actuator response. Low pass filters were chosen here so that the design could remain as generic as possible while still accurately describing what a hypersonic vehicle would be able to achieve. Since the hypersonic vehicle being considered in this study is only a conceptual vehicle based off of previous designs and concepts, it is assumed that modeling the actuators in this way is suitable. From the actuator dynamics, the cutoff frequency for δe and δc is roughly 20 Hz, while the cutoff frequency for φ and Ad is 10 Hz. This will limit the frequency response of the control effort such that the response lies within a realistic range. Additionally, table 2.2 shows the saturation limits that will be imposed upon the control efforts. 2.3 Model Linearization Now that a derivation of the nonlinear model has been presented, it is necessary to discuss the linearization of the model. When applying an H∞ LPV controller, the nonlinear system is turned into a finite number of linearized models for a set of given parameters to the system. Linear controllers are then synthesized at each of these linearized points. For this particular study, 49 total linear plant models will be derived based upon the variation of the velocity and 36 Table 2.2: Actuator Saturation Limits Actuator δe δc φ Ad Upper Limit π 6 π 9 .77 1 Lower Limit π − 12 − π9 .1 .01 altitude of the vehicle. The details of this parameterization will be discussed in chapter 3. This section will not set out to discuss the process of synthesizing controllers, but it will discuss the process for linearization. This will be important to cover before an open loop analysis of the hypersonic vehicle dynamics can be performed. For this study, the system is linearized for a given Mach number and altitude. With these two values, a trim equilibrium is obtained by setting up an optimization problem to get the trim conditions for the given Mach and altitude. Once the trim conditions have been obtained, and linearized model can be generated easily. To obtain the trim conditions, lower and upper bounds are set for the control surfaces, and then a set of equalities is specified to minimize the constrained nonlinear multivariable function [39]. The goal is to find the minimum of a nonlinear multivariable function, min f (x, u) subject to the following conditions are satisfied x,u Aeq · " # x u = beq λ≤x≤λ (2.93) (2.94) In this case, f (x) is going to be the set of equations defined by 2.41-2.48. The input to f (x) will be the state vector x which is defined as, x = [Vt α Q h θ η1 η̇1 η2 η̇2 η3 η̇3 ]T (2.95) and the control effort which will be defined as, h iT u = δe δ c φ A d (2.96) as well as the φn defined in equation 2.9. The following values are used for Aeq and beq respectively. Aeq = [0 − 1 0 0 1 0 0 0 0 0 0 0 0 0 0] 37 (2.97) beq = 0 (2.98) The upper and lower bounds for the optimization are chosen such that the system states remain within a feasible range, and so that the control effort is within the saturation limits. In addition to being with the saturation limits for the actuators, it was found that reducing the range of the optimization for the linearization improved the performance of the closed loop system when switching between controllers. This was largely due to the trim conditions for the actuators being relatively close to one another. Since this is the case, the following bounds were chosen for the system. λ = Vt 3π 3π 10π 0 h 3 0 1 0 1 0 0 .5 .8 180 180 180 T −5π .1 .6 λ = Vt 0 0 h 0 − 1 0 − 1 0 − 1 0 0 180 (2.99) T (2.100) Once these values have been set, it is possible to run the optimization for the system to obtain the vector describing the trim conditions for the 11 plant states and the 4 control states of the system. The resulting trim conditions are then used to generate the linearized plant. To generate the linearized plant from the nonlinear system, assume that the nonlinear function takes the form of, ẋ = f (x, u) = ∂f ∂f (x − x) + (u − u) ∂x x̄,ū ∂u x̄,ū (2.101) This equation represents the equation of motion for the system where A= ∂f ∂x x̄,ū (2.102) B= ∂f ∂u x̄,ū (2.103) and, It is possible to approximate A and B by the following two equations − f x − ∆x 2 ,u ∆x f x, u + ∆u − f x, u − ∆u ∂f 2 2 = ∂u ∆u f x+ ∂f = ∂x ∆x 2 ,u (2.104) (2.105) Equations 2.104 and 2.105 both hold true as lim , so as long as (∆x, ∆u) is sufficiently small, ∆x→0 the linearized plants hold true. Similarly, the C matrix for the linearized system will just be identity, and the D matrix will be null. Now that the linearized system and trim conditions 38 have been obtained for a given Mach and altitude, it is possible to synthesize a controller for the system. Before looking at control synthesis however, it is beneficial to look at the open loop dynamics for the system. 2.4 Open Loop Analysis This section will investigate the open loop dynamics of the hypersonic vehicle. It will not be feasible to look at every possible plant for the system as there are an infinite number of linearized plants that can be generated. For this reason, only five plants will be examined in detail. The range for Mach in this study will be from Mach 7 to Mach 9, and the range for altitude will be from 70,000 feet to 90,000 feet. Therefore, the five linearized plants that will be considered will be for the minimum of this range, the maximum of this range, and the mid point of this range. This will help the reader to gain some understanding of the open loop characteristics of the hypersonic vehicle across the range of operation. First, look at the natural frequencies of the hypersonic vehicle in table 2.3. This table shows the first three natural frequencies for the vibration of the vehicle. Note that the ωn3 is about 98 Hz. It is unlikely that any of the vehicles actuators will be operating at a frequency that high. This means that it is unlikely that higher order modes of vibration will be excited by the control effort, so at least from this aspect it is suitable to only retain the first three modes of vibration. Table 2.3: Natural Frequencies for Hypersonic Vehicle Natural ωn1 ωn2 ωn3 Frequencies 20.3468 49.2371 97.7239 Tables 2.4 and 2.5 show the Eigenvalues of the flexible open loop plants and the rigid body open loop plants respectively. All open loop systems are unstable. The Eigenvalues show that the rigid body plant behaves differently from the flexible body plant. There is a significant difference between the poles of the rigid body system and the flexible body system for a given system state. There are also significant differences between the poles of the three systems in this open loop analysis for a given state. 39 In addition to the Eigenvalues for the two systems, the Bode plots for the open loop systems were analyzed. The Bode plots listed in this section show the open loop characteristics for the cases where the altitude is 70,000 feet and the speed is Mach 7, the altitude is 70,000 feet and the speed is Mach 9, the altitude is 80,000 feet and the Mach is 8, the altitude is 90,000 feet and the Mach is 7, and the altitude is 90,000 feet and the Mach is 9. It can be seen from figures 2.4(a)-2.9(b) that there is a significant difference in the open loop dynamics between the rigid body model, and the flexible body model. The Bode plots show that there is a difference in the location and magnitude of the peaks between the rigid body model and the flexible body model. This verifies the results of the study completed by Chavez and Schmidt which discuss the importance of including the flexible effects of the hypersonic vehicle into the vehicle dynamics [12]. This study will investigate the ability of the robust control algorithm to control a flexible body plant using a controller synthesized both with and without the flexible effects. In the case of rigid body controller only rigid body effects will be retained and, the flexible effects of the plant will be treated as disturbances to the system. This will be discussed more in later chapters. 40 Table 2.4: Flexible Open Loop Eigenvalues 70,000 ft Mach 7 -1.9548 + 97.6861 i -1.9548 - 97.6861 i -0.9845 + 49.6968 i -0.9845 - 49.6968 i -0.4385 + 24.0045 i -0.4385 - 24.0045 i -3.7647 3.6020 0.0003 + 0.0329 i 0.0003 - 0.0329 i -0.0016 70,000 ft Mach 9 -1.9557 + 97.3151 i -1.9557 - 97.3151 i -0.9866 + 49.4139 i -0.9866 - 49.4139 i -0.4595 + 24.2929 i -0.4595 - 24.2929 i -6.2512 6.1054 -0.0013 0 + 0.0310 i 0 - 0.0310 i 80,000 ft Mach 8 -1.9549 + 97.5144 i -1.9549 - 97.5144 i -0.9852 + 49.4036 i -0.9852 - 49.4036 i -0.4272 + 22.6697 i -0.4272 - 22.6697 i -4.3964 4.2802 -0.0012 0 + 0.0337 i 0 - 0.0337 i 90,000 ft Mach 7 -1.9546 + 97.5733 i -1.9546 - 97.5733 i -0.9849 + 49.3411 i -0.9849 - 49.3411 i -0.4136 + 21.5416 i -0.4136 - 21.5416 i 3.3422 -3.4190 -0.0008 0 + 0.0374 i 0 - 0.0374 i 90,000 ft Mach 9 -1.9546 + 97.5336 i -1.9546 - 97.5336 i -0.9850 + 49.3954 i -0.9850 - 49.3954 i -0.4174 + 22.0907 i -0.4174 - 22.0907 i -4.0229 3.9454 -0.0008 0 + 0.0364 i 0 - 0.0364 i Table 2.5: Rigid Open Loop Eigenvalues 70,000 ft Mach 7 -4.4110 4.1850 -0.0017 0.0004 + 0.0338 i 0.0004 - 0.0338 i 70,000 ft Mach 9 -7.0407 6.7837 -0.0014 0 + 0.0394 i 0 - 0.0394 i 80,000 ft Mach 8 -4.6792 4.5209 -0.0012 0 + 0.0398 i 0 - 0.0398 i 41 90,000 ft Mach 7 3.4596 -3.5501 -0.0009 0 + 0.0393 i 0 - 0.0393 i 90,000 ft Mach 9 -4.2686 4.1695 -0.0008 0 + 0.0390 i 0 - 0.0390 i Flexible System Elevator to Velocity Bode Plot 100 Rigid System Elevator to Velocity Bode Plot 150 70,000 ft at Mach 7 80,000 ft at Mach 8 100 90,000 ft at Mach 9 50 70,000 ft at Mach 9 90,000 ft at Mach 7 50 70,000 ft at Mach 7 0 80,000 ft at Mach 8 0 90,000 ft at Mach 9 70,000 ft at Mach 9 −50 −2 10 90,000 ft at Mach 7 −1 0 10 10 1 10 2 10 3 10 −50 −2 10 4 10 200 200 150 150 100 100 50 50 0 −50 −2 10 −1 10 −1 10 10 0 10 1 10 0 10 2 10 1 10 3 10 2 10 4 3 10 3 10 3 10 3 10 3 10 0 −1 0 10 10 1 10 2 10 3 10 −50 −2 10 4 10 10 (a) Flexible Body 4 (b) Rigid Body Figure 2.4: Bode plot of δe to Vt for the open loop plant Flexible System Fuel Equivalence Ratio to Velocity Bode Plot 80 60 40 40 20 0 −20 −40 −60 −2 10 Rigid System Fuel Equivalence Ratio to Velocity Bode Plot 80 60 20 0 70,000 ft at Mach 7 80,000 ft at Mach 8 −20 90,000 ft at Mach 9 70,000 ft at Mach 9 −40 90,000 ft at Mach 7 −1 0 10 10 1 10 2 10 3 10 −60 −2 10 4 10 −60 −20 −80 −40 70,000 ft at Mach 7 80,000 ft at Mach 8 90,000 ft at Mach 9 70,000 ft at Mach 9 90,000 ft at Mach 7 −1 10 −1 10 10 0 10 1 10 0 10 2 10 1 10 2 10 4 −60 −100 −80 −120 −100 −140 −160 −2 10 −120 −1 0 10 10 1 10 2 10 3 10 −140 −2 10 4 10 10 (a) Flexible Body 4 (b) Rigid Body Figure 2.5: Bode plot of φ to Vt for the open loop plant Flexible System Diffuser Area Ratio to Velocity Bode Plot 100 50 50 0 0 70,000 ft at Mach 7 70,000 ft at Mach 7 80,000 ft at Mach 8 −50 80,000 ft at Mach 8 −50 90,000 ft at Mach 9 70,000 ft at Mach 9 −100 −2 10 −1 10 90,000 ft at Mach 9 70,000 ft at Mach 9 90,000 ft at Mach 7 0 10 1 10 2 10 3 10 −100 −2 10 4 10 200 200 150 150 100 100 50 50 0 −50 −2 10 Flexible System Diffuser Area Ratio to Velocity Bode Plot 100 90,000 ft at Mach 7 −1 10 −1 10 10 0 10 1 10 0 10 2 10 1 10 2 10 4 0 −1 10 0 10 1 10 2 10 3 10 −50 −2 10 4 10 (a) Flexible Body 10 (b) Rigid Body Figure 2.6: Bode plot of Ad to Vt for the open loop plant 42 4 Flexible System Elevator to Altitude Bode Plot 150 150 100 50 50 0 0 70,000 ft at Mach 7 −50 80,000 ft at Mach 8 −50 90,000 ft at Mach 9 −100 −150 −2 10 Rigid System Elevator to Altitude Bode Plot 200 100 70,000 ft at Mach 9 −100 90,000 ft at Mach 7 −1 0 10 10 1 10 2 10 3 10 −150 −2 10 4 10 350 70,000 ft at Mach 7 80,000 ft at Mach 8 90,000 ft at Mach 9 70,000 ft at Mach 9 90,000 ft at Mach 7 −1 10 −1 10 10 0 10 1 10 0 10 2 10 1 10 3 10 2 10 4 3 10 3 10 3 10 3 10 3 10 200 300 150 250 200 100 150 100 50 50 0 −2 10 −1 0 10 10 1 10 2 10 3 10 0 −2 10 4 10 10 (a) Flexible Body 4 (b) Rigid Body Figure 2.7: Bode plot of δe to h for the open loop plant Flexible System Fuel Equivalence Ratio to Altitude Bode Plot 150 100 50 50 0 0 70,000 ft at Mach 7 −50 70,000 ft at Mach 7 −50 80,000 ft at Mach 8 90,000 ft at Mach 9 −100 −150 −2 10 Rigid System Fuel Equivalence Ratio to Altitude Bode Plot 150 100 70,000 ft at Mach 9 90,000 ft at Mach 7 −1 0 10 80,000 ft at Mach 8 90,000 ft at Mach 9 −100 10 1 10 2 10 3 10 −150 −2 10 4 10 400 70,000 ft at Mach 9 90,000 ft at Mach 7 −1 10 −1 10 10 0 10 1 10 0 10 2 10 1 10 2 10 4 200 300 100 200 100 0 0 −100 −100 −200 −200 −300 −2 10 −1 0 10 10 1 10 2 10 3 10 −300 −2 10 4 10 10 (a) Flexible Body 4 (b) Rigid Body Figure 2.8: Bode plot of φ to h for the open loop plant Flexible System Diffuser Area Ratio to Altitude Bode Plot 150 100 50 50 0 −50 −100 −150 −200 −2 10 Rigid System Diffuser Area Ratio to Altitude Bode Plot 150 100 0 −50 70,000 ft at Mach 7 80,000 ft at Mach 8 −100 90,000 ft at Mach 9 70,000 ft at Mach 9 −150 90,000 ft at Mach 7 −1 10 0 10 1 10 2 10 3 10 −200 −2 10 4 10 800 400 600 300 70,000 ft at Mach 7 80,000 ft at Mach 8 90,000 ft at Mach 9 70,000 ft at Mach 9 90,000 ft at Mach 7 −1 10 −1 10 10 0 10 1 10 0 10 2 10 1 10 2 10 4 200 400 100 200 0 0 −200 −2 10 −100 −1 10 0 10 1 10 2 10 3 10 −200 −2 10 4 10 (a) Flexible Body 10 (b) Rigid Body Figure 2.9: Bode plot of Ad to h for the open loop plant 43 4 From table 2.4, it can be seen that the last six Eigenvalues represent the flexible effects of the system. The first two values are the Eigenvalues associated with the first mode, the third and fourth Eigenvalues represent the the second mode, and the fifth and sixth Eigenvalues represent the third mode. From here it is possible to see that the real part of the Eigenvalue starts to get very small. This means that the flexible effects of the higher order modes will not be as great as the first mode of vibration. This seems to be a valid assumption from the basic open loop analysis, and can be verified through closed loop simulation. Figure 2.10 shows the modes of vibration for the first three flexible modes of vibration using the assumed modes method. First Three Mode Shapes 0.3 0.2 Displacement in Feet 0.1 0 −0.1 −0.2 −0.3 First Mode Second Mode Third Mode −0.4 0 10 20 30 40 50 60 Distance in Feet 70 80 90 100 Figure 2.10: Hypersonic Vehicle Mode Shapes 2.5 Conclusions This chapter has briefly outlined the derivation of the equations of motion for the hypersonic vehicle, and the basis for flexibility modeling in the hypersonic vehicle. A brief discussion of the aerodynamics was included as well as linearization and an open loop analysis. The actuator dynamics were also included in this study as well. The results of the section show that the model is a high fidelity closed form solution to the hypersonic vehicle dynamics. There are several aspects of modeling that have still not been discussed though [6, 46, 54, 14]. Amongst these issues is thermal effects on the hypersonic vehicle system. No account for the expansion 44 or the change of stiffness of the vehicle during flight has been accounted for in this model. Though the thermal effects of hypersonic flight do in fact have an effect on the system, it can be difficult to model these thermal effects. Instead of modeling these effects in this study, it has been determined that an investigation of the control systems robustness to certain physical parameter uncertainties will be considered instead. The uncertainties that will be considered will be the mass of the vehicle, the moment of inertia for the vehicle, and the air properties for air at a given altitude (i.e. temperature, pressure, and density). These values may not completely encompass the complete thermal effects on the vehicle, but it is considered to be sufficient for a preliminary study. This chapter has also made some additional assumptions as well. The flexibility of the aircraft has been approximated using the assumed modes approach with only three modes of vibration. This yields results which have been determined to be accurate enough for this situation, but in order to get an increased amount of accuracy in the calculations of mode shapes, either an increase in the number of modes accounted for must applied, or a finite element analysis will be necessary. The problem is that it is hard to use a control algorithm on a finite element analysis because this numerical technique does not yield a closed form equation of motion. Future research in this area may yield an improved analysis of the flexibility of the vehicle, but currently this approximation is thought to be sufficient. Similarly, increasing the number of modes considered in the calculations makes the control synthesis more difficult from a computational standpoint. Current computer hardware limitations make solving higher ordered systems more difficult using the control method suggested in this study. Specifically, the increased order of the plant creates more LMI’s that must be solved. Another issue with this particular model is that it does not always yield a solution to certain aerodynamic situations. There are not always real roots to some of the polynomials that must be solved for using this method. This cause some discontinuities in the flight envelope of the vehicle. Some of these discontinuities are a result of the hypersonic vehicle’s physical flight limitations while others depend more upon the assumptions made to ensure that the vehicle is operating near its operating flight conditions [8, 7, 40, 54]. This study will not investigate the entire operable flight envelope, but future work should in fact investigate the necessary limitations on actuators and flight conditions to ensure that stable flight is always obtainable. Additionally, it should be noted that many simplifications and assumptions have been made to calculate the different hypersonic flow properties. The best method for calculating the flow properties for the hypersonic vehicle is to use computational fluid dynamics, but this method of numerical analysis is not well suited for feedback control synthesis, therefore alternate methods must be used for the control design process as suggested here [6]. 45 In spite of some of the current limitations, the model used for this study is in fact a rather good progression from previously proposed models for hypersonic vehicles. The work in the area of hypersonic vehicle dynamics continues to progress [6]. This particular model was chosen because of its ability to accurately estimate the true hypersonic vehicle dynamics while still maintaining a closed form equation of motion for the purpose of applying a feedback control system. Though it is not exact, this particular model seems to be the one of the best closed form description of the vehicle dynamics currently available. 46 Chapter 3 Full State Feedback Control for Hypersonic Vehicle The previous chapters have discussed the theory of H∞ LPV control design techniques and hypersonic vehicle dynamics. This chapter will discuss the application of the H∞ LPV controller to a specific hypersonic vehicle model. The focus of this chapter will be the synthesis and simulation of a full state feedback controller for the hypersonic vehicle. This chapter will look at the velocity tracking and altitude tracking of the vehicle. Comparisons will be made between the H∞ LPV controllers and optimal linear H∞ controller design. The synthesis conditions will be analyzed, and the ideal case will be simulated. The effects of uncertainty in the system will be investigated as well. As described in chapters 1 and 2, the continuous system will be discretized into a finite number of points. At each of these points, the system will be linearized and a controller will be synthesized at each of the subsequent points. The goal is to design a set of LPV controllers such that the stable region of each controller overlaps the other neighboring controllers to yield a set of controllers that have a controllable region that encompasses the entire parameter space. This can be seen in figure 3.1. For this study, the parameters chosen for the LPV synthesis will be the Mach number and the altitude of the vehicle. 3.1 Control Synthesis Before the simulation results can be evaluated, it is necessary to design the controller for the hypersonic vehicle. The theory for H∞ LPV control design has been discussed in chapter 1. This section will demonstrate how this theory is directly applied to the hypersonic vehicle for the full state feedback case with consideration for both the velocity tracking case and the altitude 47 Altitude Linearized HSV Stable Region for LPV Controller Mach Figure 3.1: Parameterized Space with Linearized Hypersonic Vehicle Grid tracking case. Consideration will be given to determine what the parameter variation rates and griding structures should be in order to improve the robust performance of the controlled hypersonic vehicle. 3.1.1 Velocity Tracking First the velocity tracking case will be considered. The state vector for the flexible body case is the same as in equation 2.95. This basic description of the linearized plant is the starting point for designing a controller to track velocity. The state vector for the flexible hypersonic vehicle has been defined as, xp = [Vt α Q h θ η1 η̇1 η2 η̇2 η3 η̇3 ]T (3.1) By linearizing the nonlinear hypersonic vehicle model at specified equilibrium conditions, one can determine the following state space model for the linearized plant. ẋp = Ap xp + Bp u (3.2) yp = Cp xp (3.3) Using this state space system, it will be necessary to augment to the plant dynamics at each trim condition with both the actuator states in equations 2.90 and 2.91 along with the a new 48 state to describe the velocity tracking state. Figure 3.2 shows the open loop interconnection for the augmented system. In this figure, P is the plant which has been linearized about a set of P(1) u P Wact P(2:11) Wact (1 : 4) ref + - e 1 s e Figure 3.2: Open Loop Interconnected System For Velocity Tracking trim conditions and Wact is the actuator dynamics defined in 2.90 and 2.91. P is the linearized dynamics. Notice from figure 3.2 that the difference between the reference velocity and the actual system velocity is integrated and fed through to the controller. This integral state enforces the system to track the reference velocity. The open loop interconnected plant will now be referred to as, " ss Polic = Aolic Bolic w · Colic w · Dolic # (3.4) The open loop system for H∞ LPV control synthesis is defined in equations 1.8-1.10. Using T R this set of equations for the system, the state vector for Polic is,x = xp δe δc φ Ad (ref − v) . R T The controlled output from Polic becomes, y = [δe δc φ Ad ]T where, e = v (ref − v) and u = [δe δc φ Ad ]T . 49 For the purpose of synthesis, Colic and Dolic in equation 3.4 are multiplied by a weighting function W . This weight function is defined as, 1 0 0 0 0 0 .5 0 0 0 0 0 1000 0 0 W = 0 0 0 1000 0 0 0 0 0 316.2778 0 0 0 0 0 0 0 0 0 0 1000 (3.5) For this study, a constant weighting function was chosen to penalize the control effort. These values were chosen ad hoc for this study through a process of trial and error. The reader should note that even though these weighting functions do work for the system, they may not be the optimal choice. Using a frequency based weighting function may even prove to provide better results, but for the purposes of this study, it was assumed that a constant set of weighting functions would suffice. In this particular weighting function, only the diagonal terms are nonzero. The first two terms are applied to the error states in the system, and the last four terms are applied to the control efforts. In this weighting function, large values add a higher penalty to the particular value. By selecting these weighting functions, the controller will penalize the control efforts because of the high weightings associated with these states. It is beneficial to choose high values for the weighting function on the states associated with the actuators in the system because it is important to keep the actuators from saturating their limits. In order to aid in this pursuit, high weighting functions are chosen because they will have a strong penalty to the actuator effort when synthesizing a controller. Respectively, it can be seen that the error states are chosen to be much smaller because it will be desirable for the error dynamics to be very small. In order for this to occur, the penalties applied to these states should be small so that the synthesized controller will have more effect on these states. Therefore, these values are chosen to be small such that the desired performance can be achieved. Now the open loop interconnected plant has been established for a single linearized point for a given Mach number and altitude. By extension, this can be applied to all of the chosen trim conditions for the hypersonic vehicle. The Mach number and altitude correspond to the parameters ρ1 and ρ2 from the LPV synthesis discussed in chapter 1. A set of open loop interconnected plants has been developed, it is now possible to start synthesizing a set of LPV H∞ controllers for the system. To do this, first it will be necessary to parameterize R(ρ). To this end, it is necessary to determine the basis function for R(ρ). 50 Assume that R(ρ) is parameterized in the following form, R (ρ) = R0 + ρ1 R1 + ρ2 R2 (3.6) Notice that for this study, R(ρ) has been chosen as a simple linear parameter dependent function. From equation 1.41, the following basis function vector f (ρ) takes the form of, f (ρ) = [1 ρ1 ρ2 ] (3.7) for all parameter points. With the basis function vector and its gradient defined, it is also necessary to define a set of bounds for the parameter variation rate, ν. For this dissertation, ν is defined as |ρ̇| ≤ ν (3.8) where ν is a constant vector representing both the upper and lower bound for the parameter variation rate. The first term describes the limitation on how quickly the velocity of the aircraft can change, and the second term describes how quickly the altitude can change. Now that the basis function and constants have been set, all of the criteria is met to solve for R(ρ) and γ. This is done by solving equation 1.28 using efficient LMI techniques. Now that R(ρ) and γ have been solved, the full state feedback LPV controller with H∞ optimization can now be found. This calculation is done simply by applying all of the known state information for the open loop interconnected plant, and the values calculated for R(ρ) and γ to the full state feedback equation 1.29. F (ρ) is the state feedback control gains for the closed loop system. Now that the H∞ LPV controller for the air-breathing hypersonic vehicle has been established, it will be informative to investigate the effects of changing the parameter variation rates as well as the number of griding points. To this end, various controllers were synthesized to determine the optimal values for each of these parameters for the hypersonic vehicle. The results of this parametric study will be used to synthesize a controller, and then analyze the response of the resulting closed-loop system. The first parameter that will be considered is the parameter variation rate limit, ν, defined in equation 3.8. The parameter variation rate puts a limitation on how quickly the parameters in the system can change with respect to time. Typically for H∞ LPV problems, there is a trade off between how robust the system is versus the performance of the system. The selection of the parameter variation rate limit has an effect on this tradeoff. It will therefore be beneficial to 51 investigate the effects of the parameter variation rate limit, ν, on the H∞ performance variable, γ, for the system. Assuming that the system has seven trim points for the Mach number and seven trim points for the altitude for a total of 49 linear trimmed plants with a velocity range from Mach 7 to Mach 9 and an altitude range from 70,000 feet to 90,000 feet with all trim conditions evenly spaced in the range, an H∞ LPV controller can be synthesized with different parameter variation rate limits. The results of these control synthesis problems can be seen in table 3.1. The lower the parameter variation rate limits are, the more robust the controller is. However, these limitations are also imposed on the system such that the change in the Mach number and altitude with respect to time are both limited. It is therefore necessary to choose a parameter variation rate limit that gives the desired balance between performance and robustness. From table 3.1, it can be seen that the γ performance variable does not change significantly for the different cases listed. The lower parameter variation rate limits do in fact have a lower γ performance variable which indicates an increase in the robust capabilities of the controller, but for the full state feedback velocity tracking case, it is not a significant difference. These results show that the difference between the robust capabilities of the different controllers synthesized is small. Because of these results, the case where ν = [.5 200]T is chosen for this study since it will have a balance between robust capabilities and system performance. Table 3.1: γ performance for different parameter variation rates Parameter variation rate limits ν H∞ performance variable γ [.1 50]T 2.2201 [.3 200] T 2.2219 [.5 200] T 2.2224 [.7 200] T 2.2227 [.5 100]T 2.2222 T 2.2225 T 2.2235 [.5 300] [1 500] Now that the parameter variation rate limits have been selected for the control synthesis, it will be important to look at the effect that changing the number of griding points in the parameter space will have on the synthesis of the system. To accomplish this task, the parameter variation rate limits will be specified as ν = [.5 200]T , which was discussed previously. The 52 goal of this study will be to evaluate the effect of additional griding points on the system’s robust capabilities. Typically in H∞ LPV control problems, having a denser griding in the parameter space will increase the H∞ γ performance value [30, 31]. Figure 3.3 shows the different parameter values for the Mach number and altitude for the different controller griding numbers considered in this study. For this study, it is arbitrarily assumed that the desired operating range is from 70,000 feet to 90,000 feet in altitude and from Mach 7 to Mach 9. Since the hypersonic vehicle is a nonlinear system, using different parameter points for the purpose of linearization can lead to drastically different synthesis results, even for the same grid size and spacing. The selection of certain parameter points may even prove to be outside of the operating region since the operating envelope is not continuous over the entire parameter space [40]. This arises from the fact that the operating envelope is, as McRuer put it, filled with holes like swiss cheese. Care should be taken when synthesizing a controller using this method to ensure that each of the linearized trim points exist inside the operating envelope for the vehicle, and not in one of these holes of inoperability. This can be difficult since there is no guidance to ensure the operating envelope of the hypersonic vehicle. Altitude 90,000 86,667 7x7 83,333 6x6 80,000 5x5 76,667 4x4 73,333 9 8.67 8.33 8 7.67 7.33 7 70,000 Mach Figure 3.3: Parameterized Space with Linearized Controller Grid for Different Cases (Note Larger Blocks Are Inclusive of Smaller Blocks) 53 The results for these control synthesis problems can be seen in table 3.2. In this table it can be seen that the H∞ γ performance value actually decreases with less points. For each of the three systems shown in table 3.2, there are a total of 409 optimization variables (OV’s) to be solved for in the LMI’s. As the number of griding points in the parameter space increases, the number of LMI’s also increases. The increasing number of constraints on the problem make the optimization problem more difficult to solve. The results in table 3.2, show that as the number of griding points in the parameter space are increased, the γ performance value is worse. It is important to choose a grid density such that the linearized controllers have valid operating ranges that cover the entire parameter space while an attempt is made to reduce the computational complexity of the problem. There is currently no method for ensuring that the entire parameter space lies within a controllable region of one of the linearized controllers, but it is possible to verify that the system controller will keep the vehicle within the operational limits through nonlinear simulation. The griding structure chosen for this study was the 7×7 structure since it exhibited the highest griding density that could be synthesized while maintaining robust performance capabilities. It can be seen from table 3.2 that the difference between the γ performance values is relatively low. The difference between the γ performance value for the 4 × 4 case and the 7 × 7 case was only about .3. Therefore, the 7 × 7 griding structure was chosen since it will balance robust capabilities with parameter space controllability. The 7 × 7 griding structure will result in 49 linearized trim points, and each of these trim points will have a controller synthesized. Table 3.2: γ performance for different number of griding points Grid dimensions 4×4 Number of LMI’s 80 OV’s 409 H∞ performance γ 1.9155 5×5 125 409 2.0331 6×6 180 409 2.1187 7×7 245 409 2.2224 From the design consideration mentioned previously, the final controller that will be synthesized for the velocity tracking case will have a parameter variation rate limit of [.5 200]T , a 7 × 7 grid structure, a range in altitude from 70,000 feet to 90,000 feet, and a range in speed from Mach 7 to Mach 9. The resulting 49 controllers which are synthesized are indeed stable. The eigenvalues for five selected trim conditions for the closed loop system can be seen in table 3.3. 54 From this table, it can be seen that these three systems are stable because the real portion of the eigenvalues are all negative. 55 Table 3.3: Closed Loop Eigenvalues for Selected Velocity Tracking Trim Conditions 70,000 ft Mach 7 -10991.836 -1.964 + 97.373 i -1.964 - 97.373 i -93.001 -1.181 + 49.235 i -1.181 - 49.235 i -25.035 + 8.526 i -25.035 - 8.526 i -2.728 + 22.913 i -2.728 - 22.913 i -13.645 -10.031 -4.400 -0.038 -0.189 -0.224 70,000 ft Mach 9 -1720491.692 -23655.982 -1.961 + 97.383 i -1.961 - 97.383 i -79.466 -1.410 + 49.157 i -1.410 - 49.157 i -44.219 -3.347 + 23.973 i -3.347 - 23.973 i -14.988 -10.020 -4.8189 -0.043 -0.217 + 0.025 i -0.217 - 0.025 i 80,000 ft Mach 8 -1.957 + 97.491 i -1.957 - 97.491 i -69.327 -1.104 + 49.393 i -1.104 - 49.393 i -47.519 -1.793 + 22.640 i -1.793 - 22.640 i -22.924 -17.810 -12.559 -10.038 -3.934 -0.226 -0.083 -0.151 56 90,000 ft Mach 7 -1696964.033 -117.883 -1.963 + 97.547 i -1.963 - 97.547 i -1.043 + 49.281 i -1.043 - 49.281 i -26.667 -1.207 + 21.421 i -1.207 - 21.421 i -7.822 -10.664 -10.409 -2.391 -0.916 -0.070 -0.240 90,000 ft Mach 9 -581069.170 -560.183 -2.041 + 97.443 i -2.041 - 97.443 i -75.427 -1.289 + 49.104 i -1.289 - 49.104 i -26.051 -1.854 + 22.071 i -1.854 - 22.071 i -9.937 -10.119 -2.633 + 0.606 i -2.633 - 0.606 i -0.102 -0.243 3.1.2 Altitude Tracking Now that the velocity tracking control synthesis has been established for the full state feedback control, the altitude tracking case will be considered. Again, it is necessary to look at the state space equations for a single linearized trim point so that the control synthesis can then be extended to all trim points considered in a given range. For this case, the open loop plant dynamics are the same as described in equations 3.1-3.3. Using this definition of the state space system, it is possible to augment the open loop plant to create the open loop interconnected plant seen in figure 3.4. P(1:3) u Wact P P(4) P(5:11) Wact (1 : 4) ref + - e 1 s e Figure 3.4: Open Loop Interconnected System For Altitude Tracking Figure 3.4 shows the open loop interconnected plant block diagram. In this figure, P is the plant which has been linearized about a set of trim conditions just as in equation 3.4. Wact is the set of actuator dynamics defined in equations 2.90 and 2.91. The difference between this system and the open loop interconnected system described in the velocity tracking section is that this system uses the integral of the error between the reference altitude signal and the actual plant altitude. The velocity tracking case used the integral of the difference between the reference command signal and the actual plant velocity. This integral state is what accomplishes the tracking in the system. The open loop interconnected plant can now be defined as Polic which is shown in equation 3.4. It is again necessary to apply a weighting function to Polic . The weight function used for this control synthesis problem is the same as defined in equation 4.7. The same weighting function is used here even though the error states are different. Though the error states are different, the desired penalties being applied to the actuator efforts and the error dynamics still remain the same. Therefore, the same weighting function will be used for 57 the altitude tracking case as was used for the velocity tracking case. The open loop system for H∞ LPV control synthesis is defined in equations 1.8-1.10. Using this set of equations for the hypersonic vehicle system described in this section, the state T R vector for Polic becomes, x = xp δe δc φ Ad (ref − h) . The output from Polic becomes, R T y = [δe δc φ Ad ]T where, e = h (ref − h) and the control input is u = [δe δc φ Ad ]T . This concludes the creation of the open loop interconnected plant for a single set of trim conditions for the linearized plant for a given Mach and altitude. This open loop interconnection can now be extended to all of the desired trim conditions in the operating range for the hypersonic vehicle. Again, the Mach number and altitude are the scheduling parameters ρ1 and ρ2 described in chapter 1. Now that a set of open loop interconnected plants has been developed, it is possible to perform the H∞ LPV control synthesis as was described in section 3.1.1. As before, it will be necessary to find R(ρ) and γ. The same assumption for the parameterization of R(ρ) will be made. R(ρ) will be a basis function in the form of equation 3.6. Using this form, and from equation 1.43, the basis function vector f (ρ) will be the same as equation 4.11 for all parameter points. Before R and γ can be calculated, it will be necessary to impose parameter variation rate limitations on the system. The definition of these limitations can be seen in equation 3.8. The upper and lower bound on the parameter variation rates is of the same magnitude for this study. For the hypersonic vehicle there will be two parameters, therefore there will be two parameter variation rate limitations, [ν1 ν2 ]T . ν1 will represent the bound on how quickly the the velocity of the aircraft can change (specifically how quickly the Mach number can change), and ν2 will describe how quickly the altitude can change. Once ν has been selected, the criteria is set to solve for R (ρ) and γ using equation 1.28 through LMI optimization. With a solution for R (ρ) and γ, the full state feedback LPV controller with H∞ optimization can now be found. This is done using the know state information for the open loop interconnected plant and the values for R (ρ) and γ and applying these to equation 1.29. From this equation, F (ρ) will contain the gains for the closed loop system. This will conclude the method for synthesizing H∞ LPV state-feedback control for an air breathing hypersonic vehicle. As with the velocity tracking case, it will be informative to investigate the effects of changing the parameter variation rate limits as well as the number of griding points to the robust performance of the H∞ LPV controller. In order to measure the impact of the differing the 58 parameter variation rate limits and the number of griding points on the robust capabilities of the system, different controllers were synthesized using different values to compare their γ performance variables. The γ performance variable should be lower for systems with increased robust capabilities. Once the different controllers have been synthesized, the best set of values for the number of griding points and the parameter variation rate limits will be chosen and used for simulation of the hypersonic vehicle response to a reference signal. First, consider the case of varying the parameter variation rate limit, ν, defined in equation 3.8. The parameter variation rate limits puts an upper and lower bound on how quickly the parameters in the system can change with respect to time. As discussed earlier in this chapter, there is typically a tradeoff between the robust capabilities for a system versus the performance of the system for H∞ controllers. For this study, five different controllers were synthesized with different values for ν. Each of these controllers will have a range from Mach 7 to Mach 9 in velocity and a range from 70,000 feet to 90,000 feet in altitude with a 7 × 7 griding structure containing 49 evenly spaced controllers over the range. The resulting γ values from synthesizing these different controllers can be seen in table 3.4. These results show that by raising the parameter variation rate limits, the resulting robust performance variable is also higher. This indicates that the system behaves as predicted. The lower the parameter variation rate limitations are on the system, the better the robust capabilities are. However, it is not just a simple case of choosing the controller with the best robust capabilities. The system performance needs must also be considered. Because of this, it was decided that the case where ν = [.5 200]T will be considered. This value presents a good balance over the range of values chosen for the study, and it can be seen from table 3.4 that the difference between the γ performance variables is very small. This being the case, it was decided that the case chosen exhibits the balance between performance and robust capabilities desired for this study. Now that the parameter variation rate limits have been set, it will be beneficial to look at the effects of changing the number of griding points to the robust capabilities of the system. As discussed earlier with the velocity tracking case, it is necessary to use the same trim conditions when comparing different grid sizes as changing the trim points will have an effect on the system which does not show a clear or distinct trend. For the altitude tracking case, the same grid points will be used as described in figure 3.3. The results of synthesizing these different controllers using ν = [.5 200]T can be seen in table 3.5. From this table, it can be seen that using less grid points yields a lower γ performance value, and thus an increase to the robust capabilities of the system. This is due to there being less LMI’s to synthesize. For each of the cases in table 3.5, there are 409 total LMI variables. By adding additional LMI’s, the system adds constraints to these variables. This makes it more difficult to optimize the LPV controller. 