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Piloted simulation study comparing classical and robust flight control design methods

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PILOTED SIMULATION STUDY COMPARING CLASSICAL
AND ROBUST FLIGHT CONTROL
DESIGN METHODS
A THESIS
Presented to the Department of Mechanical and Aerospace Engineering
California State University, Long Beach
In Partial Fulfillment
of the Requirements for the Degree
Master of Science in Aerospace Engineering
Committee Members:
Bei Lu, Ph.D. (Chair)
Ramin Esfandiari, Ph.D.
C. Barclay Gilpin, Ph.D.
College Designee:
Mahyar Amouzegar, Ph.D.
By Daniel Joseph Alvarez
B.S., 2005, California State Polytechnic University, Pomona
December 2009
UMI Number: 1481782
All rights reserved
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a note will indicate the deletion.
UMT
Dissertation Publishing
UMI 1481782
Copyright 2010 by ProQuest LLC.
All rights reserved. This edition of the work is protected against
unauthorized copying under Title 17, United States Code.
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ABSTRACT
PILOTED SIMULATION STUDY COMPARING CLASSICAL
AND ROBUST FLIGHT CONTROL
DESIGN METHODS
By
Daniel Joseph Alvarez
December 2009
Flight controllers were developed for the F-16 VISTA aircraft using classical
PID control design methods in addition to Hoc and linear parameter varying (LPV)
robust design methods. Desktop analysis and simulation were used to quantitatively
assess and compare the performance of each controller. A piloted simulation study
was performed to gather qualitative pilot ratings and comments to validate or refute
the results and conclusions which were based on the preliminary desktop analysis.
Pilot-vehicle describing functions and wavelet analysis techniques were also used to
analyze the flight test data gathered during the simulation as a supplement to the
quantitative analysis data.
ACKNOWLEDGEMENTS
I would like to start off with a word of appreciation and thanks to Dr. Bei Lu
for her assistance, guidance, and support throughout this research effort.
To Dave Klyde, Dr. Chi-Ying Liang, Dr. Ed Bachelder, and all of my
colleagues at STI, thank you for the knowledge you have shared and the time you
have donated to help make this project a success. I could not have succeeded without
the help of the many mentors I have at STI and I look forward to repaying my debts
of gratitude through many years of friendship and professional collaboration.
To Dave Mitchell, thank you for your patience and help in the simulation
portion of the project. Your participation was a critical element of the research and
the experience was thoroughly enjoyable.
To my parents, thank you for teaching me that hard work will always be
rewarded. There were many nights that I doubted myself, but it was the drive and
ambition that you instilled in me that kept me going and for that I thank you.
And finally, to the most understanding and beautiful woman I will ever meet
in my life, a very heartfelt thank you to my wife LeeAnn for everything she has done
to make not only this project but everything that I have accomplished in life possible.
You have always had faith in me and I will never be able to repay you for all that you
have done to make me the person that I am today.
iii
TABLE OF CONTENTS
Page
ACKNOWLEDGEMENTS
Hi
LIST OF TABLES
vii
LIST OF FIGURES
viii
LIST OF ABBREVIATIONS
xi
CHAPTER
1. INTRODUCTION
1
Motivation and Objectives
Related Literature
1
3
Thesis Outline
5
2. AIRCRAFT MODEL ANALYSIS
7
Basic Modes of Longitudinal Motion
Analysis of F-16 Longitudinal Model
Notes and Observations
7
8
11
3. CLASSICAL CONTROL DESIGN
12
Design Criteria
Controller Design
Control Architecture
Design Synthesis
Nonlinear Simulation Results
Notes and Observations
4. Hoo CONTROL DESIGN
Control Theory
Design Criteria
Controller Design
12
13
13
14
17
18
20
20
22
23
iv
Page
CHAPTER
Control Architecture
Design Synthesis
Linear Simulation Results
Nonlinear Simulation Results
Scheduling Considerations
Notes and Observations
23
24
27
29
31
32
5. LPV CONTROL DESIGN
33
Control Theory
Design Criteria
Controller Design
Control Architecture
Linear Simulation Results
Nonlinear Simulation Results
Notes and Observations
33
36
36
36
38
40
42
6. COMPARISON OF CONTROL DESIGNS
43
Simulation Considerations
Performance Metrics
Linear Simulation Comparison
Overshoot
Angle of Attack
Steady-State Error
Nonlinear Simulation Comparison
Notes and Observations
43
43
45
45
47
47
48
49
7. SIMULATION TEST PLAN
50
Test Objectives
Test Plan
Flight Conditions
Sum-of-Sines Tracking Task
Controller Assessment
Evaluation Procedure
Pilot Debrief Questionnaire
50
51
51
53
57
57
58
8. FLIGHT TEST RESULTS AND ANALYSIS
Overview
Analysis Methods
Wavelet Transform Analysis
Compensatory Control Describing Functions
v
59
59
61
61
65
CHAPTER
Page
Pilot-Vehicle System Metrics
Pilot Technique and Rating Philosophy
Pilot Technique
Rating Philosophy
Handling Qualities Ratings and Comments
Pilot Comments
Handling Qualities Ratings
Pilot Questionnaire
Time Histories and Scalograms
Pilot-Vehicle Describing Functions
Aggressive Tracking
Pilot Performance and Ratings
Pilot Behavior and Technique
9. SUMMARY AND CONCLUSIONS
67
68
68
71
72
72
73
77
84
90
97
97
100
108
Project Summary
108
Observations and Conclusions
110
APPENDICES
112
A. SIMULATOR DESCRIPTION
113
B. PILOT RATING SCALES
123
C. PILOT DEBRIEF QUESTIONNAIRE
126
REFERENCES
128
vi
LIST OF TABLES
TABLE
Page
1. Rigid Body Longitudinal Mode Variation with Airspeed
9
2. Open-Loop Short Period Dynamics and Stability Margins
15
3. Closed-Loop Short Period Dynamics and Performance Metrics
17
4. Weighting Functions for Hoc Control Design
26
5. Hoc Closed-Loop Performance Metrics
27
6. LPV Closed-Loop Performance Metrics
38
7. Closed-Loop Performance Metrics for Various Designs
45
8. Trim Conditions for Simulation
52
9. SOS Command Signal for Pitch
56
10. Pilot Comments for Pitch Sum-of-Sines Tracking Evaluations
74
11. Pilot-Vehicle System Metrics, Pilot 2/250 kt
98
12. Pilot-Vehicle System Metrics, Pilot 2/110 kt
99
13. Pilot Comments for Pitch Sum-of-Sines Tracking Evaluations
103
14. Comparison of Task Performance for Aggressive Tracking
