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

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ABSTRACT
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 . . . . .
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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 . . . . . . . . . . . . . . . . . . . . . .
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. 1
. 1
. 10
. 17
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Force Equations
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19
19
23
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27
35
36
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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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Chapter 4 Output Feedback Control
4.1 Control Synthesis . . . . . . . . .
4.1.1 Velocity Tracking . . . . .
4.1.2 Altitude Tracking . . . .
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. 114
. 114
. 115
. 126
for Hypersonic
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v
Vehicle
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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 . . . . . . . . . . . . . . . . .
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Implementation
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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 . . . . . . .
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32
37
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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 . . . . . . . . . . . . . . . . . . . . . . . . . .
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52
54
56
60
60
62
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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
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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 .
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. 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 . . . . . . .
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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
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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
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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
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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 .
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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.
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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.
It would also be beneficial to continue to investigate the uncertainties that exist in the
hypersonic vehicle. Chavez and Schmidt have done work in this area, but it is necessary to
continue this work to obtain an improved understanding of the modeling issues and limitations
that exist with air-breathing hypersonic flight [13]. By knowing the bounds on the uncertainty in
the model or by making these bounds smaller, the overall performance of the hypersonic vehicle
could be improved. Future work in modeling will help to generate a better understanding of
the system uncertainties and how they should be modeled.
206
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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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