59 Table 3.4: γ performance for different parameter variation rates Parameter variation rate limits ν H∞ performance variable γ [.1 50]T 1.7510 [.3 200] T 1.7554 [.5 200] T 1.7582 [.7 200] T 1.7599 [.5 100] T 1.7579 [.5 300] T 1.7584 T 1.7623 [1 500] At the same time, it is not a good idea to choose the most robust controller for a given situation. Though the smaller griding size has a better robust performance variable, it also has a smaller operating range. It is necessary for the designer to choose a grid such that it encompasses the desired operating range while still maintaining a decent level of robust capabilities. Keeping this in mind, it can be seen from the table that the γ performance variables are relatively close together. In fact, the difference between the maximum and the minimum value is roughly .11. Since this is the case, it was decided that the 7 × 7 griding structure would be the best choice since it provides the largest operating range for the system. Table 3.5: γ performance for different number of griding points Grid dimensions 4×4 Number of LMI’s 80 OV’s 409 H∞ performance γ 1.6475 5×5 125 409 1.7454 6×6 180 409 1.7541 7×7 245 409 1.7582 Similar to the velocity tracking case, the altitude tracking case will have a parameter variation rate limit of [.5 200]T , a 7 × 7 grid structure, a range in altitude from 70,000 feet to 90,000 feet, and a range in speed from Mach 7 to Mach 9. The resulting 49 controllers which are synthesized are indeed stable. The eigenvalues for three selected trim conditions for the closed loop system 60 can be seen in table 3.6. From this table, it can be seen that these three systems are stable because the real portion of the eigenvalues are all negative. 61 Table 3.6: Closed Loop Eigenvalues for Selected Altitude Tracking Trim Conditions 70,000 ft Mach 7 -1800960.696 -172.812 -2.0451 + 97.365 i -2.045 - 97.365 i -1.499 + 49.093 i -1.499 - 49.093 i -35.165 -8.815 + 21.561 i -8.815 - 21.561 i -0.108 -0.527 + 0.214 i -0.527 - 0.214 i -5.131 + 4.693 i -5.131 - 4.693 i -11.799 -10.106 70,000 ft Mach 9 -510512.920 -354.039 -1.960 + 97.352 i -1.960 - 97.352 i -1.622 + 49.173 i -1.622 - 49.173 i -44.007 -9.668 + 22.180 i -9.668 - 22.180 i -0.053 -0.101 -1.566 -4.534 + 3.441 i -4.534 - 3.441 i -11.494 -10.067 80,000 ft Mach 8 -142.116 -1.979 + 97.495 i -1.979 - 97.495 i -67.601 -1.231 + 49.327 i -1.231 - 49.327 i -27.682 -4.073 + 22.343 i -4.073 - 22.343 i -0.071 -0.541 -0.782 -4.481 + 2.951 i -4.481 - 2.951 i -11.615 -10.078 62 90,000 ft Mach 7 -2597364.289 -270.661 -1.967 + 97.555 i -1.967 - 97.555 i -1.028 + 49.296 i -1.028 - 49.296 i -1.978 + 21.800 i -1.978 - 21.800 i -0.0197 + 0.027 i -0.0197 - 0.027 i -1.619 -2.353 -14.125 -10.340 + 0.763 i -10.340 - 0.763 i -10.152 90,000 ft Mach 9 -685206.838 -489.171 -1.910 + 97.426 i -1.910 - 97.426 i -0.969 + 49.272 i -0.969 - 49.272 i -26.782 -2.872 + 21.770 i -2.872 - 21.770 i -0.045 -0.445 -1.735 + 0.655 i -1.735 - 0.655 i -5.134 -11.839 -10.070 3.1.3 Summary of Control Synthesis The controller synthesis discussed in this chapter was calculated using Matlab 2008a on a Dell Precision T5400 with an Intel Xeon processor operating at 2.33 GHz per core with 16 gigabytes of RAM. The operating system for this computer was Windows Xp 64-bit edition. Solving the LMI’s can be very computationally intensive, so a computer with a lot of processing power and RAM is recommended. Synthesis typically took between four and six hours depending upon the conditions, and other programs running on the computer. The results of the control synthesis resulted in stable closed loop plants. The stability of each controller was evaluated by looking at the Eigenvalues of each closed loop plant. Both the velocity and altitude tracking cases exhibited stable closed loop systems for all 49 controllers. Tables 3.3 and 3.6 show the Eigenvalues at five linearized trim conditions, but all of the trim conditions proved to be stable. There are some differences that should be noted between the velocity and altitude tracking cases. From the results previously listed, it can be seen that the H∞ LPV controller synthesized for the altitude tracking case has a lower γ performance variable than the one synthesized for the velocity tracking case. In fact comparing the values from tables 3.1 and 3.4 as well as 3.2 and 3.5, it can be seen that the resulting γ performance variable is always smaller in value for the altitude tracking case than it is for the velocity tracking case as long as the same weighting functions are used. This suggests that the system is more sensitive to disturbances in the velocity than it is to disturbances in the altitude of the hypersonic vehicle. This would seem to make sense given that the velocity of the vehicle is largely dependent upon the thrust generated by the scramjet, and the thrust generated by the scramjet causes a pitching moment which in turn effects the combustion in the scramjet, and thus its thrust. The altitude is not as highly coupled in the system as the velocity is. The resulting controllers that will be simulated in this chapter are represented in table 3.7. Again it can be seen that the altitude tracking displays a greater potential for robust capabilities since the γ performance variable is smaller than it is for the velocity tracking case. The rest of this chapter will be dedicated to discussing the results of simulation for the two controllers listed in this table. 3.2 Linear Control Analysis Now that the synthesis for an H∞ LPV controller has been developed, it will be beneficial to see how this type of controller differs from a linear H∞ controller. The purpose of this section 63 Table 3.7: H∞ γ Performance Values ν Grid Size γ Performance Variable Velocity Tracking [.5 200]T 7×7 2.2224 Altitude Tracking [.5 200]T 7×7 1.7582 is to compare the response of a linear H∞ controller to the response of the H∞ LPV controller at a single point. For both simulations, the linear plant model will be used. This test will be conducted for both the velocity tracking case and the altitude tracking case. Since the H∞ LPV controller has a total of 49 linear controllers, five controllers at different trim conditions will be selected to represent the range. These five trim conditions will be for the hypersonic vehicle operating at 70,000 feet at Mach 7, 90,000 feet at Mach 7, 80,000 feet at Mach 8, 70,000 feet at Mach 9, and 90,000 feet at Mach 9. These trim conditions were chosen because they represent the minimum of the range, the median of the range, and the maximum of the range for the H∞ LPV control synthesis. A linear H∞ controller will be synthesized at the same set of trim conditions, and both systems will be subjected to a step input. The results of this comparison will give a greater understanding of how the LPV controller differs from the H∞ controller. It will also provide a baseline evaluation for the H∞ LPV controller. 3.2.1 Velocity Tracking For this case, two different control algorithms will be simulated at each of the five trim conditions. The simulation for this section will differ slightly from what was discussed in the simulation section of this chapter. Instead of implementing a simulation which uses the nonlinear plant and a switching controller, a linearized plant at the trim conditions will be implemented along with a single controller. Additionally, there will be no control effort saturation limits in this test. Though the saturation limits for the actuators have been neglected, the actuator dynamics are still included for the system. The difference will be the control gains, F , being applied to the system. For each controller at each trim condition, the system will initially start with all of the system and actuator states at 0. The vehicle will then be subjected to a 50 sf2t step input. Since the plant is linear, the error codes built into the nonlinear system for failure of proper engine combustion will not be included for the linear system. This simply means that though the nonlinear plant does not have a continuous operational envelope, the linear plant 64 will [40]. The different γ values for the different controllers can be seen in table 3.8. From this table, it can be seen that the H∞ controllers have a better γ performance variable than the H∞ LPV controller. This is due to the fact that the H∞ LPV controller is synthesized over the entire parameter space while linear H∞ control provides the best performance for a single operating condition. The result is that the system is not quite as robust as a single H∞ controller designed for one trim condition. Note that the upper trim point has the largest γ value for all of the H∞ controllers. This results from the particular trim condition. Intuitively, the hypersonic vehicle has a more difficult time operating at high Mach numbers, thus the controller is not as robust at higher trim conditions [40]. Table 3.8: H∞ γ Performance Values Trim Conditions 70,000 ft at Mach 7 90,000 ft at Mach 7 80,000 ft at Mach 8 70,000 ft at Mach 9 90,000 ft at Mach 9 H∞ 1.4674 2.1291 1.7776 1.3404 2.0579 H∞ LPV 1.9364 2.1765 2.0233 1.5798 2.2017 The results of the simulations for the different trim conditions can be seen in figures 3.5-3.10 respectively. From these results, it can be seen in figures 3.5(a) and 3.5(b) that for a given trim condition, the linear H∞ controller has a slightly better performance from the aspect of settling time for the velocity. It should be noted that though the system is slower with the proposed H∞ LPV control design, it is not significantly slower. This would suggest that the H∞ LPV control technique will in fact be a suitable control method for the hypersonic vehicle over a large range of motion for the given model and assumptions being made. Additionally, it should be noted in figures 3.5(c) and 3.5(d) that the angle of attack for the H∞ LPV controller has greater spikes in value than that seen with the H∞ controller. The H∞ controller responds slower and results in a smoother curve for the angle of attack. Similarly, the pitch rate and the pitch attitude (seen in figures 3.5 and 3.6) of the vehicle have larger spikes for the H∞ LPV controller than they do for the H∞ controller. These spikes correspond to the larger magnitude spikes in the control efforts seen for the H∞ LPV controller in figures 3.8 and 65 3.9. Both systems settle to relatively the same value for a given set of trim conditions for the angle of attack, pitch rate, and pitch attitude respectively. The altitude of the system can be seen in figure 3.6. Note how in this figure it shows that the altitude for the H∞ LPV controller is less than the value for the H∞ controller. Essentially, an H∞ LPV control algorithm minimizes the amount of control effort used by the hypersonic vehicle. In this case, it means that in order to reduce the amount of control effort needed to track the reference velocity, the vehicle sacrifices its altitude. The H∞ LPV controller has a lower altitude at any given time after the beginning of the simulation. Figures 3.6, 3.7 and 3.8 show the flexible modes of the vehicle for the different simulations. From these figures it can be seen that the magnitudes of the H∞ LPV controller are just slightly larger than what is seen with the H∞ controller. This is due to the larger magnitudes in the angle of attack, pitch attitude, and pitch rate for the hypersonic vehicle. Because there are differences in the resulting steady state conditions for some of the rigid body dynamics, there are differences in the flexible states as well. Figures 3.8 and 3.9 shows the control effort for the hypersonic vehicle. From these figures, it can be seen that the H∞ controller has larger magnitudes than the H∞ LPV controller for all of the control efforts. Both controllers have similar curves, but the magnitudes are larger for the H∞ controller case. Note also how the control efforts do not settle to the same steady state conditions for the two controllers. This is due to the difference of the other system states. Figure 3.10 also shows the integration of the error for the hypersonic vehicle. Each of these curves is different because the tracking error is slightly different for each case. This contributes to the different steady state values for the control efforts seen in this figure. It should be noted here that since the performance of the H∞ controller is slightly better than the performance of the H∞ LPV controller from the standpoint of settling time, the integration of the error at steady state conditions is less for the H∞ controller. 66 Velocity 50 45 45 40 40 35 Reference Velocity 30 Single H infinity LPV 70,000 ft Mach 7 Single H infinity LPV 90,000 ft Mach 7 Single H infinity LPV 80,000 ft Mach 8 25 Single H infinity LPV 70,000 ft Mach 9 Single H infinity LPV 90,000 ft Mach 9 20 35 Reference Velocity 30 Linear H infinity 90,000 ft Mach 7 10 5 5 10 15 20 Time in Seconds 25 30 0 35 Linear H infinity 70,000 ft Mach 9 Linear H infinity 90,000 ft Mach 9 15 5 Linear H infinity 80,000 ft Mach 8 20 10 0 Linear H infinity 70,000 ft Mach 7 25 15 0 Velocity 55 50 Velocity in Feet per Second Velocity in Feet per Second 55 0 5 (a) Velocity LPV x 10 15 20 Time in Seconds 25 30 35 (b) Velocity H∞ Angle of Attack −3 5 10 Angle of Attack −3 5 0 x 10 0 Angle of Attack in Radians Angle of Attack in Radians Single H infinity LPV 70,000 ft Mach 7 Single H infinity LPV 90,000 ft Mach 7 Single H infinity LPV 80,000 ft Mach 8 −5 Single H infinity LPV 70,000 ft Mach 9 Single H infinity LPV 90,000 ft Mach 9 −10 −5 Linear H infinity 70,000 ft Mach 7 −10 Linear H infinity 90,000 ft Mach 7 Linear H infinity 80,000 ft Mach 8 Linear H infinity 70,000 ft Mach 9 Linear H infinity 90,000 ft Mach 9 −15 −20 −15 0 5 10 15 20 Time in Seconds 25 30 −20 35 0 5 (c) Angle of Attack LPV 0.01 0.005 0 Pitch Rate in Radians per Second Pitch Rate in Radians per Second 0.01 −0.005 −0.01 Single H infinity LPV 70,000 ft Mach 7 Single H infinity LPV 90,000 ft Mach 7 Single H infinity LPV 80,000 ft Mach 8 Single H infinity LPV 70,000 ft Mach 9 −0.02 25 30 35 Pitch Rate 0.015 0.005 −0.015 15 20 Time in Seconds (d) Angle of Attack H∞ Pitch Rate 0.015 10 Single H infinity LPV 90,000 ft Mach 9 0 −0.005 −0.01 −0.015 −0.02 −0.025 −0.025 −0.03 −0.03 Linear H infinity 70,000 ft Mach 7 Linear H infinity 90,000 ft Mach 7 Linear H infinity 80,000 ft Mach 8 Linear H infinity 70,000 ft Mach 9 Linear H infinity 90,000 ft Mach 9 −0.035 0 5 10 15 20 Time in Seconds 25 30 −0.035 35 (e) Pitch Rate LPV 0 5 10 15 20 Time in Seconds 25 (f) Pitch Rate H∞ Figure 3.5: Velocity Tracking Step Response 67 30 35 Altitude 100 Altitude 1 Single H infinity LPV 70,000 ft Mach 7 0 Linear H infinity 70,000 ft Mach 7 Single H infinity LPV 90,000 ft Mach 7 Linear H infinity 90,000 ft Mach 7 Single H infinity LPV 80,000 ft Mach 8 Single H infinity LPV 70,000 ft Mach 9 −100 Linear H infinity 70,000 ft Mach 9 Single H infinity LPV 90,000 ft Mach 9 Linear H infinity 90,000 ft Mach 9 −1 −200 Altitude in Feet −300 Altitude in Feet Linear H infinity 80,000 ft Mach 8 0 −400 −500 −600 −2 −3 −4 −700 −5 −800 −900 0 5 10 15 20 Time in Seconds 25 30 −6 35 0 5 (a) Altitude LPV 10 15 20 Time in Seconds Pitch Attitude Pitch Attitude −0.005 Single H infinity LPV 70,000 ft Mach 7 Linear H infinity 70,000 ft Mach 7 Single H infinity LPV 90,000 ft Mach 7 Pitch Attitude in Radians Pitch Attitude in Radians 35 0 −0.005 Single H infinity LPV 80,000 ft Mach 8 Single H infinity LPV 70,000 ft Mach 9 −0.01 Single H infinity LPV 90,000 ft Mach 9 −0.015 −0.02 Linear H infinity 90,000 ft Mach 7 Linear H infinity 80,000 ft Mach 8 Linear H infinity 70,000 ft Mach 9 −0.01 Linear H infinity 90,000 ft Mach 9 −0.015 −0.02 0 5 10 15 20 Time in Seconds 25 30 −0.025 35 0 5 (c) Pitch Attitude LPV 10 25 30 35 30 35 First Modal Coordinate 0.1 0 0 First Modal Coordinate 0.1 −0.1 −0.2 Single H infinity LPV 70,000 ft Mach 7 Single H infinity LPV 90,000 ft Mach 7 −0.3 15 20 Time in Seconds (d) Pitch Attitude H∞ First Modal Coordinate First Modal Coordinate 30 (b) Altitude H∞ 0 −0.025 25 Single H infinity LPV 80,000 ft Mach 8 −0.1 −0.2 −0.3 Single H infinity LPV 70,000 ft Mach 9 Single H infinity LPV 90,000 ft Mach 9 −0.4 −0.4 −0.5 −0.5 Linear H infinity 70,000 ft Mach 7 Linear H infinity 90,000 ft Mach 7 Linear H infinity 80,000 ft Mach 8 Linear H infinity 70,000 ft Mach 9 Linear H infinity 90,000 ft Mach 9 0 5 10 15 20 Time in Seconds 25 30 35 0 (e) η1 LPV 5 10 15 20 Time in Seconds (f) η1 H∞ Figure 3.6: Velocity Tracking Step Response 68 25 Derivative of First Modal Coordinate 8 Derivative of First Modal Coordinate 8 Linear H infinity 70,000 ft Mach 7 6 6 4 4 Linear H infinity 90,000 ft Mach 7 Linear H infinity 80,000 ft Mach 8 Derivative of First Modal Coordinate Derivative of First Modal Coordinate Linear H infinity 70,000 ft Mach 9 2 0 −2 −4 −6 Linear H infinity 90,000 ft Mach 9 2 0 −2 −4 −6 Single H infinity LPV 70,000 ft Mach 7 Single H infinity LPV 90,000 ft Mach 7 Single H infinity LPV 80,000 ft Mach 8 −8 −8 Single H infinity LPV 70,000 ft Mach 9 Single H infinity LPV 90,000 ft Mach 9 −10 0 5 10 15 20 Time in Seconds 25 30 −10 35 0 5 10 (a) η̇1 LPV 25 30 35 25 30 35 25 30 35 (b) η̇1 H∞ Second Modal Coordinate Second Modal Coordinate 0.05 Second Modal Coordinate 0.05 Second Modal Coordinate 15 20 Time in Seconds 0 0 Linear H infinity 70,000 ft Mach 7 Linear H infinity 90,000 ft Mach 7 Linear H infinity 80,000 ft Mach 8 Linear H infinity 70,000 ft Mach 9 Linear H infinity 90,000 ft Mach 9 −0.05 −0.05 Single H infinity LPV 70,000 ft Mach 7 Single H infinity LPV 90,000 ft Mach 7 Single H infinity LPV 80,000 ft Mach 8 Single H infinity LPV 70,000 ft Mach 9 Single H infinity LPV 90,000 ft Mach 9 0 5 10 15 20 Time in Seconds 25 30 35 0 5 10 (c) η2 LPV (d) η2 H∞ Derivative of Second Modal Coordinate 2.5 15 20 Time in Seconds Derivative of Second Modal Coordinate 2.5 Linear H infinity 70,000 ft Mach 7 2 2 1.5 1.5 Linear H infinity 90,000 ft Mach 7 Linear H infinity 80,000 ft Mach 8 Derivative of Second Modal Coordinate Derivative of Second Modal Coordinate Linear H infinity 70,000 ft Mach 9 1 0.5 0 −0.5 −1 −1.5 Linear H infinity 90,000 ft Mach 9 1 0.5 0 −0.5 −1 −1.5 Single H infinity LPV 70,000 ft Mach 7 Single H infinity LPV 90,000 ft Mach 7 Single H infinity LPV 80,000 ft Mach 8 −2 −2 Single H infinity LPV 70,000 ft Mach 9 Single H infinity LPV 90,000 ft Mach 9 −2.5 0 5 10 15 20 Time in Seconds 25 30 −2.5 35 (e) η̇2 LPV 0 5 10 15 20 Time in Seconds (f) η̇2 H∞ Figure 3.7: Velocity Tracking Step Response 69 Third Modal Coordinate 0.01 Third Modal Coordinate 0.01 Linear H infinity 70,000 ft Mach 7 Linear H infinity 90,000 ft Mach 7 Linear H infinity 80,000 ft Mach 8 0.005 0.005 Linear H infinity 70,000 ft Mach 9 Linear H infinity 90,000 ft Mach 9 0 Third Modal Coordinate Third Modal Coordinate 0 −0.005 Single H infinity LPV 70,000 ft Mach 7 Single H infinity LPV 90,000 ft Mach 7 Single H infinity LPV 80,000 ft Mach 8 −0.01 −0.005 −0.01 Single H infinity LPV 70,000 ft Mach 9 Single H infinity LPV 90,000 ft Mach 9 −0.015 −0.02 −0.015 0 5 10 15 20 Time in Seconds 25 30 −0.02 35 0 5 10 (a) η3 LPV 25 30 35 25 30 35 25 30 35 (b) η3 H∞ Derivative of Third Modal Coordinate 1.5 15 20 Time in Seconds Derivative of Third Modal Coordinate 1.5 Linear H infinity 70,000 ft Mach 7 Linear H infinity 90,000 ft Mach 7 Linear H infinity 80,000 ft Mach 8 Linear H infinity 70,000 ft Mach 9 1 Derivative of Third Modal Coordinate Derivative of Third Modal Coordinate 1 Single H infinity LPV 70,000 ft Mach 7 Single H infinity LPV 90,000 ft Mach 7 Single H infinity LPV 80,000 ft Mach 8 0.5 Single H infinity LPV 70,000 ft Mach 9 Single H infinity LPV 90,000 ft Mach 9 0 −0.5 −1 Linear H infinity 90,000 ft Mach 9 0.5 0 −0.5 0 5 10 15 20 Time in Seconds 25 30 −1 35 0 5 10 (c) η̇3 LPV (d) η̇3 H∞ Elevator Control Effort 0.02 15 20 Time in Seconds Elevator Control Effort 0.02 Linear H infinity 70,000 ft Mach 7 Linear H infinity 90,000 ft Mach 7 Linear H infinity 80,000 ft Mach 8 Linear H infinity 70,000 ft Mach 9 0.01 Elevator Angle in Radians Elevator Angle in Radians 0.01 0 −0.01 −0.02 Linear H infinity 90,000 ft Mach 9 0 −0.01 −0.02 Single H infinity LPV 70,000 ft Mach 7 Single H infinity LPV 90,000 ft Mach 7 Single H infinity LPV 80,000 ft Mach 8 −0.03 −0.03 Single H infinity LPV 70,000 ft Mach 9 Single H infinity LPV 90,000 ft Mach 9 0 5 10 15 20 Time in Seconds 25 30 35 0 (e) Elevator LPV 5 10 15 20 Time in Seconds (f) Elevator H∞ Figure 3.8: Velocity Tracking Step Response 70 Canard Control Effort 0.02 0 0 Single H infinity LPV 70,000 ft Mach 7 −0.02 Single H infinity LPV 90,000 ft Mach 7 Canard Angle in Radians Canard Angle in Radians Canard Control Effort 0.02 Single H infinity LPV 80,000 ft Mach 8 Single H infinity LPV 70,000 ft Mach 9 Single H infinity LPV 90,000 ft Mach 9 −0.04 −0.06 −0.02 −0.04 Linear H infinity 70,000 ft Mach 7 Linear H infinity 90,000 ft Mach 7 Linear H infinity 80,000 ft Mach 8 −0.06 Linear H infinity 70,000 ft Mach 9 Linear H infinity 90,000 ft Mach 9 −0.08 −0.1 −0.08 0 5 10 15 20 Time in Seconds 25 30 −0.1 35 0 5 (a) Canard LPV 10 15 20 Time in Seconds 0.3 0.3 0.25 Single H infinity LPV 70,000 ft Mach 7 Single H infinity LPV 70,000 ft Mach 9 Single H infinity LPV 90,000 ft Mach 9 0.15 Linear H infinity 70,000 ft Mach 9 Linear H infinity 90,000 ft Mach 9 0.1 0.05 0.05 0 0 10 15 20 Time in Seconds 25 30 −0.05 35 Linear H infinity 80,000 ft Mach 8 0.15 0.1 5 Linear H infinity 90,000 ft Mach 7 0.2 Single H infinity LPV 80,000 ft Mach 8 Throttle Ratio Throttle Ratio Linear H infinity 70,000 ft Mach 7 Single H infinity LPV 90,000 ft Mach 7 0 0 (c) Fuel Equivalence Ratio LPV 10 15 20 Time in Seconds 0.015 0.015 0.01 0.01 0.005 0.005 0 −0.005 −0.01 30 35 0 −0.005 −0.01 Single H infinity LPV 70,000 ft Mach 7 Linear H infinity 70,000 ft Mach 7 Single H infinity LPV 90,000 ft Mach 7 −0.015 Linear H infinity 90,000 ft Mach 7 −0.015 Single H infinity LPV 80,000 ft Mach 8 Linear H infinity 80,000 ft Mach 8 Single H infinity LPV 70,000 ft Mach 9 Linear H infinity 70,000 ft Mach 9 Single H infinity LPV 90,000 ft Mach 9 −0.02 25 Diffuser Area Ratio Control Effort 0.02 Diffuser Area Ratio Diffuser Area Ratio 5 (d) Fuel Equivalence Ratio H∞ Diffuser Area Ratio Control Effort 0.02 35 Throttle Control Effort 0.25 −0.05 30 (b) Canard H∞ Throttle Control Effort 0.2 25 0 5 10 15 20 Time in Seconds 25 30 Linear H infinity 90,000 ft Mach 9 −0.02 35 (e) Diffuser Area Ratio LPV 0 5 10 15 20 Time in Seconds 25 (f) Diffuser Area Ratio H∞ Figure 3.9: Velocity Tracking Step Response 71 30 35 Integration of the Error 250 200 Integration of the Error (Velocity) Integration of the Error (Velocity) 200 Single H infinity LPV 70,000 ft Mach 7 Single H infinity LPV 90,000 ft Mach 7 Single H infinity LPV 80,000 ft Mach 8 150 Single H infinity LPV 70,000 ft Mach 9 Single H infinity LPV 90,000 ft Mach 9 100 50 0 Integration of the Error 250 150 Linear H infinity 70,000 ft Mach 7 100 Linear H infinity 90,000 ft Mach 7 Linear H infinity 80,000 ft Mach 8 Linear H infinity 70,000 ft Mach 9 Linear H infinity 90,000 ft Mach 9 50 0 5 10 15 20 Time in Seconds 25 30 0 35 (a) Integral of Tracking Error LPV 0 5 10 15 20 Time in Seconds 25 30 35 (b) Integral of Tracking Error H∞ Figure 3.10: Velocity Tracking Step Response 3.2.2 Altitude Tracking For the altitude tracking case, two different control algorithms will be simulated at each of the five trim conditions. As with the velocity tracking case, a linearized plant at the trim conditions will be implemented along with a single controller. Additionally, there will be no control effort saturation limits implemented in this test. Though the saturation limits for the actuators have been neglected, the actuator dynamics are still included for the system. The difference will be the control gains, F , being applied to the system. For each controller at each trim condition, the system will initially start with all of the system and actuator states at 0. The vehicle will then be subjected to a 200 fst step input. Since the plant is linear, the error codes built into the nonlinear system for failure of proper engine combustion will not be included for the linear system. This simply means that though the nonlinear plant does not have a continuous operational envelope, the linear plant will [40]. The different γ values for the different controllers can be seen in table 3.9. From this table, it can be seen that the H∞ controllers have a better γ performance variable than the H∞ LPV controller. This is due to the fact that the H∞ LPV controller is synthesizing more LMI’s and more variables than the H∞ controller is. This results in a more constrained problem which is more difficult to optimize. The result is that the system is not quite as robust as a single H∞ controller designed for one trim condition. Note that, as seen with the velocity tracking case, the upper trim point has the largest γ value for all of the H∞ controllers. This results from the particular trim conditions. The hypersonic vehicle (and specifically the scramjet) has a more difficult time operating at high Mach numbers, thus the controller is not as robust at higher 72 trim conditions [40]. Table 3.9: H∞ γ Performance Values Trim Conditions 70,000 ft at Mach 7 90,000 ft at Mach 7 80,000 ft at Mach 8 70,000 ft at Mach 9 90,000 ft at Mach 9 H∞ 1.3885 1.6350 1.4599 1.3890 1.5427 H∞ LPV 1.5861 1.7001 1.6415 1.5918 1.6632 The altitude tracking results of the simulations for the different trim conditions can be seen in figures 3.11-3.16 respectively. As with the velocity tracking case, figures 3.11(a) and 3.11(b) show that for a given trim condition the linear H∞ controller has a slightly better performance from the aspect of settling time for the altitude tracking. It should be noted that though the system is slower with the proposed H∞ LPV control design, it is not significantly slower. This would suggest that the H∞ LPV control technique will in fact be a suitable control method for the hypersonic vehicle over a large range of motion for the given model and assumptions being made. Figures 3.11(c) and 3.11(d) show the angle of attack for the hypersonic vehicle for the two controllers. From these figures, it can be seen that the H∞ controller has a larger magnitude spike in the angle of attack than the H∞ LPV controller. This is true for all trim conditions for the system. Additionally, it can be seen that the magnitude of the spikes in the pitch rate and the pitch attitude seen in figures 3.11(e), 3.11(f), 3.12(c), and 3.12(d) show the same characteristic. Figures 3.12(a) and 3.12(b) show the velocity curve for the hypersonic vehicle. For the velocity tracking case, there was a correlation between the velocity of the system and the altitude of the system. For the altitude tracking case, this does not have as strong of a correlation. From this plot, it can be seen that as the simulation progresses, the velocity of the vehicle increases. This would indicate that there is not a strong correlation between the altitude of the hypersonic vehicle and the altitude for this case. It does show that there is a change in the velocity. The nonlinear simulation will provide more insight into this correlation. 73 Figures 3.12, 3.13 and 3.14 show the flexible modes of the hypersonic vehicle. These modes are directly correlated to the flexibility of the vehicle. As with the angle of attack, the pitch rate, and the pitch attitude for the vehicle, the flexible modes of the system have larger magnitudes for the H∞ controller than there is for the H∞ LPV controller. There is strong coupling between the angle of attack, the pitch rate, and the pitch attitude states and the flexibility of the vehicle. This being the case, it is no surprise that the H∞ controller has a high magnitude than the H∞ LPV controller. It should be noted that the flexibility in the vehicle is stable, so the transients in the plots damp out to constant values over time as the other system states reach steady state values. Figures 3.14 and 3.15 show the control effort of the vehicle. It can be seen in this figure that the elevator and canard control efforts have a large spike in the system near the start of the simulation. These large spikes are the cause of the spikes seen in the angle of attack, pitch rate, and pitch attitude of the vehicle. Note that the magnitude of the spikes for the elevator is larger for the H∞ controller than they are for the H∞ LPV controller. For the canard, this is the opposite. Also, it can be seen in the fuel equivalence ratio that the magnitude of these initial spikes is larger for the H∞ LPV controller than it is for the H∞ controller. The diffuser area ratio has a very small magnitude spike in the response. This control effort is relatively constant over the course of the simulation. Additionally, figure 3.16 shows the integration of the error signal. Since the H∞ controller has a faster response time than the H∞ LPV controller, the integration of the error is smaller at the end of the simulation for the H∞ controller. 74 Altitude Altitude 200 200 150 Altitude in Feet Altitude in Feet 150 Reference Altitude Single H infinity LPV 70,000 ft Mach 7 Single H infinity LPV 90,000 ft Mach 7 Single H infinity LPV 80,000 ft Mach 8 100 Single H infinity LPV 70,000 ft Mach 9 Single H infinity LPV 90,000 ft Mach 9 50 Reference Altitude Linear H infinity 70,000 ft Mach 7 Linear H infinity 90,000 ft Mach 7 Linear H infinity 80,000 ft Mach 8 100 Linear H infinity 70,000 ft Mach 9 Linear H infinity 90,000 ft Mach 9 50 0 0 0 5 10 15 20 Time in Seconds 25 30 35 0 5 (a) Altitude LPV Single H infinity LPV 70,000 ft Mach 7 0.04 Angle of Attack in Radians Angle of Attack in Radians 30 35 0.06 Single H infinity LPV 90,000 ft Mach 7 Single H infinity LPV 80,000 ft Mach 8 Single H infinity LPV 70,000 ft Mach 9 Single H infinity LPV 90,000 ft Mach 9 0.02 0 −0.02 Linear H infinity 70,000 ft Mach 7 0.04 Linear H infinity 90,000 ft Mach 7 Linear H infinity 80,000 ft Mach 8 Linear H infinity 70,000 ft Mach 9 Linear H infinity 90,000 ft Mach 9 0.02 0 −0.02 0 5 10 15 20 Time in Seconds 25 30 −0.04 35 0 5 (c) Angle of Attack LPV 10 15 20 Time in Seconds 25 30 35 (d) Angle of Attack H∞ Pitch Rate 0.25 Pitch Rate 0.25 0.2 0.2 0.15 Pitch Rate in Radians per Second 0.15 Pitch Rate in Radians per Second 25 Angle of Attack 0.08 0.06 −0.04 15 20 Time in Seconds (b) Altitude H∞ Angle of Attack 0.08 10 Single H infinity LPV 70,000 ft Mach 7 Single H infinity LPV 90,000 ft Mach 7 0.1 Single H infinity LPV 80,000 ft Mach 8 Single H infinity LPV 70,000 ft Mach 9 Single H infinity LPV 90,000 ft Mach 9 0.05 0 Linear H infinity 70,000 ft Mach 7 Linear H infinity 70,000 ft Mach 9 Linear H infinity 90,000 ft Mach 9 0 −0.05 −0.1 −0.1 5 10 15 20 Time in Seconds 25 30 35 Linear H infinity 80,000 ft Mach 8 0.05 −0.05 0 Linear H infinity 90,000 ft Mach 7 0.1 0 (e) Pitch Rate LPV 5 10 15 20 Time in Seconds 25 (f) Pitch Rate H∞ Figure 3.11: Altitude Tracking Step Response 75 30 35 Velocity 40 40 30 30 20 10 Linear H infinity 70,000 ft Mach 7 Linear H infinity 90,000 ft Mach 7 Linear H infinity 80,000 ft Mach 8 Linear H infinity 70,000 ft Mach 9 Linear H infinity 90,000 ft Mach 9 20 10 Single H infinity LPV 70,000 ft Mach 7 0 Velocity 50 Velocity in Feet per Second Velocity in Feet per Second 50 0 Single H infinity LPV 90,000 ft Mach 7 Single H infinity LPV 80,000 ft Mach 8 Single H infinity LPV 70,000 ft Mach 9 Single H infinity LPV 90,000 ft Mach 9 −10 0 5 10 15 20 Time in Seconds 25 30 −10 35 0 5 (a) Velocity LPV 30 35 0.07 0.06 0.06 0.05 0.05 Single H infinity LPV 70,000 ft Mach 7 Pitch Attitude in Radians Single H infinity LPV 90,000 ft Mach 7 Single H infinity LPV 80,000 ft Mach 8 0.04 Single H infinity LPV 70,000 ft Mach 9 Single H infinity LPV 90,000 ft Mach 9 0.03 0.02 0.01 Linear H infinity 70,000 ft Mach 7 Linear H infinity 90,000 ft Mach 7 Linear H infinity 80,000 ft Mach 8 0.04 Linear H infinity 70,000 ft Mach 9 Linear H infinity 90,000 ft Mach 9 0.03 0.02 0.01 0 0 −0.01 −0.01 0 5 10 15 20 Time in Seconds 25 30 −0.02 35 0 5 (c) Pitch Attitude LPV 10 15 20 Time in Seconds 25 30 35 (d) Pitch Attitude H∞ First Modal Coordinate 4 First Modal Coordinate 4 Linear H infinity 70,000 ft Mach 7 Linear H infinity 90,000 ft Mach 7 Linear H infinity 80,000 ft Mach 8 3 3 Linear H infinity 70,000 ft Mach 9 Linear H infinity 90,000 ft Mach 9 Single H infinity LPV 70,000 ft Mach 7 2 2 Single H infinity LPV 90,000 ft Mach 7 First Modal Coordinate First Modal Coordinate Pitch Attitude in Radians 25 Pitch Attitude 0.08 0.07 −0.02 15 20 Time in Seconds (b) Velocity H∞ Pitch Attitude 0.08 10 Single H infinity LPV 80,000 ft Mach 8 Single H infinity LPV 70,000 ft Mach 9 Single H infinity LPV 90,000 ft Mach 9 1 1 0 0 −1 −1 −2 0 5 10 15 20 Time in Seconds 25 30 −2 35 (e) η1 LPV 0 5 10 15 20 Time in Seconds (f) η1 H∞ Figure 3.12: Altitude Tracking Step Response 76 25 30 35 Derivative of First Modal Coordinate 30 20 20 Derivative of First Modal Coordinate Derivative of First Modal Coordinate Derivative of First Modal Coordinate 30 10 0 −10 −20 10 0 −10 −20 Single H infinity LPV 70,000 ft Mach 7 Linear H infinity 70,000 ft Mach 7 Single H infinity LPV 90,000 ft Mach 7 −30 Linear H infinity 90,000 ft Mach 7 −30 Single H infinity LPV 80,000 ft Mach 8 Linear H infinity 80,000 ft Mach 8 Single H infinity LPV 70,000 ft Mach 9 Linear H infinity 70,000 ft Mach 9 Single H infinity LPV 90,000 ft Mach 9 −40 0 5 10 15 20 Time in Seconds 25 30 Linear H infinity 90,000 ft Mach 9 −40 35 0 5 10 (a) η̇1 LPV 0.2 0.15 0.15 Single H infinity LPV 70,000 ft Mach 7 Single H infinity LPV 80,000 ft Mach 8 Single H infinity LPV 70,000 ft Mach 9 Single H infinity LPV 90,000 ft Mach 9 0.05 0 −0.05 Linear H infinity 70,000 ft Mach 7 Linear H infinity 90,000 ft Mach 7 Second Modal Coordinate Second Modal Coordinate Single H infinity LPV 90,000 ft Mach 7 0.1 Linear H infinity 80,000 ft Mach 8 0.1 Linear H infinity 70,000 ft Mach 9 Linear H infinity 90,000 ft Mach 9 0.05 0 −0.05 −0.1 −0.1 −0.15 −0.15 0 5 10 15 20 Time in Seconds 25 30 −0.2 35 0 5 10 (c) η2 LPV 4 Derivative of Second Modal Coordinate 6 4 2 0 Single H infinity LPV 70,000 ft Mach 7 Single H infinity LPV 90,000 ft Mach 7 Single H infinity LPV 80,000 ft Mach 8 −4 Single H infinity LPV 70,000 ft Mach 9 Single H infinity LPV 90,000 ft Mach 9 Linear H infinity 90,000 ft Mach 7 Linear H infinity 80,000 ft Mach 8 25 30 −10 35 (e) η̇2 LPV Linear H infinity 70,000 ft Mach 9 Linear H infinity 90,000 ft Mach 9 −8 15 20 Time in Seconds Linear H infinity 70,000 ft Mach 7 −4 −8 10 35 0 −2 −6 5 30 2 −6 0 25 Derivative of Second Modal Coordinate 8 6 −2 15 20 Time in Seconds (d) η2 H∞ Derivative of Second Modal Coordinate 8 Derivative of Second Modal Coordinate 35 0.25 0.2 −10 30 Second Modal Coordinate 0.3 0.25 −0.2 25 (b) η̇1 H∞ Second Modal Coordinate 0.3 15 20 Time in Seconds 0 5 10 15 20 Time in Seconds (f) η̇2 H∞ Figure 3.13: Altitude Tracking Step Response 77 25 30 35 Third Modal Coordinate 0.05 Third Modal Coordinate 0.05 Linear H infinity 70,000 ft Mach 7 Linear H infinity 90,000 ft Mach 7 0.04 0.04 0.03 0.03 Linear H infinity 80,000 ft Mach 8 Linear H infinity 70,000 ft Mach 9 Linear H infinity 90,000 ft Mach 9 Single H infinity LPV 70,000 ft Mach 7 Third Modal Coordinate Third Modal Coordinate Single H infinity LPV 90,000 ft Mach 7 Single H infinity LPV 80,000 ft Mach 8 Single H infinity LPV 70,000 ft Mach 9 0.02 Single H infinity LPV 90,000 ft Mach 9 0.01 0.02 0.01 0 0 −0.01 −0.01 −0.02 0 5 10 15 20 Time in Seconds 25 30 −0.02 35 0 5 10 (a) η3 LPV 25 30 35 25 30 35 25 30 35 (b) η3 H∞ Derivative of Third Modal Coordinate 2.5 15 20 Time in Seconds Derivative of Third Modal Coordinate 2.5 Linear H infinity 70,000 ft Mach 7 2 2 1.5 1.5 Linear H infinity 90,000 ft Mach 7 Linear H infinity 80,000 ft Mach 8 Derivative of Third Modal Coordinate Derivative of Third Modal Coordinate Linear H infinity 70,000 ft Mach 9 1 0.5 0 −0.5 −1 −1.5 Linear H infinity 90,000 ft Mach 9 1 0.5 0 −0.5 −1 −1.5 Single H infinity LPV 70,000 ft Mach 7 Single H infinity LPV 90,000 ft Mach 7 Single H infinity LPV 80,000 ft Mach 8 −2 −2 Single H infinity LPV 70,000 ft Mach 9 Single H infinity LPV 90,000 ft Mach 9 −2.5 0 5 10 15 20 Time in Seconds 25 30 −2.5 35 0 5 10 (c) η̇3 LPV (d) η̇3 H∞ Elevator Control Effort 0.3 15 20 Time in Seconds Elevator Control Effort 0.3 Linear H infinity 70,000 ft Mach 7 0.25 0.25 0.2 0.2 Linear H infinity 90,000 ft Mach 7 Linear H infinity 80,000 ft Mach 8 Linear H infinity 70,000 ft Mach 9 Linear H infinity 90,000 ft Mach 9 Single H infinity LPV 90,000 ft Mach 7 Elevator Angle in Radians Elevator Angle in Radians Single H infinity LPV 70,000 ft Mach 7 0.15 Single H infinity LPV 80,000 ft Mach 8 Single H infinity LPV 70,000 ft Mach 9 Single H infinity LPV 90,000 ft Mach 9 0.1 0.05 0.15 0.1 0.05 0 0 −0.05 −0.05 −0.1 0 5 10 15 20 Time in Seconds 25 30 −0.1 35 (e) Elevator LPV 0 5 10 15 20 Time in Seconds (f) Elevator H∞ Figure 3.14: Altitude Tracking Step Response 78 Canard Control Effort Canard Control Effort Linear H infinity 70,000 ft Mach 7 0.3 0.3 0.25 0.25 Linear H infinity 90,000 ft Mach 7 Linear H infinity 80,000 ft Mach 8 Linear H infinity 70,000 ft Mach 9 Linear H infinity 90,000 ft Mach 9 Single H infinity LPV 90,000 ft Mach 7 0.2 Canard Angle in Radians Canard Angle in Radians Single H infinity LPV 70,000 ft Mach 7 Single H infinity LPV 80,000 ft Mach 8 Single H infinity LPV 70,000 ft Mach 9 Single H infinity LPV 90,000 ft Mach 9 0.15 0.1 0.05 0.2 0.15 0.1 0.05 0 0 −0.05 −0.05 0 5 10 15 20 Time in Seconds 25 30 35 0 5 (a) Canard LPV 15 20 Time in Seconds 25 30 35 (b) Canard H∞ Throttle Control Effort 0.25 10 Throttle Control Effort 0.25 Single H infinity LPV 70,000 ft Mach 7 Single H infinity LPV 90,000 ft Mach 7 0.2 0.2 Single H infinity LPV 80,000 ft Mach 8 Single H infinity LPV 70,000 ft Mach 9 Single H infinity LPV 90,000 ft Mach 9 0.15 0.15 0.1 Linear H infinity 70,000 ft Mach 7 0.1 Linear H infinity 90,000 ft Mach 7 Throttle Ratio Throttle Ratio Linear H infinity 80,000 ft Mach 8 0.05 0 −0.05 −0.1 −0.1 −0.15 −0.15 0 5 10 15 20 Time in Seconds 25 30 −0.2 35 Linear H infinity 90,000 ft Mach 9 0 −0.05 −0.2 Linear H infinity 70,000 ft Mach 9 0.05 0 (c) Fuel Equivalence Ratio LPV 10 15 20 Time in Seconds 25 30 35 (d) Fuel Equivalence Ratio H∞ Diffuser Area Ratio Control Effort 0.01 5 Diffuser Area Ratio Control Effort 0.01 Single H infinity LPV 70,000 ft Mach 7 0.008 0.008 Single H infinity LPV 90,000 ft Mach 7 Single H infinity LPV 80,000 ft Mach 8 Single H infinity LPV 70,000 ft Mach 9 0.006 0.006 Single H infinity LPV 90,000 ft Mach 9 0.004 Diffuser Area Ratio Diffuser Area Ratio 0.004 0.002 0 −0.002 0.002 0 −0.002 −0.004 −0.004 −0.006 −0.006 −0.008 −0.008 Linear H infinity 70,000 ft Mach 7 Linear H infinity 90,000 ft Mach 7 Linear H infinity 80,000 ft Mach 8 Linear H infinity 70,000 ft Mach 9 Linear H infinity 90,000 ft Mach 9 −0.01 0 5 10 15 20 Time in Seconds 25 30 −0.01 35 (e) Diffuser Area Ratio LPV 0 5 10 15 20 Time in Seconds 25 (f) Diffuser Area Ratio H∞ Figure 3.15: Altitude Tracking Step Response 79 30 35 Integration of the Error 700 600 600 500 Integration of the Error (Velocity) Integration of the Error (Velocity) Integration of the Error 700 Single H infinity LPV 70,000 ft Mach 7 Single H infinity LPV 90,000 ft Mach 7 Single H infinity LPV 80,000 ft Mach 8 Single H infinity LPV 70,000 ft Mach 9 400 Single H infinity LPV 90,000 ft Mach 9 300 200 100 0 500 Linear H infinity 70,000 ft Mach 7 Linear H infinity 90,000 ft Mach 7 Linear H infinity 80,000 ft Mach 8 Linear H infinity 70,000 ft Mach 9 400 Linear H infinity 90,000 ft Mach 9 300 200 100 0 5 10 15 20 Time in Seconds 25 30 0 35 (a) Integral of Tracking Error LPV 0 5 10 15 20 Time in Seconds 25 30 35 (b) Integral of Tracking Error H∞ Figure 3.16: Altitude Tracking Step Response 3.3 Nonlinear HSV Analysis and LPV Control Implementation This section will discuss the different aspects of how to simulate the response of the nonlinear hypersonic vehicle using the previously synthesized controllers. Both altitude and velocity tracking will be considered in this section. There will be a short discussion of the simulation setup as well as the uncertainties that will be analyzed, and the switching algorithm that is implemented. 