104
15. Baseline Feel System Characteristics
120
16. Longitudinal Axis Flight Control System Parameters
122
vii
LIST OF FIGURES
FIGURE
Page
1. Longitudinal transfer function pole/zero map
7
2. Pole/zero migration with airspeed for low speed, high a trim cases
9
3. Flight envelope trim conditions
10
4. Control loop architecture
13
5. Loop closure supporting plots, 160 kt. flight condition
16
6. Linear (blue) and nonlinear (red) response to pulse input
19
7. Hoo generic block diagram
20
8. Weighted open-loop interconnection for HOT control of F-16 aircraft
24
9. Linear (blue) and ideal rate (dashed) response to pulse input, 110 kt
29
10. Linear (blue) and nonlinear (red) response to pulse input, 120 kt
30
11. LPV generic block diagram
34
12. Linear (blue) and ideal rate (dashed) response to pulse input, 110 kt
40
13. Linear (blue) and ideal rate (dashed) response to pulse input, 400 kt
40
14. Linear (blue) and nonlinear (red) response to pulse input, 120 kt
41
15. Closed-loop bandwidth and phase delay for various flight controllers
45
16. Comparison of linear responses to pulse input, 110 kt
46
17. Comparison of nonlinear responses to pulse input, 180 kt
49
18. Example pitch axis SOS command signal
56
19. Transfer function estimation using wavelet transforms
62
viii
FIGURE
Page
20. Comparison of frame sizes for windowed Fourier and wavelet
transforms
63
21. Transforms of an example time series
64
22. Compensatory control scenario
66
23. Time histories and scalograms for stick input, PID controller at 250
kt
70
24. Pilot ratings and tracking performance, 400 kt. flight condition
78
25. Pilot ratings and tracking performance, 250 kt. flight condition
79
26. Pilot ratings and tracking performance, 160 kt. flight condition
80
27. Pilot ratings and tracking performance, 110 kt. flight condition
81
28. Elevator saturation encountered during sum-of-sines tracking, PID
controller at 110 kt
82
29. Predictability questionnaire responses, pilot 1
85
30. Predictability questionnaire responses, pilot 2
85
31. Rapidity questionnaire responses, pilot 1
86
32. Rapidity questionnaire responses, pilot 2
86
33. Overshoot questionnaire responses, pilot 1
87
34. Overshoot questionnaire responses, pilot 2
87
35. PlO/bobble questionnaire responses, pilot 1
88
36. PlO/bobble questionnaire responses, pilot 2
88
37. Time histories and scalograms for elevator deflection, 110 kt. flight
condition (pilot 1)
38. Time histories and scalograms for elevator deflection, 110 kt. flight
condition (pilot 2)
39. Time histories and scalograms for elevator deflection, 400 kt. flight
condition (pilot 1)
ix
92
93
94
FIGURE
Page
40. Time histories and scalograms for elevator deflection, 400 kt. flight
condition (pilot 2)
95
41. PVS describing function variation amongst controllers, pilot 2/250 kt
98
42. PVS describing function variation amongst controllers, pilot 2/110 kt
99
43. Pilot rating comparison for aggressive pilot technique, 250 kt
102
44. Pilot rating comparison for aggressive pilot technique, 110 kt
102
45. PVS frequency response for Hx controller at 250 kt. (aggressive)
105
46. Comparison of compensatory and aggressive pilot technique, Ha/250
kt
47. Time history and scalogram comparison for compensatory and
aggressive tracking runs
105
106
48. Divergent PIO encountered during aggressive tracking, Hoo controller
at 110 kt
107
49. Pilot-in-the-loop simulator elements
114
50. STI simulator with McFadden control loader and projected display
115
51. Projected head-up display
116
52. McFadden series 292A 2-axis fighter stick
117
53. McFadden electronic control box and related wiring
118
54. Feel system elements
119
55. Longitudinal axis control system elements
121
56. Cooper-Harper ratings scale
124
57. PIO tendency scale
125
x
LIST OF ABBREVIATIONS
Abbreviation
Definition
AFFTC
Air Force Flight Test Center
CH
Cooper-Harper
deg.
Degree(s)
deg/s
Degrees per Second
DFRC
Dryden Flight Research Center
ft.
Foot (Feet)
GM
Gain Margin
HQ
Handling Qualities
HQR
Handling Qualities Rating
HUD
Head-up Display
Hz.
Hertz
in.
Inch(es)
K
Pitch Rate Gain
kt.
Knot(s)
lb.
Pound(s)
LMI
Linear Matrix Inequality
LPV
Linear Parameter Varying
MAC
Mean Aerodynamic Chord
XI
MIL-STD
Military Standard
mils
Milliradians
MIMO
Multi Input, Multi Output
NASA
National Aeronautics and Space Administration
PID
Proportional-Integral-Derivative
PIO
Pilot-Induced Oscillation
PIOR
PIO Tendency Rating
PM
Phase Margin
PSD
Power Spectral Density
PVS
Pilot/Vehicle System
q
Pitch Rate
rad/s
Radians per Second
RMS
Root Mean Square
sec.
Second(s)
SISO
Single Input, Single Output
SOS
Sum-of-Sines
STI
Systems Technology, Inc.
t
Time
TPS
Test Pilot School
VISTA
Variable In-flight Simulator Training Aircraft
VSS
Variable Stability Simulator
VTRUE
True Airspeed
a
Angle of Attack
xn
8e
Elevator Deflection
Si0n
Longitudinal Stick Input
5th
Throttle Input
0
Pitch Attitude
xp
Equivalent Time Delay
C,
Damping Ratio
COBW
Bandwidth Frequency
©n
Natural Frequency
coxo
Crossover Frequency
xiii
CHAPTER 1
INTRODUCTION
Motivation and Objectives
Flight control engineering can be broken into two distinct but inherently
interrelated subdivisions: control design and analysis and the study of aircraft
handling qualities. Control designers and analysts focus mostly on the vehicle itself.
Different control methods are developed and implemented to ensure a robust and
well-performing system considering only the end-to-end vehicle response as the
primary performance metric. Handling qualities engineers, however, are concerned
with the pilot-vehicle system (PVS). It is the pilot who is the outermost feedback
mechanism of a flight control system, and it is in this loop where aircraft stability and
performance should be measured. The metrics used to gauge pilot vehicle system
performance are sometimes subjective; pilot ratings of handling qualities and
tendency of the aircraft to enter a pilot-induced oscillation (PIO) when performing a
task are two qualitative measurements of pilot opinion. These qualitative assessments
are used in combination with the quantitative metrics derived from flight test data to
assess the validity of the preliminary desktop analysis and ultimately determine the
goodness of a design.