3.3.1 Setup This subsection will discuss how to set up the simulation model using Simulink in Matlab 2008a. It takes advantage of S-functions and other built-in function blocks. Figures 3.17 and 3.18 show the closed loop system for the velocity and altitude tracking cases respectively to be implemented in Simulink. In these figures, Wact represents the actuator dynamics discussed previously. The output of the actuator dynamics feeds straight into a saturation function. The values for these saturation limits can be seen in table 2.2. Note that for the purpose of simulation, the nonlinear plant model will be used. The nonlinear plant model was provided by the Air Force Research Lab, and the equations were discussed in chapter 2 [8, 7]. Both the plant and the controller will be implemented using S-functions. The purpose of this study is to investigate the ability of the designed LPV controller to track a reference signal while still exhibiting robust capabilities. From figures 3.17 and 3.18, it can be seen that there is no uncertainty and outside disturbance added to the model. The selection 80 X(1) Nonlinear Plant Wact u X(2:11) F - + ref Figure 3.17: Block Diagram of Closed Loop System for the Velocity Tracking Case of an uncertainty model will be discussed later in this section, but this study will investigate the nominal performance of the controller. It should also be noted that F is not a single set of gains, but instead a table containing gains for the system. It will be necessary to develop some sort of algorithm to handle the switching between these different controller gains, which will be discussed in detail later in this section. For this nonlinear simulation, two different reference signals were chosen for investigation for both the velocity and altitude tracking cases. The first reference is a ramp input to the system. The second reference is a step input. These two input signals were chosen because the ramp represents a realistic input to the system and the step represents a worst case scenario for the tracking signal. This is intended to give the reader an understanding of how different input conditions can influence the output of the system. This is not an exhaustive study, but it should give some insight to understanding the results. The intent of this study is to show the application of this control technique for the high fidelity hypersonic vehicle model used, to investigate the benefits and shortcomings of this control technique, and to lay the groundwork for future research on this controller as applied to the hypersonic vehicle. Each of the inputs was chosen such that they start at the middle of the range for both Mach number and altitude, and so that they end near the end of the range of operation. This was specified so that the vehicle would start well within its operating envelope and then be moved to the edge of the operating envelope. 81 X(1:3) X(4) Nonlinear Plant Wact u X(5:11) F - + ref Figure 3.18: Block Diagram of Closed Loop System for the Altitude Tracking Case 3.3.2 Robustness Analysis The purpose of using the H∞ LPV controller is to be able to control the hypersonic vehicle over a large range of motion while exhibiting robust capabilities. Specifically, these robust capabilities are the controller’s ability to handle uncertainties in the hypersonic vehicle model. The different types of uncertainties that can be seen in hypersonic vehicles has been the topic of research for some time. Though the modeling of hypersonic vehicles continues to improve, there are still many aspects of the system that are currently undefined. Bolender has recognized the need for improvement in hypersonic vehicle modeling with his work [6]. Additionally, Chavez and Schmidt have done research to model uncertainties in a hypersonic vehicle [13]. In their work, they discuss the using different uncertainty models in the hypersonic vehicle dynamics. What may be more important to address however, is the list of the sources of uncertainties that exist in the hypersonic vehicle. For the purposes of this study, there have been many assumptions that may not hold true for the hypersonic vehicle. A few of the potential sources of system property change include thermal effects on the vehicle, fuel consumption (change in mass), and fluctuations in the atmospheric air data used [54, 14, 60]. These all have an impact on the hypersonic vehicle, and should ultimately be modeled for a full mission. This study is dealing with a generic hypersonic vehicle though, and so this information is not available. This being the case, an attempt will be made to investigate the effects of these uncertainties on the control of the hypersonic vehicle. To accomplish this, it is assumed that the thermal effects of the system change the vehicles moment of inertia as well as the length of the vehicle, that the mass of the vehicle is changed 82 by fuel consumption, and that the air density, pressure, and temperature from the table lookup may not be accurate [44]. The goal of adding perturbations to the system for this study is not necessarily to model the effects of uncertainties in the system, but rather it is an attempt to see the effects that modeling error has on the performance of the robust controller. Since this is the case, the emphasis for the study will be to look at the results of changing these parameters as opposed to developing accurate perturbation models. This being said, each of the previously mentioned parameters (air density, air pressure, air temperature, vehicle length, and vehicle moment of inertia) were increased by 5% from their nominal values with the exception of the vehicle mass. This is assumed to capture any changes in the model due to heating and inaccurate air property tables. The value of the vehicle mass was decreased by 5% from the nominal value as this more accurately represents the fuel consumption that would take place during hypersonic flight. All of these perturbations will be applied in the nonlinear plant block in figures 3.17 and 3.18. Simulations will be run for cases perturbed and nominal. 3.3.3 LPV Control Switching Algorithm Up to this point, all the different aspects of the simulation have been covered except for the problem of how the system will switch between the different linearized controllers. There are a total of 49 controllers, and there will be a need to switch from one controller to another as the vehicle moves through the parameter space. There are some different ways of handling the switching of the controllers for LPV systems. One can implement a linear interpolation of the controllers, a blending of controllers, or even a digital switching algorithm. The design process for this controller shows that each controller gives the control output in terms of a change from the nominal trim values for a given linearized controller. If the trim values for the plant states and control forces are denoted by x̄ and ū, then the following equations define the systems states and control effort. x = x̄ + ∆x (3.9) u = ū + ∆u (3.10) Since the trim conditions are known for the system from the control synthesis and the plant states, x, are known for the system from the nonlinear plant model, it will be possible to calculate the control effort u needed. This can be done by applying the control law as seen in 83 the following equation. u = ū + F ∆x (3.11) where ∆u = F ∆x. Because of equation 3.11, it will be favorable to use a digital switching technique as it will introduce the least amount of changes to the trim values. It was also discovered during the course of this research that implementing an interpolation method is unfavorable because the system has continuity issues. When interpolating the controller, it is also necessary to interpolate the set of trim conditions x̄ and ū as well. This can pose some stability issues since, as previously stated, the interpolated set of trim conditions may not exist, or at the very least may not be accurately represented by a linear interpolation. It is also difficult to insure that the resulting controller gains are stable if this calculation is done online. This being the case, the decision was made to use a digital switching algorithm due to its simplicity for implementation as well as its computational benefits. In addition to the problem of establishing a new set of trim values, there is also an issue with the integral of the error in the state vector. When switching from one controller to the next, it is important to have a method for reseting this value. Otherwise, there is a risk that the system will run into an integral windup state which can lead to saturation in the control efforts, or even cause the system to leave the range of operability. This can be a serious problem for the controller. This situation is not quite the same as the problem discussed in the work by Groves et al [27]. Their work describes a situation where the control effort is saturated by the linear controller. With no account in the control synthesis for the saturation of the control inputs, there is a windup that can cause instability in the system. In this dissertation, the problem of windup is a consequence of the integration of the error building up to large values at new trim conditions. This too will lead to the saturation of the control efforts, but the solution technique suggested by Groves et al. will not be applicable for this case. To counteract this effect, a method for reseting the integral state at the instant the controller switched from one value to the next was needed. It was proposed that the best way to implement this was to simply reset the integral state such that the change in the control effort was minimized at the switching instant. Since the integral state does not physically represent anything that occurs in the system, it is acceptable to artificially change this value. This is often done with simple PID systems [45]. To reset the integral of the error, a constrained linear least-squares problem was used. The following equations are used for the least-squares method [39]. min kCx − dk22 x 84 (3.12) such that, Ax ≤ b (3.13) where, C = F (:, 16) b= h π 6 π 9 d = −ū − F (:, 1 : 15) · ∆x(1 : 15) # " F (:, 16) A= −F (:, 16) # " iT ū + F (:, 1 : 15) · ∆x(1 : 15) π π − .77 1 12 9 −.1 0 −ū − F (:, 1 : 15) · ∆x(1 : 15) The result of this minimization problem gives the new value for the integration of the error in which the change in the control effort has been minimized. This small change in the control effort keeps the system from saturating the control efforts at the time of switching, and helps to maintain stability in the nonlinear simulation. It has been established that a digital switching technique will be implemented. Additionally, an algorithm for handling the integral windup that occurs in the system has also been established. The remaining problem is to determine when switching controllers needs to take place during the simulation. Since there are two parameters from the H∞ LPV control synthesis problem, Mach number and altitude, the parameter space can be described by a two dimensional envelope. For the purpose of this study, it was decided that the best possible time to implement the switch from one controller to the next was if the system reached the next trim condition along a given parameter. To illustrate this, figure 3.19 shows that the system is designed such that the controllable region of one trim point overlaps the subsequent trim conditions closest to it in all directions. Although it is not possible to ensure that this is the case during synthesis, this can be validated for a given reference command through simulation. Figure 3.20 shows the switching conditions for the system along the Mach number axis. This figure illustrates the concept of switching once the threshold of the next trim condition has been met by the system. The idea of this implementation is that the hypersonic vehicle states will be as close to the trim conditions as possible so that any disturbance caused by the controller switch will have minimal effect on the system. This usually results in the controller being switched due to either Mach or altitude, and not both simultaneously. 85 Linearized Controller Altitude Linearized Controllable Region Mach Figure 3.19: Controllable Region for Linearized Controller in 2D Parameter Space Switching Threshold lines Altitude HSV Flight Path Linear Controller Mach Figure 3.20: Switching Threshold for Linearized Controller in 2D Parameter Space Along the Mach Number Axis 86 3.4 Nonlinear Simulation Results This section will examine at the results from simulating the nonlinear hypersonic vehicle. For both the velocity and altitude tracking cases, two different command signals will be considered, a ramp input and a multiple step input. For each of these cases, there will plots for the system states, the actuators, and the integration of the error. There will also be a plot showing the controller switching times. Table A.1 in appendix A shows the Mach and altitude trim conditions for a given reference controller number. This will help the reader interpret the meaning of the controller switching plots. 3.4.1 Velocity Tracking This subsection will look at the results of the two command signals for the velocity tracking case. For these two inputs, the initial conditions for the system were set to be the trim conditions for Mach 8 at 80,000 feet (with controller 25). For each of the two inputs, the plots show the system responses for both the perturbed and nominal systems as indicated by the legends. Ramp Response The tracking signal used for this case starts at a velocity of 7, 819.6 fst and has a slope of 20 sf2t for a duration of 60 seconds. After the 60 second interval, the slope of the ramp is 0 sf2t for a total simulation time of 90 seconds. The results from the simulation can be seen in figures 3.21, 3.22, and 3.23. Figure 3.21 shows the rigid body states of the vehicle, figure 3.22 shows the flexible modes of the vehicle, and figure 3.23 shows the control effort as well as the reference controller number. Figure 3.21(a) shows the reference velocity and the actual velocity of the hypersonic vehicle for the perturbed and nominal cases. It can be seen from this figure that the vehicle does in fact track the desired velocity. There are some small fluctuations in the velocity at the switching points for the perturbed case. The controller switching can be seen in figure 3.23(e). This figure shows the different controller numbers used with respect to time. Each time the controller reference number changes, the system switches from one set of linear controller gains to another set of linear controller gains. It can be seen that the perturbed case switches at different times than the nominal case. Also note how the two cases use different controller numbers. Figure 3.21(b) shows the angle of attack for the hypersonic vehicle. It is important that the angle of attack should stay relatively close to zero. Otherwise, the vehicle could potentially lose the proper conditions needed for combustion in the scramjet. The figure shows that this value 87 is in fact relatively small. The maximum value is about 4.6◦ , and the minimum value is roughly −0.6◦ . This is within a reasonable range for this study. It can also be seen from this figure that the curve has some abrupt changes. These occur at the switching points in the system. The spikes caused by switching from one controller to the next are slightly more severe in the case with perturbation in the system. It can be seen that the maximum values for the angle of attack occur at or near the switching points. Figures 3.21(c) and 3.21(e) show the pitch rate and pitch attitude of the hypersonic vehicle respectively. It can be seen from these figures that the pitch attitude is relatively small in magnitude. There are large spikes in the pitch attitude which correspond to the switching conditions. These spikes are in fact amplified in the pitch rate (which is rightly so since this is the time derivative of the pitch attitude). It is important that the pitch rate stay relatively small for the vehicle because of the couplings between the pitch attitude and the propulsion system discussed in chapter 2. The pitch attitude stays within a range from −1.1◦ to 4◦ for this simulation. As this angle changes, there are effects upon both the propulsion of the vehicle in terms of thrust and the bending of the vehicle. This bending can be seen in figure 3.22. When the pitch attitude jumps, the vehicle experiences the largest deflections. These deflections oscillate at high frequencies before damping out. Figure 3.21(d) shows the altitude of the hypersonic vehicle. This state is important to look at because the result shows the optimality of the controller. As the velocity increases, the altitude drops. This drop is from 80,000 feet to about 70,000 feet over the course of the simulation. This is not a favorable characteristic to have in the hypersonic vehicle, but it does make sense in the context of the simulation. The H∞ LPV controller is designed such that the hypersonic vehicle will achieve the velocity tracking desired with the minimum amount of energy amplification possible. That is to say that it will achieve this with the least amount of control effort needed. This means that the system is trading its potential energy for kinetic energy in order to put less control effort into the system. As the vehicle loses altitude, it gains airspeed. Note that the perturbed case falls slightly slower than the nominal case. This is most likely due to the fact that the mass is lower for the perturbed case, thus the thrust generated by the controller generates more lift than the vehicle needs to maintain the controlled altitude. This results in a slightly higher altitude for the perturbed case at the end of the simulation. Figure 3.21(f) shows the integration of the error for the system. From this figure, it can be seen that the actual error continues to grow throughout the simulation. The adjusted integration looks more like a sawtooth wave. The adjusted integration of the error is the product of the optimization described previously that minimizes the amount of change in the actuator control 88 effort. It should be noted that this value is not simply reset to zero. In some instances, this value is greater after the optimization. It is important to reset the integration of the error in the system, otherwise the actuators will saturate in time, and thus lead to the potential for instability. Similar to the other control efforts, the elevator and canard control efforts, seen in figures 3.23(a) and 3.23(b), have spikes at the switching times. It is important to keep the magnitude of these spikes relatively low since saturation can occur and possibly lead to instability. It should be noted that for both the elevator and canard, the case with perturbation in the system shows a transient at the beginning of the simulation. This indicates that the elevator and canard are effected more directly by the perturbation in the system than the other control efforts. This is due to the fact that these control surfaces are at the fore and aft ends of the vehicle respectively. These locations are where the moment of inertia for the vehicle is the smallest, and consequently this is where the maximum deflections take place. Since there are uncertainties in the system which affect the flexible modes of the system, the initial trim values are off. The other actuators are not as greatly effected, but the difference in the location of the displacement of the vehicle seems to show a greater effect on these control surfaces. Since the flexible mode is stable, this transient damps out, and the vehicle seems to operate without any additional anomaly throughout the rest of the simulation. It can be seen however that the perturbed case does in fact have spikes with slightly larger magnitudes at the switching times. The effects of pitch on the thrust of the vehicle is seen by the fuel equivalence ratio as seen in figure 3.23(c). Again, the spikes in the system correspond to the switching times. The fuel equivalence ratio saturates when the pitch rate has large positive spikes. This occurs because the angle of attack spikes high at these locations. For large angles of attack, the shock wave is not on the lip of the cowl door. This means that the hypersonic vehicle has sub-optimal conditions for scramjet combustion. In order to maintain the thrust needed for the vehicle to track the reference velocity, the amount of fuel in the combustion chamber must be increased, thus an increase in the fuel equivalence ratio. Additionally, the diffuser area ratio changes as a result. Figure 3.23(d) shows that the diffuser area ratio has jumps to match the spikes in the fuel equivalence ratio. These changes in the diffuser area ratio are a result of the controller trying to obtain the needed amount of thrust while using the least amount of control effort needed to accomplish this. The fuel equivalence ratio, diffuser area ratio, pitch attitude, angle of attack, and the flexible effects of the vehicle body all affect the amount of thrust produced by the scramjet engine. These plots verify the coupling between the different states of the system. It can also be seen that the perturbed case performs slightly worse than the nominal case. 89 Velocity Angle of Attack 9000 0.08 0.07 Angle of Attack Perturbed 8800 Angle of Attack Nominal Angle of Attack in Radians Velocity in Feet per Second 0.06 8600 8400 8200 0.05 0.04 0.03 0.02 0.01 8000 0 Reference Velocity Actual Velocity Perturbed Actual Velocity Nominal 0 10 20 30 40 50 Time in Seconds 60 70 80 −0.01 90 0 10 (a) Velocity 30 40 50 Time in Seconds 60 70 80 90 (b) Angle of Attack Pitch Rate Altitude 4 8 x 10 7.9 0.15 Pitch Rate Perturbed Altitude Perturbed 7.8 Pitch Rate Nominal 0.1 Altitude Nominal 7.7 0.05 Altitude in Feet Pitch Rate in Radians per Second 20 0 −0.05 7.6 7.5 7.4 7.3 7.2 −0.1 7.1 −0.15 0 10 20 30 40 50 Time in Seconds 60 70 80 90 0 10 20 (c) Pitch Rate 30 40 50 Time in Seconds 60 70 80 90 (d) Altitude Pitch Attitude Integration of the Error 0.07 5000 0.06 4500 Pitch Attitude Perturbed Pitch Attitude Nominal 0.05 Integration of the Error (Velocity) 4000 Pitch Attitude in Radians 0.04 0.03 0.02 0.01 0 3500 3000 Adjusted Integration of Error Perturbed Actual Integration of Error Perturbed 2500 Adjusted Integration of Error Nominal Actual Integration of Error Nominal 2000 1500 1000 −0.01 500 −0.02 0 0 10 20 30 40 50 Time in Seconds 60 70 80 90 0 (e) Pitch Attitude 10 20 30 40 50 Time in Seconds 60 70 (f) Integral of Tracking Error Figure 3.21: Velocity Tracking Ramp Response 90 80 90 First Modal Coordinate Derivative of First Modal Coordinate First Modal Coordinate Perturbed Derivative of First Modal Coordinate Perturbed 50 First Modal Coordinate Nominal 4 Derivative of First Modal Coordinate Nominal 40 30 Derivative of First Modal Coordinate First Modal Coordinate 3 2 1 0 20 10 0 −10 −20 −30 −40 −1 0 10 20 30 40 50 Time in Seconds 60 70 80 90 0 10 20 30 (a) η1 Second Modal Coordinate 70 80 90 70 80 90 70 80 90 Derivative of Second Modal Coordinate Derivative of Second Modal Coordinate Perturbed 20 Second Modal Coordinate Nominal 0.1 Derivative of Second Modal Coordinate Nominal Derivative of Second Modal Coordinate 15 0 Second Modal Coordinate 60 (b) η̇1 Second Modal Coordinate Perturbed 0.2 40 50 Time in Seconds −0.1 −0.2 −0.3 −0.4 −0.5 10 5 0 −5 −10 −0.6 −15 −0.7 0 10 20 30 40 50 Time in Seconds 60 70 80 90 0 10 20 30 (c) η2 40 50 Time in Seconds 60 (d) η̇2 Third Modal Coordinate Derivative of Third Modal Coordinate 0 Third Modal Coordinate Perturbed Third Modal Coordinate Nominal −0.02 Derivative of Third Modal Coordinate Nominal Derivative of Third Modal Coordinate 3 −0.04 Third Modal Coordinate Derivative of Third Modal Coordinate Perturbed 4 −0.06 −0.08 2 1 0 −1 −2 −0.1 −3 −0.12 −4 0 10 20 30 40 50 Time in Seconds 60 70 80 90 0 (e) η3 10 20 30 40 50 Time in Seconds (f) η̇3 Figure 3.22: Velocity Tracking Ramp Response 91 60 Elevator Control Effort Canard Control Effort Elevator Control Effort Perturbed 0.25 Elevator Control Effort Nominal −0.05 0.2 Canard Angle in Radians Elevator Angle in Radians −0.1 0.15 0.1 0.05 −0.15 −0.2 −0.25 0 −0.3 −0.05 Canard Control Effort Perturbed Canard Control Effort Nominal −0.1 0 10 20 30 40 50 Time in Seconds 60 70 80 90 0 10 20 30 (a) Elevator 40 50 Time in Seconds 60 70 80 90 70 80 90 (b) Canard Throttle Control Effort Diffuser Area Ratio Control Effort Diffuser Area Ratio Control Effort Perturbed Diffuser Area Ratio Control Effort Nominal 0.98 0.7 0.96 0.6 Diffuser Area Ratio 0.4 0.92 0.9 0.88 0.86 0.3 0.84 0.2 0.82 Throttle Control Effort Perturbed Throttle Control Effort Nominal 0.1 0 10 20 30 0.8 40 50 Time in Seconds 60 70 80 90 0 10 (c) Fuel Equivalence Ratio 20 30 40 50 Time in Seconds Controller Number 44 Controller Number Perturbed 42 Controller Number Nominal 40 38 36 34 32 30 28 26 0 10 60 (d) Diffuser Area Ratio 46 Controller Number Throttle Ratio 0.94 0.5 20 30 40 50 Time in Seconds 60 70 80 90 (e) Controller Reference Number Figure 3.23: Velocity Tracking Ramp Response 92 Multiple Step Response The tracking signal used for this case starts at a velocity of 7, 819.6 fst and has six step inputs of 100 fst every 40 seconds starting at zero seconds. After 200 seconds, the reference velocity is held at 8, 419.6 fst for the remainder of the simulation for a total simulation time of 90 seconds. The results from the simulation can be seen in figures 3.24, 3.25, and 3.26. Figure 3.24 shows the rigid body states of the vehicle, figure 3.25 shows the flexible modes of the vehicle, and figure 3.26 shows the control effort as well as the reference controller number. The step command is a very challenging input for the hypersonic vehicle. It is not possible for the system to respond beyond a certain speed because of the bounds which are imposed upon the parameter variation rates. Because of these limitations, the vehicle was not able to remain within an operational range when given a single large step command. This is the reason for the multiple steps being used in this simulation. This is a worst case scenario for the hypersonic vehicle, and it is not recommended to command a step input for an actual system. Figure 3.24(a) shows the velocity of the hypersonic vehicle with respect to the reference velocity. From this figure, it can be seen that the velocity of the vehicle tracks the reference velocity, but it produces a time lag in the system. After each new step is introduced into the simulation, the slope of the velocity of the vehicle increases. This increase in acceleration is driven by the sudden increase in error between the reference velocity and the actual velocity in the system which can be seen in figure 3.24(f). As this error is reduced, the acceleration drops. It is less obvious to see the effects of switching for this case because the velocity behavior is effected by the step commands. Figure 3.24(b) shows the angle of attack for the hypersonic vehicle with a multiple step input. From this figure, it can be seen that there are spikes in the angle of attack not only when the controller switches (see figure 3.26(e)), but also when a new step command is added to the system. It appears as though the step input acts as a perturbation to the system, and then is damped out over time. It is important that the magnitude of the angle of attack remains small as it will have an effect on the propulsion system. For this simulation, the angle of attack has a maximum of about 4.0◦ and a minimum of roughly −1.0◦ . These values are acceptable for hypersonic vehicle operation. It can be seen from this figure that there is a slight difference in the angle of attack between the perturbed case and the nominal case. There is a significant amount of fluctuation in both cases during the transient portions of the response, but during the steady state portion, the two systems seem to approach relatively the same value. 93 Figures 3.24(c) and 3.24(e) show the pitch rate and pitch attitude respectively. As with the angle of attack, the pitch rate and pitch attitude are effected by the step inputs to the system. It can be seen that the step input produces a spike in both the pitch rate and the pitch attitude. This is important to look at since large values in the pitch attitude have an effect on the thrust and flexible effects of the vehicle [12]. Figure 3.25 shows the flexible modes for the hypersonic vehicle. From these figures it can be seen that both the switching times and the times at the step inputs have affected the flexible modes of the system. These will translate into displacements of the forebody and aft body of the vehicle. These displacements along with the spikes in the pitch attitude and the angle of attack cause the scramjet combustion to be effected. Figure 3.24(d) shows the altitude of the hypersonic vehicle. Again this value decreases with time as the velocity of the vehicle increases. The nature of the H∞ LPV controller dictates that the least amount of control effort is to be spent in order to achieve the desired results, which in this case means that the potential energy of the vehicle (the altitude) will be traded to achieve the desired increase in velocity. This is not a desired result, but it is reasonable given the setup used for the controller in the system. Figure 3.24(f) shows the integration of the error for the hypersonic vehicle. In this figure, both the actual and corrected integration of the error are shown for both the perturbed and nominal cases. It can be seen that the adjusted integration of the error is reset to a new value when the controller switches. It should also be noted that this is not the only discontinuity in the figure. There is also a bit of a jump when the step commands are added into the system. This sudden increase in the commanded velocity causes the tracking error in the system to increase instantaneously, thus the integration of the error also increases at the same instant. Figures 3.26(a) and 3.26(b) show the response of the elevator and canard. These two control efforts have spikes when the step inputs are put into the system as well as when the controller switches. These spikes correspond with the spikes seen in the flexible effects of the system in figure 3.25. The flexing of the hypersonic vehicle causes the forces imposed on the vehicle by the canard and elevator to be changed dependent upon their new locations. This is accounted for in the control synthesis, and it causes these actuators to have a slightly higher frequency than the other actuators in the system. It should be noted that there is a slight difference between the response of the perturbed system and the nominal system for these two actuators. This difference is due to the change in flexibility of the system with perturbed physical parameters. Figure 3.25 shows the flexible modes of the hypersonic vehicle. From these plots, it can be seen that the perturbed case has a slightly different response than the nominal case. Because of 94 this, the two cases have slightly different deflection values at different times. This results in the elevator and canard being in physically different places at different times, which in turn means that the force generated by these control surfaces is different for the two cases. Figures 3.26(c) and 3.26(d) show the fuel equivalence ratio and the diffuser area ratio. These two quantities control the thrust produced by the vehicle. Since the amount of air coming into the scramjet is not directly controlled, these control efforts will have to be changed to maintain or increase the thrust of the vehicle. The figures show that there are spikes in these control efforts corresponding to the times when the controller switches, and when the step inputs are applied to the system. The fuel equivalence ratio controls the amount of fuel entering the combustion chamber, and thus is directly related to the thrust of the vehicle. As the vehicle begins to respond, the error between the command velocity and the actual velocity is reduced. As this difference is reduced, the amount of thrust required is also reduced. If the step is too great in magnitude, this control effort will be saturated for a time period that is too long, and will cause instability in the system. The simulation will terminate under this condition because the scramjet will leave its operational range in order to try and achieve the desired acceleration of the hypersonic vehicle, and there will not be enough control authority to prevent this from occurring. From the figures mentioned, it can be seen that there is a slight difference in the values for the perturbed case and the nominal case. 95 Velocity Angle of Attack 8400 0.07 0.06 8300 Angle of Attack in Radians Velocity in Feet per Second 0.05 8200 8100 Reference Velocity Actual Velocity Perturbed Actual Velocity Nominal 0.04 0.03 0.02 8000 0.01 0 7900 Angle of Attack Perturbed Angle of Attack Nominal −0.01 0 50 100 Time in Seconds 150 200 0 50 (a) Velocity 100 Time in Seconds 150 200 (b) Angle of Attack Pitch Rate Altitude 4 8 x 10 7.9 0.15 Altitude Perturbed Altitude Nominal 0.1 7.7 Altitude in Feet Pitch Rate in Radians per Second 7.8 0.05 7.6 7.5 7.4 0 7.3 7.2 −0.05 Pitch Rate Perturbed 7.1 Pitch Rate Nominal 0 50 100 Time in Seconds 150 200 0 50 (c) Pitch Rate 100 Time in Seconds 150 200 (d) Altitude Integration of the Error Pitch Attitude 2500 Adjusted Integration of Error Perturbed Actual Integration of Error Perturbed 0.05 Adjusted Integration of Error Nominal Actual Integration of Error Nominal 2000 Integration of the Error (Velocity) Pitch Attitude in Radians 0.04 0.03 0.02 0.01 1500 1000 0 500 −0.01 Pitch Attitude Perturbed Pitch Attitude Nominal −0.02 0 50 100 Time in Seconds 150 0 200 0 (e) Pitch Attitude 50 100 Time in Seconds 150 200 (f) Integral of Tracking Error Figure 3.24: Velocity Tracking Step Response 96 First Modal Coordinate Derivative of First Modal Coordinate 30 2 Derivative of First Modal Coordinate Nominal 1.5 20 Derivative of First Modal Coordinate 1 First Modal Coordinate Derivative of First Modal Coordinate Perturbed 0.5 0 −0.5 −1 First Modal Coordinate Perturbed First Modal Coordinate Nominal −1.5 10 0 −10 −20 −2 −30 −2.5 0 50 100 Time in Seconds 150 200 0 50 (a) η1 100 Time in Seconds 150 200 (b) η̇1 Second Modal Coordinate Derivative of Second Modal Coordinate 0.1 15 Derivative of Second Modal Coordinate Perturbed Derivative of Second Modal Coordinate Nominal 0 Derivative of Second Modal Coordinate 10 Second Modal Coordinate −0.1 −0.2 −0.3 −0.4 −0.5 5 0 −5 −10 Second Modal Coordinate Perturbed Second Modal Coordinate Nominal −0.6 −15 0 50 100 Time in Seconds 150 200 0 50 (c) η2 150 200 (d) η̇2 Third Modal Coordinate 0 100 Time in Seconds Derivative of Third Modal Coordinate Derivative of Third Modal Coordinate Perturbed 4 Derivative of Third Modal Coordinate Nominal −0.02 3 Third Modal Coordinate Perturbed Third Modal Coordinate Nominal Derivative of Third Modal Coordinate Third Modal Coordinate −0.04 −0.06 −0.08 −0.1 −0.12 2 1 0 −1 −2 −3 −4 −0.14 0 50 100 Time in Seconds 150 200 0 (e) η3 50 100 Time in Seconds (f) η̇3 Figure 3.25: Velocity Tracking Step Response 97 150 200 Elevator Control Effort Canard Control Effort −0.08 0.15 Elevator Control Effort Perturbed Canard Control Effort Perturbed −0.1 Elevator Control Effort Nominal Canard Control Effort Nominal Canard Angle in Radians Elevator Angle in Radians −0.12 0.1 0.05 −0.14 −0.16 −0.18 −0.2 0 −0.22 −0.24 0 50 100 Time in Seconds 150 200 0 50 (a) Elevator 150 200 (b) Canard Throttle Control Effort Diffuser Area Ratio Control Effort 1 0.55 0.98 0.5 0.96 0.45 0.94 Diffuser Area Ratio 0.4 0.35 0.3 0.92 0.9 0.88 0.25 0.86 0.2 0.84 0.82 0.15 Throttle Control Effort Perturbed Diffuser Area Ratio Control Effort Perturbed Throttle Control Effort Nominal 0 50 100 Time in Seconds 150 Diffuser Area Ratio Control Effort Nominal 0.8 200 0 50 (c) Fuel Equivalence Ratio 100 Time in Seconds Controller Number Controller Number Perturbed Controller Number Nominal 34 32 30 28 26 24 0 150 (d) Diffuser Area Ratio 36 Controller Number Throttle Ratio 100 Time in Seconds 50 100 Time in Seconds 150 200 (e) Controller Reference Number Figure 3.26: Velocity Tracking Step Response 98 200 3.4.2 Altitude Tracking This subsection will look at the results of the two command signals for the altitude tracking case. For these two inputs, the initial conditions for the system were set to be the trim conditions for Mach 8 at 80,000 feet (with controller 25). For each of the two inputs, the plots show the system responses both perturbed and nominal systems as indicated by the legends. Ramp Response The tracking signal used for this case starts at an altitude of 80,000 and has a slope of 120 fst for a duration of 60 seconds. After the 60 second interval, the slope of the ramp is 0 fst for a total simulation time of 90 seconds. The results from the simulation can be seen in figures 3.27, 3.28, and 3.29. Figure 3.27 shows the rigid body states of the vehicle, figure 3.28 shows the flexible modes of the vehicle, and figure 3.29 shows the control effort as well as the reference controller number. Figure 3.27(a) shows the velocity of the hypersonic vehicle. The velocity of the vehicle is not constant for the altitude tracking case with a ramp input. The velocity actually changes quite drastically. It can be seen if figure 3.29(e) that the controller switches at about 31.1 seconds and 58.9 seconds. Notice that there appears to be a cusp point in the velocity at the switching times. The velocity as a whole increases over the duration of the run. This could be an unfavorable characteristic for the vehicle if a desired velocity profile is to be kept. It should be noted that the velocity shown in the figure is in terms of feet per second, but the scheduling parameter for the velocity is the Mach number. Though the velocity is changing by relatively large amounts (approximately 1030 fst ), the controller switching is only changing based upon the change in altitude. This means that the Mach number for the vehicle stays withing ±.33 of Mach 8. The difference between the perturbed case and the nominal case is almost negligible for the velocity. Figure 3.27(b) shows the angle of attack for the hypersonic vehicle. The angle of attack for this simulation stays relatively small. The maximum value is about 2.9◦ , and the minimum value is about 0.3◦ . It should be noted that the range of change for the angle of attack is smaller with the altitude tracking ramp case than it is with the velocity tracking ramp case. There are large spikes in the angle of attack when the controller switches. Over time these spikes settle out to steady state conditions. Note here that the perturbed case has slightly lower angles of attack than the nominal case. 99 Figure 3.27(c) and 3.27(e) show the pitch rate and pitch attitude of the hypersonic vehicle respectively. These values are important since they, along with the angle of attack and the flexible effects of the vehicle, have a great impact on the scramjet propulsion system [12]. It can be seen that there are spikes in the pitch attitude of the vehicle initially as well as at the switching conditions. There are corresponding spikes in the pitch rate at these times as well. These values stay relatively small. The maximum pitch attitude is roughly 2.9◦ , and the minimum is roughly 0.3◦ . These values are relatively low. As a result, the flexible effects in figure 3.28 are not as large as what was seen in the velocity tracking case. Note that for the figures discussed in this paragraph, the perturbed cases and the nominal cases are relatively close to each other. Figure 3.27(f) shows the integration of the error for the hypersonic vehicle. This value is reset at the switching points as seen with the velocity tracking cases. As before, the adjusted integration looks similar to a sawtooth wave. Note that the perturbed case and the nominal case match up almost exactly. This would make sense because the altitude tracking plots show that the two systems have the same response. Figure 3.27(d) shows the altitude of the vehicle. This figure shows that the altitude is initially at 80,000 feet, and it reaches 87,200 feet by the end of the simulation. Note that there is a small amount of lag between the reference command and the response of the actual altitude of the vehicle. This slow response is typical for robust controllers. It should also be noted that the curve is very smooth. There is not as much effect from the switching of the controller on the altitude for this case as there is on the velocity for the velocity tracking case. This would signify that the altitude change is not as sensitive as the velocity change of the vehicle is. This is also supported by the fact that the perturbation in the system seems to have no effect on the systems ability to track the altitude. Both the perturbed and nominal cases have almost the same response. Figures 3.29(a) and 3.29(b) show the elevator and canard responses respectively. Both of these responses have spikes that occur initially and at the switching points. There are some high frequency transients that occur after these spikes, but the responses quickly damp out. For the elevator, the perturbed case is slightly lower than the nominal case. This is not the case with the canard as there is almost no difference between the two cases in this figure. It can be seen if figures 3.29(c) and 3.29(d) that the fuel equivalence ratio and the diffuser area ratio are smaller than the values for the velocity tracking cases. There are spikes in the fuel equivalence ratio at the switching points. It should also be noted that the fuel equivalence 100 ratio saturates its lower bound limit. This would indicate that the vehicle is producing more thrust than is needed. These saturations occur just after the switching points, and correspond to the drops in velocity seen after switching. Figure 3.29(e) shows that the controllers switch at the same time and to the same controller reference numbers for the two systems. There is no difference between the perturbed case and the nominal case. This would solidify the statements made earlier that the perturbation has a greater effect on the velocity tracking of the vehicle than it does the altitude tracking. 101 Velocity Angle of Attack 0.045 0.04 Angle of Attack Perturbed 8000 Angle of Attack Nominal Angle of Attack in Radians Velocity in Feet per Second 0.035 7950 7900 0.03 0.025 0.02 0.015 0.01 0.005 Velocity Perturbed 7850 Velocity Nominal 0 −0.005 0 10 20 30 40 50 Time in Seconds 60 70 80 90 0 10 (a) Velocity 30 40 50 Time in Seconds 60 70 80 90 (b) Angle of Attack Pitch Rate Altitude 4 x 10 8.7 0.08 8.6 Pitch Rate Perturbed Pitch Rate Nominal 0.06 8.5 0.04 Altitude in Feet Pitch Rate in Radians per Second 20 0.02 0 8.4 8.3 8.2 −0.02 8.1 Reference Altitude Actual Altitude Perturbed Actual Altitude Nominal 0 10 20 30 40 50 Time in Seconds 60 70 80 8 90 0 10 20 (c) Pitch Rate 30 40 50 Time in Seconds 60 70 80 90 60 70 80 90 (d) Altitude Pitch Attitude Integration of the Error 4 x 10 0.05 Adjusted Integration of Error Perturbed Actual Integration of Error Perturbed Adjusted Integration of Error Nominal 0.045 Actual Integration of Error Nominal 2 Integration of the Error (Velocity) Pitch Attitude in Radians 0.04 0.035 0.03 0.025 0.02 0.015 1.5 1 0.5 0.01 Pitch Attitude Perturbed Pitch Attitude Nominal 0.005 0 10 20 30 40 50 Time in Seconds 60 70 80 0 90 (e) Pitch Attitude 0 10 20 30 40 50 Time in Seconds (f) Integral of Tracking Error Figure 3.27: Altitude Tracking Ramp Response 102 First Modal Coordinate Derivative of First Modal Coordinate 30 2 20 Derivative of First Modal Coordinate 2.5 First Modal Coordinate 1.5 1 0.5 10 0 −10 −20 0 First Modal Coordinate Perturbed Derivative of First Modal Coordinate Perturbed First Modal Coordinate Nominal Derivative of First Modal Coordinate Nominal −30 −0.5 0 10 20 30 40 50 Time in Seconds 60 70 80 90 0 10 20 30 (a) η1 40 50 Time in Seconds 60 70 80 90 70 80 90 (b) η̇1 Second Modal Coordinate Derivative of Second Modal Coordinate 3 0 Derivative of Second Modal Coordinate Second Modal Coordinate 2 −0.05 −0.1 Second Modal Coordinate Perturbed Second Modal Coordinate Nominal −0.15 1 0 −1 −2 −3 Derivative of Second Modal Coordinate Perturbed Derivative of Second Modal Coordinate Nominal −4 −0.2 0 10 20 30 40 50 Time in Seconds 60 70 80 90 0 10 20 30 (c) η2 40 50 Time in Seconds 60 (d) η̇2 Third Modal Coordinate Derivative of Third Modal Coordinate 0.01 1.5 0 Derivative of Third Modal Coordinate 1 Third Modal Coordinate −0.01 −0.02 −0.03 −0.04 0.5 0 −0.5 −1 −1.5 Third Modal Coordinate Perturbed Derivative of Third Modal Coordinate Perturbed −2 Third Modal Coordinate Nominal −0.05 0 10 20 30 40 50 Time in Seconds 60 70 80 90 Derivative of Third Modal Coordinate Nominal 0 (e) η3 10 20 30 40 50 Time in Seconds (f) η̇3 Figure 3.28: Altitude Tracking Ramp Response 103 60 70 80 90 Elevator Control Effort Canard Control Effort 0.26 0.15 0.24 0.1 0.2 Canard Angle in Radians Elevator Angle in Radians 0.22 0.18 0.16 0.14 0.05 0 −0.05 0.12 0.1 −0.1 Elevator Control Effort Perturbed 0.08 Canard Control Effort Perturbed Elevator Control Effort Nominal 0 10 20 30 40 50 Time in Seconds 60 70 80 Canard Control Effort Nominal 90 0 10 20 30 (a) Elevator 40 50 Time in Seconds 60 70 80 90 (b) Canard Throttle Control Effort Diffuser Area Ratio Control Effort 0.98 0.6 0.96 0.94 Diffuser Area Ratio 0.92 0.4 Throttle Control Effort Perturbed Throttle Control Effort Nominal 0.9 0.88 0.86 0.3 0.84 0.82 0.2 0.8 Diffuser Area Ratio Control Effort Perturbed 0.78 0.1 0 10 20 30 40 50 Time in Seconds 60 70 80 90 Diffuser Area Ratio Control Effort Nominal 0 10 (c) Fuel Equivalence Ratio 20 30 40 50 Time in Seconds Controller Number 26.8 Controller Number Perturbed 26.6 Controller Number Nominal 26.4 26.2 26 25.8 25.6 25.4 25.2 25 0 10 60 70 (d) Diffuser Area Ratio 27 Controller Number Throttle Ratio 0.5 20 30 40 50 Time in Seconds 60 70 80 90 (e) Controller Reference Number Figure 3.29: Altitude Tracking Ramp Response 104 80 90 Multiple Step Response The tracking signal used for this case starts at an altitude of 80,000 and has 6 step inputs of 650 feet every 20 seconds starting at 0 seconds. After 100 seconds, the input is held constant at 83,900 feet for a total simulation time of 130 seconds. The results from the simulation can be seen in figures 3.30, 3.31, and 3.32. Figure 3.30 shows the rigid body states of the vehicle, figure 3.31 shows the flexible modes of the vehicle, and figure 3.32 shows the control effort as well as the reference controller number. Figure 3.30(a) shows the velocity of the hypersonic vehicle. It can be seen from this figure that there are large changes when the steps are input to the vehicle. Each time that a step command is input to the system, the velocity undergoes a sharp increase or a sharp decrease. There is also an abrupt change when the controller switches. Figure 3.32(e) shows the controller number with respect to time. It can be seen that there is a switch at about 105 seconds. This switch corresponds to the change in velocity that occurs at the same time. As with the ramp case, the multiple step input has a velocity that fluctuates over the course of the simulation. The maximum value for the velocity is about 8060 fst , and the minimum is about 7800 fst . There is no switch based off of the Mach number which means that the Mach ranges from 7.67 to 8.33. It can be seen that the velocity is slightly lower for the nominal case over the majority of the simulation than it is for the perturbed case. Figure 3.30(b) shows the angle of attack for the hypersonic vehicle. There are spikes in the angle of attack that occur initially as well as at the points where the step inputs are applied. Note that the controller switch does not have much of an effect on the angle of attack. It can be seen from the figure that the perturbed case has almost no appreciable difference from the nominal case. The range for the angle of attack is from roughly −2.3◦ to 9.0◦ . This is getting to be a large angle of attack for a hypersonic vehicle, but the vehicle still remains within the operational range. Figures 3.30(c) and 3.30(e) show the pitch rate and the pitch attitude respectively. From these two figures, it can be seen that, as with the angle of attack, there are spikes in the graphs that occur initially and when the step inputs are applied to the system. The spikes for this simulation are rather large, but fortunately, they do not last for a long duration. For this simulation, the hypersonic vehicle is undergoing large angles of attack, large pitch rates, and large pitch attitudes. This results in large vehicle deflections as indicated by the plots of the flexible modes in figure 3.31. For the pitch rate, pitch attitude, the flexible modes, the fuel equivalence ratio, and the diffuser area ratio, the perturbed case and the nominal case yield very similar responses. The perturbed case does have slightly higher magnitudes, but this 105 difference is relatively small. It should also be noted that there is a spike that occurs in all of the system states and control efforts except the altitude just after 20 seconds in the simulation. This spike is caused by the change in the angle of attack and the pitch attitude. At around 21 seconds, both of these values begin to go back to zero, which is the optimal angle for the scramjet operation. As the optimum propulsion is achieved, less fuel is needed to produce the desired amount of thrust. This reduction in the required fuel causes the fuel equivalence ratio to drop until it saturates. Since the hypersonic vehicle is highly coupled, this saturation has an effect on the other system states. The result is a set of spikes seen in the different system states just after 20 seconds. The duration is short because the controller quickly adjusts to pull the fuel equivalence ratio out of the saturation region. The perturbed system has a more pronounced of a spike in the system states than the nominal system does. This is driven by the different physical properties for the two cases. Figure 3.30(d) shows the altitude of the vehicle with the six step inputs. For this simulation, the vehicle manages to track the altitude smoothly. The altitude of the vehicle starts the simulation at 80,000 feet, and it ends the simulation at 83,900 feet. Note that the system response is relatively slow for each step added to the system due to weak coupling between the altitude dynamics and the rest of the hypersonic vehicle dynamics. It takes almost ten seconds for the vehicle to reach steady state conditions after a step input. It should also be noted that there is almost no difference between the perturbed case and the nominal case. This means that the perturbation in the physical parameters have little to no effect on the altitude tracking of the vehicle. Figure 3.30(f) shows the integration of the error between the reference altitude and the actual altitude for the vehicle. It can be seen that there is a jump in this error every time a step is input to the system. The adjusted integration of the error can be seen in the figure as well. Note that this value is reset when the controller switches. There is no appreciable difference between the perturbed case and the nominal case for this figure. This makes sense because there is no difference between the altitude for the two cases. Figures 3.32(a) and 3.32(b) show the elevator and canard responses respectively. As seen with some of the other system states, these two control efforts have large spikes in their graphs at the points in time that correspond with step inputs to the system or switching in the controller. This is to be expected since the error in the system is high after the step input is applied to the system, and thus it commands a larger control effort to reduce this error. Additionally, after the controller switches from one set of gains to another, there is a change in the trim point for the elevator and canard. This new trim point can often result in a spike in the graph. 106 The duration of all of these spikes is relatively low, and the system seems to stabilize relatively quickly. These control efforts do not see saturation during the course of this simulation which is also favorable. It should be noted that there is a spike just after 20 seconds have passed in the simulation as was seen previously with other system states and the other control efforts. This is due to the saturation of the fuel equivalence ratio discussed previously. Figures 3.32(c) and 3.32(d) show the fuel equivalence ratio and the diffuser area ratio for the hypersonic vehicle. Throughout the course of this simulation, the fuel equivalence ratio in conjunction with the diffuser area ratio, attempt to adjust the scramjet conditions for the change in the location of the shock wave. As discussed previously, the fuel equivalence ratio has spikes to match those spikes seen in the angle of attack, pitch rate, and the pitch attitude for the hypersonic vehicle. Similarly, the diffuser area ratio also has these spikes. The saturation in the control efforts of the system is a cause for concern. In this simulation, the vehicle is able to maintain flight throughout the time duration evaluated, but as can be seen at about 21 seconds into the simulation, the saturation of the fuel equivalence ratio has an undesirable effect on the system states. For this simulation, the vehicle was able to recover to stable flight, but under different conditions the saturation of control efforts could lead to instability. 107 Velocity Angle of Attack 8050 0.14 0.12 8000 Angle of Attack in Radians Velocity in Feet per Second 0.1 7950 7900 0.08 0.06 0.04 0.02 0 7850 −0.02 Velocity Perturbed Angle of Attack Perturbed Velocity Nominal 0 20 40 60 80 Time in Seconds 100 Angle of Attack Nominal −0.04 120 0 20 (a) Velocity 40 60 80 Time in Seconds 100 120 (b) Angle of Attack Pitch Rate Altitude 4 x 10 0.4 8.35 0.3 8.25 Altitude in Feet Pitch Rate in Radians per Second 8.3 0.2 0.1 0 8.2 8.15 −0.1 8.1 −0.2 8.05 −0.3 Reference Altitude Actual Altitude Perturbed Pitch Rate Perturbed Actual Altitude Nominal Pitch Rate Nominal 0 20 40 60 80 Time in Seconds 100 8 120 0 20 (c) Pitch Rate 40 60 80 Time in Seconds 100 120 (d) Altitude Pitch Attitude Integration of the Error 0.16 12000 0.14 0.12 Integration of the Error (Velocity) 10000 Pitch Attitude in Radians 0.1 0.08 0.06 0.04 8000 Adjusted Integration of Error Perturbed Actual Integration of Error Perturbed Adjusted Integration of Error Nominal 6000 Actual Integration of Error Nominal 4000 0.02 0 2000 Pitch Attitude Perturbed −0.02 Pitch Attitude Nominal 0 20 40 60 80 Time in Seconds 100 0 120 (e) Pitch Attitude 0 20 40 60 80 Time in Seconds 100 (f) Integral of Tracking Error Figure 3.30: Altitude Tracking Step Response 108 120 First Modal Coordinate Derivative of First Modal Coordinate 8 50 Derivative of First Modal Coordinate First Modal Coordinate 6 4 2 0 0 −50 Derivative of First Modal Coordinate Perturbed Derivative of First Modal Coordinate Nominal −100 −2 First Modal Coordinate Perturbed First Modal Coordinate Nominal −4 −150 0 20 40 60 80 Time in Seconds 100 120 0 20 40 (a) η1 60 80 Time in Seconds 100 120 100 120 (b) η̇1 Second Modal Coordinate Derivative of Second Modal Coordinate 1.5 60 Second Modal Coordinate Perturbed Second Modal Coordinate Nominal 40 Derivative of Second Modal Coordinate Second Modal Coordinate 1 0.5 0 −0.5 20 0 −20 −40 Derivative of Second Modal Coordinate Perturbed Derivative of Second Modal Coordinate Nominal −1 −60 0 20 40 60 80 Time in Seconds 100 −80 120 0 20 40 (c) η2 (d) η̇2 Third Modal Coordinate Derivative of Third Modal Coordinate 30 0.3 Third Modal Coordinate Perturbed 0.2 Derivative of Third Modal Coordinate Perturbed 20 Third Modal Coordinate Nominal Derivative of Third Modal Coordinate Third Modal Coordinate 60 80 Time in Seconds 0.1 0 −0.1 Derivative of Third Modal Coordinate Nominal 10 0 −10 −20 −0.2 0 20 40 60 80 Time in Seconds 100 120 0 (e) η3 20 40 60 80 Time in Seconds (f) η̇3 Figure 3.31: Altitude Tracking Step Response 109 100 120 Elevator Control Effort Canard Control Effort 0.4 0.3 0.35 0.2 Canard Angle in Radians Elevator Angle in Radians 0.3 0.25 0.2 0.15 0.1 0.1 0 −0.1 0.05 0 Elevator Control Effort Perturbed Canard Control Effort Perturbed Elevator Control Effort Nominal Canard Control Effort Nominal −0.2 −0.05 −0.1 0 20 40 60 80 Time in Seconds 100 120 0 20 40 (a) Elevator 60 80 Time in Seconds 100 120 100 120 (b) Canard Throttle Control Effort Diffuser Area Ratio Control Effort 0.98 Diffuser Area Ratio Control Effort Perturbed Diffuser Area Ratio Control Effort Nominal 0.7 0.96 0.94 0.6 Diffuser Area Ratio 0.4 0.9 0.88 0.86 0.84 0.3 0.82 Throttle Control Effort Perturbed Throttle Control Effort Nominal 0.2 0.8 0.78 0.1 0 20 40 60 80 Time in Seconds 100 120 0 20 (c) Fuel Equivalence Ratio 40 60 80 Time in Seconds (d) Diffuser Area Ratio Controller Number 26 25.9 25.8 25.7 Controller Number Throttle Ratio 0.92 0.5 25.6 Controller Number Perturbed Controller Number Nominal 25.5 25.4 25.3 25.2 25.1 25 0 20 40 60 80 Time in Seconds 100 120 (e) Controller Reference Number Figure 3.32: Altitude Tracking Step Response 110 3.5 Conclusions This chapter has discussed the method for synthesizing and simulating an H∞ LPV controller for a hypersonic vehicle. It has applied both velocity tracking and altitude tracking to the vehicle and displayed the difference between perturbed and nominal cases. The results for the simulations were plotted and displayed. This section will seek to draw some deeper meaning from the results of control synthesis and simulation. From the control synthesis study, it can be seen that choosing the appropriate range and the proper trim conditions can be crucial to the controller design process. It will be beneficial for future work to investigate the operational range and limitations of air-breathing hypersonic flight. For this study, it was concluded that an evenly spaced grid containing 49 total controllers with a range from Mach 7 to Mach 9 and an altitude from 70,000 feet to 90,000 feet would be the best option. This option was determined to be the best due to its large operational range and its robust capabilities with current available computational power. It can be seen from the control synthesis that the altitude tracking has a better H∞ optimization value. This would suggest that the hypersonic vehicle is more robust for the altitude tracking case than it is for the velocity tracking case. This would make sense because of the coupled nature of the system states. The velocity is directly linked to the amount of thrust produced by the scramjet, and as such it is coupled to the angle of attack, the flexibility of the system and the pitch attitude of the vehicle. The altitude is not directly affected by the scramjet, and so it makes sense that it is not as sensitive to perturbation in the system. This chapter has shown the effect that controller switching has on the hypersonic vehicle. It can be seen from the results shown that switching between controllers can cause the system states to display sharp changes and transients. This is an interesting phenomenon. It results from the difference in the trim values with the different controllers. Though the algorithm is designed to reduce the amount of change in the actuator forces, the other system states are not bounded. When the system states of the vehicle spike, it requires that the resulting control force is increased as a result. So even though the change in the actuators is minimized at the time when the controller switches, it is not minimized in the short period of time directly following the switch. This period of time is driven by the system states and the controller gains. It is difficult for the system to minimize the controller switching effect because of the sensitivity of the system and the highly coupled nature of scramjet powered hypersonic flight. One of the main things that should be noted from this chapter is that the controller is optimized such that it uses the least amount of control effort possible to achieve the desired 111 tracking results. This means that for velocity tracking, the altitude drops continually over the course of simulation to meet the need for an increase in velocity. This was not a desired result. It is unlikely that a mission for the hypersonic vehicle would accept such a large change in the altitude. Similarly, it can be seen that the altitude tracking case does not maintain a constant velocity. In fact there is a relatively large fluctuation in the velocity of the vehicle for the altitude tracking case. Again, it is unlikely that this would be a desired mission scenario. Because of this, there is a need to develop a combined velocity and altitude tracking case. However, this is not truly possible since these are competing design objectives. The system cannot be optimized for velocity tracking with altitude tracking simultaneously, so it will be necessary to develop some alternative method for controlling the vehicle. The next chapter will discuss such a method. The simulation in this chapter has also shown that there is a correlation between the angle of attack and the flexible states of the vehicle. As the angle of attack increases, the motions for the flexible modes increase. This is an important relationship to understand because both of these values play into the efficiency of the scramjet engine. As the vehicle deflection increases (both at the tip of the forebody and the aft of the vehicle), the amount of thrust generated by the vehicle is affected. This is because both the flow into and out of the scramjet is changed. As the nose of the vehicle moves, the location of the shock wave moves as well. This can mean that the scramjet is receiving more or less air than it needs. The angle of attack also plays into this as the turning angle of the air passing through the shock wave is related to the angle of attack. This can also effect the speed of the air entering the scramjet. On the aft of the vehicle, the deflection affects the speed of the air leaving the scramjet since the aft of the vehicle along with the shear layer create an external expansion of the exhaust gases. As the position of the tail of the vehicle and the shear layer move, the amount of expansion of the exhaust gases change. This change results in different speeds of the exhaust, and different lift and drag coefficients. This effect is also propagated to the pitching moment as well. It is important that the value of the angle of attack stay relatively small so that the scramjet stays within an operational range. This chapter has also focused strictly on the full state feedback case. This assumes that there are perfect measurements of all the system states being fed to the controller. This may not be a good assumption to make since there are several system states that will be difficult or impossible to measure. For this reason, it will be preferable to apply output feedback to the system so that the unmeasurable states can be accounted for, and noise can be added into the measured states accordingly. 112 Though there are some assumptions made in this chapter that may not accurately represent the actual system, the results from this study show the characteristics of an H∞ LPV controller. It can be seen that the amount of perturbation added to the system for this study is within a reasonable range, and that it does not have a significant affect on the velocity or altitude tracking state. The perturbation does affect other system states, but these states stay within a reasonable range for hypersonic flight. 113 Chapter 4 Output Feedback Control for Hypersonic Vehicle Chapter 3 discussed the process for synthesizing a full state H∞ LPV controller and simulating it for the hypersonic vehicle. This chapter will focus on synthesizing and simulating an output feedback H∞ LPV controller for the hypersonic vehicle. It is important to look at the output feedback case because it is a more realistic representation of how a control system could be applied to the hypersonic vehicle. This is due to the fact that it is not always possible to measure all of states of a given system. This chapter will study the results of both velocity tracking and altitude tracking. An effort will be made to resolve the issues seen with the full state feedback systems with regard to altitude changing dramatically during velocity tracking, and velocity changing dramatically during altitude tracking. Additionally, the effects of flexibility will be studied in the system. The effects of disturbance to the system as well as uncertainty in the system will be investigated. The results of synthesis and simulation will be analyzed, and the major points will be concluded at the end of this chapter. 4.1 Control Synthesis This section will discuss the output feedback control synthesis procedure for both the velocity tracking case and the altitude tracking case. This section will also look at the differences between the synthesis used for the rigid body system versus the flexible body system. It is important to look at the output feedback case because it is a more realistic model of the hypersonic vehicle. It may not always be feasible or desirable to measure all the states of the system because of physical or fiscal limitations. For the purposes of this study, it is assumed that the only 114 states that can be measured are the five system states that are not directly associated with the flexibility of the hypersonic vehicle. The six states that represent the modes of vibration for the vehicle will be treated as the estimated states inside the control algorithm. The reader should note that there may be states that are being measured in this study that may not be realistically measured in a true design scenario. This is simply an attempt to establish a method for applying this new control theory to the hypersonic vehicle. Any desired changes in instrumentation can be applied by the designer when synthesizing a controller for a hypersonic vehicle. Here it will be assumed that the vehicle has five measurable states. 4.1.1 Velocity Tracking First, the velocity tracking case will be considered. Both the flexible body case, and the rigid body cases will be considered in this section. It is important to look at the effects of the flexible body modes on the system. By making the comparison between the rigid body controller and the flexible body controller, it will be possible to determine whether it will be necessary to include the flexible body dynamics in the control synthesis. This subsection will discuss the steps needed to synthesize a controller for both the flexible and rigid body models. For this model, the linearized plant model will be derived as seen in chapter 2. The resulting state vectors for the linearized plant will be, xp,f = [V α Q h θ η1 η̇1 η2 η̇2 η3 η̇3 ]T (4.1) xp,r = [V α Q h θ]T (4.2) and where the subscripts f and r in xp,∗ denote the flexible and rigid bodies respectively. Note that there are eleven states in the flexible model while there are only five in the rigid body model. For the purpose of this study, the first five states will be assumed to be measurable and the six flexible states will not be measurable for the flexible body case. The rigid body case is still considered to be an output feedback case, even though all of the states are measurable, since the five measured rigid body states will have sensor noise added into them. These assumptions were made on the basis that it would be difficult to actually measure the flexible modes of vibration for the hypersonic vehicle since the modes are dependent upon the displacement of the vehicle body itself with respect to the rigid body, and on the principle that there would be some noise present in the sensors. Measuring the flexibility of the vehicle would require many sensors and complex calculations to be analyzed in the loop. This may not be economical or feasible for hypersonic flight. It is understood that these are not the only states that may be 115 difficult to measure or calculate, but to simplify the synthesis, these assumptions are enforced. Using state vector in equations 4.1 and 4.2, the following state space systems are set up for the linearized plant. ẋp,∗ = Ap,∗ · xp,∗ + Bp,∗ · u (4.3) yp,∗ = Cp,∗ · xp,∗ (4.4) where the ∗ in equations 4.3 and 4.4 are the general expression for the flexible and rigid body cases respectively. To synthesize a controller for the output feedback velocity tracking case, it will be necessary to define the open loop interconnected plant model at each trim condition. Augmenting the linearized plant model with the actuator states in equations 2.90 and 2.91 along with an integral state and a proportional state will give the open loop interconnected plant for the output feedback case. This model can be seen in the block diagrams in figure 4.1. In this figure, it can be seen that the system is augmented with the actuator states and a weighted disturbance block. In figure 4.1(a), it can be seen that the linearized plant P only shows five states being output from the block. This block does in fact output all eleven states for the flexible body case, but since the first five are the only states that will be measured, the last six states have been omitted from the block diagram. For the rigid body case, there will only be five states in P. Additionally, Wact is the actuator dynamics from equations 2.90 and 2.91. In this figure, P is defined as, " ss P = Ap,∗ Bp,∗ # Cp,∗ Dp,∗ (4.5) As with the full state feedback case, the error between the reference velocity of the system and the actual velocity of the vehicle is integrated and added as a state of the system. The main difference is the ordering of the states that results from the augmentation of the system. Figure 4.1(b) shows the order of the states for Pact . Notice that there are six error states in the system, and six outputs. The outputs from the system are the five measured states, and the integral of the error. The error states show that there are two internal states based off of the system states. The result is that there is an integral and a proportional feedback in the system. The integral feedback causes the system to track the velocity of the system, but the proportional state has been setup such that it loosely regulates the altitude. The term loosely is used because the gain on this state is unity. This means that the altitude will never actually converge to a value, it simply will penalize any change in the altitude. The simulation will 116 P(4) e Pact Wact (1) Wact (2) u u P(1) P(2) P(3) P(4) P(5) P Wact Pact ref Wact (4) P(1) P(2) P(3) P(4) y P(5) Wact (1 : 4) ref + e - 1 s e e (a) Plant with Actuator and Integral Augmentation (b) Pact u Polic Polic(1 : 6) Pact (1 : 6) u ref e Wact (3) Pact Pact (7 : 12) Polic ref + - Polic(7 : 12) d d Wd (c) Plant with Weighted Disturbance (d) Open Loop Interconnected Plant Figure 4.1: Open Loop Interconnected System For Velocity Tracking Output Feedback verify this later. Pact is now defined as, Aact B1act B2act ss Pact = W · C1act W · D11act W · D12act C2act D21act D22act 117 (4.6) where w is a weighting function defined as, 1 0 0 0 0 0 .5 0 0 0 0 0 1000 0 0 W = 0 0 0 1000 0 0 0 0 0 316.2778 0 0 0 0 0 0 0 0 0 0 1000 (4.7) This weighting is used to penalize the output of controller so that the saturation limits will not be exceeded. These values were chosen ad hoc for this study through a process of trial and error. It should be noted that these may not be the ideal values for the weighting function, but these values do work for this particular system. A frequency dependent weight function could prove to yield an even better result than the constant weight function used in this study, but this is beyond the scope of this research. In this particular weighting function, only the diagonal terms are non-zero. The first two terms are applied to the error states in the system, and the last four terms are applied to the control efforts. In this weighting function, large values add a higher penalty to the particular value. By selecting these weighting functions, the controller will penalize the control efforts because of the high weightings associated with these states. It is beneficial to choose high values for the weighting function on the states associated with the actuators in the system because it is important to keep the actuators from saturating their limits. In order to aid in this pursuit, high weighting functions are chosen because they will have a strong penalty to the actuator effort when synthesizing a controller. Respectively, it can be seen that the error states are chosen to be much smaller because it will be desirable for the error dynamics to be very small. In order for this to occur, the penalties applied to these states should be small so that the synthesized controller will have more effect on these states. Therefore, these values are chosen to be small such that the desired performance can be achieved. Figrue 4.1(c) shows the interconnection of Pact with the weighted disturbance Wd . In this figure, d is a vector with six disturbance states which is defined as, d = [d1 d2 d3 d4 d5 d6 ]T (4.8) These disturbances are added to the measured state as well as the integration of the error state. The disturbances on the measured state are intended to represent measurement noise in the system, and the disturbance added to the integration of the error state represents numerical error in the integration process. Though the disturbance for the integration of the error was 118 included in the control synthesis, this will not be added into the simulation as this value would be very small. Wd is a weighting function designed to penalize the disturbance to the system. It is defined as, .01 0 0 0 0 0 .01 0 0 0 0 0 .01 0 0 Wd = 0 0 0 .01 0 0 0 0 0 .01 0 0 0 0 0 0 0 0 0 0 .01 (4.9) With this weighting function defined, the open loop interconnected plant can be constructed. The open loop interconnected plant, Polic can be seen in figure 4.1(d). This figure shows that the system has the control effort u, the reference velocity, and the disturbance to the system as inputs. There are a total of 12 outputs from the plant, and there are 16 states to the T R system. The state vector for Polic is, x = xp,∗ δe δc φ Ad (ref − v) . The output from T R T R Polic becomes, y = v α Q h θ (ref − v) where e = h (ref − v) δe δc φ Ad , and u = [δe δc φ Ad ]T . Note that the output y contains the disturbed measurements for the vehicle. This concludes the setup of the open loop interconnected plant for the velocity tracking case at a single linearized trim condition for a given Mach number and altitude for both the flexible and rigid body cases. By extension, this method can be applied to all of the chosen trim conditions for the hypersonic vehicle. The Mach and altitude correspond to the parameters ρ1 and ρ2 from the LPV synthesis discussed in chapter 1. Now that a set of open loop interconnected plants has been generated, it is possible to synthesize a set of LPV H∞ controllers for the hypersonic vehicle. In chapter 3, the full state feedback case was discussed. The method used for full state feedback only required that R(ρ) and γ be determined. For the output feedback case, there are three LMI’s used for the synthesis as seen in equations 1.32-1.34. Before these LMI’s can be solved, matrix functions R(ρ) and S(ρ) must be parameterized. For this study, R(ρ) was chosen as a simple linear parameterdependent function while S(ρ) was chosen as a constant value. Since S(ρ) was chosen to be constant, the controller gains will depend only on ρ and not ρ̇. The basis function vectors f (ρ) and g(ρ) take the form of f (ρ) = [1 ρ1 ρ2 ] (4.10) g(ρ) = [1] (4.11) 119 for all parameter points. Therefore, R(ρ) and S(ρ) are parameterized as R (ρ) = R0 + ρ1 R1 + ρ2 R2 and S (ρ) = S0 . Now that the basis function vectors have been defined, it is necessary to define a set of bounds for the parameter variation rate, ν. For this dissertation, ν is defined as, |ρ̇| ≤ ν (4.12) where ν is a constant vector representing both the upper and lower bound for the parameter variation rate. ν will be a vector with two terms in it. The first term will represent the limitation on how quickly the Mach number of the hypersonic vehicle can change. The second term describes how quickly the altitude of the hypersonic vehicle can change. Now that the basis functions and the constants have all been set for the control synthesis problem, all of the criteria is met to solve for R(ρ), S(ρ), and γ. This is accomplished by solving equations 1.32-1.34 using efficient LMI techniques. Once R(ρ), S(ρ), and γ have been solved, the H∞ LPV output feedback controller gains can now be calculated. This calculation is accomplished by using the known values for R(ρ), S(ρ), and γ as well as the open loop interconnected plant data and applying that to equations 1.35-1.40. The result is a set of output feedback controller gains for the hypersonic vehicle. With the H∞ LPV controller for the hypersonic vehicle established, it will be beneficial to look at the effects of using different parameter variation rates and different numbers of gridding points have on the control synthesis problem. To accomplish this, different controllers were synthesized to determine the optimal values for each of these parameters for the hypersonic vehicle. The results of this parametric study will be used to synthesize a controller, and then analyze the response of the resulting closed-loop system. First, consider the parameter variation rate limit ν defined in equation 4.12. The parameter variation rate puts a limitation on how quickly the parameters in the system can change with respect of time. There is usually a tradeoff between the robust capabilities and the performance associated with H∞ LPV control problems. The selection of the parameter variation rate limit has an effect on this tradeoff, so it is important to choose this variable such that the system has the desired aspects of performance and robustness. Therefore, it is beneficial to investigate the effects of the parameter variation rate limit, ν, on the H∞ performance variable, γ, for the system. Assuming that the hypersonic vehicle has seven trim points for the Mach number and seven trim points for the altitude for a total of 49 linear trimmed plants with a velocity range from Mach 7 to Mach 9 and an altitude range from 70,000 feet to 90,000 feet with all trim conditions evenly spaced in the range, an H∞ LPV controller can be synthesized with 120 different parameter variation rate limits. Table 4.1 shows the results of this study. This table shows that the lower parameter variation rate limits yield lower γ values which translates to increased robust capabilities for the system. The tradeoff, however, is in the fact that there are now lower rates imposed upon the rate of change in Mach number and altitude for the hypersonic vehicle. Therefore, it will be necessary to choose a parameter variation rate limit that is a balance between the performance and robustness of the vehicle. For this study, it was decided that the case where ν = [.1 200]T would provide the best balance between performance and robustness. Table 4.1: γ performance for different parameter variation rates Parameter variation rate limits ν Flexible Body γ Rigid Body γ [.01 200]T 112.7380 83.2507 T [.05 200] 112.9252 83.4346 [.1 200] T 113.2146 83.6931 [.2 200] T 113.6599 84.1999 [.3 200] T 114.2606 84.7131 [.4 200] T 115.2945 85.1583 [.5 200] T 116.9904 85.6793 [.1 50] T 113.1878 84.1295 [.1 100] T 113.1951 84.1639 [.1 300] T 113.2558 84.2772 [.1 400] T 113.4078 84.4805 [.1 500] T 114.3578 84.9415 [.5 500] T 117.9836 86.6760 Using the selected parameter variation rate limits, it will also be important to look at the effect of changing the number of griding points in the parameter space will have on the synthesis of the system. The purpose of this study will be to evaluate the effect of additional griding points on the system’s robust capabilities. Typically in H∞ LPV control problems, having a denser griding in the parameter space will decrease the H∞ γ performance value [30, 31]. For the purposes of this study however, the spacing of the grid points will not be changed since choosing different trim conditions would result in a different set of linearized plants. This 121 would yield the potential for numerical error in the system. To avoid this, the griding density will be tested by choosing a large set of trim conditions, and then removing trim conditions from the parameter space. This should result in an increase in the γ performance variable with a larger set of trim conditions since this will mean additional LMI’s will be considered in the optimization. Figure 3.3 shows the different parameter values for the Mach number and altitude for the different controller griding numbers considered in this study. For this study, it is arbitrarily assumed that the desired operating range is from 70,000 feet to 90,000 feet in altitude and from Mach 7 to Mach 9. Since the hypersonic vehicle is a nonlinear system, using different parameter points for the purpose of linearization can lead to drastically different synthesis results, even for the same grid size and spacing. Table 4.2 shows the result of the griding point study. In this table, it can be seen that the H∞ γ performance variable decreases with less points. For each of the flexible body cases, there were 545 variables to be solved for in the optimization process. For the rigid body case, there were 221 optimization variables (OV’s) to be solved for. Since the number of variables stays constant for each case, an increase in the number of LMI’s will cause increased constraints to be opposed on the optimization. From the table, it can be seen that less griding points equates to less LMI’s. Additionally, the rigid body cases have smaller γ performance values than the flexible body cases. Since there are less variables to solve for in the rigid body cases, it is possible to achieve better solutions. The rigid body problem is a more simple case, therefore the optimization results are improved over the flexible body case. Though the smaller griding sizes may yield a better robust performance, it is many times preferable to use a larger grid to ensure that the entire parameter space is covered in the controllable region of the set of linear controllers that have been synthesized. For this reason, the 7 × 7 grid was chosen such that there would be a total of 49 different linear controllers over the entire parameter space. From Table 4.2: γ performance for different number of gridding points Grid dimension γ Flexible Body LMI’s OV’s γ Rigid Body LMI’s OV’s 4×4 93.4078 96 545 61.1913 96 221 5×5 110.4690 150 545 69.3434 150 221 6×6 110.7171 216 545 77.5342 216 221 7×7 113.2146 294 545 84.1999 294 221 122 the design considerations mentioned previously in this subsection, the final controller that will be synthesized for the purpose of simulating the output feedback velocity tracking case will have parameter variation rate limits of [.1 200]T , a 7 × 7 grid structure, a range in altitude from 70,000 feet to 90,000 feet, and a range in speed from Mach 7 to Mach 9. The resulting 49 controllers which are synthesized are indeed stable. The eigenvalues for five selected trim conditions for the closed loop system can be seen in tables 4.3 and 4.4 for the flexible and rigid body models respectively. From these tables, it can be seen that the real portion of all of the eigenvalues are in fact negative. This ensures that the controllers developed are in fact stable. 