Though the theories used to design and analyze control systems have
advanced in the past fifty years, the control methods and architectures employed on
1
many modern aircraft have remained largely the same. Classical proportionalintegral-derivative (PID) control systems are the standard for most aircraft due to
their simple structure. Today, the existence of heavily instrumented test aircraft
allows for complex control systems to be simulated in flight. In recent years, robust
control design methods have been employed in an attempt to address the problem of
aircraft instability in flight due to the lack of robustness offered by classical control.
The research on robust flight control has offered valid control solutions which
balance stability and performance for many platforms. However, most have typically
addressed only one component of the flight control problem, focusing on the design
of a controller and a subsequent desktop analysis. Of the few studies that have
extended the analysis and controller assessment into the pilot vehicle system through
piloted simulation or flight test, almost all have been in an attempt to prove the
viability of the robust controller. Even fewer have compared the performance of a
flight controller designed with robust design methods against a classically-designed
controller, and only one comparison in terms of pilot opinion and the ability to
perform a mission-oriented flying task in a flight test could be found.
One of the first questions that arises when presented with a system upgrade is
whether or not the new system will offer improvement over the legacy system it is
replacing. The emergence of robust flight controllers lends itself to this question, and
paper studies exist comparing the systems in detail, only a select few extend those
conclusions into the realm of piloted simulation. Of the research available where
piloted simulation is employed to solicit pilot opinion of a robust controller, very few
offer a comparison against the system architecture it is intending to improve upon.
2
The objective of this thesis is to provide a direct comparison of classical and
robust control design methods using both desktop analysis and a piloted simulation
handling qualities study and to draw conclusions as to which design methodology
offers superior closed loop performance in a compensatory tracking task.
This research attempts to solidify three areas of flight control engineering
which have existed largely independently for the past few decades: the use of robust
control design techniques to design flight controllers for high performance aircraft,
the comparison of classical and robust control design methodologies, and the use of
piloted simulation to append the quantitative analysis result with qualitative pilot
opinion.
Related Literature
Robust control techniques are not novel or innovative, but their application to
aircraft flight control is something that has emerged in recent decades, brought forth
by the need for highly complex systems able to accommodate highly capable
airframes. Research in this area has been largely dedicated to the design and analysis
of robust flight controllers for existing high performance aircraft, such as the work of
Lu, Wu, and Kim [1][2] and Shin, Balas, and Kaya [3][4] which utilized the F-16 as a
test vehicle to employ linear parameter varying (LPV) control design techniques and
assess their merit. Similarly, Morse and Ossman [5] developed a model following
reconfigurable flight control system for the F-16 which attempted to improve the
robustness of the aircraft in the event of a failure. Thesis studies gave insight and
validation to robust control techniques when applied to a high performance aircraft
and assessed the performance and stability of the developed controllers, but none
3
directly compared the resulting performance with that of a classically-designed
controller.
Dixon, Taylor, and Shaban [6] and McCowan [7] have explored and
compared the two design methodologies, but the work of Dixon, Taylor, and Shaban
utilizes a non-aerospace application. Though this work can be roughly correlated to
an aircraft controller, it is impossible to draw a conclusion as to the goodness of the
resulting system or methods used in an aircraft without a aerospace-specific analysis.
McCowan offers a direct comparison between classical and modern control design
techniques as applied to an aircraft. His research focused on reduction of disturbance
effects by way of classical and modern disturbance rejection controllers. This gave
the insight necessary to begin making direct comparisons of the various control
design methodologies in reference to aircraft applications, but the results were based
on desktop simulation and did not include the final piece of the flight control system,
the pilot.
Piloted simulation has long been used to assess pilot-vehicle performance and
to gather pilot opinion data. Sheldon and Osmon [8] extended the analysis of a flight
control system designed for the F-16 using quantitative feedback theory to include a
high fidelity manned simulation. It was found that the resulting system did not meet
level 1 handling qualities, but it did not provide a comparison of system performance
to that of a classically designed controller for the same aircraft.
Of all documented research, only one source was found which compared the
stability and performance of classical and robust control using flight test results.
Edwards [9] developed several controllers using classical and robust methods and
4
conducted a flight test using the Calspan VSS II Learjet. This research successfully
brought the three areas of flight control engineering together, but it was lacking a
handling qualities study in which pilots gave handling qualities and PIO tendency
ratings in an assessment of each controller.
The research that has been done to compare classical and robust control
design techniques has largely focused on desktop analysis, while the large majority of
available research involving the use of flight test and simulation involves testing the
validity and performance of just one type of controller. Though research exists that
employs flight test to compare classical and robust flight control designs, the
assessment was merely "pass/fail" and a formal handling qualities study was not
performed. The research presented in this thesis hopes to expand and improve upon
the comparisons made and conclusions drawn by Edwards.
Thesis Outline
A detailed outline of this thesis is as follows:
Chapter 2 presents a survey of the F-16 VISTA model, which is used as the
test vehicle for the study.
Chapter 3 documents the development of a proportional-integral (PI)
controller using the superaugmentation principles outlined in [10] and analyzes both
linear and nonlinear simulation results of the closed loop system.
Chapter 4 gives a brief introduction of Hoo control theory and presents the
design of an FL controller. Linear and nonlinear simulation results are examined and
a preliminary analysis of the closed-loop vehicle performance is performed.
5
Chapter 5 outlines the theory and methodology behind the LPV controller and
details the design procedure. As with the other two designs, linear and nonlinear
simulation results are presented and analyzed.
Chapter 6 offers a systematic comparison of the three closed-loop systems.
Included in this chapter is a survey of the closed-loop system metrics for the three
controller designs as well as linear and nonlinear time response comparisons detailing
the differences between the various systems in the time domain.
Chapter 7 presents a detailed test plan which describes the evaluation task and
gives a brief background on piloted simulation and handling qualities analysis.
Evaluation procedures are also given in detail.
Chapter 8 gives the results of the flight test and subsequent handling qualities
analysis.
Chapter 9 offers a summary and draws final conclusions.