123 Table 4.3: Closed Loop Eigenvalues for Selected Flexible Body Velocity Tracking Trim Conditions 70,000 ft Mach 7 -1523769.7 70,000 ft Mach 9 -15091871.8 80,000 ft Mach 8 -27447.9 90,000 ft Mach 7 -27447.9 90,000 ft Mach 9 -27447.9 -146317.2 -279492.9 -6065.8 -6065.8 -6065.8 -27242.3 -27427.7 -4388.1 -4388.1 -4388.1 -4440.8 -3292.5 -874.1 -873.6 -874.3 -1600.0 -1225.0 -127.5 -120.3 + 1.7 i -125.9 -399.8 -313.9 -109.0 -120.3 - 1.7 i -110.3 -94.9 -2.5 + 97.6 i -2.6 + 97.5 i -2.6 + 97.5 i -2.6 + 97.5 i -2.3 + 97.4 i -2.5 - 97.6 i -2.6 - 97.5 i -2.6 - 97.5 i -2.6 - 97.5 i -2.3 - 97.4 i -2.0 + 97.6 i -2.0 + 97.5 i -2.0 + 97.4 i -2.0 + 97.4 i -2.0 + 97.4 i -2.0 - 97.6 i -2.0 - 97.5 i -2.0 - 97.4 i -2.0 - 97.4 i -2.0 - 97.4 i -79.6 -56.3 + 18.5 i -67.6 -56.6 + 23.5 i -77.8 -1.4 + 49.2 i -56.3 - 18.5 i -1.9 + 49.5 i -56.6 - 23.5 i -1.7 + 49.0 i -1.4 - 49.2 i -1.8 + 49.0 i -1.9 - 49.5 i -1.8 + 49.0 i -1.7 - 49.0 i -1.0 + 49.3 i -1.8 - 49.0 i -1.2 + 49.5 i -1.8 - 49.0 i -1.1 + 49.3 i -1.0 - 49.3 i -1.0 + 49.4 i -1.2 - 49.5 i -1.0 + 49.3 i -1.1 - 49.3 i -29.3 + 7.7 i -1.0 - 49.4 i -41.7 -1.0 - 49.3 i -25.7 + 8.8 i -29.3 - 7.7 i -1.1 + 22.5 i -19.0 + 19.2 i -1.4 + 21.9 i -25.7 - 8.8 i -7.7 + 22.2 i -1.1 - 22.5 i -19.0 - 19.2 i -1.4 - 21.9 i -8.4 + 22.9 i -7.7 - 22.2 i -6.8 + 22.1 i -4.0 + 25.0 i -6.9 + 21.8 i -8.4 - 22.9 i -0.9 + 21.6 i -6.8 - 22.1 i -4.0 - 25.0 i -6.9 - 21.8 i -1.7 + 22.9 i -0.9 - 21.6 i -22.7 -1.6 + 22.9 i -25.4 -1.7 - 22.9 i -19.1 -21.0 + 3.6 i -1.6 - 22.9 i -22.7 -20.2 -0.4 + 0.4 i -21.0 - 3.6 i -21.9 -19.9 -11.7 + 1.6 i -0.4 - 0.4 i -0.2 -13.9 + 1.5 i -0.2 -11.7 - 1.6 i -0.4 -0.5 + 0.4 i -13.9 - 1.5 i -0.4 + 0.4 i -6.1 + 0.9 i -0.2 -0.5 - 0.4 i -3.8 -0.4 - 0.4 i -6.1 - 0.9 i -4.6 + 1.6 i -1.3 -1.1 + 0.6 i -1.0 -5.0 -4.6 - 1.6 i -5.4 + 2.2 i -1.1 - 0.6 i -4.0 + 1.88 i -0.6 + 0.3 i -14.3 -5.4 - 2.2 i -0.2 -4.0 - 1.8 i -0.6 - 0.3 i -13.2 -13.4 -0.6 -12.0 -0.2 -11.0 -10.7 -11.5 -10.7 -10.0 -10.0 -10.0 -10.0 -10.0 124 Table 4.4: Closed Loop Eigenvalues for Selected Rigid Body Velocity Tracking Trim Conditions 70,000 ft Mach 7 -20.0 70,000 ft Mach 9 -1398017.1 80,000 ft Mach 8 -5869.0 90,000 ft Mach 7 -5869.0 90,000 ft Mach 9 -5869.0 -16432.1 -4026.1 -4105.0 -4105.0 -4105.0 -7980.3 -4425.6 -4425.6 -4425.6 -4425.6 -3949.1 -3302.3 -342.9 -343.7 -342.7 -4425.9 -260.5 -2.0 + 97.5 i -2.0 + 97.3 i -2.0 + 97.5 i -845.8 -2.0 + 97.6 i -2.0 - 97.5 i -2.0 - 97.3 i -2.0 - 97.5 i -290.8 -2.0 - 97.6 i -83.0 -84.5 -82.2 -132.1 -79.5 -60.6 -1.1 + 49.7 i -63.7 -2.0 + 97.7 i -1.0 + 49.4 i -1.0 + 49.5 i -1.1 - 49.7 i -1.0 + 49.4 i -2.0 - 97.7 i -1.0 - 49.4 i -1.0 - 49.5 i -53.1 -1.0 - 49.4 i -1.1 + 49.8 i -0.4 + 21.5 i -36.8 -0.2 + 24.1 i -37.5 -1.1 - 49.8 i -0.4 - 21.5 i -0.3 + 22.6 i -0.2 - 24.1 i -30.0 -32.7 -30.7 -0.3 - 22.6 i -35.0 + 3.5 i -0.4 + 22.0 i -0.1 + 23.9 i -21.6 + 2.5 i -30.9 -35.0 - 3.5 i -0.4 - 22.0 i -0.1 - 23.9 i -21.6 - 2.5 i -22.5 -11.6 + 10.4 i -21.7 -19.5 + 2.8 i -0.3 + 0.6 i -19.1 -11.6 - 10.4 i -19.4 -19.5 - 2.8 i -0.3 - 0.6 i -13.8 -21.0 -16.5 -5.9 + 6.4 i -0.3 + 0.1 i -5.6 + 3.2 i -18.4 -3.5 + 3.3 i -5.9 - 6.4 i -0.3 - 0.1 i -5.6 - 3.2 i -13.8 -3.5 - 3.3 i -0.2 -3.9 + 2.1 i -0.2 -2.4 -0.3 + 0.5 i -0.3 + 0.4 i -3.9 - 2.1 i -0.4 + 0.5 i -1.1 + 0.9 i -0.3 - 0.5 i -0.3 - 0.4 i -18.4 -0.4 - 0.5 i -1.1 - 0.9 i -0.25 -3.1 -15.9 -1.0 -0.2 -0.7 -10.0 + 0.3 i -12.4 -10.3 + 0.5 i -0.8 -10.8 -10.0 - 0.3 i -10.8 -10.3 - 0.5 i -10.4 -10.6 -10.0 -10.0 -10.0 -10.0 -10.0 125 4.1.2 Altitude Tracking Now that the velocity tracking control synthesis has been established, this subsection will look at the altitude tracking case for both the flexible body and rigid body cases. By looking at flexible and rigid body cases, it will be possible to ascertain whether it will be necessary to include the flexible body dynamics in the control synthesis. This subsection will discuss the steps needed to synthesize a controller for both the flexible and rigid body models. As seen in the previous subsection, the linearized plant model will be derived as seen in chapter 2. The open loop plant dynamics are the same as seen in equations 4.1-4.4. As seen in the previous subsection, there are eleven states in the flexible plant model, and five states in the rigid body model. Again, the first five states will be assumed to be measurable and the six flexible states will not be measurable for the flexible body case. The rigid body case is still considered to be an output feedback case, as before even though all of the states are measurable because the five measured rigid body states will have sensor noise added to them. In order to synthesize a controller for the output feedback altitude tracking case, it will be necessary to define the open loop interconnected plant model at each trim condition. To do this, the linearized plant must be augmented with the actuator states in equations 2.90 and 2.91 along with an integral and proportional state in order to achieve the desired altitude tracking. Figure 4.2 shows the open loop interconnected plant block diagrams that achieve this. As seen with the velocity tracking case, the system is augmented with the actuator states and a weighted disturbance block. Figure 4.2(a) shows only five states being output from the linearized plant P . This block does in fact have eleven states, but since the flexible states are not measured, they are not shown in the interconnection. For the rigid body case however, there will only be five states in the linearized plant P . Additionally, Wact is the actuator dynamics from equations 2.90 and 2.91. P is defined the same here as it is in equation 4.5. Note that as seen with the velocity tracking case, the integral of the difference between the reference altitude and the actual altitude is added as a system state. Figure 4.2(b) shows the inputs and outputs of for Pact . Note that there are two inputs and twelve outputs. The first six outputs are error states, and the last six outputs are outputs of the system. The outputs from the system here are the five measured states, and the integral of the error. The error states show that there are two internal states based off of the system states. This results in a proportional and integral feedback state in the system. This two states will cause the system to track to altitude of the vehicle while maintaining a relatively small change in the velocity of the vehicle. Pact is now defined as in equation 4.6, and w is defined in equation 4.7. The weight function used for this control synthesis problem is the same as 126 P(1) e Pact Wact (1) Wact (2) u u P(1) P(2) P(3) P(4) P(5) P Wact ref Pact Wact (4) P(1) P(2) P(3) P(4) y P(5) Wact (1 : 4) ref + e - 1 s e e (a) Plant with Actuator and Integral Augmentation (b) Pact u Polic Polic(1 : 6) Pact (1 : 6) u ref e Wact (3) Pact Pact (7 : 12) ref + - Polic Polic(7 : 12) d d Wd (c) Plant with Weighted Disturbance (d) Open Loop Interconnected Plant Figure 4.2: Open Loop Interconnected System For Altitude Tracking Output Feedback defined in equation 4.7. The same weighting function is used here even though the error states are different. Though the error states are different, the desired penalties being applied to the actuator efforts and the error dynamics still remain the same. Therefore, the same weighting function will be used for the altitude tracking case as was used for the velocity tracking case. Figure 4.2(c) shows the interconnection of Pact with the wighted disturbance Wd . In this figure, d is defined the same as it is in equation 4.8. This is a vector of six disturbances that will be added to the measured states as well as the integration of the error state. This is intended to represent the sensor noise that would be seen in the system and the numerical error associated with the integration of the error. Though the disturbance for the integration of the 127 error was included in the control synthesis, this will not be added into the simulation as this value would be very small. The disturbance vector d is multiplied by a weighting function Wd . Wd is defined in equation 4.9. The purpose of this weighting function is to penalize the effect of the disturbance in the system for the sake of simulation. The open loop interconnected plant, Polic can be seen in figure 4.2(d). This figure shows that the system has the control effort u, the reference altitude, and the disturbance to the system as inputs. There are a total of 12 outputs from the plant, and there are 16 states to the T R system. The state vector for Polic is, x = xp,∗ δe δc φ Ad (ref − h) . The output from T R T R Polic becomes, y = v α Q h θ (ref − h) where e = v (ref − h) δe δc φ Ad , and u = [δe δc φ Ad ]T . Note that the output y contains the disturbed measurements for the vehicle. This concludes the setup of the open loop interconnected plant for the altitude tracking case at a single linearized trim condition for a given Mach number and altitude for both the flexible body and rigid body cases. By extension, this method can be applied to all of the chosen trim conditions for the hypersonic vehicle. The Mach number and altitude correspond to the parameters ρ1 and ρ2 from the LPV synthesis discussion in chapter 1. Now that a set of open loop interconnected plants has been generated, it is possible to synthesize a set of LPV H∞ controllers for the hypersonic vehicle. The process for the altitude tracking case is the same as it was for the velocity tracking case in the previous subsection. First, it is necessary to establish the parameterized forms of R (ρ) and S (ρ). The altitude tracking case has the same parameterized form as seen in equations ?? and ?? for the velocity tracking case. As was seen before, R (ρ) is a simple linear parameter-dependent function, and S (ρ) is a constant value. As a result, the controller gains will depend only on ρ and not ρ̇. The basis function vectors for the altitude tracking case are the same as those seen in equations 4.10 and 4.11. With these basis functions and vectors defined, it is necessary to define a set of bounds for the parameter variation rate, ν. The definition of ν can be seen in equation 4.12. Now that ν and the basis function vectors have been set for the control synthesis problem, it is possible to solve for R (ρ), S (ρ), and γ using equations 1.32-1.34 and efficient LMI techniques. With R (ρ), S (ρ), and γ solved for, the H∞ LPV output feedback controller gains can be calculated for the altitude tracking case. These gains are calculated by applying R (ρ), S (ρ), and γ to equations 1.35-1.40. The results from this set of equations is a state space representation 128 of the output feedback controller for the altitude tracking case of the air-breathing hypersonic vehicle. The method for synthesizing an H∞ LPV controller has now been established. This method uses several different parameter inputs for the purpose of synthesis. Changing these parameters has an effect on the resulting control synthesis. It will be beneficial to a controls designer to see the effects of changing the different parameters that are in the control synthesis problem. Specifically, this study will look at the effects of changing the parameter variation rate limits and the number of gridding points used in the synthesis. The results of this study will be used to effectively evaluate what the different parameters used for the synthesis problem in this dissertation should be for the altitude tracking output feedback controller for the hypersonic vehicle. The first parameter that will be considered is the parameter variation rate limit, ν, defined in equation 4.12. As mentioned previously, this parameter variation rate limit puts a limitation on how quickly the parameters can change in the system with respect to time. It can be important to select this variation rate limit wisely because there can be an important tradeoff between performance and the robust capabilities of the system. If the parameter variation rate limit is set to be a very large value, then the system may respond to the command signal more quickly, but it will also have less robustness. The converse of this is also true. To see this trend, a simple study was conducted to show the relationship between different ν values and the γ performance variable. Table 4.5 shows the results of the parametric study for the hypersonic vehicle. This study assumes that the hypersonic vehicle has seven trim points for the Mach number and seven trim points for the altitude for a total of 49 linear trimmed plants with a velocity range from Mach 7 to Mach 9 and an altitude range from 70,000 feet to 90,000 feet with all trim conditions evenly spaced in the range. Using this set of plants, an H∞ LPV controller was synthesized using a different ν value for both the flexible and rigid body cases. The results of this table show that as the parameter variation rate limits get smaller, the γ performance variable gets smaller. A smaller γ performance variable means that the resulting controller has increased robust capabilities. By investigating this table, it was decided that the case where ν = [.1 200]T would provide the best balance between performance and robustness for the hypersonic vehicle. Using the selected parameter variation rate limits, it is possible to investigate the effects of changing the number of griding points in the parameter space. The purpose of this study 129 Table 4.5: γ performance for different parameter variation rates Parameter variation rate limits ν Flexible Body γ Rigid Body γ [.01 200]T 105.4755 96.5305 T [.05 200] 106.0167 96.9756 [.1 200] T 107.2043 97.7403 [.2 200] T 107.6150 98.2592 [.3 200] T 108.6442 99.0030 [.4 200] T 109.6327 99.6791 [.5 200] T 110.9550 100.1782 [.1 50]T 107.0061 97.0839 [.1 100] T 107.0892 97.2123 [.1 300] T 107.4375 97.5096 [.1 400]T 108.0371 97.9823 [.1 500] T 109.8784 98.4479 [.5 500] T 113.3821 100.8012 will be to investigate the effects on the γ performance variable that result from an increase or a decrease in the total number of linearized trim points considered for the control synthesis problem. In order to make a valid comparison however, it is necessary to ensure that, even though the total number of trim points is being changed, the Mach number and altitude of each trim condition used is the same in all of the cases considered. This means that the range of the hypersonic vehicle will be smaller for lower gridding numbers. Figure 3.3 shows this setup. It can be seen from this figure that four different cases will be considered. The smallest gridding case will be a 4 × 4, and the largest will be a 7 × 7. The same trim points will be used in each case, but the larger cases will have more trim points that the subsequent smaller cases will not include. The reasoning behind this has to due with the linearization that is taking place. Choosing different trim points would result in a different set of linearized plants. This would yield the potential for numerical error in the system. Table 4.6 shows the results of the gridding point study. This table indicates that the H∞ γ performance variable decreases with less points. For each of the flexible body cases, there are 545 variables to be solved for in the optimization process. For the rigid body cases, there are 221 optimization variables to solve for. Since the number of variables is constant for all 130 cases, an increase in the number of LMI’s will cause increased constraints to be opposed on the optimization. The table shows that less gridding points equates to fewer LMI’s. It should also be noted that the rigid body cases have smaller γ values than the flexible body cases because there are less optimization variables to solve for. The lower number of optimization variables make it possible to solve for a more optimal solution to the problem which results in a lower γ performance variable. Though the smaller gridding cases yield better robust capabilities, they also have a smaller operational range. This being the case, it was decided that the 7 × 7 grid would give the desired operational range. Table 4.6: γ performance for different number of gridding points Grid dimension Flexible Body γ LMI’s OV’s Rigid Body γ LMI’s OV’s 4×4 78.8801 96 545 68.9623 96 221 5×5 88.2682 150 545 78.1467 150 221 6×6 97.2745 216 545 89.3062 216 221 7×7 107.2043 294 545 97.7403 294 221 The final controller synthesized for this dissertation used a parameter variation rate limit of [.1 200]T and a grid size of 7 × 7 for a total of 49 trim points and controllers. The range used is from Mach 7 to Mach 9, and from 70,000 feet to 90,000 feet. The eigenvalues for five selected trim conditions can be seen in tables 4.7 and 4.8 for both the flexible and rigid body cases. From these tables, it can be seen that the real portion of all the eigenvalues are in fact negative. This ensures that the controllers developed are indeed stable. 131 Table 4.7: Closed Loop Eigenvalues for Selected Flexible Body Altitude Tracking Trim Conditions 70,000 ft Mach 7 -20.0 70,000 ft Mach 9 -2545621.3 80,000 ft Mach 8 -4666.3 90,000 ft Mach 7 -4666.3 90,000 ft Mach 9 -4666.3 -99996.0 -4192.9 -4134.0 -4134.0 -4134.0 -9738.7 -3988.4 -3988.4 -3988.4 -3988.4 -4453.9 -2539.4 -861.3 -861.3 -861.3 -2137.5 -844.1 -89.8 -92.3 -88.7 -3988.3 -2.0 + 97.6 i -2.0 + 97.5 i -2.0 + 97.5 i -2.0 + 97.5 i -144.1 -2.0 - 97.6 i -2.0 - 97.5 i -2.0 - 97.5 i -2.0 - 97.5 i -2.0 + 97.6 i -2.0 + 97.6 i -2.0 + 97.5 i -2.0 + 97.3 i -2.0 + 97.5 i -2.0 - 97.6 i -2.0 - 97.6 i -2.0 - 97.5 i -2.0 - 97.3 i -2.0 - 97.5 i -2.0 + 97.7 i -77.4 -1.2 + 49.1 i -1.2 + 49.3 i -1.2 + 49.1 i -2.0 - 97.7 i -1.1 + 49.2 i -1.2 - 49.1 i -1.2 - 49.3 i -1.2 - 49.1 i -60.1 -1.1 - 49.2 i -1.0 + 49.4 i -1.1 + 49.4 i -1.0 + 49.4 i -1.1 + 49.7 i -1.0 + 49.3 i -1.0 - 49.4 i -1.1 - 49.4 i -1.0 - 49.4 i -1.1 - 49.7 i -1.0 - 49.3 i -45.1 -48.0 -45.9 -1.3 + 49.3 i -32.6 -1.8 + 22.6 i -0.6 + 24.9 i -32.1 -1.3 - 49.3 i -2.0 + 21.9 i -1.8 - 22.6 i -0.6 - 24.9 i -1.6 + 22.8 i -39.9 -2.0 - 21.9 i -0.7 + 22.7 i -1.7 + 21.2 i -1.6 - 22.8 i -0.7 + 24.3 i -0.6 + 21.6 i -0.7 - 22.7 i -1.7 - 21.2 i -0.9 + 21.9 i -0.7 - 24.3 i -0.6 - 21.6 i -27.1 + 4.4 i -18.0 + 12.9 i -0.9 - 21.9 i -2.6 + 23.4 i -22.7 + 0.6 i -27.1 - 4.4 i -18.0 - 12.9 i -25.6 -2.6 - 23.4 i -22.7 - 0.6 i -24.2 -23.2 -21.8 -19.1 + 1.4 i -17.0 -17.4 + 1.0 i -18.6 + 4.5 i -18.6 + 1.7 i -19.1 - 1.4 i -15.2 -17.4 - 1.0 i -18.6 - 4.5 i -18.6 - 1.7 i -4.4 + 3.7 i -0.5 + 0.7 i -0.1 -17.4 -2.6 + 2.4 i -4.4 - 3.7 i -0.5 - 0.7 i -0.4 -1.0 + 1.9 i -2.6 - 2.4 i -0.1 -0.1 -0.8 + 0.7 i -1.0 - 1.9 i -0.1 -0.7 + 0.6 i -0.1 -0.8 - 0.7 i -0.1 -0.3 -0.7 - 0.6 i -3.7 + 0.1 i -3.9 + 1.9 i -0.9 -0.6 + 0.7 i -1.2 -3.7 - 0.1 i -3.9 - 1.9 i -0.6 -0.6 - 0.7 i -10.4 -11.9 -10.6 -11.3 -10.5 -9.9 -10.5 -10.3 -10.3 -10.4 -10.1 -10.0 -10.0 -10.0 -10.0 132 Table 4.8: Closed Loop Eigenvalues for Selected Rigid Body Altitude Tracking Trim Conditions 70,000 ft Mach 7 -20.0 70,000 ft Mach 9 -80153.6 80,000 ft Mach 8 -1513.2 90,000 ft Mach 7 -1513.2 90,000 ft Mach 9 -1513.2 -2280.5 -1391.1 -181.9 -181.9 -181.9 -1252.1 -181.9 -2.0 + 97.5 i -1.9 + 97.3 i -2.0 + 97.5 i -181.9 -2.0 + 97.6 i -2.0 - 97.5 i -1.9 - 97.3 i -2.0 - 97.5 i -2.0 + 97.7 i -2.0 - 97.6 i -77.5 -79.0 -76.2 + 2.5 i -2.0 - 97.7 i -81.1 -61.4 + 8.4 i -66.5 + 21.8 i -76.2 - 2.5 i -81.6 -1.0 + 49.4 i -61.4 - 8.4 i -66.5 - 21.8 i -1.0 + 49.4 i -0.9 + 49.8 i -1.0 - 49.4 i -0.9 + 49.4 i -0.8 + 49.6 i -1.0 - 49.4 i -0.9 - 49.8 i -0.3 + 21.6 i -0.9 - 49.4 i -0.8 - 49.6 i -0.2 + 22.1 i -35.2 -0.3 - 21.6 i -35.0 -35.2 -0.2 - 22.1 i -32.6 + 10.0 i -31.0 + 11.3 i -0.3 + 22.6 i 0.5 + 24.0 i -32.2 + 10.9 i -32.6 - 10.0 i -31.0 - 11.3 i -0.3 - 22.6 i 0.5 - 24.0 i -32.2 - 10.9 i -0.1 + 23.5 i -33.0 -10.3 + 11.6 i -6.5 + 19.2 i -34.9 -0.1 - 23.5 i -3.2 + 2.4 i -10.3 - 11.6 i -6.5 - 19.2 i -0.4 + 0.7 i -19.3 -3.2 - 2.4 i -0.5 + 0.7 i -1.3 + 1.2 i -0.4 - 0.7 i -16.0 + 6.3 i -0.1 -0.5 - 0.7 i -1.3 - 1.2 i -0.1 -16.0 - 6.3 i -0.2 -0.1 -0.1 -0.3 -0.1 -0.4 + 0.6 i -0.3 -0.7 -5.0 + 2.2 i -0.7 + 0.5 i -0.4 - 0.6 i -3.5 -1.0 -5.0 - 2.2 i -0.7 - 0.5 i -18.4 -20.4 + 0.1 i -19.0 -18.4 -1.3 -20.2 -20.4 - 0.1 i -20.2 -20.2 -3.5 -20.0 -19.9 -20.0 -20.0 -6.6 -10.3 -9.5 -9.0 -10.0 -10.0 + 0.1 i -10.0 -10.1 -10.0 -10.0 + 0.1 i -10.0 - 0.1 i -10.0 -10.0 + 0.1 i -10.0 -10.0 - 0.1 i -10.0 -10.0 -10.0 - 0.1 i -10.0 -10.0 133 4.1.3 Summary of Control Synthesis The control synthesis discussed in this chapter was calculated using Matlab 2008a on a Dell Precision T5400 with an Intel Xeon processor operating at 2.33 GHz per core with 16 gigabytes of RAM. The operating system was Windows XP 64-bit edition. Solving the LMI’s can be very taxing on the computer due to the iterative process of the optimization, and the large number of calculations involved with a system this complex. Synthesis typically took between four and six hours depending upon the conditions, and other programs running on the computer. It should be noted that a computer with an insufficient amount of RAM will not be able to synthesize controllers for the hypersonic vehicle model used in this dissertation. The results of the control synthesis discussed in this section resulted in stable closed loop plants. The stability of each of the linear controllers for the different cases synthesized can be verified by looking at the eigenvalues of the closed loop plants. Both the velocity and altitude tracking cases exhibited stable closed loop eigenvalues for both the flexible and rigid body cases. Tables 4.3, 4.4, 4.7, and 4.8 show the eigenvalues of five different trim points for the different cases considered in this section. All of the eigenvalues for the 49 controllers used in each control synthesis problem in fact had stable controllers, but only a sample of those results were shown for brevity. There are some important differences to note with this control synthesis study. For instance, there is a difference in the H∞ LPV controller γ performance between the flexible and rigid body cases. Tables 4.5, 4.6, 4.1, and 4.2 show that the flexible body cases have higher γ performance values than the rigid body cases. This falls in line with the expectations set forth before the study was conducted. It makes sense that the rigid body cases would have lower γ values because these cases are simpler than the more complex flexible body cases, and as a result they have fewer optimization variables to solve for. This means that there is less complexity in the system with the same number of constraints. This makes the optimization easier for the computer. Similarly, it should be noted that there are differences between the γ performance values for the velocity and altitude tracking cases. From the results listed previously in this section, it can be seen that the H∞ LPV controllers synthesized for the altitude tracking cases have lower γ performance values than the ones synthesized for the velocity tracking cases. In fact, by comparing tables 4.5, 4.6, 4.1, and 4.2, it can be seen that the resulting γ performance is always smaller in value for the altitude tracking cases than it is for the velocity tracking cases as long as the same weighting functions are used. This reinforces the results seen in chapter 3 that came to the same conclusion. It seems as though the hypersonic vehicle is more sensitive 134 to disturbances in the velocity of the vehicle than it is to disturbances in the altitude of the hypersonic vehicle. Again, this aligns with the ideology that velocity of the vehicle is largely dependent upon the thrust generated by the scramjet, and that the thrust of the scramjet causes a pitching moment which in turn effects the combustion in the scramjet, and thus the thrust it generates. Since the altitude is not as highly coupled in the hypersonic vehicle system, it is not as sensitive to disturbance. Table 4.9: H∞ γ Performance Values ν Grid Size γ Performance Variable Velocity Tracking Flexible Body [.1 200]T 7×7 113.2146 Velocity Tracking Rigid Body [.1 200]T 7×7 83.6931 Altitude Tracking Flexible Body [.1 200]T 7×7 107.2043 Altitude Tracking Rigid Body [.1 200]T 7×7 97.7403 The resulting controllers that will be simulated in this chapter are represented in table 4.9. From this table it can again be seen that the altitude tracking cases exhibit a greater potential for robust capabilities because of their smaller γ performance values. Additionally, the same conclusion can be drawn about the rigid body cases since they too have smaller γ performance values than the flexible body cases. It should be noted here though that even though the rigid body cases have smaller γ performance values, they may not indeed prove to be as robust because of the fact that they will be applied to a flexible body plant model. This means that all of the flexible effects in the system will be treated as disturbance to the system, so this set of controllers will see more disturbance to the system than the flexible cases will. So even though the γ performance values are lower, it does not mean that the simulated system performance will necessarily be better than that of the flexible body case. It simply means that under the controller design constraints, the controllers exhibit a lower γ performance value. 135 The four different controllers that will be investigated in this chapter have now been synthesized. The rest of this chapter will discuss the results of simulation for the four controllers listed in table 4.9. A baseline test will be conducted in the next section to determine the characteristics of a linear control algorithm as compared to the H∞ LPV algorithm, and the rest of this chapter will discuss the results of the controllers synthesized in this section to different tracking signals. 4.2 Linear Control Analysis Now that the synthesis for an H∞ LPV controller has been developed, it will be beneficial to see how this type of controller differs from a linear H∞ controller. The purpose of this section, as in chapter 3, is to compare the response of a linear H∞ controller to the response of the H∞ LPV controller at a single point. For both simulations, the linear plant model will be used. This test will be conducted for both the velocity tracking case and the altitude tracking case. For simplicity, only the flexible body cases will be examined here. Since the H∞ LPV controller has a total of 49 linear controllers, five controllers at different trim conditions will be selected to represent the range. These five trim conditions will be for the hypersonic vehicle operating at 70,000 feet at Mach 7, 90,000 feet at Mach 7, 80,000 feet at Mach 8, 70,000 feet at Mach 9, and 90,000 feet at Mach 9. These trim conditions were chosen because they represent the minimum of the range, the median of the range, and the maximum of the range for the H∞ LPV control synthesis. A linear H∞ controller will be synthesized at the same set of trim conditions, and both systems will be subjected to a step input. The results of this comparison will give a greater understanding of how the LPV controller differs from the H∞ controller. It will also provide a baseline evaluation for the H∞ LPV controller. 4.2.1 Velocity Tracking For this case, two different control algorithms will be simulated at each of the five trim conditions. The simulation for this section will differ slightly from what was discussed in the simulation section of this chapter. Instead of implementing a simulation which uses the nonlinear plant and a switching controller, a linearized plant at the trim conditions will be implemented along with a single controller. Additionally, there will be no control effort saturation limits in this test. Though the saturation limits for the actuators have been neglected, the actuator dynamics are still included for the system. The difference will be the controller, K, being applied to the system. For each controller at each trim condition, the system will initially start with all of the system and actuator states at 0. The vehicle will then be subjected to a 50 sf2t step input. Since the plant is linear, the error codes built into the nonlinear system for failure of proper 136 engine combustion will not be included for the linear system. This simply means that though the nonlinear plant does not have a continuous operational envelope, the linear plant will [40]. The different γ values for the different controllers can be seen in table 4.10. From this table, it can be seen that the H∞ LPV controller γ performance is higher than for the H∞ controller. These results are similar to what was seen in 3. This result is due to the fact that the H∞ LPV controller is synthesized over the entire parameter space while the linear H∞ controller provides the best performance for a single operating condition. Table 4.10: H∞ γ Performance Values Trim Conditions 70,000 ft at Mach 7 90,000 ft at Mach 7 80,000 ft at Mach 8 70,000 ft at Mach 9 90,000 ft at Mach 9 H∞ 28.3030 60.8467 43.6681 39.3117 63.6928 H∞ LPV 61.3070 64.6023 54.2059 70.4895 75.8167 The results of the simulations for the different trim conditions can be seen in figures 4.3-4.8 respectively. From these results, it can be seen in figures 4.3(a) and 4.3(b) that for a given trim condition, the linear H∞ controller has a slightly better performance from the aspect of settling time for the velocity. It should be noted that though the system is slower with the proposed H∞ LPV control design, it is not significantly slower. This would suggest that the H∞ LPV control technique will in fact be a suitable control method for the hypersonic vehicle over a large range of motion for the given model and assumptions being made. Also note that there is a slight overshoot in the H∞ controller that is not present in the H∞ LPV controller. Additionally, it should be noted in figures 4.3(c) and 4.3(d) that the angle of attack for the H∞ LPV controller has slightly greater spikes in value than that seen with the H∞ controller. The H∞ controller responds slower and results in a smoother curve for the angle of attack. Similarly, the pitch rate and the pitch attitude (seen in figures 4.3(e), 4.3(f), 4.4(c), and 4.4(d)) of the vehicle have larger spikes for the H∞ LPV controller than they do for the H∞ controller. These spikes correspond to the larger magnitude spikes in the control efforts seen for the H∞ LPV controller in figures 4.6 and 4.7. Both systems settle to relatively the same value for a given set of trim conditions for the angle of attack, pitch rate, and pitch attitude respectively. 137 The altitude of the system can be seen in figures 4.4(a) and 4.4(a). These figures show that though the altitude fluctuates over the course of the simulation, it reaches a steady state condition before the end of the simulation. This is the result of adding in the unity gain regulatory state on the altitude for the velocity tracking case. It should also be noted that there is a larger range of fluctuations found in the H∞ LPV controller than in the H∞ controller. Figures 4.4, 4.5, and 4.6 show the flexible modes of the vehicle for the different simulations. From these figures it can be seen that the magnitudes of the H∞ controller are just slightly larger than what is seen with the H∞ LPV controller for the first mode of vibration. The other modes are very close in value. From chapter 2, it is known that the first mode of vibration is the dominant mode. It makes sense then that this mode would have the most dramatic difference between the two controllers. This difference is due to the slightly larger magnitudes in the angle of attack, pitch attitude, and pitch rate for the hypersonic vehicle. Because there are differences in the resulting steady state conditions for some of the rigid body dynamics, there are differences in the flexible states as well. Figures 4.6 and 4.7 shows the control effort for the hypersonic vehicle. From these figures, it can be seen that the H∞ controller has larger magnitudes than the H∞ LPV controller for the elevator. For the canard and the diffuser area ratio, the H∞ LPV controller has a slightly larger magnitude. For the fuel equivalence ratio, the two controllers produce very similar curves. Figure 4.8 also shows the integral of the error for the two different controllers. From this figure, it can be seen that the two different linear controllers yield very similar plots. The performance of the two linear controllers is very similar with the H∞ controller having a slightly better performance on the tracking state. 138 Velocity 60 50 40 Velocity in Feet per Second Velocity in Feet per Second 50 Reference Velocity Single H infinity LPV 70,000 ft Mach 7 Single H infinity LPV 90,000 ft Mach 7 30 Single H infinity LPV 80,000 ft Mach 8 Single H infinity LPV 70,000 ft Mach 9 Single H infinity LPV 90,000 ft Mach 9 20 10 0 Velocity 60 40 Reference Velocity Linear H infinity 70,000 ft Mach 7 Linear H infinity 90,000 ft Mach 7 30 Linear H infinity 80,000 ft Mach 8 Linear H infinity 70,000 ft Mach 9 Linear H infinity 90,000 ft Mach 9 20 10 0 5 10 15 20 Time in Seconds 25 30 0 35 0 5 (a) Velocity LPV x 10 0 −0.5 Angle of Attack in Radians 0 −1 −1.5 Single H infinity LPV 70,000 ft Mach 7 Single H infinity LPV 90,000 ft Mach 7 Single H infinity LPV 80,000 ft Mach 8 −2 x 10 Single H infinity LPV 70,000 ft Mach 9 −1.5 Linear H infinity 70,000 ft Mach 7 Linear H infinity 90,000 ft Mach 7 Linear H infinity 80,000 ft Mach 8 −2 Linear H infinity 70,000 ft Mach 9 Linear H infinity 90,000 ft Mach 9 −2.5 −2.5 −3 −3 −3.5 −3.5 0 5 10 15 20 Time in Seconds 25 30 35 0 5 (c) Angle of Attack LPV x 10 0 0 Pitch Rate in Radians per Second 1 −1 −2 Single H infinity LPV 70,000 ft Mach 7 Single H infinity LPV 90,000 ft Mach 7 Single H infinity LPV 80,000 ft Mach 8 Single H infinity LPV 70,000 ft Mach 9 Single H infinity LPV 90,000 ft Mach 9 25 35 Linear H infinity 70,000 ft Mach 7 30 (e) Pitch Rate LPV Linear H infinity 80,000 ft Mach 8 Linear H infinity 90,000 ft Mach 9 −6 35 Linear H infinity 90,000 ft Mach 7 Linear H infinity 70,000 ft Mach 9 −5 15 20 Time in Seconds x 10 −3 −5 10 30 −2 −4 5 25 −1 −4 0 15 20 Time in Seconds Pitch Rate −3 2 1 −3 10 (d) Angle of Attack H∞ Pitch Rate −3 Pitch Rate in Radians per Second 35 −1 Single H infinity LPV 90,000 ft Mach 9 −6 30 0.5 −0.5 2 25 Angle of Attack −3 1 0.5 Angle of Attack in Radians 15 20 Time in Seconds (b) Velocity H∞ Angle of Attack −3 1 10 0 5 10 15 20 Time in Seconds 25 (f) Pitch Rate H∞ Figure 4.3: Velocity Tracking Step Response 139 30 35 Altitude 25 Altitude 25 Single H infinity LPV 70,000 ft Mach 7 Single H infinity LPV 90,000 ft Mach 7 20 20 Single H infinity LPV 80,000 ft Mach 8 Single H infinity LPV 70,000 ft Mach 9 Single H infinity LPV 90,000 ft Mach 9 15 15 10 10 Altitude in Feet Altitude in Feet Linear H infinity 70,000 ft Mach 7 5 Linear H infinity 70,000 ft Mach 9 Linear H infinity 90,000 ft Mach 9 0 0 −5 −10 −10 0 5 10 15 20 Time in Seconds 25 30 −15 35 Linear H infinity 80,000 ft Mach 8 5 −5 −15 Linear H infinity 90,000 ft Mach 7 0 5 (a) Altitude LPV x 10 x 10 30 35 0.5 0 −0.5 Pitch Attitude in Radians 0 −0.5 −1 −1.5 Single H infinity LPV 70,000 ft Mach 7 Single H infinity LPV 90,000 ft Mach 7 Single H infinity LPV 80,000 ft Mach 8 −2 Single H infinity LPV 70,000 ft Mach 9 −1 −1.5 Linear H infinity 70,000 ft Mach 7 Linear H infinity 90,000 ft Mach 7 Linear H infinity 80,000 ft Mach 8 −2 Linear H infinity 70,000 ft Mach 9 Single H infinity LPV 90,000 ft Mach 9 Linear H infinity 90,000 ft Mach 9 −2.5 −2.5 −3 −3 −3.5 −3.5 0 5 10 15 20 Time in Seconds 25 30 35 0 5 (c) Pitch Attitude LPV 10 15 20 Time in Seconds 25 30 35 (d) Pitch Attitude H∞ First Modal Coordinate 0.25 First Modal Coordinate 0.25 0.2 0.2 0.15 0.15 Single H infinity LPV 70,000 ft Mach 7 Linear H infinity 70,000 ft Mach 7 Single H infinity LPV 90,000 ft Mach 7 First Modal Coordinate First Modal Coordinate 25 Pitch Attitude −3 1 0.5 Pitch Attitude in Radians 15 20 Time in Seconds (b) Altitude H∞ Pitch Attitude −3 1 10 Single H infinity LPV 80,000 ft Mach 8 0.1 Single H infinity LPV 70,000 ft Mach 9 Single H infinity LPV 90,000 ft Mach 9 0.05 Linear H infinity 90,000 ft Mach 7 Linear H infinity 70,000 ft Mach 9 Linear H infinity 90,000 ft Mach 9 0.05 0 0 −0.05 −0.05 −0.1 Linear H infinity 80,000 ft Mach 8 0.1 −0.1 0 5 10 15 20 Time in Seconds 25 30 35 0 (e) η1 LPV 5 10 15 20 Time in Seconds (f) η1 H∞ Figure 4.4: Velocity Tracking Step Response 140 25 30 35 Derivative of First Modal Coordinate 0.8 Derivative of First Modal Coordinate 0.8 Single H infinity LPV 70,000 ft Mach 7 Single H infinity LPV 90,000 ft Mach 7 0.6 0.6 Single H infinity LPV 80,000 ft Mach 8 Single H infinity LPV 70,000 ft Mach 9 Single H infinity LPV 90,000 ft Mach 9 0.4 Derivative of First Modal Coordinate Derivative of First Modal Coordinate 0.4 0.2 0 −0.2 0.2 0 −0.2 −0.4 −0.4 −0.6 −0.6 −0.8 −0.8 Linear H infinity 70,000 ft Mach 7 Linear H infinity 90,000 ft Mach 7 Linear H infinity 80,000 ft Mach 8 Linear H infinity 70,000 ft Mach 9 Linear H infinity 90,000 ft Mach 9 0 5 10 15 20 Time in Seconds 25 30 35 0 5 10 (a) η̇1 LPV 35 0.02 0.015 0.015 Single H infinity LPV 70,000 ft Mach 7 Single H infinity LPV 80,000 ft Mach 8 Single H infinity LPV 70,000 ft Mach 9 0.01 Single H infinity LPV 90,000 ft Mach 9 0.005 Linear H infinity 80,000 ft Mach 8 −0.005 −0.005 10 15 20 Time in Seconds 25 30 Linear H infinity 90,000 ft Mach 9 0.005 0 5 Linear H infinity 70,000 ft Mach 9 0.01 0 0 Linear H infinity 70,000 ft Mach 7 Linear H infinity 90,000 ft Mach 7 Second Modal Coordinate Single H infinity LPV 90,000 ft Mach 7 Second Modal Coordinate 30 Second Modal Coordinate 0.025 0.02 −0.01 25 (b) η̇1 H∞ Second Modal Coordinate 0.025 15 20 Time in Seconds −0.01 35 0 5 10 (c) η2 LPV 25 30 35 (d) η2 H∞ Derivative of Second Modal Coordinate 0.25 15 20 Time in Seconds Derivative of Second Modal Coordinate 0.25 Single H infinity LPV 70,000 ft Mach 7 Single H infinity LPV 90,000 ft Mach 7 0.2 0.2 Single H infinity LPV 80,000 ft Mach 8 Single H infinity LPV 70,000 ft Mach 9 Single H infinity LPV 90,000 ft Mach 9 0.15 Derivative of Second Modal Coordinate Derivative of Second Modal Coordinate 0.15 0.1 0.05 0 −0.05 Linear H infinity 80,000 ft Mach 8 −0.1 10 15 20 Time in Seconds 25 30 −0.2 35 (e) η̇2 LPV Linear H infinity 90,000 ft Mach 9 −0.05 −0.15 5 Linear H infinity 70,000 ft Mach 9 0 −0.1 0 Linear H infinity 90,000 ft Mach 7 0.05 −0.15 −0.2 Linear H infinity 70,000 ft Mach 7 0.1 0 5 10 15 20 Time in Seconds (f) η̇2 H∞ Figure 4.5: Velocity Tracking Step Response 141 25 30 35 Third Modal Coordinate −3 6 x 10 5 4 3 3 Single H infinity LPV 70,000 ft Mach 7 Single H infinity LPV 90,000 ft Mach 7 Third Modal Coordinate Third Modal Coordinate x 10 5 4 Single H infinity LPV 80,000 ft Mach 8 2 Single H infinity LPV 70,000 ft Mach 9 Single H infinity LPV 90,000 ft Mach 9 1 0 Linear H infinity 90,000 ft Mach 7 0 −2 −3 −3 10 15 20 Time in Seconds 25 30 −4 35 Linear H infinity 70,000 ft Mach 9 Linear H infinity 90,000 ft Mach 9 −2 5 Linear H infinity 80,000 ft Mach 8 1 −1 0 Linear H infinity 70,000 ft Mach 7 2 −1 −4 Third Modal Coordinate −3 6 0 5 10 (a) η3 LPV 25 30 35 (b) η3 H∞ Derivative of Third Modal Coordinate 0.03 15 20 Time in Seconds Derivative of Third Modal Coordinate 0.03 Single H infinity LPV 70,000 ft Mach 7 Single H infinity LPV 90,000 ft Mach 7 Single H infinity LPV 80,000 ft Mach 8 0.02 0.02 Single H infinity LPV 70,000 ft Mach 9 Derivative of Third Modal Coordinate Derivative of Third Modal Coordinate Single H infinity LPV 90,000 ft Mach 9 0.01 0 −0.01 0.01 0 −0.01 −0.02 −0.02 −0.03 −0.03 Linear H infinity 70,000 ft Mach 7 Linear H infinity 90,000 ft Mach 7 Linear H infinity 80,000 ft Mach 8 Linear H infinity 70,000 ft Mach 9 Linear H infinity 90,000 ft Mach 9 0 5 10 15 20 Time in Seconds 25 30 35 0 5 10 (c) η̇3 LPV 30 35 Elevator Control Effort 0.01 0.008 0.008 0.006 0.006 0.004 0.004 Elevator Angle in Radians Elevator Angle in Radians 25 (d) η̇3 H∞ Elevator Control Effort 0.01 15 20 Time in Seconds 0.002 0 −0.002 −0.004 Linear H infinity 70,000 ft Mach 7 Linear H infinity 90,000 ft Mach 7 Linear H infinity 80,000 ft Mach 8 0.002 Linear H infinity 70,000 ft Mach 9 Linear H infinity 90,000 ft Mach 9 0 −0.002 −0.004 −0.006 −0.006 Single H infinity LPV 70,000 ft Mach 7 Single H infinity LPV 90,000 ft Mach 7 Single H infinity LPV 80,000 ft Mach 8 −0.008 −0.008 Single H infinity LPV 70,000 ft Mach 9 Single H infinity LPV 90,000 ft Mach 9 −0.01 0 5 10 15 20 Time in Seconds 25 30 −0.01 35 (e) Elevator LPV 0 5 10 15 20 Time in Seconds (f) Elevator H∞ Figure 4.6: Velocity Tracking Step Response 142 25 30 35 Canard Control Effort 0.02 0.015 0.015 Single H infinity LPV 70,000 ft Mach 7 Single H infinity LPV 90,000 ft Mach 7 Single H infinity LPV 80,000 ft Mach 8 Single H infinity LPV 70,000 ft Mach 9 Single H infinity LPV 90,000 ft Mach 9 0.005 0 −0.005 Linear H infinity 70,000 ft Mach 7 0.01 Canard Angle in Radians Canard Angle in Radians 0.01 −0.01 Canard Control Effort 0.02 Linear H infinity 90,000 ft Mach 7 Linear H infinity 80,000 ft Mach 8 Linear H infinity 70,000 ft Mach 9 Linear H infinity 90,000 ft Mach 9 0.005 0 −0.005 0 5 10 15 20 Time in Seconds 25 30 −0.01 35 0 5 (a) Canard LPV 15 20 Time in Seconds 0.12 Single H infinity LPV 70,000 ft Mach 7 0.1 Linear H infinity 70,000 ft Mach 7 0.1 Single H infinity LPV 90,000 ft Mach 7 Linear H infinity 90,000 ft Mach 7 Single H infinity LPV 80,000 ft Mach 8 Linear H infinity 80,000 ft Mach 8 Single H infinity LPV 70,000 ft Mach 9 0.08 Throttle Ratio Throttle Ratio 35 0.14 0.12 Single H infinity LPV 90,000 ft Mach 9 0.06 0.04 0.02 0.02 0 0 −0.02 5 10 15 20 Time in Seconds 25 30 35 Linear H infinity 90,000 ft Mach 9 0.06 0.04 0 Linear H infinity 70,000 ft Mach 9 0.08 −0.02 0 5 (c) Fuel Equivalence Ratio LPV x 10 10 15 20 Time in Seconds 25 30 35 (d) Fuel Equivalence Ratio H∞ Diffuser Area Ratio Control Effort −3 Diffuser Area Ratio Control Effort −3 1 0.5 0.5 0 0 Diffuser Area Ratio Diffuser Area Ratio 30 Throttle Control Effort 0.16 0.14 1 25 (b) Canard H∞ Throttle Control Effort 0.16 10 −0.5 −1 x 10 −0.5 Linear H infinity 70,000 ft Mach 7 −1 Linear H infinity 90,000 ft Mach 7 Linear H infinity 80,000 ft Mach 8 Linear H infinity 70,000 ft Mach 9 −1.5 Linear H infinity 90,000 ft Mach 9 −1.5 Single H infinity LPV 70,000 ft Mach 7 Single H infinity LPV 90,000 ft Mach 7 −2 −2 Single H infinity LPV 80,000 ft Mach 8 Single H infinity LPV 70,000 ft Mach 9 Single H infinity LPV 90,000 ft Mach 9 −2.5 0 5 10 15 20 Time in Seconds 25 30 −2.5 35 (e) Diffuser Area Ratio LPV 0 5 10 15 20 Time in Seconds 25 (f) Diffuser Area Ratio H∞ Figure 4.7: Velocity Tracking Step Response 143 30 35 Integration of the Error 350 300 300 250 Integration of the Error (Velocity) Integration of the Error (Velocity) Integration of the Error 350 Single H infinity LPV 70,000 ft Mach 7 Single H infinity LPV 90,000 ft Mach 7 Single H infinity LPV 80,000 ft Mach 8 Single H infinity LPV 70,000 ft Mach 9 200 Single H infinity LPV 90,000 ft Mach 9 150 250 200 Linear H infinity 70,000 ft Mach 9 100 50 50 0 5 10 15 20 Time in Seconds 25 30 0 35 (a) Integral of Tracking Error LPV Linear H infinity 90,000 ft Mach 7 Linear H infinity 80,000 ft Mach 8 100 0 Linear H infinity 70,000 ft Mach 7 150 Linear H infinity 90,000 ft Mach 9 0 5 10 15 20 Time in Seconds 25 30 35 (b) Integral of Tracking Error H∞ Figure 4.8: Velocity Tracking Step Response 4.2.2 Altitude Tracking For the altitude tracking case, two different control algorithms will be simulated at each of the five trim conditions. As with the velocity tracking case, a linearized plant at the trim conditions will be implemented along with a single controller. Additionally, there will be no control effort saturation limits implemented in this test. Though the saturation limits for the actuators have been neglected, the actuator dynamics are still included for the system. The difference will be the control gains, K, being applied to the system. For each controller at each trim condition, the system will initially start with all of the system and actuator states at 0. The vehicle will then be subjected to a 200 fst step input. Since the plant is linear, the error codes built into the nonlinear system for failure of proper engine combustion will not be included for the linear system. This simply means that though the nonlinear plant does not have a continuous operational envelope, the linear plant will [40]. The different γ values for the different controllers can be seen in table 4.11. From this table, it can be seen that the H∞ controllers have a better γ performance variable than the H∞ LPV controller. This is due to the fact that the H∞ LPV controller is synthesizing more LMI’s and more variables than the H∞ controller is. This results in a more constrained problem which is more difficult to optimize. The result is that the system is not quite as robust as a single H∞ controller designed for one trim condition. The altitude tracking results of the simulations for the different trim conditions can be seen in figures 4.9-4.14 respectively. As with the velocity tracking case, figures 4.9(a) and 4.9(b) 144 Table 4.11: H∞ γ Performance Values Trim Conditions 70,000 ft at Mach 7 90,000 ft at Mach 7 80,000 ft at Mach 8 70,000 ft at Mach 9 90,000 ft at Mach 9 H∞ 14.7571 32.6148 21.8591 19.3016 32.7719 H∞ LPV 100.3374 67.6847 31.6570 107.2033 48.0301 show that for a given trim condition the linear H∞ controller has a slightly better performance from the aspect of settling time for the altitude tracking. It should be noted that though the system is slower with the proposed H∞ LPV control design, it is not significantly slower. This would suggest that the H∞ LPV control technique will in fact be a suitable control method for the hypersonic vehicle over a large range of motion for the given model and assumptions being made. Additionally, it can be observed that there is a larger peak overshoot associated with the H∞ controller. Figures 4.9(c) and 4.9(d) show the angle of attack for the hypersonic vehicle for the two controllers. From this figure, it can be seen that the H∞ controller has a slightly larger magnitude spike in the angle of attack than the H∞ LPV controller. This is true for all trim conditions for the system. Additionally, it can be seen that the magnitude of the spikes in the pitch rate and the pitch attitude seen in figures 4.9(e), 4.9(f), 4.10(c), and 4.10(d) show the same characteristic. This would make sense because there are high frequency response characteristics seen in the control efforts seen in figures 4.12 and 4.13. The control efforts drive these system states, thus there are small magnitude spikes. Figures 4.10(a) and 4.10(b) show the velocity curve for the hypersonic vehicle. From this figure, it can be seen that the velocities for the H∞ controller settle out to a steady state condition by the end of the simulation. The H∞ LPV controller continues to increase over the course of the simulation. This is not desirable, but the amount of change over the course of the simulation is not very large. This small growth is acceptable in that it will not cause the vehicle to move out of the controllable region in a short period of time. It would appear that the regulation state on the velocity does not work well with the H∞ LPV controller, but the nonlinear simulation will give more insight into this relationship. Figures 4.10, 4.11, and 4.12 show the flexible modes of the hypersonic vehicle. These modes are directly correlated to the flexibility of the vehicle. These values are slightly larger for the 145 H∞ controller than for the H∞ LPV controller. These results are very close in value which shows that the altitude tracking case for output feedback has less effect on the flexibility of the hypersonic vehicle than the velocity tracking case does. For this reason, the performance of the two linear controllers matches up closely for this case whereas it did not for the velocity tracking case. There is a slight difference between the controllers, but the difference is minimal. Figures 4.12 and 4.13 show the control effort of the vehicle. It can be seen in this figure that the elevator and canard control efforts have a large spike in the system near the start of the simulation. These large spikes are the cause of the spikes seen in the angle of attack, pitch rate, and pitch attitude of the vehicle. It can be seen in the fuel equivalence ratio that the magnitude of this control effort is larger for the H∞ LPV controller than it is for the H∞ controller. The diffuser area ratio response is almost identical for the two different controllers. The diffuser area ratio is relatively constant over the course of the simulation. Additionally, figure 4.14 shows the integration of the error signal. Since the H∞ controller has a faster response time than the H∞ LPV controller, the integration of the error is smaller at the end of the simulation for the H∞ controller. 146 Altitude Altitude 200 200 150 Altitude in Feet Altitude in Feet 150 Reference Altitude Single H infinity LPV 70,000 ft Mach 7 Single H infinity LPV 90,000 ft Mach 7 Single H infinity LPV 80,000 ft Mach 8 100 Single H infinity LPV 70,000 ft Mach 9 Single H infinity LPV 90,000 ft Mach 9 50 Reference Altitude Linear H infinity 70,000 ft Mach 7 Linear H infinity 90,000 ft Mach 7 Linear H infinity 80,000 ft Mach 8 100 Linear H infinity 70,000 ft Mach 9 Linear H infinity 90,000 ft Mach 9 50 0 0 0 5 10 15 20 Time in Seconds 25 30 35 0 5 (a) Altitude LPV Angle of Attack in Radians Angle of Attack in Radians 0.02 Single H infinity LPV 70,000 ft Mach 7 Single H infinity LPV 90,000 ft Mach 7 Single H infinity LPV 80,000 ft Mach 8 35 Single H infinity LPV 70,000 ft Mach 9 0.01 Single H infinity LPV 90,000 ft Mach 9 0 Linear H infinity 90,000 ft Mach 7 Linear H infinity 80,000 ft Mach 8 −0.02 −0.02 10 15 20 Time in Seconds 25 30 −0.03 35 Linear H infinity 90,000 ft Mach 9 0 −0.01 5 Linear H infinity 70,000 ft Mach 9 0.01 −0.01 0 Linear H infinity 70,000 ft Mach 7 0 5 (c) Angle of Attack LPV 0.04 Pitch Rate in Radians per Second 0.05 0.04 Single H infinity LPV 70,000 ft Mach 7 Single H infinity LPV 90,000 ft Mach 7 Single H infinity LPV 80,000 ft Mach 8 0.02 15 20 Time in Seconds 25 30 35 Pitch Rate 0.06 0.05 0.03 10 (d) Angle of Attack H∞ Pitch Rate 0.06 Pitch Rate in Radians per Second 30 0.03 0.02 Single H infinity LPV 70,000 ft Mach 9 Single H infinity LPV 90,000 ft Mach 9 0.01 0 −0.01 −0.02 0.03 Linear H infinity 70,000 ft Mach 7 Linear H infinity 90,000 ft Mach 7 Linear H infinity 80,000 ft Mach 8 0.02 Linear H infinity 70,000 ft Mach 9 Linear H infinity 90,000 ft Mach 9 0.01 0 −0.01 −0.02 −0.03 −0.04 25 Angle of Attack 0.04 0.03 −0.03 15 20 Time in Seconds (b) Altitude H∞ Angle of Attack 0.04 10 −0.03 0 5 10 15 20 Time in Seconds 25 30 −0.04 35 (e) Pitch Rate LPV 0 5 10 15 20 Time in Seconds 25 (f) Pitch Rate H∞ Figure 4.9: Altitude Tracking Step Response 147 30 35 Velocity Velocity 40 35 35 30 30 25 25 Velocity in Feet per Second Velocity in Feet per Second 40 20 15 10 5 Linear H infinity 70,000 ft Mach 7 Linear H infinity 90,000 ft Mach 7 Linear H infinity 80,000 ft Mach 8 20 Linear H infinity 70,000 ft Mach 9 Linear H infinity 90,000 ft Mach 9 15 10 5 0 0 Single H infinity LPV 70,000 ft Mach 7 Single H infinity LPV 90,000 ft Mach 7 Single H infinity LPV 80,000 ft Mach 8 −5 −5 Single H infinity LPV 70,000 ft Mach 9 Single H infinity LPV 90,000 ft Mach 9 −10 0 5 10 15 20 Time in Seconds 25 30 −10 35 0 5 (a) Velocity LPV 0.03 Single H infinity LPV 70,000 ft Mach 7 35 Single H infinity LPV 80,000 ft Mach 8 Single H infinity LPV 70,000 ft Mach 9 0.02 Single H infinity LPV 90,000 ft Mach 9 0.01 Linear H infinity 80,000 ft Mach 8 −0.01 −0.01 10 15 20 Time in Seconds 25 30 −0.02 35 Linear H infinity 90,000 ft Mach 9 0.01 0 5 Linear H infinity 70,000 ft Mach 9 0.02 0 0 Linear H infinity 70,000 ft Mach 7 Linear H infinity 90,000 ft Mach 7 Pitch Attitude in Radians Pitch Attitude in Radians Single H infinity LPV 90,000 ft Mach 7 0 5 (c) Pitch Attitude LPV 10 15 20 Time in Seconds 25 30 35 (d) Pitch Attitude H∞ First Modal Coordinate 1.5 First Modal Coordinate 1.5 1 1 Single H infinity LPV 70,000 ft Mach 7 Linear H infinity 70,000 ft Mach 7 Single H infinity LPV 90,000 ft Mach 7 First Modal Coordinate First Modal Coordinate 30 0.04 0.03 Single H infinity LPV 80,000 ft Mach 8 0.5 Single H infinity LPV 70,000 ft Mach 9 Single H infinity LPV 90,000 ft Mach 9 0 −0.5 −1 25 Pitch Attitude 0.05 0.04 −0.02 15 20 Time in Seconds (b) Velocity H∞ Pitch Attitude 0.05 10 Linear H infinity 90,000 ft Mach 7 Linear H infinity 80,000 ft Mach 8 0.5 Linear H infinity 70,000 ft Mach 9 Linear H infinity 90,000 ft Mach 9 0 −0.5 0 5 10 15 20 Time in Seconds 25 30 −1 35 (e) η1 LPV 0 5 10 15 20 Time in Seconds (f) η1 H∞ Figure 4.10: Altitude Tracking Step Response 148 25 30 35 Derivative of First Modal Coordinate 4 3 2 Derivative of First Modal Coordinate 3 Derivative of First Modal Coordinate Derivative of First Modal Coordinate 4 Single H infinity LPV 70,000 ft Mach 7 Single H infinity LPV 90,000 ft Mach 7 Single H infinity LPV 80,000 ft Mach 8 Single H infinity LPV 70,000 ft Mach 9 1 Single H infinity LPV 90,000 ft Mach 9 0 −1 2 1 0 −1 Linear H infinity 70,000 ft Mach 7 −2 −2 −3 −3 Linear H infinity 90,000 ft Mach 7 Linear H infinity 80,000 ft Mach 8 Linear H infinity 70,000 ft Mach 9 Linear H infinity 90,000 ft Mach 9 0 5 10 15 20 Time in Seconds 25 30 35 0 5 10 (a) η̇1 LPV 0.02 0.01 0.01 Second Modal Coordinate Second Modal Coordinate 30 35 Second Modal Coordinate 0.03 0.02 0 −0.01 Single H infinity LPV 70,000 ft Mach 7 Single H infinity LPV 90,000 ft Mach 7 −0.02 25 (b) η̇1 H∞ Second Modal Coordinate 0.03 15 20 Time in Seconds Single H infinity LPV 80,000 ft Mach 8 Single H infinity LPV 70,000 ft Mach 9 0 −0.01 −0.02 Single H infinity LPV 90,000 ft Mach 9 −0.03 −0.03 −0.04 −0.04 −0.05 −0.05 Linear H infinity 70,000 ft Mach 7 Linear H infinity 90,000 ft Mach 7 Linear H infinity 80,000 ft Mach 8 Linear H infinity 70,000 ft Mach 9 Linear H infinity 90,000 ft Mach 9 0 5 10 15 20 Time in Seconds 25 30 35 0 5 10 (c) η2 LPV 25 30 35 25 30 35 (d) η2 H∞ Derivative of Second Modal Coordinate 1.5 15 20 Time in Seconds Derivative of Second Modal Coordinate 1.5 Linear H infinity 70,000 ft Mach 7 Linear H infinity 90,000 ft Mach 7 Linear H infinity 80,000 ft Mach 8 1 1 Linear H infinity 70,000 ft Mach 9 Derivative of Second Modal Coordinate Derivative of Second Modal Coordinate Linear H infinity 90,000 ft Mach 9 0.5 0 −0.5 Single H infinity LPV 70,000 ft Mach 7 −1 0.5 0 −0.5 −1 Single H infinity LPV 90,000 ft Mach 7 Single H infinity LPV 80,000 ft Mach 8 Single H infinity LPV 70,000 ft Mach 9 Single H infinity LPV 90,000 ft Mach 9 −1.5 0 5 10 15 20 Time in Seconds 25 30 −1.5 35 (e) η̇2 LPV 0 5 10 15 20 Time in Seconds (f) η̇2 H∞ Figure 4.11: Altitude Tracking Step Response 149 Third Modal Coordinate −3 10 x 10 Third Modal Coordinate −3 10 8 x 10 8 6 6 Single H infinity LPV 90,000 ft Mach 7 4 Third Modal Coordinate Third Modal Coordinate Single H infinity LPV 70,000 ft Mach 7 Single H infinity LPV 80,000 ft Mach 8 Single H infinity LPV 70,000 ft Mach 9 Single H infinity LPV 90,000 ft Mach 9 2 0 4 2 0 −2 −2 −4 −4 −6 −6 Linear H infinity 70,000 ft Mach 7 Linear H infinity 90,000 ft Mach 7 Linear H infinity 80,000 ft Mach 8 Linear H infinity 70,000 ft Mach 9 Linear H infinity 90,000 ft Mach 9 0 5 10 15 20 Time in Seconds 25 30 35 0 5 10 (a) η3 LPV 15 20 Time in Seconds 25 30 35 25 30 35 (b) η3 H∞ Derivative of Third Modal Coordinate Derivative of Third Modal Coordinate Linear H infinity 70,000 ft Mach 7 Linear H infinity 90,000 ft Mach 7 0.1 Linear H infinity 80,000 ft Mach 8 0.1 Linear H infinity 70,000 ft Mach 9 Derivative of Third Modal Coordinate Derivative of Third Modal Coordinate Linear H infinity 90,000 ft Mach 9 0.05 0 Single H infinity LPV 70,000 ft Mach 7 −0.05 Single H infinity LPV 90,000 ft Mach 7 Single H infinity LPV 80,000 ft Mach 8 Single H infinity LPV 70,000 ft Mach 9 Single H infinity LPV 90,000 ft Mach 9 −0.1 −0.15 −0.2 0.05 0 −0.05 −0.1 −0.15 0 5 10 15 20 Time in Seconds 25 30 −0.2 35 0 5 10 (c) η̇3 LPV (d) η̇3 H∞ Elevator Control Effort 0.06 0.05 0.04 0.04 Single H infinity LPV 70,000 ft Mach 7 0.03 Single H infinity LPV 90,000 ft Mach 7 Elevator Angle in Radians Elevator Angle in Radians Elevator Control Effort 0.06 0.05 Single H infinity LPV 80,000 ft Mach 8 Single H infinity LPV 70,000 ft Mach 9 0.02 Single H infinity LPV 90,000 ft Mach 9 0.01 0 Linear H infinity 80,000 ft Mach 8 15 20 Time in Seconds 25 30 −0.03 35 (e) Elevator LPV Linear H infinity 90,000 ft Mach 9 0 −0.02 10 Linear H infinity 70,000 ft Mach 9 0.01 −0.01 5 Linear H infinity 90,000 ft Mach 7 0.02 −0.02 0 Linear H infinity 70,000 ft Mach 7 0.03 −0.01 −0.03 15 20 Time in Seconds 0 5 10 15 20 Time in Seconds (f) Elevator H∞ Figure 4.12: Altitude Tracking Step Response 150 25 30 35 Canard Control Effort Canard Control Effort 0.02 0.015 0.015 0.01 0.01 Canard Angle in Radians Canard Angle in Radians 0.02 0.005 0 −0.005 0.005 0 −0.005 −0.01 −0.01 Single H infinity LPV 70,000 ft Mach 7 Linear H infinity 70,000 ft Mach 7 Single H infinity LPV 90,000 ft Mach 7 −0.015 Linear H infinity 90,000 ft Mach 7 −0.015 Single H infinity LPV 80,000 ft Mach 8 Linear H infinity 80,000 ft Mach 8 Single H infinity LPV 70,000 ft Mach 9 Linear H infinity 70,000 ft Mach 9 Single H infinity LPV 90,000 ft Mach 9 −0.02 0 5 10 15 20 Time in Seconds 25 30 Linear H infinity 90,000 ft Mach 9 −0.02 35 0 5 (a) Canard LPV 15 20 Time in Seconds 25 30 35 (b) Canard H∞ Throttle Control Effort 0.05 10 Throttle Control Effort 0.05 Single H infinity LPV 70,000 ft Mach 7 Single H infinity LPV 90,000 ft Mach 7 Single H infinity LPV 80,000 ft Mach 8 0.04 0.04 Single H infinity LPV 70,000 ft Mach 9 Single H infinity LPV 90,000 ft Mach 9 0.03 0.03 Linear H infinity 70,000 ft Mach 7 Linear H infinity 90,000 ft Mach 7 Linear H infinity 70,000 ft Mach 9 Throttle Ratio Throttle Ratio Linear H infinity 80,000 ft Mach 8 0.02 0.01 0.01 0 0 −0.01 Linear H infinity 90,000 ft Mach 9 0.02 −0.01 0 5 10 15 20 Time in Seconds 25 30 35 0 5 (c) Fuel Equivalence Ratio LPV 1.5 1.5 1 1 0.5 0.5 0 Single H infinity LPV 70,000 ft Mach 7 Single H infinity LPV 90,000 ft Mach 7 −0.5 Single H infinity LPV 80,000 ft Mach 8 x 10 Linear H infinity 90,000 ft Mach 7 Linear H infinity 80,000 ft Mach 8 Linear H infinity 70,000 ft Mach 9 Linear H infinity 90,000 ft Mach 9 −1 −1 −1.5 −1.5 5 10 15 20 Time in Seconds 25 30 35 Linear H infinity 70,000 ft Mach 7 −0.5 Single H infinity LPV 90,000 ft Mach 9 0 30 0 Single H infinity LPV 70,000 ft Mach 9 −2 25 Diffuser Area Ratio Control Effort −3 2 Diffuser Area Ratio Diffuser Area Ratio x 10 15 20 Time in Seconds (d) Fuel Equivalence Ratio H∞ Diffuser Area Ratio Control Effort −3 2 10 −2 35 (e) Diffuser Area Ratio LPV 0 5 10 15 20 Time in Seconds 25 (f) Diffuser Area Ratio H∞ Figure 4.13: Altitude Tracking Step Response 151 30 35 Integration of the Error 900 900 800 800 700 700 600 500 Single H infinity LPV 70,000 ft Mach 7 400 Single H infinity LPV 90,000 ft Mach 7 Single H infinity LPV 80,000 ft Mach 8 Single H infinity LPV 70,000 ft Mach 9 300 Single H infinity LPV 90,000 ft Mach 9 600 500 Linear H infinity 70,000 ft Mach 7 400 Linear H infinity 80,000 ft Mach 8 Linear H infinity 70,000 ft Mach 9 200 100 100 0 5 10 15 20 Time in Seconds 25 30 0 35 (a) Integral of Tracking Error LPV Linear H infinity 90,000 ft Mach 7 300 200 0 Integration of the Error 1000 Integration of the Error (Velocity) Integration of the Error (Velocity) 1000 Linear H infinity 90,000 ft Mach 9 0 5 10 15 20 Time in Seconds 25 30 35 (b) Integral of Tracking Error H∞ Figure 4.14: Altitude Tracking Step Response 4.3 Nonlinear HSV Analysis and LPV Control Implementation This section will outline the procedure used to simulate the response of the nonlinear hypersonic vehicle using the previously synthesized controllers. This section will explain the different caveats of the simulation process including the basic setup, switching algorithm, and disturbances found in the system. Both altitude tracking and velocity tracking will be discussed as well as both the rigid body and flexible body cases. 4.3.1 Setup This subsection will discuss how to set up and simulate the closed loop hypersonic vehicle model using Simulink and Matlab 2008a. In order to build the model, S-functions and other built in Simulink functions will be utilized. This dissertation will assume that the reader has a working knowledge of this software. The block diagrams for the velocity tracking and altitude tracking can be seen in figures 4.15 and 4.16 respectively. It should be noted here that there is no difference in the block diagram for the flexible and rigid body cases. This is due to the fact that the measured states are the rigid body states, and the flexible states are the estimated states in the system. In the two figures shown, Wact represents the actuator dynamics discussed previously. There is also a saturation function that is applied to the actuators. The saturation levels for these actuators can be seen in table 2.2. For the purpose of simulation, the full nonlinear plant model will be used for both the flexible body cases as well as the rigid body cases. This will show the flexible 152 X(1) Nonlinear Plant X(2:5) d(1:5) + Wact u K - + ref Figure 4.15: Block Diagram of Closed Loop System for the Velocity Tracking Case states response when a rigid body controller is used. These effects will help to establish the differences between the flexible and rigid body controllers. As a result it will be possible to draw some conclusions as to whether the rigid body controller is a valid assumption to make when implementing an H∞ LPV controller to the hypersonic vehicle. The nonlinear plant used in the simulation is the same model as provided in chapter 2 [8, 7]. Both the plant and the controller, K, will be implemented using S-functions. It should be noted that K is not a single set of gains, but a table of controllers. It will be necessary to develop an algorithm to switch between the controllers appropriately. For this nonlinear simulation, two different reference signals were chosen for investigation for each controller designed. A total of 8 different simulations will be presented in this dissertation. The first reference signal is a ramp input to the system. The second reference signal is a multistep input to the system. These two inputs were chosen because the ramp represents a realistic input to the system, and the multi-step input represents a worst case scenario for the tracking signal. These two reference signals were chosen ad hoc, and are by no means considered to be the results of an exhaustive study. It is merely intended to give the reader an understanding of how different input conditions can influence the output of the system. Each of the inputs were chosen such theat they start at the middle of the range for both Mach number and altitude, and so that they end near the end of the range of operation. This was specified so that the vehicle would start well within the operating envelope and then be moved to the edge of the operating envelope. 153 X(1:3) X(4) Nonlinear Plant X(5) d(1:5) + Wact u K - + ref Figure 4.16: Block Diagram of Closed Loop System for the Altitude Tracking Case 4.3.2 Robustness Analysis The purpose for implementing an H∞ LPV controller is to control the hypersonic vehicle over a large range of motion while exhibiting robust capabilities. Robust capabilities are the system’s ability to handle uncertainties and perturbations to the system. Uncertainties are considered to be things that may exist in the physical system that are not properly modeled. Perturbations are outside disturbance that comes into the system. The uncertainties and perturbations that can be applied to the hypersonic vehicle have been the topic of research for quite some time. The hypersonic vehicle is very difficult to model [6]. This being the case, it will be important to investigate some aspects of uncertainty and perturbation in the hypersonic vehicle model used in this dissertation. As discussed in chapter 3, there are many assumptions that have been made for the hypersonic vehicle model used in this dissertation that may not be valid. Some of these potential sources of error include thermal effects on the vehicle, fuel consumption (a change in mass), and fluctuations in the atmospheric air data used [54, 14, 60]. These all have an impact on the hypersonic vehicle and should ultimately be modeled for a full mission. This dissertation is dealing with a generic hypersonic vehicle, and so this information is not available. Since this is the case, an attempt to investigate the effects of these uncertainties on the control of the hypersonic vehicle will be made. To accomplish this, it is assumed that the thermal effects of the system change the vehicle’s moment of inertia as well as the length of the vehicle. Additionally, it is assumed that the mass of the vehicle is changed by fuel consumption, and that the air density, pressure, and temperature from the table lookup may not be accurate [44]. 154 In addition to system uncertainty, it is important to look at outside disturbances to the hypersonic vehicle as well. There are many outside things that could affect the hypersonic vehicle, but in order to simplify the model, only the sensor noise will be used for this dissertation. Note that in figures 4.15 and 4.16 there is a disturbance, d, added into the measurement states. This disturbance is supposed to represent the sensor noise present in the system. Since a hypersonic vehicle is a high performance vehicle, and given its relatively large price tag, it is suitable to assume that very high quality sensors would be used in such a system. By making this assumption, it is now suitable to say that the sensor noise would be relatively small. To simulate the sensor noise, random number blocks were used in Simulink. Table 4.12 shows the values used to simulate the disturbances in the Simulink model. Table 4.12: Sensor noise variance and seed values State Seed Variance V 23 .01 ft/s α 1 .00035 radians Q 314159265 .001 rad/s h 1.23 × 106 1 ft θ 61 .001 radians The goal of adding perturbations to the system for this study is not necessarily to model the effects of uncertainties in the system, but rather it is an attempt to see the effects that modeling error has on the performance of the robust controller. Since this is the case, the emphasis for the study will be to look at the results of changing these parameters as opposed to developing accurate perturbation models. This being said, each of the previously mentioned parameters (air density, air pressure, air temperature, vehicle length, and vehicle moment of inertia) were increased by 5% from their nominal values with the exception of the vehicle mass. This is assumed to capture any changes in the model due to heating and inaccurate air property tables. The value of the vehicle mass was decreased by 5% from the nominal value as this more accurately represents the fuel consumption that would take place during hypersonic flight. All of these perturbations will be applied in the nonlinear plant block in figures 4.15 and 4.16 along with the outside disturbances mentioned previously. Simulations will be run for cases perturbed and nominal, and an attempt will be made to draw any significant conclusions that 155 can be made from the results. 4.3.3 LPV Control Switching Algorithm All of the different aspects of simulating the H∞ LPV controller have been covered at this point except for the problem of switching between the different linear controllers. There are a total of 49 linear controllers that have been designed for the hypersonic vehicle model, and there is a need to switch between the different controllers as the vehicle moves through the parameter space. There are several different ways of handling this switching as was mentioned in chapter 3. Some switching algorithms included interpolation, blending, and digital switching just to name a few. The design process for the H∞ LPV controller shows that each controller gives the control output in terms of a change from the nominal trim values for a given linearized controller. If the trim values for the plant states and control forces are denoted by x̄ and ū, then the following equations define the system states and control effort. x = x̄ + ∆x (4.13) u = ū + ∆u (4.14) Since the trim conditions are known for the system from the control synthesis and the plant states, x, are known for the system from the nonlinear plant model, it will be possible to calculate the control effort, u, needed. This can be done by applying the control law as seen in the following equations. " # ∆x ẋk = Ak xk + Bk R e " #! ∆x u = ū + Ck xk + Dk R e (4.15) (4.16) Where the controller, K, is defined as, " ss K= Ak Bk Ck Dk # (4.17) and xk represents the internal states in the output feedback system. Note that even though the five measured states from the system are with respect to a set of trim conditions, the integral of the error between the reference signal and the tracking state of the vehicle is not. This is because the reference signal does not change with respect to a set of trim conditions. Because 156 of equation 4.16, it will be beneficial to use a digital switching algorithm so that the least amount of trim condition changes can be imposed upon the system. As discussed in chapter 3, there is a problem with using interpolation methods that results from continuity issues. When interpolating the controller, it is also necessary to interpolate the set of trim conditions x̄ and ū as well. This can pose some stability issues since, as previously stated, the interpolated set of trim conditions may not exist, or at the very least may not be accurately represented by a linear interpolation. It is also difficult to insure that the resulting controller gains are stable if this calculation is done online. This being the case, the decision was made to use a digital switching algorithm due to its simplicity for implementation as well as its computational benefits. In addition to the problem of establishing a new set of trim values, there is also an issue with the integral of the error in the state vector. When switching from one controller to the next, it is important to have a method for reseting this value. Otherwise, there is a risk that the system will run into an integral windup state which can lead to saturation in the control efforts, or even cause the system to leave the range of operability. This can be a serious problem for the controller. This situation is not quite the same as the problem discussed in the work by Groves et al [27]. Their work describes a situation where the control effort is saturated by the linear controller. With no account in the control synthesis for the saturation of the control inputs, there is a windup that can cause instability in the system. In this dissertation, the problem of windup is a consequence of the integration of the error building up to large values at new trim conditions. This too will lead to the saturation of the control efforts, but the solution technique suggested by Groves et al. will not be applicable for this case. To counteract this effect, a method for reseting the integral state at the instant the controller switched from one value to the next was needed. In chapter 3, it was proposed that the best way to implement this was to simply reset the integral state such that the change in the control effort was minimized at the switching instant. Since the integral state does not physically represent anything that occurs in the system, it is acceptable to artificially change this value. This is often done with simple PID systems [45]. For the output feedback case however, there are even more states that do not physically represent system states. These internal controller states, which are used in estimating the non-measured states, can also be reset to gain additional control over the integral windup problem. To this end, a suitable switching algorithm was designed to alleviate this issue. To reset the internal system states, xk , and the integral of the error in the system, a set of R LMI’s were derived such that a new set of values for xk and e are solved to minimize the change in the control effort u. Assuming that Vcl = xTcl P xcl ≥ 0 The LMI’s used are listed as 157 follows " Vcl γ uT+ u+ − γI x− p xk # − Vcl >0 + x+ p xk (4.18) ≥0 ulower ≤ u+ ≤ uupper (4.19) (4.20) The results of this minimization problem gives the new vector x̃k which contains the new set of internal states that will minimize the control effort after switching. This small change in the control effort keeps the system from saturating the actuators at the time of the switching, and helps to maintain stability in the nonlinear simulation. It has been established that a digital switching technique will be implemented. Additionally, an algorithm for handling the integral windup that occurs in the system has also been established. The remaining problem is to determine when switching controllers needs to take place during the simulation. Since there are two parameters from the H∞ LPV control synthesis problem, Mach number and altitude, the parameter space can be described by a two dimensional envelope. As discussed in chapter 3, it was decided that the best possible time to implement the switch from one controller to the next was if the system reached the next trim condition along a given parameter. Figure 3.19 shows that the system is designed such that the controllable region of one trim point overlaps the subsequent trim conditions closest to it in all directions. Although it is not possible to ensure that this is the case during synthesis, this can be validated for a given reference command through simulation. Figure 3.20 shows the switching conditions for the system along the Mach number axis. This figure illustrates the concept of switching once the threshold of the next trim condition has been met by the system. The idea of this implementation is that the hypersonic vehicle states will be as close to the trim conditions as possible so that any disturbance caused by the controller switch will have minimal effect on the system. This usually results in the controller being switched due to either Mach or altitude, and not both simultaneously. 4.4 Nonlinear Simulation Results This section will examine at the results from simulating the nonlinear hypersonic vehicle. For the velocity and altitude tracking cases with both flexible and rigid body controllers, two different command signals will be considered, a ramp input and a multiple step input. For each of these cases, there will plots for the system states, the actuators, and the integration of the error. Additionally, the flexible modes of vibration will be plotted both for the flexible 158 and rigid body cases to show the flexibility of the hypersonic vehicle. There will also be a plot showing the controller switching times. Table A.1 in appendix A shows the Mach and altitude trim conditions for a given reference controller number. This will help the reader interpret the meaning of the controller switching plots. 4.4.1 Unstable Linear Control To further motivate this research, this subsection will show the results of applying a linear H∞ controller to the nonlinear plant. It will be seen that using a single linear controller over a large operational range will result in a failure of the linear controller. For the purposes of the simulation conducted in this subsection, the linear H∞ controller synthesized for the trim conditions at Mach 8 and 80,000 feet in altitude. The ramp case with no perturbation added to the flexible body system for both the velocity tracking and altitude tracking cases will be shown in this section. These cases were chosen because they are the easiest cases to control. If the linear controller fails for these cases, then it will not be able to handle more difficult cases such as a step command, a rigid body controller, or a case where perturbation is included in the system. Velocity Tracking The tracking signal used for this case starts at a velocity of 7, 819.6 fst and has a slope of 20 sf2t for a duration of 60 seconds. After the 60 second interval, the slope of the ramp is 0 sf2t for the remainder of the simulation time. The results from the simulation can be seen in figures 4.17-4.19. From this set of plots, it can be seen that the simulation fails at about 59 seconds. Figure 4.17 shows the rigid body states for the velocity tracking case. From this set of plots, it can be seen that the system does seem to track the command velocity until the simulation fails. This simulation failure is a result of the angle of attack, pitch attitude and pitch rate becoming very large towards the end of this simulation. These large values cause the roots to the polynomials used to solve the flow through the scramjet to become imaginary. This simply means that there is flame-out in the scramjet. The model used in this dissertation is not equipped to handle this situation, and as a result the simulation is terminated and deemed a failure. Figure 4.18 shows the flexible states of the system. It can be seen from these plots that the flexible states of the system become very large towards the end of the simulation. This would indicate that the controller is having more difficulty in maintaining stability as the hypersonic 159 vehicle tracks farther away from the trim conditions. It will be necessary to implement a different control algorithm to handle this extended range of control. Figure 4.19 shows the control effort of the vehicle. It can be seen from this set of plots that the control effort is growing in both magnitude and frequency towards the end of the simulation. This growth is due to the vehicle reaching the limitation of its operational envelope. It is likely that the velocity tracking state would go unstable in a short period of time were it not the failure of the model that caused a termination of the simulation. 160 Angle of Attack Velocity 0.021 9000 Actual Velocity Reference Velocity 0.02 8800 0.019 Angle of Attack in Radians Velocity in Feet per Second 0.018 8600 8400 8200 0.017 0.016 0.015 0.014 0.013 8000 0.012 0.011 0 10 20 30 Time in Seconds 40 50 0 10 (a) Velocity Flexible 20 30 Time in Seconds 40 50 (b) Angle of Attack Flexible Pitch Rate Altitude 4 8 x 10 0.07 0.06 7.999 7.998 0.04 0.03 Altitude in Feet Pitch Rate in Radians per Second 0.05 0.02 0.01 0 7.997 7.996 7.995 −0.01 7.994 −0.02 −0.03 7.993 0 10 20 30 Time in Seconds 40 50 0 10 (c) Pitch Rate Flexible 20 40 50 (d) Altitude Flexible Pitch Attitude Integration of the Error 0.02 5000 0.019 4500 0.018 4000 Integration of the Error (Velocity) Pitch Attitude in Radians 30 Time in Seconds 0.017 0.016 0.015 0.014 3500 3000 2500 2000 0.013 1500 0.012 1000 500 0.011 0 10 20 30 Time in Seconds 40 0 50 (e) Pitch Attitude Flexible 0 10 20 30 Time in Seconds 40 50 (f) Integral of Tracking Error Flexible Figure 4.17: Velocity Tracking Ramp Response Unstable 161 First Modal Coordinate Derivative of First Modal Coordinate 200 8 150 6 100 Derivative of First Modal Coordinate First Modal Coordinate 250 10 4 2 0 −2 −4 50 0 −50 −100 −150 −200 −6 −250 −8 0 10 20 30 Time in Seconds 40 50 0 10 (a) η1 Flexible 30 Time in Seconds 40 50 (b) η̇1 Flexible Second Modal Coordinate Derivative of Second Modal Coordinate 0.8 30 0.6 20 Derivative of Second Modal Coordinate 0.4 Second Modal Coordinate 20 0.2 0 −0.2 −0.4 10 0 −10 −20 −30 −0.6 −40 −0.8 −50 0 10 20 30 Time in Seconds 40 50 0 10 (c) η2 Flexible 20 30 Time in Seconds 40 50 (d) η̇2 Flexible Third Modal Coordinate Derivative of Third Modal Coordinate −0.02 2 Derivative of Third Modal Coordinate −0.04 Third Modal Coordinate −0.06 −0.08 −0.1 −0.12 −0.14 0 −2 −4 −0.16 −6 −0.18 −0.2 0 10 20 30 Time in Seconds 40 −8 50 (e) η3 Flexible 0 10 20 30 Time in Seconds 40 (f) η̇3 Flexible Figure 4.18: Velocity Tracking Ramp Response Unstable 162 50 Elevator Control Effort Canard Control Effort −0.05 0.2 −0.1 0.15 −0.15 Canard Angle in Radians Elevator Angle in Radians 0.1 0.05 0 −0.05 −0.2 −0.25 −0.1 −0.3 −0.15 −0.35 −0.2 0 10 20 30 Time in Seconds 40 50 0 10 (a) Elevator Flexible 20 30 Time in Seconds 40 50 (b) Canard Flexible Throttle Control Effort Diffuser Area Ratio Control Effort 0.8 1.1 1 0.798 0.9 0.796 Diffuser Area Ratio Throttle Ratio 0.8 0.7 0.6 0.794 0.792 0.5 0.4 0.79 0.3 0.788 0.2 0 10 20 30 Time in Seconds 40 50 0 (c) Fuel Equivalence Ratio Flexible 10 20 30 Time in Seconds 40 50 (d) Diffuser Area Ratio Flexible Figure 4.19: Velocity Tracking Ramp Response 163 Altitude Tracking The tracking signal used for this case starts at 80,000 feet and has a slope of 100 fst for the first 70 seconds and then a slope of 0 fst for the remainder of the simulation. The results from the simulation can be seen in figures 4.20-4.22. From this set of plots, it can be seen that the simulation fails at about 46 seconds. Figure 4.20 shows the rigid body states for the altitude tracking case. From this set of plots, it can be seen that the system does seem to track the command altitude until the simulation fails. This simulation failure is a result of the angle of attack and pitch attitude becoming very large in magnitude towards the end of this simulation. These large values cause the roots to the polynomials used to solve the flow through the scramjet to become imaginary. This simply means that there is flame-out in the scramjet. As stated previously, the model used in this dissertation is not equipped to handle this situation, and as a result the simulation is terminated and deemed a failure. Figure 4.21 shows the flexible states of the system. It can be seen from these plots that the flexible states of the system become very large towards the end of the simulation. This would indicate that the controller is having more difficulty in maintaining stability as the hypersonic vehicle tracks farther away from the trim conditions. This helps to reinforce the previous statements that a different control algorithm will be needed for a large operational envelope. Figure 4.22 shows the control effort of the vehicle. It can be seen from this set of plots that the control effort increases heading towards saturation near the end of the simulation. This growth is due to the vehicle reaching the limitation of its operational envelope. It is likely that the altitude tracking state would go unstable in a short period of time were it not the failure of the model that caused a termination of the simulation. 