6
CHAPTER 2
AIRCRAFT MODEL ANALYSIS
Basic Modes of Longitudinal Motion
In the longitudinal axis, the dynamics of an aircraft are characterized by two
oscillatory motions, one which is lightly damped and has a long oscillatory period
("phugoid") and one which is heavily damped and has a very short oscillatory period
("short period"). Phugoid motion occurs at constant angle of attack (a) and primarily
involves changes in altitude and airspeed while short period motion occurs at nearly
constant airspeed and involves changes in angle of attack.
WSP
0.75
X
I"
0.25
op
X
1/T82
-0.5
1/T61
-e
e~
0
Real
FIGURE 1. Longitudinal transfer function pole/zero map.
7
0.5
A pole/zero map is shown in figure 1 for a typical longitudinal transfer
function. The subscripts P and SP denote phugoid and short period mode parameters
while Tei and Te2 are the transfer function zeros. In the figure, the phugoid pole is at
a much lower frequency and more lightly damped than the short period pole.
The nonlinear equations of motion for the aircraft can be linearized in pitch
and represented in state-space form. In this formulation, the longitudinal equations
are ordinary differential equations with constant coefficients. These coefficients
represent the aerodynamic stability derivatives and mass and inertia characteristics of
the airplane. The state variables which represent the longitudinal motion are the
aircraft body velocities u and w, pitch rotational rate q, and pitch attitude 6.
Analysis of F-16 Longitudinal Model
The F-16 is a relaxed static stability aircraft, meaning that its bare airframe
dynamics possess very lightly damped or unstable poles for most of the flight
envelope. This problem tends to become complicated at higher angles of attack and
lower speeds, which is an indicator of potential issues during control development.
Non-minimum phase zeros are also present for some flight conditions, leading to a
difficult control problem in certain corners of the flight envelope.
In this research, the bare airframe dynamics were analyzed for the 12,000 ft.
altitude cases only. Transfer functions for all of the chosen flight conditions
corresponding to a 12,000 ft. flight altitude were generated with pitch rate, pitch
attitude, and angle of attack as outputs and the poles and zeros were plotted. Pole and
zero migrations for very low speed, high a cases are shown in figure 2 and the
8
dynamics for the four flight conditions of interest are summarized in table 1. The
symbols t, and con represent the mode damping ratio and natural frequency.
Increasing Airspeed
0.75
|
0.5
Increasing Airspeed
0.25
X.i
0^-
-1
~©
©-©—©-0.5
0
Real
0.5
FIGURE 2. Pole/zero migration with airspeed for low speed, high a trim cases.
TABLE 1. Rigid Body Longitudinal Mode Variation with Airspeed
VTRUE (kt)
Kp.fflnp]
[<^SP,G>nSP]
1/Tei
1/T92
110
[0.2392,0.214]
[0.3717,0.8013]
( 0.00202)
(0.2661)
160
[0.06331,0.1524]
[0.4753, 1.06]
( 0.00737)
(0.3925)
250
[ 0.06356, 0.09673 ]
[0.5007,1.44]
(0.01884)
(0.5709)
400
[0.1354,0.05838]
[ 0.5497, 2.039 ]
(0.01917)
(0.9211)
9
The plot shows the tendency of the phugoid mode to migrate toward, but not
cross into, the right half-plane as airspeed increases. Recall that for increasing
airspeed, angle of attack decreases, leading to the conclusion that high angles of
attack and/or low airspeeds have a stabilizing effect on longitudinal static stability.
There are also non-minimum phase zeros for the lower speed cases, though at 110 kt.
the Te) zero is in the left half-plane.
Flight conditions near 200 kt. can be defined as transitional; that is, they exist
in a locus of high power, high a trim conditions below those that would be
considered normal, level flight for the F-16. Much like the low speed flight regime of
a rotorcraft, low speed aerodynamic phenomena could possibly have an effect on
vehicle dynamics for the configuration of the F-16 being considered in this work.
Shown in figure 3, these transitional flight conditions exhibit odd trim behavior,
which may be model anomalies, or may be valid mathematical representations of the
aircraft at these transitional flight conditions.
RIM
35
30
O) 25
« 20
15
H
a
10
5
0
150
200
250
VTRuE(kt)
FIGURE 3. Flight envelope trim conditions.
10
300
350
400
The F-16 is advertised as having relaxed static stability, but the analysis
shows that it is only in these transitional cases that right half-plane poles exist. For
both lower and higher speeds, the phugoid is lightly damped but still exhibits stable
behavior. Because the phugoid mode is typically controlled by the pilot closure, the
classical control design task will focus on short period damping and frequency and
will use methods established in [10].
Notes and Observations
In order to include high a trim conditions in the control design, an altitude of
12,000 ft. was chosen as the baseline flight condition. This allows for a range of trim
angles of attack from 1-30 degrees and a variation of trim airspeed of 110-400 kt. in
which to select flight conditions for control design and flight simulation.
11
CHAPTER 3
CLASSICAL CONTROL DESIGN
Design Criteria
An advantage of classical control design methods is simplicity. Many systems
can be adequately compensated with first or second order controllers. It is desirable
for the compensator design to be of lower order to keep complexity to a minimum
while still providing adequate closed-loop control of the aircraft.
For the current design, proportional-integral (PI) controller in the form of
f
*,
P
s+—
T
v
(3.1)
i
is used to draw the short period dynamics toward a more desirable state and forces the
phugoid dynamics into the 1/Te zeros through the placement of the 1/Tq compensator
zero. The gain kq is scheduled on airspeed and is selected for each flight condition to
provide a pitch bandwidth of between 3.75 and 4.25 rad/s and is scheduled on
airspeed between different flight conditions. Stability is not an issue with this design,
as all flight conditions and selected gains provide adequate stability with additional
margin.
The short period dynamics are the dominant mode in the closed-loop system,
as the phugoid is driven into two real poles at the 1/Tei and 1/Te2 open-loop zeros.
The 1/Tq zero is placed above the rigid body dynamics but is located below the
12
desired crossover frequency, giving it the desired "k/s-like" characteristic in the
crossover region. The term "k/s-like" refers to the behavior of a response which
exhibits the characteristics of a single integrator 1/s with a pure gain k. This is
desirable as the pilot-vehicle crossover model states that an aircraft which exhibits
this behavior is likely to require less pilot compensation and therefore induce less
workload [11]. For our chosen flight envelope, the open-loop short period dynamics
range from 0.80 rad/s at 110 kt. up to 2.04 rad/s at 400 kt., and the 1/Tq compensator
zero is chosen to be 3 rad/s.
Controller Design
Control Architecture
The system is a pitch rate command/rate feedback control loop. A 10 Hz,
critically damped second order actuator model and ±25 degree elevator position limit
were added to more completely model the aircraft response to pitch rate commands.