164 Velocity Angle of Attack 0.04 7810 0.03 0.02 Angle of Attack in Radians Velocity in Feet per Second 7800 7790 7780 7770 0.01 0 −0.01 −0.02 7760 −0.03 7750 0 5 10 15 20 25 Time in Seconds 30 35 40 45 0 5 (a) Velocity Flexible 10 15 20 25 Time in Seconds 30 35 40 45 (b) Angle of Attack Flexible Pitch Rate Altitude 4 x 10 8.45 8.4 0.01 0.005 8.3 Altitude in Feet Pitch Rate in Radians per Second 8.35 0 8.25 8.2 8.15 −0.005 8.1 Actual Altitude Reference Altitude −0.01 8.05 0 5 10 15 20 25 Time in Seconds 30 35 40 8 45 0 5 (c) Pitch Rate Flexible 10 15 30 35 40 45 40 45 (d) Altitude Flexible Pitch Attitude Integration of the Error 14000 0.05 12000 Integration of the Error (Velocity) 0.04 0.03 Pitch Attitude in Radians 20 25 Time in Seconds 0.02 0.01 0 −0.01 10000 8000 6000 4000 2000 −0.02 0 5 10 15 20 25 Time in Seconds 30 35 40 0 45 (e) Pitch Attitude Flexible 0 5 10 15 20 25 Time in Seconds 30 35 (f) Integral of Tracking Error Flexible Figure 4.20: Altitude Tracking Ramp Response Unstable 165 First Modal Coordinate Derivative of First Modal Coordinate 1 3.4 3.2 0.8 Derivative of First Modal Coordinate 3 First Modal Coordinate 2.8 2.6 2.4 2.2 2 1.8 0.6 0.4 0.2 0 1.6 −0.2 1.4 0 5 10 15 20 25 Time in Seconds 30 35 40 45 0 5 10 (a) η1 Flexible 15 20 25 Time in Seconds 30 35 40 45 35 40 45 35 40 45 (b) η̇1 Flexible Second Modal Coordinate Derivative of Second Modal Coordinate 0.15 0.15 Derivative of Second Modal Coordinate Second Modal Coordinate 0.1 0.05 0 0.1 0.05 0 −0.05 −0.1 −0.05 −0.15 −0.1 0 5 10 15 20 25 Time in Seconds 30 35 40 45 0 5 10 (c) η2 Flexible 20 25 Time in Seconds 30 (d) η̇2 Flexible Third Modal Coordinate Derivative of Third Modal Coordinate 0.03 0.03 0.02 Derivative of Third Modal Coordinate 0.02 Third Modal Coordinate 15 0.01 0 −0.01 0.01 0 −0.01 −0.02 −0.02 −0.03 0 5 10 15 20 25 Time in Seconds 30 35 40 45 0 (e) η3 Flexible 5 10 15 20 25 Time in Seconds 30 (f) η̇3 Flexible Figure 4.21: Altitude Tracking Ramp Response Unstable 166 Elevator Control Effort Canard Control Effort 0.3 0.35 0.25 0.2 Canard Angle in Radians Elevator Angle in Radians 0.3 0.25 0.15 0.1 0.05 0.2 0 −0.05 0.15 0 5 10 15 20 25 Time in Seconds 30 35 40 45 0 5 (a) Elevator Flexible 10 15 20 25 Time in Seconds 30 35 40 45 35 40 45 (b) Canard Flexible Throttle Control Effort Diffuser Area Ratio Control Effort 0.83 0.8 0.825 0.7 Diffuser Area Ratio Throttle Ratio 0.82 0.6 0.5 0.815 0.81 0.4 0.805 0.3 0.2 0.8 0 5 10 15 20 25 Time in Seconds 30 35 40 45 0 (c) Fuel Equivalence Ratio Flexible 5 10 15 20 25 Time in Seconds 30 (d) Diffuser Area Ratio Flexible Figure 4.22: Altitude Tracking Ramp Response Unstable 4.4.2 Stable Velocity Tracking This subsection will look at the results of the two command signals for the velocity tracking flexible and rigid body cases. For these two inputs, the initial conditions for the system were set to be the trim conditions for Mach 8 at 80,000 feet (with controller 25). For each of the two inputs, the plots show the system responses for both the perturbed and nominal systems as indicated by the legends. Ramp Response The tracking signal used for this case starts at a velocity of 7, 819.6 fst and has a slope of 20 sf2t for a duration of 60 seconds. After the 60 second interval, the slope of the ramp is 0 sf2t for a total simulation time of 100 seconds. The results from the simulation for both the flexible and 167 rigid body cases can be seen in figures 4.23-4.28. Figure 4.23 shows the velocity, angle of attack and the pitch rate for both the flexible and rigid body cases. From this figure, the perturbed and nominal cases for both the flexible and rigid body controllers can be seen. It can be seen that for both the flexible and rigid body cases, the velocity tracking is achieved at essentially the same rate. It should also be noted that the perturbed and nominal cases have the same velocity curve. This shows that the velocity tracking is successful for each case both with and without perturbations in the system. This would suggest that a rigid body controller could be used for the velocity tracking case. Figures 4.23(c) and 4.23(d) show the angle of attack for the flexible and rigid body cases respectively. These figures show that the maximum angle of attack for the flexible body case is roughly 2.6◦ , while the maximum angle of attack for the rigid body case is roughly 2.0◦ . There are some minor differences between the responses seen between the angle of attack for the rigid and flexible body systems, but what should be noted is the difference seen between the perturbed and nominal cases. It appears as though the hypersonic vehicle angle of attack is very sensitive to perturbation in the system. The propagation of noise is very evident from these figures. These results are not desirable, but it may be acceptable since there are no constraints put on the angle of attack in the control synthesis. Figures 4.23(e) and 4.23(f) show the pitch rate for the flexible and rigid body cases respectively. As seen with the angle of attack, the pitch rate of the vehicle is similar for the flexible and rigid body cases, but this state is also sensitive to perturbation in the system. The maximum pitch rate for the flexible body case is approximately 4.6◦ , while the maximum pitch rate for the rigid body case is roughly 3.4◦ . Again it is not desirable to have such an influence from the perturbation of the system, but the highly coupled nature of the hypersonic vehicle makes this a difficult thing to achieve. The velocity tracking is not effected by the noise in these other states however, so this simulation could be considered a success. Figure 4.24 shows the altitude, pitch attitude and the integration of the error for both the flexible and rigid body cases. From this figure, the perturbed and nominal cases for both the flexible and rigid body controllers can be seen. It can be seen that for both the flexible and rigid body cases, the altitude of the hypersonic vehicle is close. In fact, the steady state conditions for the two simulations run have a difference of about 35 feet. It can also be seen that the effects of perturbation in the system have only a small effect on the altitude of the vehicle. This is partially due to the fact that there is a unity gain regulating the altitude tracking. Since this state is now included as a regulation state in the control synthesis, it is more robust to 168 perturbation in the system. This figure shows a successful altitude regulation state which is different from what was seen in chapter 3 where the altitude went down as the velocity went up in the system. This proves that there is a correlation between the velocity and altitude of the system. It also shows that the change in altitude can be minimized during velocity tracking by using a unity proportional gain on the altitude in the control synthesis. Figures 4.24(c) and 4.24(d) show the pitch attitude for the flexible and rigid body cases respectively. From these two plots, it can be seen that the perturbation in the system has a large effect on the pitch attitude of the hypersonic vehicle for both the flexible and rigid body cases. The flexible body case has a slightly larger pitch attitude than the rigid body case does. The maximum value of the pitch attitude for the flexible body case is approximately 2.6◦ while the maximum value for the rigid body case is approximately 2.0◦ . Figures 4.24(e) and 4.24(e) show the integration of the error for the flexible and rigid body cases respectively. From these two figures, it can be seen that the flexible body case accumulates a slightly larger amount of error, but that the adjusted integration of the error is approximately the same for the two cases. Also, it should be noted that the integration of the error is reset when switching takes place in the controller. Both the flexible and rigid body cases are switching at roughly the same times. This would indicate that rigid body assumptions would be valid for controlling the vehicle during velocity tracking. Figures 4.25-4.26 show the flexible modes of the hypersonic vehicle for both the flexible and rigid body cases. It can be seen from these figures that the perturbation in the system has a very large effect on the flexibility of the hypersonic vehicle for both the flexible and rigid body cases. Even though the rigid body controller does not include the flexible states of the hypersonic vehicle in the synthesis, the flexible states still exist in the nonlinear plant used. These are plotted during the simulation. From these figures it can be seen that the first mode of vibration has a response that is similar in magnitude for the flexible and rigid body cases. The derivative of the first mode of vibration however shows that the flexible body case has a higher value. For the higher order modes of vibration and their respective derivatives, the flexible body case has higher values than those seen in the rigid body case. This would signify that the flexible body case does have a slightly larger deflections due to vibration, but since the first mode is the dominant mode, the differences are not significant. This would support the idea that the rigid body controller is suitable for flexible body velocity tracking case. Figure 4.27 shows the elevator deflection angle, the canard deflection angle and the fuel equivalence ratio for the flexible and rigid body cases. From this plot, it can be seen that the 169 perturbed cases have larger frequencies and magnitudes on their responses than the nominal cases for both the flexible and rigid body cases. Even though there is a high frequency for the perturbed cases, they still fall within the defined bandwidth limitations discussed in chapter 2. Note how the perturbed cases oscillate around the nominal case. It appears as though the mean of the perturbed case is the nominal case. Figures 4.27(a) and 4.27(b) show the response of the elevator control effort. This plot shows that the flexible body case has a slightly larger range of motion. The rigid body case operates between 2.3◦ and 12.6◦ where the flexible body case operates between −1.7◦ and 12.0◦ . Also note that there is a different initial trim value for the flexible and rigid body cases. Figures 4.27(c) and 4.27(d) show the response of the canard control effort. As seen with the elevator, the canard plots show that the flexible body case has a slightly larger range of motion. The rigid body case operates between 2.3◦ and −9.2◦ where the flexible body case operates between 0◦ and −18.3◦ . Also note that there is a different initial trim value for the flexible and rigid body cases. Figures 4.27(e) and 4.27(f) show the response of the fuel equivalence ratio for the flexible and rigid body cases respectively. These two cases have responses that are similar in value. This would make sense seeing as how the velocity and altitude of the vehicle were roughly the same as well. The fuel equivalence ratio is the control effort that is most directly linked to the thrust of the vehicle, so this relationship falls in line with the previous results. Also note how the perturbation has less of an effect on the fuel equivalence ratio as compared to the elevator and canard control efforts. Figures 4.28(a) and 4.28(b) show the response of the diffuser area ratio for the flexible and rigid body cases respectively. These two figures show that the perturbation present in the system has very little effect on the diffuser area ratio. It should be noted however that the response seen in the flexible body case is completely different from the one seen in the rigid body case. Figures 4.28(c) and 4.28(d) show the controller reference numbers for the flexible and rigid body cases respectively. It should be noted in figures 4.23-4.28 that there are spikes or small discontinuities that take place at about 20 seconds, 40 seconds and 60 seconds into the simulation. These spikes are the results of controller switching. It can be seen from figures 4.28(c) and 4.28(d) that the different cases all switch at approximately the same time, and that they all use the same controllers. 170 Velocity Velocity 9200 9000 9000 8800 8800 Velocity in Feet per Second Velocity in Feet per Second 9200 8600 8400 8200 8600 8400 8200 8000 8000 Reference Velocity Reference Velocity Actual Velocity Perturbed Actual Velocity Perturbed Actual Velocity Nominal 7800 0 10 20 30 40 50 60 Time in Seconds 70 80 90 Actual Velocity Nominal 7800 100 0 10 20 (a) Velocity Flexible 40 50 60 Time in Seconds 70 80 90 100 80 90 100 90 100 (b) Velocity Rigid Angle of Attack 0.045 30 Angle of Attack 0.045 0.04 0.04 0.035 0.035 0.03 Angle of Attack in Radians Angle of Attack in Radians Angle of Attack Perturbed 0.025 0.02 0.015 0.01 Angle of Attack Nominal 0.03 0.025 0.02 0.015 0.01 0.005 0.005 Angle of Attack Perturbed Angle of Attack Nominal 0 0 10 20 30 40 50 60 Time in Seconds 70 80 90 0 100 0 10 (c) Angle of Attack Flexible 30 40 Pitch Rate Perturbed Pitch Rate Nominal 0.06 Pitch Rate in Radians per Second Pitch Rate in Radians per Second Pitch Rate Nominal 0.08 0.06 0.04 0.02 0 −0.02 0.04 0.02 0 −0.02 −0.04 −0.04 −0.06 −0.06 −0.08 70 Pitch Rate 0.1 Pitch Rate Perturbed 0.08 50 60 Time in Seconds (d) Angle of Attack Rigid Pitch Rate 0.1 20 0 10 20 30 40 50 60 Time in Seconds 70 80 90 −0.08 100 (e) Pitch Rate Flexible 0 10 20 30 40 50 60 Time in Seconds 70 (f) Pitch Rate Rigid Figure 4.23: Velocity Tracking Ramp Response 171 80 Altitude 4 8.005 x 10 Altitude 4 8.005 x 10 Altitude Perturbed 8.004 8.004 8.003 8.003 8.002 8.002 Altitude in Feet Altitude in Feet Altitude Nominal 8.001 8.001 8 8 7.999 7.999 Altitude Perturbed Altitude Nominal 7.998 7.997 7.998 0 10 20 30 40 50 60 Time in Seconds 70 80 90 7.997 100 0 10 20 (a) Altitude Flexible 40 50 60 Time in Seconds 70 80 90 100 80 90 100 (b) Altitude Rigid Pitch Attitude 0.045 30 Pitch Attitude 0.045 0.04 0.04 0.035 0.035 0.03 Pitch Attitude in Radians Pitch Attitude in Radians Pitch Attitude Perturbed 0.025 0.02 0.015 0.01 Pitch Attitude Nominal 0.03 0.025 0.02 0.015 0.01 0.005 0.005 Pitch Attitude Perturbed Pitch Attitude Nominal 0 0 10 20 30 40 50 60 Time in Seconds 70 80 90 0 100 0 10 (c) Pitch Attitude Flexible 20 30 7000 7000 6000 6000 5000 Adjusted Integration of Error Perturbed Actual Integration of Error Perturbed Adjusted Integration of Error Nominal Actual Integration of Error Nominal 3000 5000 Adjusted Integration of Error Perturbed Actual Integration of Error Perturbed Actual Integration of Error Nominal 3000 2000 1000 1000 10 20 30 40 50 60 Time in Seconds 70 80 90 0 100 (e) Integral of Tracking Error Flexible Adjusted Integration of Error Nominal 4000 2000 0 70 Integration of the Error 8000 Integration of the Error (Velocity) Integration of the Error (Velocity) Integration of the Error 0 50 60 Time in Seconds (d) Pitch Attitude Rigid 8000 4000 40 0 10 20 30 40 50 60 Time in Seconds 70 80 90 (f) Integral of Tracking Error Rigid Figure 4.24: Velocity Tracking Ramp Response 172 100 First Modal Coordinate 2.5 First Modal Coordinate 2.5 2 2 First Modal Coordinate Perturbed First Modal Coordinate Nominal 1.5 1.5 1 First Modal Coordinate First Modal Coordinate 1 0.5 0 0.5 0 −0.5 −0.5 −1 −1 −1.5 −1.5 First Modal Coordinate Perturbed First Modal Coordinate Nominal −2 0 10 20 30 40 50 60 Time in Seconds 70 80 90 −2 100 0 10 20 30 (a) η1 Flexible 30 30 20 20 10 0 −10 −20 80 90 100 10 0 −10 −20 −30 −30 Derivative of First Modal Coordinate Perturbed Derivative of First Modal Coordinate Perturbed Derivative of First Modal Coordinate Nominal −40 70 Derivative of First Modal Coordinate 40 Derivative of First Modal Coordinate Derivative of First Modal Coordinate 50 60 Time in Seconds (b) η1 Rigid Derivative of First Modal Coordinate 40 40 0 10 20 30 40 50 60 Time in Seconds 70 80 90 Derivative of First Modal Coordinate Nominal −40 100 0 10 20 30 (c) η̇1 Flexible 40 50 60 Time in Seconds 70 80 90 100 (d) η̇1 Rigid Second Modal Coordinate Second Modal Coordinate Second Modal Coordinate Perturbed Second Modal Coordinate Nominal 0.4 0.4 Second Modal Coordinate Perturbed Second Modal Coordinate Nominal 0.2 Second Modal Coordinate Second Modal Coordinate 0.2 0 −0.2 0 −0.2 −0.4 −0.4 −0.6 −0.6 −0.8 0 10 20 30 40 50 60 Time in Seconds 70 80 90 −0.8 100 (e) η2 Flexible 0 10 20 30 40 50 60 Time in Seconds (f) η2 Rigid Figure 4.25: Velocity Tracking Ramp Response 173 70 80 90 100 Derivative of Second Modal Coordinate 30 Derivative of Second Modal Coordinate 30 Derivative of Second Modal Coordinate Perturbed Derivative of Second Modal Coordinate Nominal 20 Derivative of Second Modal Coordinate Derivative of Second Modal Coordinate 20 10 0 −10 −20 10 0 −10 −20 Derivative of Second Modal Coordinate Perturbed Derivative of Second Modal Coordinate Nominal −30 0 10 20 30 40 50 60 Time in Seconds 70 80 90 −30 100 0 10 20 30 (a) η̇2 Flexible 50 60 Time in Seconds 70 80 90 100 (b) η̇2 Rigid Third Modal Coordinate 0.04 40 Third Modal Coordinate 0.04 Third Modal Coordinate Perturbed Third Modal Coordinate Nominal 0.02 0 0 −0.02 −0.02 Third Modal Coordinate Third Modal Coordinate 0.02 −0.04 −0.06 −0.08 −0.04 −0.06 −0.08 −0.1 −0.1 −0.12 −0.12 −0.14 −0.14 −0.16 0 10 20 30 40 50 60 Time in Seconds 70 80 90 −0.16 100 Third Modal Coordinate Perturbed Third Modal Coordinate Nominal 0 10 20 30 (c) η3 Flexible 50 60 Time in Seconds 70 80 90 100 (d) η3 Rigid Derivative of Third Modal Coordinate 10 40 Derivative of Third Modal Coordinate 10 Derivative of Third Modal Coordinate Perturbed Derivative of Third Modal Coordinate Nominal 8 8 4 2 0 −2 −4 −6 Derivative of Third Modal Coordinate Nominal 4 2 0 −2 −4 −6 −8 −10 Derivative of Third Modal Coordinate Perturbed 6 Derivative of Third Modal Coordinate Derivative of Third Modal Coordinate 6 −8 0 10 20 30 40 50 60 Time in Seconds 70 80 90 −10 100 (e) η̇3 Flexible 0 10 20 30 40 50 60 Time in Seconds (f) η̇3 Rigid Figure 4.26: Velocity Tracking Ramp Response 174 70 80 90 100 Elevator Control Effort Elevator Control Effort 0.25 0.25 Elevator Control Effort Perturbed Elevator Control Effort Perturbed Elevator Control Effort Nominal Elevator Control Effort Nominal 0.2 Elevator Angle in Radians Elevator Angle in Radians 0.2 0.15 0.1 0.15 0.1 0.05 0.05 0 0 −0.05 0 10 20 30 40 50 60 Time in Seconds 70 80 90 −0.05 100 0 10 20 (a) Elevator Flexible 40 50 60 Time in Seconds 70 80 90 100 (b) Elevator Rigid Canard Control Effort 0.05 30 Canard Control Effort 0.05 0 0 Canard Control Effort Perturbed Canard Control Effort Nominal −0.05 Canard Angle in Radians Canard Angle in Radians −0.05 −0.1 −0.15 −0.2 −0.1 −0.15 Canard Control Effort Perturbed Canard Control Effort Nominal −0.2 −0.25 −0.25 −0.3 −0.3 0 10 20 30 40 50 60 Time in Seconds 70 80 90 100 0 10 20 (c) Canard Flexible 30 50 60 Time in Seconds 70 80 90 100 (d) Canard Rigid Throttle Control Effort Throttle Control Effort 0.6 0.6 0.5 0.5 0.4 0.4 Throttle Ratio Throttle Ratio 40 0.3 0.2 0.3 0.2 Throttle Control Effort Perturbed Throttle Control Effort Nominal 0.1 0.1 Throttle Control Effort Perturbed Throttle Control Effort Nominal 0 0 10 20 30 40 50 60 Time in Seconds 70 80 90 0 100 (e) Fuel Equivalence Ratio Flexible 0 10 20 30 40 50 60 Time in Seconds 70 80 90 (f) Fuel Equivalence Ratio Rigid Figure 4.27: Velocity Tracking Ramp Response 175 100 Diffuser Area Ratio Control Effort 0.95 Diffuser Area Ratio Control Effort 0.95 0.9 0.9 0.85 0.85 0.8 0.8 Diffuser Area Ratio Diffuser Area Ratio Diffuser Area Ratio Control Effort Perturbed 0.75 Diffuser Area Ratio Control Effort Perturbed 0.7 Diffuser Area Ratio Control Effort Nominal 0.65 0.75 0.7 0.65 0.6 0.6 0.55 0.55 0.5 Diffuser Area Ratio Control Effort Nominal 0 10 20 30 40 50 60 Time in Seconds 70 80 90 0.5 100 0 (a) Diffuser Area Ratio Flexible 20 30 40 50 60 Time in Seconds 90 100 45 Controller Number Perturbed Controller Number Perturbed Controller Number Nominal Controller Number Controller Number Nominal Controller Number 80 Controller Number 50 45 40 35 30 25 70 (b) Diffuser Area Ratio Rigid Controller Number 50 10 40 35 30 0 10 20 30 40 50 60 Time in Seconds 70 80 90 25 100 (c) Controller Reference Number Flexible 0 10 20 30 40 50 60 Time in Seconds 70 80 90 100 (d) Controller Reference Number Rigid Figure 4.28: Velocity Tracking Ramp Response 176 Multi-Step Response The tracking signal used for this case starts at a velocity of 7, 819.6 fst and has 6 step inputs of 150 sf2t every 40 seconds starting at a time of zero seconds. After an interval of 200 seconds, the input is constant for an additional 40 seconds giving a total simulation time of 240 seconds. The results from the simulation for both the flexible and rigid body cases can be seen in figures 4.29-4.34. Figure 4.29 shows the velocity, angle of attack and the pitch rate for both the flexible and rigid body cases. From this figure, the perturbed and nominal cases for both the flexible and rigid body controllers can be seen. As with the ramp case, it can be seen that for both the flexible and rigid body cases, the velocity tracking is achieved at essentially the same rate. It should also be noted that the perturbed and nominal cases have the same velocity curve. This shows that the velocity tracking is successful for each case both with and without perturbations in the system. This suggests that a rigid body controller could be used for the velocity tracking case. Figures 4.29(c) and 4.29(d) show the angle of attack for the flexible and rigid body cases respectively. These figures show that the maximum angle of attack for the flexible body case is roughly 2.8◦ , while the maximum angle of attack for the rigid body case is roughly 2.0◦ . As seen with the ramp input, there are some minor differences between the responses seen between the angle of attack for the rigid and flexible body systems, but what should be noted is the difference seen between the perturbed and nominal cases. As stated previously, the hypersonic vehicle angle of attack is very sensitive to perturbation in the system. The propagation of noise is evident and though these results are not desirable, it is acceptable since there are no constraints put on the angle of attack in the control synthesis. Figures 4.29(e) and 4.29(f) show the pitch rate for the flexible and rigid body cases respectively. As seen with the ramp input, the pitch rate of the vehicle is similar for the flexible and rigid body cases, but this state is also sensitive to perturbation in the system. The maximum pitch rate for the flexible body case is approximately 4.6◦ , while the maximum pitch rate for the rigid body case is roughly 4.0◦ . It is not desirable to have such an influence from the perturbation of the system, but the highly coupled nature of the hypersonic vehicle makes this difficult to achieve. The velocity tracking is not effected by the noise in these other states however, so this simulation is considered a success. Figure 4.30 shows the altitude, pitch attitude and the integration of the error for both the flexible and rigid body cases for the multi-step input. From this figure, the perturbed and 177 nominal cases for both the flexible and rigid body controllers can be seen. As with the ramp input, the altitude of the hypersonic vehicle is close for both the flexible and rigid body cases. It can also be seen that the effects of perturbation in the system have only a small effect on the altitude of the vehicle. Again, this is due to the fact that there is a unity gain regulating the altitude tracking. This figure shows a successful altitude regulation state which is different from what was seen in chapter 3 where the altitude went down as the velocity went up in the system. This reinforces the results seen with the ramp input that shows the change in altitude can be minimized during velocity tracking by using a unity proportional gain on the altitude in the control synthesis. Figures 4.30(c) and 4.30(d) show the pitch attitude for the flexible and rigid body cases respectively. From these two plots, it can be seen that the perturbation in the system has a large effect on the pitch attitude of the hypersonic vehicle for both the flexible and rigid body cases. This makes sense given the responses seen with the pitch rate seen in figures 4.29(e) and 4.29(f). The flexible body case has a slightly larger pitch attitude than the rigid body case does. The maximum value of the pitch attitude for the flexible body case is approximately 2.7◦ while the maximum value for the rigid body case is approximately 1.9◦ . Figures 4.30(e) and 4.30(e) show the integration of the error for the flexible and rigid body cases respectively. From these two figures, it can be seen that the flexible body case accumulates a slightly larger amount of error, but that the adjusted integration of the error is approximately the same for the two cases. Also, it should be noted that the integration of the error is reset when switching takes place in the controller. Both the flexible and rigid body cases are switching at roughly the same times. This would indicate that rigid body assumptions would be valid for controlling the vehicle during velocity tracking. Additionally note that the curve for the integration is not a smooth curve as seen with the ramp input. This is due to the multi-step input. As each new step is input into the system, there is a sharp turn in the integration of the error. This occurs because the step introduces an instantaneous error at the time it is applied. Since the steps are applied as the system is beginning to reach steady state conditions, the plot of the integration takes the form seen here. Figures 4.31-4.32 show the flexible modes of the hypersonic vehicle for both the flexible and rigid body cases. It can be seen from these figures that the perturbation in the system has a very large effect on the flexibility of the hypersonic vehicle for both the flexible and rigid body cases. Even though the rigid body controller does not include the flexible states of the hypersonic vehicle in the synthesis, the flexible states still exist in the nonlinear plant used. These are plotted during the simulation. From these figures it can be seen that the first mode 178 of vibration has a response that is similar in magnitude for the flexible and rigid body cases. The derivative of the first mode of vibration however shows that the flexible body case has a higher value. For the higher order modes of vibration and their respective derivatives, the flexible body case has higher values than those seen in the rigid body case. This would indicate that the flexible body case does have a slightly larger deflections due to vibration, but since the first mode is the dominant mode, the differences are not significant. This would support the idea that the rigid body controller is suitable for flexible body velocity tracking case. These results follow the trends seen with the ramp input case. Figure 4.33 shows the elevator deflection angle, the canard deflection angle and the fuel equivalence ratio for the flexible and rigid body cases. From this plot, it can be seen that the perturbed cases have larger frequencies and magnitudes on their responses than the nominal cases for both the flexible and rigid body cases. Even though there is a high frequency for the perturbed cases, they still fall within the defined bandwidth limitations discussed in chapter 2. Note how the perturbed cases oscillate around the nominal case just as seen with the ramp input case. It appears as though the mean of the perturbed case is the nominal case. Figures 4.33(a) and 4.33(b) show the response of the elevator control effort. This plot shows that the flexible body case has a slightly larger range of motion. The rigid body case operates between 2.3◦ and 13.8◦ where the flexible body case operates between −2.9◦ and 14.3◦ . Also note that there is a different initial trim value for the flexible and rigid body cases. Figures 4.33(c) and 4.33(d) show the response of the canard control effort. As seen with the elevator, the canard plots show that the flexible body case has a slightly larger range of motion. The rigid body case operates between 1.7◦ and −9.2◦ where the flexible body case operates between −1.1◦ and −17.2◦ . Also note that there is a different initial trim value for the flexible and rigid body cases. Figures 4.33(e) and 4.33(f) show the response of the fuel equivalence ratio for the flexible and rigid body cases respectively. These two cases have responses that are similar in value. This would make sense seeing as how the velocity and altitude of the vehicle were roughly the same as well. The fuel equivalence ratio is the control effort that is most directly linked to the thrust of the vehicle, so this relationship falls in line with the previous results. Also note how the perturbation has less of an effect on the fuel equivalence ratio as compared to the elevator and canard control efforts. 179 Figures 4.34(a) and 4.34(b) show the response of the diffuser area ratio for the flexible and rigid body cases respectively. These two figures show that the perturbation present in the system has very little effect on the diffuser area ratio. It should be noted however that, as seen with the ramp input, the response seen in the flexible body case is completely different from the one seen in the rigid body case. Figures 4.34(c) and 4.34(d) show the controller reference numbers for the flexible and rigid body cases respectively. It should be noted in figures 4.29-4.34 that there are spikes or small discontinuities that take place at about 80 seconds and 160 seconds into the simulation. These spikes are the results of controller switching. There are additional spikes and discontinuities that occur every 40 seconds for the first 200 seconds of the simulation. These spikes and discontinuities are from the step inputs applied to the system. It can be seen from figures 4.34(c) and 4.34(d) that the different cases all switch at approximately the same time, and that they all use the same controllers. 180 Velocity Velocity 8800 8700 8700 8600 8600 8500 8500 Velocity in Feet per Second Velocity in Feet per Second 8800 8400 8300 8200 8100 8400 8300 8200 8100 8000 8000 Reference Velocity 7900 Reference Velocity 7900 Actual Velocity Perturbed Actual Velocity Perturbed Actual Velocity Nominal 7800 0 50 100 Time in Seconds 150 Actual Velocity Nominal 7800 200 0 50 (a) Velocity Flexible 0.04 0.035 0.035 Angle of Attack in Radians Angle of Attack in Radians 0.045 0.04 0.03 0.025 0.02 0.015 0.03 0.025 0.02 0.015 0.01 0.01 0.005 0.005 Angle of Attack Perturbed Angle of Attack Perturbed Angle of Attack Nominal 0 50 100 Time in Seconds 150 Angle of Attack Nominal 0 200 0 (c) Angle of Attack Flexible 50 100 Time in Seconds 150 200 (d) Angle of Attack Rigid Pitch Rate Pitch Rate Pitch Rate Perturbed 0.08 0.08 Pitch Rate Nominal 0.06 0.06 0.04 0.04 Pitch Rate in Radians per Second Pitch Rate in Radians per Second 200 Angle of Attack 0.05 0.045 0 150 (b) Velocity Rigid Angle of Attack 0.05 100 Time in Seconds 0.02 0 −0.02 −0.04 −0.06 0.02 0 −0.02 −0.04 −0.06 −0.08 Pitch Rate Perturbed −0.08 0 50 100 Time in Seconds 150 200 Pitch Rate Nominal 0 (e) Pitch Rate Flexible 50 100 Time in Seconds 150 (f) Pitch Rate Rigid Figure 4.29: Velocity Tracking Step Response 181 200 Altitude 4 8.004 x 10 Altitude 4 8.004 x 10 Altitude Perturbed 8.002 8.002 8 8 Altitude in Feet Altitude in Feet Altitude Nominal 7.998 7.996 7.998 7.996 Altitude Perturbed 7.994 7.994 7.992 7.992 7.99 0 50 100 Time in Seconds 150 7.99 200 Altitude Nominal 0 50 (a) Altitude Flexible 0.04 0.035 0.035 Pitch Attitude in Radians Pitch Attitude in Radians 0.045 0.04 0.03 0.025 0.02 0.015 0.03 0.025 0.02 0.015 0.01 0.01 0.005 0.005 Pitch Attitude Perturbed Pitch Attitude Perturbed Pitch Attitude Nominal 0 50 100 Time in Seconds 150 Pitch Attitude Nominal 0 200 0 (c) Pitch Attitude Flexible 50 Integration of the Error 150 200 Integration of the Error Adjusted Integration of Error Perturbed 6000 Actual Integration of Error Perturbed Actual Integration of Error Perturbed Adjusted Integration of Error Nominal Adjusted Integration of Error Nominal Actual Integration of Error Nominal Actual Integration of Error Nominal 5000 Integration of the Error (Velocity) 5000 Integration of the Error (Velocity) 100 Time in Seconds (d) Pitch Attitude Rigid Adjusted Integration of Error Perturbed 6000 4000 3000 2000 1000 0 200 Pitch Attitude 0.05 0.045 0 150 (b) Altitude Rigid Pitch Attitude 0.05 100 Time in Seconds 4000 3000 2000 1000 0 50 100 Time in Seconds 150 0 200 (e) Integral of Tracking Error Flexible 0 50 100 Time in Seconds 150 200 (f) Integral of Tracking Error Rigid Figure 4.30: Velocity Tracking Step Response 182 First Modal Coordinate First Modal Coordinate 2.5 First Modal Coordinate Perturbed 2.5 2 First Modal Coordinate Nominal 2 First Modal Coordinate Perturbed First Modal Coordinate Nominal 1.5 First Modal Coordinate First Modal Coordinate 1.5 1 0.5 0 1 0.5 0 −0.5 −0.5 −1 −1 −1.5 0 50 100 Time in Seconds 150 −1.5 200 0 50 (a) η1 Flexible Derivative of First Modal Coordinate 200 Derivative of First Modal Coordinate 30 20 20 Derivative of First Modal Coordinate Derivative of First Modal Coordinate 150 (b) η1 Rigid 30 10 0 −10 −20 100 Time in Seconds 10 0 −10 −20 Derivative of First Modal Coordinate Perturbed Derivative of First Modal Coordinate Nominal −30 Derivative of First Modal Coordinate Perturbed −30 Derivative of First Modal Coordinate Nominal 0 50 100 Time in Seconds 150 200 0 50 (c) η̇1 Flexible Second Modal Coordinate Perturbed Second Modal Coordinate Nominal Second Modal Coordinate Nominal 0.2 Second Modal Coordinate Second Modal Coordinate 200 Second Modal Coordinate 0.4 Second Modal Coordinate Perturbed 0.2 0 −0.2 −0.4 −0.6 −0.8 150 (d) η̇1 Rigid Second Modal Coordinate 0.4 100 Time in Seconds 0 −0.2 −0.4 −0.6 0 50 100 Time in Seconds 150 −0.8 200 (e) η2 Flexible 0 50 100 Time in Seconds 150 (f) η2 Rigid Figure 4.31: Velocity Tracking Step Response 183 200 Derivative of Second Modal Coordinate 30 Derivative of Second Modal Coordinate 30 Derivative of Second Modal Coordinate Perturbed Derivative of Second Modal Coordinate Nominal 20 Derivative of Second Modal Coordinate Derivative of Second Modal Coordinate 20 10 0 −10 −20 10 0 −10 −20 Derivative of Second Modal Coordinate Perturbed Derivative of Second Modal Coordinate Nominal −30 0 50 100 Time in Seconds 150 −30 200 0 50 (a) η̇2 Flexible 150 Third Modal Coordinate Third Modal Coordinate Perturbed Third Modal Coordinate Perturbed 0.02 Third Modal Coordinate Nominal 0 −0.02 −0.02 Third Modal Coordinate 0 −0.04 −0.06 −0.08 −0.04 −0.06 −0.08 −0.1 −0.1 −0.12 −0.12 −0.14 −0.14 0 50 100 Time in Seconds 150 −0.16 200 0 50 (c) η3 Flexible 100 Time in Seconds 200 Derivative of Third Modal Coordinate 8 6 4 4 Derivative of Third Modal Coordinate 6 2 0 −2 −4 2 0 −2 −4 −6 −6 Derivative of Third Modal Coordinate Perturbed Derivative of Third Modal Coordinate Perturbed Derivative of Third Modal Coordinate Nominal −8 150 (d) η3 Rigid Derivative of Third Modal Coordinate 8 Derivative of Third Modal Coordinate Third Modal Coordinate Third Modal Coordinate Nominal −0.16 200 (b) η̇2 Rigid Third Modal Coordinate 0.02 100 Time in Seconds 0 50 100 Time in Seconds Derivative of Third Modal Coordinate Nominal 150 −8 200 (e) η̇3 Flexible 0 50 100 Time in Seconds 150 (f) η̇3 Rigid Figure 4.32: Velocity Tracking Step Response 184 200 Elevator Control Effort 0.2 Elevator Control Effort 0.2 Elevator Control Effort Perturbed 0.15 Elevator Angle in Radians Elevator Angle in Radians Elevator Control Effort Nominal 0.1 0.05 0.15 0.1 0.05 0 0 Elevator Control Effort Perturbed Elevator Control Effort Nominal −0.05 0 50 100 Time in Seconds 150 −0.05 200 0 50 (a) Elevator Flexible 100 Time in Seconds 150 (b) Elevator Rigid Canard Control Effort Canard Control Effort 0 0 Canard Control Effort Perturbed Canard Control Effort Perturbed Canard Control Effort Nominal −0.1 −0.15 −0.2 −0.1 −0.15 −0.2 −0.25 −0.25 −0.3 −0.3 0 50 100 Time in Seconds 150 Canard Control Effort Nominal −0.05 Canard Angle in Radians −0.05 Canard Angle in Radians 200 200 0 50 (c) Canard Flexible 100 Time in Seconds 150 200 (d) Canard Rigid Throttle Control Effort Throttle Control Effort Throttle Control Effort Perturbed 0.6 0.5 0.5 0.4 0.4 Throttle Ratio Throttle Ratio Throttle Control Effort Nominal 0.6 0.3 0.2 0.3 0.2 0.1 0.1 Throttle Control Effort Perturbed Throttle Control Effort Nominal 0 0 50 100 Time in Seconds 150 0 200 (e) Fuel Equivalence Ratio Flexible 0 50 100 Time in Seconds 150 200 (f) Fuel Equivalence Ratio Rigid Figure 4.33: Velocity Tracking Step Response 185 Diffuser Area Ratio Control Effort Diffuser Area Ratio Control Effort 0.9 0.9 Diffuser Area Ratio Control Effort Perturbed Diffuser Area Ratio Control Effort Nominal 0.85 Diffuser Area Ratio Diffuser Area Ratio 0.85 0.8 0.75 0.8 0.75 Diffuser Area Ratio Control Effort Perturbed Diffuser Area Ratio Control Effort Nominal 0.7 0.65 0.7 0 50 100 Time in Seconds 150 0.65 200 0 (a) Diffuser Area Ratio Flexible 100 Time in Seconds 150 200 (b) Diffuser Area Ratio Rigid Controller Number 40 50 Controller Number 40 Controller Number Perturbed Controller Number Perturbed Controller Number Nominal Controller Number Nominal Controller Number 35 Controller Number 35 30 25 30 0 50 100 Time in Seconds 150 25 200 (c) Controller Reference Number Flexible 0 50 100 Time in Seconds 150 200 (d) Controller Reference Number Rigid Figure 4.34: Velocity Tracking Step Response 186 4.4.3 Stable Altitude Tracking This subsection will look at the results of the two command signals for the altitude tracking case. For these two inputs, the initial conditions for the system were set to be the trim conditions for Mach 8 at 80,000 feet (with controller 25). For each of the two inputs, the plots show the system responses both perturbed and nominal systems as indicated by the legends. It will be important to note that even though it is possible to solve a rigid body controller for the altitude tracking case, it is not suitable to stabilize the system when applied to the nonlinear hypersonic vehicle plant model. The rigid body model does indeed stabilize the rigid body plant, but is unable to stabilize the nonlinear plant. For this reason, these results will be omitted, and only the flexible body results will be shown for the altitude tracking case. Ramp Response The tracking signal used for this case starts at 80,000 feet and has a slope of 100 fst for the first 70 seconds and then a slope of 0 fst for an additional 30 seconds for a total simulation time of 100 seconds. The results from the simulation for both the nominal and perturbed cases can be seen in figures 4.35-4.37. Figure 4.35(a) shows the velocity of the hypersonic vehicle. The velocity of the vehicle is not constant for the altitude tracking case with a ramp input. The variation of the velocity is not as great for this case as what was seen in the full state feedback case in chapter 3. From this figure, it can be seen that the minimum velocity for this particular case is about 7790 fst , and the maximum velocity is approximately 7960 fst . This is a relatively small variation for the speeds in the simulation. It should also be noted that there is a difference in the velocity for the perturbed and nominal cases. From the results shown in this plot, it would seem that accounting for the velocity in the control synthesis has improved the overall response of the velocity curve during altitude tracking. Note that even though there is a difference between the perturbed and nominal cases, there is no high frequency noise propagation on the velocity of the hypersonic vehicle. This would seem to indicate that the difference between the nominal and perturbed case is due to the uncertainty in the plant, and not the sensor noise added to the system. Figure 4.35(b) shows the angle of attack for the hypersonic vehicle. The angle of attack for this simulation is relatively large. The maximum value is about 7.4◦ , and the minimum value is about −13.8◦ . There are large spikes in the angle of attack when the controller switches. Over time these spikes settle out to steady state conditions. It can be seen that the perturbed case has a high frequency component to the signal. This shows that there is some noise propagation 187 through the controller. It should be noted that this propagation is greater for the altitude tracking case than it is for the velocity tracking case. This would suggest that the velocity tracking case is more robust than the altitude tracking case for output feedback. This would make sense as well since the rigid body controller for the altitude tracking case was unable to control the nonlinear hypersonic vehicle. Figure 4.35(c) shows the pitch rate of the hypersonic vehicle. The pitch rate for this simulation deg is relatively high. The maximum pitch rate is 40.1 deg s , and the minimum pitch rate is −63.0 s . There are large spikes present when the controller switches. Note that the magnitude of the spikes seen in the perturbed case are larger than those seen in the nominal case. Figure 4.35(d) shows the altitude of the vehicle. This figure shows that the altitude is initially at 80,000 feet, and it reaches 87,000 feet by the end of the simulation. Note that there is a small amount of lag between the reference command and the response of the actual altitude of the vehicle. This slow response is typical for robust controllers. It should be noted that there are two distinct bumps that occur in the altitude tracking that correspond to the controller switching times. These bumps are more exaggerated than what was seen in the velocity tracking case. Both the perturbed and nominal cases have almost the same response. Figure 4.35(e) shows the pitch attitude of the hypersonic vehicle. The pitch attitude has a large range of variation that is not seen in the velocity tracking case. The maximum pitch attitude is 8.0◦ , and the minimum pitch attitude is −13.2◦ . There are also very large spikes in the pitch attitude that correspond to the controller switching that is present in the simulation. These spikes in pitch attitude explain the large spikes seen in the pitch rate of the vehicle. Additionally, it can be seen that the high frequency noise present on the pitch attitude perturbed case is amplified in the pitch rate. This is typical when a noisy signal is differentiated. Figure 4.35(f) shows the integration of the error for the hypersonic vehicle. This value is reset at the switching points as seen with the velocity tracking cases. The adjusted integration looks similar to a sawtooth wave. Note that the perturbed case and the nominal case match up almost exactly. This would make sense because the altitude tracking plots show that the two systems have the same response. Figure 4.36 shows the flexible states of the hypersonic vehicle. There are large transient values that are present when switching takes place. Note that as the modal order goes up, the magnitude of the modal coordinate goes down. This simply means that the first mode of 188 vibration is the dominant mode. It should also be noted that the sensor noise applied to the system has a very large effect on the flexibility of the hypersonic vehicle. Figures 4.37(a) shows the elevator response. It should be noted that when the controller switches, this control effort is temporarily saturated on both the upper and lower limits. This is not a favorable response since prolonged control effort saturation can lead to system instability. However, since the duration of the saturation is relatively short, the hypersonic vehicle is able to track the reference altitude. Additionally, it should be noted that the perturbed case has a much higher frequency of oscillation due to the noise propagating through the system. This is also unfavorable, and is a sign that the controller has severe limitations on its robust capabilities. The perturbed response does however stay within the operational limits of this simulation, so the tracking is still successful. Figure 4.37(b) shows the canard response. Just as was seen in the elevator response, the canard control effort saturates at the upper and lower limit just after the switches take place in the system. Again, this is not a favorable response, but it is still within the allowable limits of the simulation. Also, note that the perturbed system has a high frequency oscillation about the nominal case. This is the propagation of the sensor noise and uncertainty in the system. Figure 4.37(c) shows the fuel equivalence ratio response. Note that the overall response of the fuel equivalence ratio is smaller for the altitude tracking case than it is for the velocity tracking case. This would make sense because the velocity tracking case requires more thrust to achieve a higher velocity. This is not necessarily the case with the altitude tracking case. Note that there are spikes in the fuel equivalence ratio when a switch in the controller takes place. Figure 4.37(d) shows the diffuser area ratio. As with the other control efforts, there are spikes that occur when controller switching takes place. Additionally, the overall value of the diffuser area ratio is smaller for the altitude tracking case than what was seen for the velocity tracking case. Note how the sensor noise and uncertainty in the system effects the perturbed case. This high frequency oscillation is not seen with the nominal case. This is not a favorable response, but is still within the operational limits of the system. Figure 4.37(e) shows that the controllers switch at the same time and to the same controller reference numbers for the two systems. There is no difference between the perturbed case and the nominal cases. This would solidify the statements made earlier that the perturbation has a greater effect on the velocity tracking of the vehicle than it does the altitude tracking. 189 Velocity 7960 7940 0.1 7920 0.05 Angle of Attack in Radians 7900 Velocity in Feet per Second Angle of Attack 0.15 7880 7860 7840 0 −0.05 −0.1 −0.15 7820 Velocity Perturbed Velocity Nominal −0.2 7800 Angle of Attack Perturbed Angle of Attack Nominal 7780 0 10 20 30 40 50 60 Time in Seconds 70 80 90 −0.25 100 0 10 (a) Velocity Flexible 40 50 60 Time in Seconds 70 80 90 100 Altitude 4 8.8 0.6 x 10 8.7 0.4 8.6 0.2 8.5 0 Altitude in Feet Pitch Rate in Radians per Second 30 (b) Angle of Attack Flexible Pitch Rate 0.8 20 −0.2 −0.4 8.4 8.3 8.2 −0.6 8.1 −0.8 8 −1 Reference Altitude Actual Altitude Perturbed Pitch Rate Perturbed Actual Altitude Nominal Pitch Rate Nominal −1.2 0 10 20 30 40 50 60 Time in Seconds 70 80 90 7.9 100 0 10 (c) Pitch Rate Flexible 30 40 50 60 Time in Seconds 70 80 90 100 80 90 100 (d) Altitude Flexible Pitch Attitude 0.15 20 Integration of the Error 4 x 10 3 Adjusted Integration of Error Perturbed Actual Integration of Error Perturbed 0.1 Adjusted Integration of Error Nominal Actual Integration of Error Nominal 2.5 Integration of the Error (Velocity) Pitch Attitude in Radians 0.05 0 −0.05 −0.1 2 1.5 1 −0.15 0.5 −0.2 Pitch Attitude Perturbed Pitch Attitude Nominal −0.25 0 10 20 30 40 50 60 Time in Seconds 70 80 90 0 100 (e) Pitch Attitude Flexible 0 10 20 30 40 50 60 Time in Seconds 70 (f) Integral of Tracking Error Flexible Figure 4.35: Altitude Tracking Ramp Response 190 First Modal Coordinate 20 Derivative of First Modal Coordinate 300 15 200 Derivative of First Modal Coordinate First Modal Coordinate 10 5 0 −5 100 0 −100 −10 −200 −15 First Modal Coordinate Perturbed Derivative of First Modal Coordinate Perturbed First Modal Coordinate Nominal −20 0 10 20 30 40 50 60 Time in Seconds 70 80 90 Derivative of First Modal Coordinate Nominal −300 100 0 10 20 (a) η1 Flexible 40 50 60 Time in Seconds 70 80 90 100 (b) η̇1 Flexible Second Modal Coordinate 2 30 Derivative of Second Modal Coordinate 100 Second Modal Coordinate Perturbed Second Modal Coordinate Nominal 1.5 80 60 Derivative of Second Modal Coordinate Second Modal Coordinate 1 0.5 0 −0.5 −1 40 20 0 −20 −40 −1.5 −60 −2 −80 −2.5 −100 Derivative of Second Modal Coordinate Perturbed Derivative of Second Modal Coordinate Nominal 0 10 20 30 40 50 60 Time in Seconds 70 80 90 100 0 10 20 (c) η2 Flexible 40 50 60 Time in Seconds 70 80 90 100 (d) η̇2 Flexible Third Modal Coordinate 0.3 30 Derivative of Third Modal Coordinate 25 Derivative of Third Modal Coordinate Perturbed Derivative of Third Modal Coordinate Nominal 20 0.2 Derivative of Third Modal Coordinate 15 Third Modal Coordinate 0.1 0 −0.1 −0.2 10 5 0 −5 −10 −0.3 −15 Third Modal Coordinate Perturbed Third Modal Coordinate Nominal −0.4 0 10 20 30 40 50 60 Time in Seconds 70 80 90 −20 100 (e) η3 Flexible 0 10 20 30 40 50 60 Time in Seconds (f) η̇3 Flexible Figure 4.36: Altitude Tracking Ramp Response 191 70 80 90 100 Elevator Control Effort 0.6 Canard Control Effort 0.4 0.5 0.3 Elevator Control Effort Perturbed Canard Control Effort Nominal 0.2 0.3 Canard Angle in Radians Elevator Angle in Radians Canard Control Effort Perturbed Elevator Control Effort Nominal 0.4 0.2 0.1 0 0 −0.1 −0.2 −0.1 −0.3 −0.2 −0.3 0.1 0 10 20 30 40 50 60 Time in Seconds 70 80 90 −0.4 100 0 10 20 (a) Elevator Flexible 40 50 60 Time in Seconds 70 80 90 100 (b) Canard Flexible Throttle Control Effort 0.6 30 Diffuser Area Ratio Control Effort 0.9 Throttle Control Effort Perturbed Throttle Control Effort Nominal 0.88 0.86 0.45 0.84 Diffuser Area Ratio 0.5 0.4 0.35 0.3 0.82 0.8 0.78 0.25 0.76 0.2 0.74 0.15 0.72 Diffuser Area Ratio Control Effort Perturbed Diffuser Area Ratio Control Effort Nominal 0.1 0 10 20 30 40 50 60 Time in Seconds 70 80 90 0.7 100 0 (c) Fuel Equivalence Ratio Flexible 10 20 30 40 50 60 Time in Seconds Controller Number 26.8 Controller Number Perturbed 26.6 Controller Number Nominal 26.4 26.2 26 25.8 25.6 25.4 25.2 25 0 10 20 70 80 90 (d) Diffuser Area Ratio Flexible 27 Controller Number Throttle Ratio 0.55 30 40 50 60 Time in Seconds 70 80 90 100 (e) Controller Reference Number Flexible Figure 4.37: Altitude Tracking Ramp Response 192 100 Multi-Step Response The tracking signal used for this case is a multi-step input. It starts at 80,000 feet and has six steps of 850 fst each every 20 seconds starting at a time of 0 seconds for the first 100 seconds of the simulation. The total simulation time is 130 seconds. The results from the simulation for both the nominal and perturbed cases can be seen in figures 4.38-4.40. Figure 4.38(a) shows the velocity of the hypersonic vehicle. The velocity of the vehicle is not constant for the altitude tracking case with a ramp input. The variation of the velocity is not as great for this case as what was seen in the full state feedback case in chapter 3. From this figure, it can be seen that the minimum velocity for this particular case is about 7785 fst , and the maximum velocity is approximately 7960 fst . This is a relatively small variation for the speeds in the simulation. It should also be noted that there is a difference in the velocity for the perturbed and nominal cases. Since the general shapes of the two different responses seems to be the same with a small offset, it could be concluded that the sensor noise does not attribute much to this difference. In fact the uncertainty applied to the system seems to be the dominating factor in the perturbed velocity response. Overall the response of the velocity seems to be rather robust to sensor noise. Figure 4.38(b) shows the angle of attack for the hypersonic vehicle. The angle of attack for this simulation is relatively large. The maximum value is about 9.2◦ , and the minimum value is about −10.3◦ . There are large spikes in the angle of attack when the controller switches as well as when the step inputs are applied to the system. These spikes settle out to steady state conditions towards the end of the simulation. It can be seen that the perturbed case has a high frequency component to the signal. This shows that there is some noise propagation through the controller. Figure 4.38(c) shows the pitch rate of the hypersonic vehicle. The pitch rate for this simulation deg is relatively high. The maximum pitch rate is 47.0 deg s , and the minimum pitch rate is −109.0 s . There are large spikes present when the controller switches and when the step inputs are applied to the system. Note that the spikes associated with the times when the controller switches are greater than those associated with the step inputs tot the system. This would suggest that the hypersonic vehicle is more sensitive to the controller switching than it is to the step inputs to the system. Figure 4.38(d) shows the altitude of the vehicle. This figure shows that the altitude is initially at 80,000 feet, and it reaches 85,100 feet by the end of the simulation. Note that there is a small amount of lag between the reference command and the response of the actual altitude of 193 the vehicle. The altitude is able to successfully track the multi-step input. It should be noted here that the nominal and perturbed cases have nearly identical responses. This shows that the robust controller is successful in tracking the desired altitude while minimizing the effects of sensor noise and parametric uncertainty on the hypersonic vehicle. Figure 4.38(e) shows the pitch attitude of the hypersonic vehicle. The pitch attitude has a large range of variation for the multi-step case. The maximum pitch attitude is 9.1◦ , and the minimum pitch attitude is −9.7◦ . There are also very large spikes in the pitch attitude that correspond to the controller switching and the step inputs in the simulation. These spikes in pitch attitude explain the spikes seen in the pitch rate of the vehicle. Additionally, it can be seen that the high frequency noise present on the pitch attitude perturbed case is amplified in the pitch rate. This is typical when a noisy signal is differentiated. Figure 4.38(f) shows the integration of the error for the hypersonic vehicle. This value is reset at the switching points as seen with the velocity tracking cases. The curved bumps in the signal are caused the step inputs into the system. The signal is not continuous because the reference signal is not a continuous function. It should be noted that perturbed and nominal cases have the exact same error. This would indicate that the altitude for both cases is also identical. Figure 4.39 shows the flexible states of the hypersonic vehicle. There are large transient values that are present when switching takes place and when step commands are input into the simulation. Note that as the modal order goes up, the magnitude of the modal coordinate goes down. This simply means that the first mode of vibration is the dominant mode. It should also be noted that the perturbation applied to the system has a very large effect on the flexibility of the hypersonic vehicle. Figures 4.40(a) shows the elevator response. It should be noted that when the controller switches, the control effort is temporarily saturated on both the upper and lower limits. This is not a favorable response since prolonged control effort saturation can lead to system instability. However, since the duration of the saturation is relatively short, the hypersonic vehicle is able to track the reference altitude. Additionally, it should be noted that the perturbed case has a much higher frequency of oscillation due to the noise propagating through the system. This is also unfavorable, and is a sign that the controller has severe limitations on its robust capabilities. The perturbed response does however stay within the operational limits of this simulation, so the tracking is still successful. The nominal case does not exhibit these same effects. 194 Figure 4.40(b) shows the canard response. The response of the canard shows similar results as what was seen with the elevator. There are spikes that occur when step inputs are applied to the system as well as when the controller switching takes place. The perturbed case exhibits high frequency oscillations that are not present in the nominal case. The canard also saturates briefly after the controller switch takes place. Despite these unfavorable effects in the control effort response, the system maintains stability and manages to successfully track the reference altitude while maintaining a relatively small variation in the velocity of the hypersonic vehicle. Figure 4.37(c) shows the fuel equivalence ratio response. The overall response of the fuel equivalence ratio is small for the multi-step case as it was for the ramp input case. Note that there are spikes in the fuel equivalence ratio when a switch in the controller takes place as well as when the step inputs are applied in the simulation. It can also be seen from this plot that the perturbed case has a much higher frequency response than the nominal case does. This would indicate that the noise and uncertainty in the system has a large effect on the control efforts in the system. Figure 4.40(d) shows the diffuser area ratio. As with the other control efforts, there are spikes that occur when controller switching takes place and when the step inputs are applied. Note how the sensor noise and uncertainty in the system effects the perturbed case. This high frequency oscillation is not seen with the nominal case. This is not a favorable response, but is still within the operational limits of the system. Figure 4.40(e) shows that the controllers switch at the same time and to the same controller reference numbers for the two systems. There is no difference between the perturbed case and the nominal cases. This would solidify the statements made earlier that the perturbation has a greater effect on the velocity tracking of the vehicle than it does the altitude tracking. Note that there is one switch that takes place in the simulation and a total of two different controllers used. This switching time correspond with the spikes seen in the different system states, and these spikes drove the decision to minimize the amount of controller switching that takes place in the system in order to help stabilize the hypersonic vehicle. 195 Velocity Angle of Attack 7960 0.15 7940 0.1 Angle of Attack in Radians Velocity in Feet per Second 7920 7900 7880 7860 0.05 0 −0.05 7840 −0.1 Velocity Perturbed 7820 Angle of Attack Perturbed Velocity Nominal Angle of Attack Nominal −0.15 7800 0 20 40 60 80 Time in Seconds 100 120 0 20 (a) Velocity Flexible 40 60 80 Time in Seconds 100 120 (b) Angle of Attack Flexible Pitch Rate Altitude 4 x 10 8.5 0.8 8.45 0.6 Pitch Rate Perturbed Pitch Rate Nominal 8.4 8.35 0.2 Altitude in Feet Pitch Rate in Radians per Second 0.4 0 −0.2 8.3 8.25 −0.4 8.2 −0.6 8.15 −0.8 8.1 −1 8.05 Reference Altitude Actual Altitude Perturbed Actual Altitude Nominal 0 20 40 60 80 Time in Seconds 100 8 120 0 20 (c) Pitch Rate Flexible 40 60 80 Time in Seconds 100 120 (d) Altitude Flexible Pitch Attitude Integration of the Error 4 x 10 0.15 2 1.8 0.1 Integration of the Error (Velocity) Pitch Attitude in Radians 1.6 0.05 0 −0.05 Pitch Attitude Perturbed Pitch Attitude Nominal 1.4 1.2 Adjusted Integration of Error Perturbed Actual Integration of Error Perturbed Adjusted Integration of Error Nominal 1 Actual Integration of Error Nominal 0.8 0.6 −0.1 0.4 0.2 −0.15 0 20 40 60 80 Time in Seconds 100 0 120 (e) Pitch Attitude Flexible 0 20 40 60 80 Time in Seconds 100 120 (f) Integral of Tracking Error Flexible Figure 4.38: Altitude Tracking Step Response 196 First Modal Coordinate Derivative of First Modal Coordinate 250 200 10 Derivative of First Modal Coordinate 150 First Modal Coordinate 5 0 −5 First Modal Coordinate Perturbed First Modal Coordinate Nominal 100 50 0 −50 −100 −150 −10 −200 Derivative of First Modal Coordinate Perturbed −15 Derivative of First Modal Coordinate Nominal −250 0 20 40 60 80 Time in Seconds 100 120 0 20 (a) η1 Flexible 40 60 80 Time in Seconds 100 120 (b) η̇1 Flexible Second Modal Coordinate Derivative of Second Modal Coordinate 2 80 1.5 Second Modal Coordinate Perturbed Derivative of Second Modal Coordinate Second Modal Coordinate Derivative of Second Modal Coordinate Perturbed 60 Second Modal Coordinate Nominal 1 0.5 0 −0.5 −1 Derivative of Second Modal Coordinate Nominal 40 20 0 −20 −40 −1.5 −60 −2 0 20 40 60 80 Time in Seconds 100 120 0 20 (c) η2 Flexible 40 60 80 Time in Seconds 100 120 (d) η̇2 Flexible Third Modal Coordinate Derivative of Third Modal Coordinate 15 0.15 10 Derivative of Third Modal Coordinate Perturbed 0.1 Derivative of Third Modal Coordinate Derivative of Third Modal Coordinate Nominal Third Modal Coordinate 0.05 0 −0.05 −0.1 −0.15 5 0 −5 −10 Third Modal Coordinate Perturbed Third Modal Coordinate Nominal −0.2 −15 −0.25 0 20 40 60 80 Time in Seconds 100 120 0 (e) η3 Flexible 20 40 60 80 Time in Seconds (f) η̇3 Flexible Figure 4.39: Altitude Tracking Step Response 197 100 120 Elevator Control Effort Canard Control Effort 0.5 0.3 0.4 Canard Control Effort Perturbed Canard Control Effort Nominal 0.2 Canard Angle in Radians Elevator Angle in Radians 0.3 0.2 0.1 0 0.1 0 −0.1 −0.2 −0.1 −0.2 Elevator Control Effort Perturbed −0.3 Elevator Control Effort Nominal 0 20 40 60 80 Time in Seconds 100 120 0 20 40 (a) Elevator Flexible 60 80 Time in Seconds 100 120 (b) Canard Flexible Throttle Control Effort Diffuser Area Ratio Control Effort 0.55 0.86 0.5 Throttle Control Effort Perturbed Diffuser Area Ratio Control Effort Perturbed 0.84 Throttle Control Effort Nominal Diffuser Area Ratio Control Effort Nominal 0.45 0.82 Diffuser Area Ratio 0.3 0.25 0.8 0.78 0.76 0.2 0.74 0.15 0.72 0.1 0 20 40 60 80 Time in Seconds 100 120 0 (c) Fuel Equivalence Ratio Flexible 20 40 60 80 Time in Seconds Controller Number 25.9 25.8 25.7 25.6 Controller Number Perturbed Controller Number Nominal 25.5 25.4 25.3 25.2 25.1 25 0 20 100 (d) Diffuser Area Ratio Flexible 26 Controller Number Throttle Ratio 0.4 0.35 40 60 80 Time in Seconds 100 120 (e) Controller Reference Number Flexible Figure 4.40: Altitude Tracking Step Response 198 120 4.5 Conclusions This chapter has discussed the method for synthesizing and simulating an output feedback H∞ LPV controller for the flexible and rigid body hypersonic vehicle models. It has applied both velocity tracking and altitude tracking to the vehicle and displayed the difference between perturbed and nominal cases. The results for the simulations were plotted and displayed. This section will seek to draw some deeper meaning from the results of control synthesis and simulation for output feedback. The control synthesis study performed in this chapter show the effects of changing the parameter variation rate as well as the operational range of the vehicle. From the study conducted, it can be seen that choosing the appropriate parameter variation rate and operational range on the hypersonic vehicle can be critical to the system performance and the robust capabilities of the controller. It is important to understand the cost to performance sacrifices that will be made in order to achieve the desired performance criteria. For this study, it was concluded that an evenly spaced grid containing 49 controllers with a range from Mach 7 to Mach 9 and an altitude from 70,000 feet to 90,000 feet would be the best option. This option was determined to be the best due to its large operational range and its robust capabilities with current available computational power. Additionally, it was decided that ν = [.1 200]T because of the cost to performance tradeoffs. It will be important for a designer to make the appropriate decisions when designing an output feedback H∞ LPV controller for the air-breathing hypersonic vehicle. It can be seen from the control synthesis that the rigid body controllers have a better robust performance than the flexible body cases do. Though this may be the case with the synthesis, it has been shown through simulation that this may not actually hold up. This is due to the fact that the rigid body controller is synthesized for a rigid body hypersonic vehicle, but then it is applied to the flexible body plant. This means that there is an additional amount of perturbation in the rigid body system that is not counted for the flexible body system. Though the control synthesis for the rigid body system are much simpler, it is not always possible to get this controller to work for the flexible body system. The rigid body controller worked for the velocity tracking simulations run, but it did not work for the altitude tracking simulations run. Further testing should be conducted along this path to have a deeper understanding of why this did not work, but it is probably best to use the most accurate model possible when dealing with hypersonic flight. This chapter has also shown a potential solution to the tradeoff between altitude and velocity when tracking one of those two system states. Chapter 3 revealed that when tracking a velocity 199 for the full state feedback case, the altitude declined steadily over the course of the simulation. The same was true for the altitude tracking case, the velocity would decrease. By using a unity gain in the proportional term on the non-tracking state, this effect was minimized. The output feedback controller exhibited only small fluctuations in altitude when tracking velocity, and small fluctuations in velocity when tracking altitude. It was also discovered that it was not possible to track velocity and altitude simultaneously using the current control structure. This chapter has also shown the effects that controller switching has on the hypersonic vehicle. From the simulation results in the previous section, it can be seen that there are sharp changes and transients in the system states as a result of switching from one controller to another. This is the same thing that was seen with the full state feedback case in chapter 3. The difference between the output feedback case and full state feedback cases is the method used to minimize this effect during the switching. This effect seems to be one of the largest limiting factors to this controller. The system is very sensitive to the switching that takes place in the controller, so every effort must be made to minimize the amount of controller switching that takes place. This is not always feasible, and thus limits the capabilities of the controller. The output feedback case does have more freedom to minimize this effect under the optimization that takes place during switching due to the large number of non-physical states that can be reset, but it is also more sensitive to the switching taking place in the system. The simulation in this chapter also show that there is a strong correlation between the angle of attack and the flexible states of the vehicle. As the angle of attack increases, the motions for the flexible modes increase. This is an imporant relationship to understand because both of these values play into the efficiency of the scramjet engine. There is also a strong coupling between the angle of attack, the pitch attitude, and the pitch rate of the hypersonic vehicle. All of these states play a role in how much air flows into the scramjet engine and as a result, they have an effect on how much thrust is produced by the scramjet engine. It is very important that the angle of attack stays relatively small so that the scramjet stays within its operational range. The controller designed in this chapter has not taken this into account, but it is something that should be considered in future work. This could simply be treated as an additional state to have a proportional feedback added to in the system as seen with the velocity regualtion during altitude tracking and the altitude regulation during velocity tracking. This would allow the control designer to penalize the angle of attack in the control synthesis to minimize the amount of fluctuation seen. This chapter has focused on the output feedback case. Even though this may be a more practical approach than the full state feedback case, it has still made some major assumptions 200 about which states can be measured. It was assumed for this study that all of the rigid body states were measureable. This may not actually be possible, but without an actual vehicle to study, these assumptions were made. It should also be noted that the sensor noise was chosen to be relatively small. This choice was made based upon the assumption that an expensive and high precision vehicle like this would require high quality sensors. It should be understood that an actual system could have different sensor noise levels that could potentially cause problems in the controller. Though there are some assumptions made in this chapter that may not accurately represent the actual system, the results from this study show the characteristics of an H∞ LPV controller. It can be seen that the amount of perturbation added to the system for this study is within a reasonable range, and that it does not have a significant affect on the tracking or regulation states in the simulation. The perturbation in the system may have a large effect on the other system variables, especially flexible body states and the control efforts, but they remain within a reasonable range for hypersonic flight. 201 Chapter 5 Conclusions This dissertation has looked at applying H∞ LPV control to the flexible air-breathing hypersonic vehicle model. Both full state feedback and output feedback were considered for the velocity tracking and altitude tracking cases. The effects of perturbation in the system were analyzed, as well as the effects of synthesizing a rigid body controller and treating the flexibility of the system as a perturbation to the system. Additionally, an open loop study was conducted to characterize the hypersonic vehicle dynamics. This chapter will seek to draw the major conclusions of the dissertation as well as outline any future work that should be conducted. 5.1 Contributions This dissertation has investigated the air-breathing hypersonic vehicle model developed by Bolender and Doman [8, 7]. From this investigation, it has been concluded that this model is the best closed form solution to hypersonic vehicle dynamics that exists currently. The open loop study performed on this model has shown that using the assumed modes approach allows for the flexibility of the vehicle to be simplified. The three modes used should be more than sufficient to model the flexibility of the hypersonic vehicle. It was shown in chapter 2 that the first mode of vibration is the dominant mode in the system. It was also discovered during the course of this open loop study that the hypersonic vehicle model is not continuous across the entire operational envelope. In the modeling of the hypersonic vehicle and the flow through the scramjet engine, complex fourth order polynomial expressions were used in the calculations to determine the thrust generated by the scramjet. There are times when there are no real roots to the polynomials that must be solved using this model, which means that even though the vehicle may operate around a certain set of flight conditions there are discontinuities that exist withing the flight envelope at certain flight conditions. This is a limitation of the model itself, and may not be indicative of the physical system. 202 This dissertation has also dealt with the control synthesis problem for the hypersonic vehicle. Two different parametric studies were conducted. The results of the full state feedback parametric study show that there is an increase in the γ performance value when the parameter variation rate increases. This holds true for both the velocity and altitude tracking cases. This same trend was also seen with the output feedback case as well. It shows that a designer must consider the desired robust performance when establishing the parameter variation rate for the controllers. There is always a cost to performance tradeoff associated with controls design. In this case the tradeoff is between the performance of the H∞ LPV controller and its robust capabilities. The study in the output feedback controller showed that the rigid body controllers always exhibited lower γ performance values than the flexible body controllers. This is because the rigid body system has fewer states and results in a smaller number of optimization variables, therefore the resulting controller will have less states and be easy to implement. This did not result in an increase in robust performance when the rigid body controllers were applied to the flexible plant because the flexibility in the plant acts as an additional perturbation to the system. Again, this is a decision that a designer must make when setting up the controls problem. There are computational benefits to using the rigid body assumptions, but there are performance sacrifices that are made as a result. The rigid body controller was not able to achieve altitude tracking for the system. It is best to use as much information about the true model of the system as possible when designing the H∞ LPV controller. The full state feedback study also showed that there was a link between the velocity and altitude states when tracking the velocity or the altitude. This was a very important discovery to make. During velocity tracking, the velocity increased as the altitude dropped steadily. Similarly, the velocity decreased steadily as the altitude increased during altitude tracking. The H∞ LPV controller is an optimal controller, therefore the controller will command the system to track the reference while using the least amount of control effort possible. This results in a tradeoff between the velocity and the altitude. It was also discovered that these two control objectives were in opposition to each other, so it is not possible to track both altitude and velocity simultaneously. For this reason, a regulation state was added to the output feedback controller to control the secondary state. This phenomenon solidifies that the controller is indeed achieving the robust performance objectives that it is supposed to achieve. During the course of study it was discovered that there are some problems that occur with the switching between the linear controllers used in the H∞ LPV controller. This problem arises from the linearization of the hypersonic vehicle about a set of trim conditions during synthesis. The control effort and the system states are defined in reference to the trim conditions, so when the simulation switches from one controller to the next, there are problems that occur. In 203 addition to this, the integration of the error is then set at a high value which may not be entirely accurate. This was accounted for by developing a two different switching algorithms. For the full state feedback case, the integral of the error was reset after switching between controllers took place such that the change in the actuator response was minimized at that time step. For the output feedback controller, all of the internal system states were reset in order to minimize the change in the actuator response at that time step. This took care of the switching problem, but it may not be the optimal solution. Additionally, the control switching algorithm was set up to ensure that the system did not rapidly switch back and forth between the two different controllers which would consequently result in a constant resetting of the integration of the error or the internal states of the system respectively. It is necessary for the integral of the error to be allowed to grow over time in order for enough control authority to be implemented to achieve tracking. The simulation results from the full state and output feedback cases both support the idea that the hypersonic vehicle is a highly coupled system. This is well known from the open loop study performed on the hypersonic vehicle. The simulation results both show that there seems to be a high sensitivity on the angle of attack for the vehicle. As the angle of attack increases for the system, the flexible modes of the system increase. This results in larger displacements for the bow of the hypersonic vehicle. The angle of attack and the location of the bow of the hypersonic vehicle are two of the most important factors in determining the amount of airflow into the mouth of the scramjet engine, and thus in determining the thrust of the scramjet. This makes it difficult to track velocity or altitude when there are high angles of attack involved. Precaution must be taken to ensure that the hypersonic vehicle remains within the valid operational range of the model. If the angle of attack becomes too great, then there no longer exists a real solution to the thrust of the hypersonic vehicle model used in this dissertation. Therefore, it is important to keep relatively low angles of attack during simulations, but this may not always be feasible using the controllers developed in this dissertation since there was no account for tracking or regulating the angle of attack during the synthesis process. The simulations in this dissertation also show that there is a capability of the H∞ LPV controller to be robust to perturbation in the system. In all of the cases presented in this dissertation, the tracking states always have a close response for the perturbed and nominal states. There is also a very similar response for the regulatory states seen in the output feedback cases. This shows that the H∞ LPV controller exhibits the desired robust qualities for the hypersonic vehicle. 204 5.2 Future Work This dissertation has explored many areas of the H∞ LPV control of the flexible air-breathing hypersonic vehicle, but has also opened the door to more exploration and research. This section will outline some of the major questions raised as a direct result of this study, and the future research that should be conducted to answer some of these questions. The model used in this dissertation is the best model available as of the time of this writing, however it does leave a lot of room for improvement. There are many assumptions that have been made during the formulation of this model that are potential areas for concern. One of the main issues is that the model is only a two dimensional model. It does not account for any lateral motion in the formulation of the equations of motion. Additionally, there are no thermal effects accounted for in this model. There is a significant amount of expansion and contraction that takes place during the flight of the vehicle due to the extreme temperatures from the scramjet engine and the drag of the vehicle. A proper fuel consumption model would also be necessary to accurately account for the change in mass of the vehicle. These two parameters will have a direct effect on the flexible nature of the hypersonic vehicle. Another big issue that needs to be resolved is the discontinuous nature of the model that has been observed. During the course of this study, it has been documented that there are certain sets of conditions under which there is no solution to the polynomials used to calculate the flow through the scramjet, and thus the thrust of the vehicle. It will be important to develop a model that has a continuous closed form solution that accurately represents the entire range of the vehicle. There is some current work being conducted to this end that may address some of these issues [15, 22, 18]. It would also be advantageous to look at the switching algorithm used in this study. The solution technique used in this study successfully stabilizes the hypersonic vehicle, but there are still many unfavorable spikes and transients that occur at or around the switching conditions. There is also a potential for actuator saturation to occur during this short time period. It would be preferable to design an algorithm that did not experience this sort of phenomenon. It may be advantageous to apply an anti-windup compensator to alleviate some of the issues that were seen when actuator saturation occurred [55]. Additionally, the angle of attack for the vehicle needs to be addressed in the control synthesis in future work. It should most likely be included as a controlled output in the system, or set up as a regulation state as was done in the output feedback case where the velocity was regulated during the altitude tracking and the altitude was regulated during the velocity tracking. By minimizing the angle of attack, there should be an increase in the efficiency of the scramjet. 205 This would also help to keep the vehicle within its operational range. Once an actual design for an air-breathing hypersonic vehicle has been obtained, it will be important to go back and substitute actual actuator dynamics into the model instead of using the low pass filters that were used for synthesis and simulation in this dissertation. It will also be necessary to obtain better information about the saturation limits of the actuators as well. Using the information obtained in this study, it would be interesting to extend this research to look at µ synthesis. Incorporating µ synthesis into the LPV framework has been done previously [56, 35]. It would involve performing the H∞ LPV synthesis multiple times within the D-K iteration framework. Essentially, it would be necessary to start by finding a set of open loop interconnected plants, and then scale these plants by pre-multiplying the scaling matrix D and post-multiplying D−1 . With the scaled set of open loop interconnected plants, it would be possible to synthesize the scaled H∞ LPV controller. With the controller K synthesized, the next step would be to minimize DF` (P, K) D−1 ≤ γ. This would give a new value for D, ∞ and the process would be iterated until an optimal controller could be synthesized [38]. Using this method with a structured uncertainty block could help to improve the performance of the hypersonic vehicle. Of course, this would require a better understanding of the uncertainties that are present in the system, but would also allow the designer to make changes in the control design to tradeoff between system uncertainty and system performance. By specifying the level of uncertainty in the system to be within a bounded region, it is possible to design a control with an increased performance over the H∞ LPV controller used in this study. Incorporating the µ synthesis and LPV control techniques would be the next logical progression in this line of controls research. 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Jones and Bartlett Publishers, third edition, 2006. 211 APPENDIX 212 Appendix A Controller Reference Number Table For Full State Feedback 213 Table A.1: Controller Reference Numbers Reference Number 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 Mach 7 7 7 7 7 7 7 7.33 7.33 7.33 7.33 7.33 7.33 7.33 7.67 7.67 7.67 7.67 7.67 7.67 7.67 8 8 8 8 Altitude (ft) 70,000 73,333.33 76,666.67 80,000 83,333.33 86,666.67 90,000 70,000 73,333.33 76,666.67 80,000 83,333.33 86,666.67 90,000 70,000 73,333.33 76,666.67 80,000 83,333.33 86,666.67 90,000 70,000 73,333.33 76,666.67 80,000 Reference Number 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 214 Mach 8 8 8 8.33 8.33 8.33 8.33 8.33 8.33 8.33 8.67 8.67 8.67 8.67 8.67 8.67 8.67 9 9 9 9 9 9 9 Altitude (ft) 83,333.33 86,666.67 90,000 70,000 73,333.33 76,666.67 80,000 83,333.33 86,666.67 90,000 70,000 73,333.33 76,666.67 80,000 83,333.33 86,666.67 90,000 70,000 73,333.33 76,666.67 80,000 83,333.33 86,666.67 90,000

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