The actual F-16 actuator models are consistent with these assumptions. The control
loop block diagram is shown in figure 4.
~^
>
K
s
( 0
s+—
T
Actuator
K
/ Saturation
FIGURE 4. Control loop architecture.
13
i
F-16
Design Synthesis
Classical frequency domain methods were used to determine the open-loop
stability margins and closed-loop short period damping and frequency for various
values of kq. Because the dynamics are dominated by the short period mode, the
bandwidth is closely associated with the short period frequency. This mode
determines the system break frequency and therefore the phase roll off that defines
the bandwidth frequency a>BwGains were selected to provide a closed-loop short period mode with a
damping ratio of 0.6-0.7 at a frequency of 4.0-5.0 rad/s, which provides a reasonably
damped overshoot. Command overshoot is acceptable as it results in a highly
responsive airplane while the relatively high bandwidth allows for pilots to
compensate for the error generated with the pilot loop closure.
A summary of the open-loop dynamics and stability margins is shown in table
2, where coxo defines the crossover frequency and gain and phase margin are shown
as PM and GM, respectively. It can be seen that stability margins are not of concern,
as a surplus of phase margin is present and gain margin is much greater than the
established minimum of 6 dB at the -180 degree phase crossover frequency GO-ISOCrossover frequencies of 5 rad/s will yield the desired bandwidth, so in addition to
selecting gain to provide preferable closed-loop short period dynamics, open-loop
crossover and the resulting closed-loop bandwidth were also critical gain selection
criteria. Gains are negative because a positive pitch rate command results in a
negative elevator deflection.
14
TABLE 2. Open-Loop Short Period Dynamics and Stability Margins
VTRUE (kt)
K
KsP,«nSp]
©xo (rad/s)
PM (deg)
GM (dB)
110
-240
[ 0.37, 0.80 ]
4.81
57.17
26.36
160
-100
[0.48, 1.06]
4.91
60.01
26.22
250
-45
[0.50, 1.44]
4.93
63.42
26.38
400
-17
[ 0.55, 2.04 ]
4.92
70.50
26.72
An example loop closure is shown in figure 5 for 160 kt., where the short
period closed-loop poles are denoted by a prime and shown as a square to distinguish
them from open-loop poles and zeros. For the 160 kt. case shown in figure 5a, the
closed-loop short period damping and frequency are [C,SP, ©nsp'] = [ 0.64, 3.81]. The
lightly damped phugoid mode is driven into the real axis, where it splits into the two
open-loop zeros to become real closed-loop poles which are dominated by the short
period dynamics of the aircraft in the closed-loop system. The frequency responses
shown in figure 5b and figure 5c show magnitude in blue and phase as a dotted red
line.
As shown in figure 5d, the response of angle of attack a initially follows the
response of pitch attitude 6, but then begins to wander as the pitch rate command is
set to zero and the system holds attitude. Investigation of the nonlinear response
shows that this is due to a loss in altitude and airspeed with the pitch rate input; the
command is to elevator only while throttle is held constant. As the aircraft pitches
15
o\
10
r
-5
'- 0 -
»
FIGURE 5. Loop closure supporting plots, 160 kt. flight condition.
c) Closed-Loop Bode Plot
Freq (rad/sec)
a) Root Locus
d) Closed-Loop Pulse Response
b) Open-Loop Bode Plot
9
10
up, it induces a higher drag coefficient, and begins to descend and slow down, thus
tilting the velocity vector upward. Because the system is holding Euler angle, this
change in the velocity vector shows up as an increase in angle of attack. This
phenomenon is of much lower magnitude at higher airspeeds as the trim pitch attitude
is not large and the held attitudes do not result in a loss of airspeed or altitude.
The complete summary of closed-loop dynamics and performance metrics is
shown in table 3. Phase delay xp, the time delay between pilot input and surface
movement, is sufficiently low for all cases and the target short period frequency and
damping and target bandwidth are all satisfied with the selected gains.
TABLE 3. Closed-Loop Short Period Dynamics and Performance Metrics
VTRUE (kt)
Kq
KsP',<°nSp']
110
-240
[0.61,3.72]
3.78
0.0192
160
-100
[0.64 3.81 ]
3.89
0.0191
250
-45
[0.66,3.91]
3.99
0.0190
400
-17
[0.68,4.14]
4.18
0.0190
COBW
(rad/s)
TP
(sec)
Nonlinear Simulation Results
The resulting control system was implemented in Simulink to test the
response using nonlinear simulation. Different from linear simulation, where the
aircraft is represented using a state-space block in Simulink, the nonlinear aircraft
17
model is in MATLAB™ m-flle format and uses a Runge-Kutta iterative method to
solve the nonlinear equations of motion.
Another main difference between the nonlinear and linear simulation models
is the inclusion of the elevator position limit. For most longitudinal maneuvering at
most flight conditions, these limits are more than adequate and are rarely reached.
However, because our flight envelope includes cases at very low airspeeds with very
large trim angles of attack, there is a large amount of negative elevator deflection
present at trim. For these low speed cases, the surface position limit is briefly
encountered, though the response to the 5 deg/s pulse input used in the desktop
simulation is not greatly affected.
It is anticipated that near the operating point, responses of the nonlinear and
linearized models will be very similar. Continuing with the analysis of the 160 kt.
case, the closed-loop time response is compared for the linear and nonlinear systems.
As expected, the response matches almost identically for this case in pitch rate and
pitch Euler attitude, but because the nonlinear model takes into account the altitude,
airspeed, and flight path, it shows a slightly larger angle of attack than the linear
model. This is likely due to divergence from the operating point of the linear model.
Notes and Observations
There is a tendency to reach surface position limits at lower airspeeds due to
large trim elevator deflections needed to maintain large a trim conditions. The gain
was lowered for these cases to reduce the possibility of hitting these saturation limits.
18
Through careful gain selection, the bandwidth and phase delay were kept
almost constant across the entire flight envelope. This will be beneficial during the
flight test phase as the aircraft will exhibit a "snappy" response to pilot inputs for the
entire flight envelope.
FIGURE 6. Linear (blue) and nonlinear (red) response to pulse input.
19
CHAPTER 4
Hoo CONTROL DESIGN
Control Theory
The objective of Hoc control design is to synthesize a stabilizing controller
which minimizes the maximum energy of the output given a bounded energy input.
The feedback system used for Hoo optimization is shown in figure 7 with plant P(s),
controller K(s), control inputs u, disturbance inputs w, controlled outputs z, and
measured outputs y.
w
z
P(s)
u
y
K(s)
FIGURE 7. Ho, generic block diagram.
The state-space model of the plant is given by:
x = Ax + B^w + B2u
z = CjX + Dnw + Dnu
y = C2x + D2lw + D22u
20
(4.1)
All matrix-valued state-space data have appropriate dimensions. The
following plant parameter assumptions are made:
1. A, B2, and C2 are stabilizable and detectable.
2. The matrices \Bl
D{2~\ and [C2 D2l] have full row rank.
3. D22 = 0.
A suboptimal Hoo control problem is formulated to find a controller K(s) such
that the closed-loop system is internally stable and the Hoo norm of the closed-loop
system (the maximum gain from w to z) is less than the parameter y, |z||2 < y\w\2,
where the controller is of the form:
u
A Bk
Ck Dk
(4.2)
In this research, the Hoo control problem was solved via a linear matrix
inequality (LMI) approach, where the solvability conditions are formulated as a set of
LMIs as opposed to the traditional algebraic Riccati equations. The following LMIs
are the synthesis conditions for the Hoc control problem, where the symbol "<"
indicates that the matrix product on the left hand side is negative definite and ">"
indicates positive semi-definite.
21
r
N„ 0
AR + RAr
DU
B?
RC
-yi
D'
ATS + SA
SB,
C,I
BTXS
-yl
£>,11
C,
D„
CXR
r
N,
R
I
I
S
B
(ND
i
-yl
T
T
<0
\
^
0
o <0
/
(4.3)
>0
NR and Ns are matrices whose columns form bases of the null spaces of
\B\
D[2 Ol and [C2 D21 0]. A continuous-time y-suboptimal Ho, problem is
solvable if and only if there exists symmetric matrices R and S such that the LMIs in
(4.3) are satisfied. The resulting R, S, and y can then be used to construct the
controller K(s) [12].
Design Criteria
A family of Ho, controllers for the flight conditions of interest was developed
via the LMI approach. For every discrete airspeed, an FL controller in the form of
(4.3) is derived using the linearized system dynamics for that trim condition in
addition to the various weighting functions associated with the desired performance
or anticipated control effort of the pilot. If the LMI conditions given in (4.3) are
solvable, then the stability of the closed-loop system for that flight condition is
guaranteed and the FLo norm of the closed-loop system is less than y. Between
different flight conditions, the control gains Ah Bh C& and D* are scheduled on
airspeed.
22
The design criteria for the FL controllers are similar to those for the classical
control design. A rapid response to control inputs with desirable overshoot and
adequate damping are of great importance as these characteristics will likely translate
to good flying qualities during the piloted simulation. The key closed-loop metrics
remain bandwidth and phase delay, with short period damping and frequency again
the dominant dynamics.
Weighting functions were chosen such that the desired performance is
achieved by the closed-loop system. Though general rules of thumb exist, each plant
differs and performance and control effort goals vary so that an exact formulation for
the selection of weighting functions is not available. Instead, the performance
weighting functions and control effort penalties are selected by inspection to design a
robust controller capable of providing adequate performance while providing the
stability guarantee that is the primary benefit of FL, control.
Controller Design
Control Architecture
A block diagram of the system interconnection for synthesizing the Ho,
controller is shown in figure 8. The F-16 linear plant dynamics contain 5 states and 2
control inputs for elevator and throttle. The outputs are aircraft velocity V, pitch rate
q, and flight path angle y. Weighting functions to simulate actuator dynamics and
measurement noise are included, in addition to weighting functions designed to
penalize the control effort and filter the error signal. These weighting functions are
given in table 4. The open-loop system including the F-16 dynamics and all
23
weighting functions is the generalized plant P(s) shown in figure 7. The inputs of the
generalized plant include a 3 element noise signal n, pitch rate command qcmd, and the
control command inputs 5™ and 5cem . The outputs of the generalized plant are the
weighted error signal ep, weighted control effort eu, and the measurement input to the
controller K.
^
^->
F-16
wn
«
e„{5}
W.
wn
< 4' o<-
Wai
<r
<T\
«{3}
^crad
WA
9ideal
FIGURE 8. Weighted open-loop interconnection for H*, control of F-16 aircraft.
Design Synthesis
The design of an Hm controller that provides satisfactory performance depends
largely on the proper selection of the weighting functions included in the generalized
plant formulation [13]. The remaining portion of the Ho, control design is largely
mathematical and can be dealt with using the MATLAB Robust Control Toolbox,
which offers tools to specify, manipulate, and numerically solve LMIs. The LMI
solver mincx is used to find the solution of the LMIs defined in (4.3) for a minimum
value of y. The linear controller K is then constructed based on the LMI solutions R,
24
S, and y and the state-space data of the generalized plant, andwill ultimately be
implemented on both the linearized and fully nonlinear plant for verification of its
performance.
Because the optimal Ho, controller synthesis is largely a mathematical
manipulation, the control problem resolves to the proper selection of weighting
functions so that the resulting linear controller delivers adequate performance on the
less predictable nonlinear plant. As listed in table 4, Wact and W„ are weighting
functions used to estimate actuator dynamics and sensor noise so that when the
controller receives sensed signals and outputs actuator commands, it can properly
handle variations from the optimum. Accordingly, the weighting function Wu is used
to penalize the control effort and filter the error such that surface saturation and rate
limits can be handled robustly.
The control problem is synthesized such that the pitch rate error from an ideal
pitch rate response is allowed to be large for frequencies above the system bandwidth
frequency and small for frequencies where the system can track commands. This is
accomplished though the lag-lead weighting filter Wp.
The ideal command weighting function Wtdeai was initially set to provide the
same command bandwidth as the classical controller, but performance was lacking
and the weighting function had to be tuned for more bandwidth in order to converge
to a satisfactory result.
25
TABLE 4. Weighting Functions for HM Control Design
Description
Expression
Symbol
10
(throttle)
5 + 10
Actuator
Wai
62.82
[0.7,62.8]
(elevator)
0.8 ft/s (velocity)
Sensor Noise
0.6 deg/s (pitch rate)
wn
0.1 deg (flight path angle)
O.lin.(^)
0.1in/s(4)
Control Effort
0.1 deg (Se)
wu
0.001 deg/s ( ^ )
0.0002 deg/s2 (Se)
Performance
wD
Ideal Command
Wideal
0.818^ + 70.71
5 + 0.7071
7.22
[0.8,7.2]
The shorthand form is defined by a(s + b)[s2 + 2C,as + a>2] = a(b)[£, a>]
26
Though this formulation should provide adequate tracking performance, it
does allow a small amount of steady-state error. This is acceptable in most cases as
typical implementations of H*, control involve flight path angle or attitude commandtype systems. However, for the current implementation, the presence of steady-state
error is extremely undesirable as it induces a small amount of pitch rate which washes
out pitch angle. Where the classical controller drives steady-state error to zero and
thus holds attitude, the H» controller allows attitude to decrease rather than hold.
This would likely be an issue in a piloted simulation comparison of the two
controllers for capture/hold-type tasks.
A summary of the closed-loop metrics for Hoo control is given in table 5.
TABLE 5. Hoo Closed-Loop Performance Metrics
VTRUE (kt)
G)Bw (rad/s)
t p (sec)
110
2.39
0.190
160
2.83
0.155
250
3.05
0.126
400
3.17
0.098
Linear Simulation Results
The family of controllers was then implemented with the linear F-16
dynamics derived previously to test performance for each flight condition. A pulse
27
rate command of 5 deg/s is implemented and the actual rate response is compared
with the commanded ideal response.
The results were as expected: at lower airspeeds where trim conditions
involve large elevator deflections and high angles of attack, the aircraft struggles to
achieve the desired ideal performance. Because this objective is much higher than
the desired bandwidth of about 4 rad/s, this is still considered adequate performance
and it compares well with the overshoot and bandwidth experienced in the closedloop classically-controlled linear simulation.
As airspeed increases and trim conditions become less extreme, more
aerodynamic force can be generated by the elevator and the response to pitch rate
commands becomes increasingly fast. For airspeeds faster than 200 kt., the response
is close to the ideal and exhibits minimal overshoot. Overshoot has been shown to be
a significant factor in flying qualities and though a minimal amount of overshoot is
acceptable for good performance, excess overshoot can have an adverse effect on
pilot ratings in rate command systems [14].
The linear response in figure 9 shows a well followed rate command with no
steady-state error and minimal overshoot. It can be seen that the rate achieved has a
much lower bandwidth than the 7.2 rad/s break frequency of the ideal command
weighting function. Because airplane bandwidth is defined as the frequency which
provides 45 degrees of phase margin to the pilot as he closes the outer pilot loop on
the closed-loop aircraft, the airplane bandwidth reported is much less than what is
desired in the problem formulation. However, the response is still considered ideal
28
for comparison to the classically-controlled response during the flight test portion of
the project due to its similar behavior and bandwidth.
«5
A
•D
o-o
—
— 45
5 . "0
| 3 5 _ ^
£
—^
30
„45
CO
CO
alpha
«40
0
1
2
3
4
5
6
Time (sec)
7
8
9
10
FIGURE 9. Linear (blue) and ideal rate (dashed) response to pulse input, 110 kt.
Similar results were seen for the other airspeeds in the defined flight
envelope. At higher airspeeds, the aircraft tracks the ideal pitch rate command much
more closely, but in addition to a slight undershoot a steady-state error is introduced
which causes attitude to slowly wash out over time. This will be noticeable to a pilot,
but will likely be negated subconsciously by way of a slight command offset through
the stick.
Nonlinear Simulation Results
The nature of a linear Hoo controller is such that the performance weighting
function allows for small amounts of steady-state error [13]. For attitude commandtype systems, this is considered acceptable as long as it remains within a certain
29
range. Because the allowable steady-state error is determined by the performance
weighting function, it was believed that adjustment of Wp would drive error to zero.
However, fine adjustments caused the optimization routine to not converge on a
solution. The weighting function was optimized to provide the best possible results,
but at higher airspeeds the steady-state pitch rate error remained and pitch attitude
was not held as precisely as at lower airspeed flight conditions.
For airspeeds below 170 kt., the nonlinear simulation matches very closely
with the linear response, except for the 120 kt. case shown in figure 10 where the
angle of attack diverges from the linear result. Because pitch rate and pitch attitude
overlay almost exactly and angle of attack is the only variable that differs, the
conclusion is that the nonlinear model has an anomalous aerodynamic lookup at this
flight condition related to changes in angle of attack with induced pitching moment.
This is a possible explanation for all flight conditions whose nonlinear simulation
results show the same mismatch in angle of attack for certain flight conditions.
7
8
9
10
FIGURE 10. Linear (blue) and nonlinear (red) response to pulse input, 120 kt.
30
Scheduling Considerations
Despite closed-loop system performance goals being similar to those for the
classically-designed controller, significant differences exist such that strengths and
weaknesses of each method will likely appear during piloted simulation. One of the
most significant limitations of the Ho, controllers designed is that they are discrete for
each flight condition, and the maneuvers planned for the simulation portion of the
project involve changes in angle of attack and airspeed. This would imply the need to
schedule Hoo gains in a manner similar to the classical controller, which is undesirable
as it departs from the stability guarantee of the robust control formulation. In this
context, the controller is implemented in the same manner and suffers from the same
shortcoming of stability not being guaranteed between design points.
Given that the advantage of stability is no longer a factor in comparison, the
Hoo controller now has the same weaknesses as the classical controller yet suffers
from additional performance shortcomings such as a steady-state rate error that
causes the attitude response to wash out and a rate response that varies from nicely
damped undershoot to lightly damped overshoot for different flight conditions.
Variations such as this imply that performance for flight conditions between design
points will encounter large transitions in the interpolated gains and are prone to suboptimal performance for these flight conditions.
31
Notes and Observations
The presence of steady-state error for certain flight conditions causes attitude
hold performance to degrade significantly and would likely be a source of poor flying
qualities during the simulation portion of the project for capture/hold-type tasks.
There is a significantly larger amount of phase delay present for the closedloop system when compared with the classical controller. A higher bandwidth for
higher airspeeds somewhat offsets the deficiency.
32
CHAPTER 5
LPV CONTROL DESIGN
Control Theory
Linear systems whose state-space data depend continuously on a time-varying
parameter/)(/) are called linear parameter varying (LPV) systems. It is assumed that
the parameter p evolves continuously over time and the trajectory of the parameter is
unknown in priori, but can be measured in real time. A generalized LPV system can
be described by:
x
A(p(t))
B(p(t))
_y.
C(/>(/))
D(p{t))
(5.1)
The LPV model can be described as a group of local descriptions of nonlinear
dynamics. There are several approaches used to obtain reliable LPV models with the
Jacobian linearization approach being the most widespread technology.
LPV control theory is a systematic gain scheduling technique which has been
used to design controllers for dynamical systems over a wide parameter envelope. It
is an extension of H» control theory in which the relationship between real-time
parameter variations and performance is explicitly taken into account. This enables a
controller to be designed for wide ranges of operating conditions with theoretical
guarantees of stability, performance, and robustness throughout the region. The
33
block diagram for the LPV control problem is shown in figure 11, where P(p) is a
generalized plant defined by the state-space equation:
A(p)
Cx{p)
C2(p)
Bx(p)
Dn(p)
D2i(p)
B2{p)
Dn{p)
D22(p)
(5.2)
The output feedback controller K(p) is of the form:
A(P)
Ck{p)
Bk(p)
Dk(p)_ y
(5.3)
The controller defined in (5.3) is also a state-space system.
w
p(p)
u
y
K(p)
FIGURE 11. LPV generic block diagram.
The LPV control approach is based on first generating a set of LMIs over the
parameter set:
34
(NR(P)
0
A(p)R + RAT(p)
RC((p)
Bx{p)
Cx{p)R
-yl
Du(p)
n(p)
-n
BTX{P)
Ns(p) oY
D
AT(p)S + SA{p)
BlT(p)S
Cx{p)
f
R
I
SBx(p)
-yl
Dn(p)
NR(p)
Cl{p)
Ns(p)
D\\p)
-yl
0
0
<0
<0 (5.4)
I\
>0
S
The LPV controller is then constructed from the solution of the LMIs.
Different from the LM1 formulation of Ho, control, the state-space matrices of the
plant are continuous functions of the scheduling parameter/). Theoretically, the
synthesis conditions in (5.4) must hold for any value of the scheduling parameter/) in
the allowable region, making it an infinitely LMI-constrained problem. In many
practical problems, the parameter dependency is dealt with using the "gridding"
approach. Assume that n discrete values of the scheduling parameter/) are selected
and are dense enough to represent the dynamics over the entire parameter region.
The LMI problem defined in (5.4) can then be expressed as 3« LMIs.
Whereas in Ho, control a single operating condition is used to construct a
single controller through an LMI minimization, the entire set of linear state-space
models is used to construct a series of LMIs which are then minimized to find the
symmetric R and S matrices used to construct the controller K(p). The controller
resulting from the LPV control approach stabilizes the closed-loop system for all
flight conditions. However, because the control synthesis was designed to include
such a large region of the flight envelope, performance is sacrificed when compared
35
to the discrete Hoo controllers in lieu of a "global" stability guarantee and simplicity of
having a single controller.
In order to improve the probability that a set ofR and S matrices will be found
and to improve the performance of the resulting controller K(p), the matrix variables
R and S are allowed to vary linearly with the scheduling parameter/? such that:
R(p) = R0+pRl
v
'
S{p) = S0+pSl
(5.5)
This allows the controller constructed from R{p) and S(p) and derived from
the complete set of LMIs more freedom to provide optimum performance for all
operating conditions while still offering the benefit of a stability guarantee.
Design Criteria
The LPV control problem is formulated following the same design criteria as
the previously described Hoo control. For the current formulation, p is selected to be
airspeed and the flight envelope is to be defined as shown in figure 3 where 14
distinct trim points between 110 and 400 kt. are selected.
Controller Design
Control Architecture
The control problem was set up identically to the Hoo problem. The
generalized plant block diagram is shown in figure 8 and described in control
architecture section of chapter 4.
Because the formulation for LPV control is a generalization of H* control, an
in-depth explanation of the LPV controller design is not necessary. The plant
dynamics, weighting functions, and LMI formulations are all identical between the
36
two; the only difference being the number of LMIs that the minimization routine
must find a solution for, as each discrete Hx problem has only 3 LMIs whereas the
LPV controller has a total of 42 LMIs for 14 flight conditions from which to find a
single set of R{p), S(p), and y. In Hoo control, R and S are unique to each flight
condition and the controller K constructed from them is also unique. In LPV control,
the controller K(p) = Diip) + Clip) {sI-Ai^p))'1 Bi£p) is scheduled based on the
varying parameter p and is universal to all flight conditions. This is the basis for the
guarantee of stability at all points in the flight envelope.
The closed-loop performance metrics of interest are again bandwidth and
phase delay, as these are the best predictors of flying qualities in piloted simulation
studies [15]. Though command following for the Hoo controller was good, the
presence of phase delay was of concern when compared to the classical controller as
the order of magnitude difference would likely manifest as degraded flying qualities.
However, the lack of significant overshoot and adequate damping showed
performance benefits which were believed to offset the performance penalties the
phase delay introduced. Because the LPV formulation is in many ways identical to
the Hoo control problem but with an expected degradation in performance, it was not
expected that the large amounts of phase delay seen with HM control would be
improved with LPV control. However, it was noted that both bandwidth and phase
delay were improved with LPV control, as shown in table 6.
37
TABLE 6. LPV Closed-Loop Performance Metrics
VTRUE
(kt)
OOBW
(rad/s)
TP
(sec)
110
3.00
0.137
160
3.29
0.093
250
3.42
0.045
400
3.46
0.038
Linear Simulation Results
A problem noted in the discussion of the Hx, simulation results was the
presence of steady-state error in pitch rate. Despite good command following
characteristics, there existed a steady-state error that would cause pitch attitude and
angle of attack to wash out over time. This occurred mostly at higher trim airspeeds.
It was discussed that part of the development of an Hoc controller defines an
acceptable steady-state error by means of a performance weighting function and that
this error is acceptable for flight path command and other attitude command-type
systems. For a rate command, attitude hold system, however, this slow loss of
commanded attitude is likely to show itself implicitly as degraded pilot ratings or
explicitly as negative pilot comments.
Though table 6 shows bandwidth is acceptable, the presence of a slight
steady-state error may affect pilot tasks if the tasks require precision capture/holdtype maneuvers. It is also possible that the error is sufficiently small and the tasks
38
sufficiently dynamic that the pilot loop closure will simply negate the commanded
rate error by way of a command bias through the stick, though it is not possible to
determine which the case is until a thorough analysis of the simulation test data is
performed.
For higher airspeeds, bandwidth increases and phase delay decreases, with the
linear response showing very good ideal command following and excellent attitude
hold characteristics. Though phase delay is again significantly larger for all flight
conditions when compared with the classical control design, the lack of overshoot
and precise command following will likely offset the deficiencies in a piloted
simulation study.
A general observation made when examining the linear time histories is that
as airspeed increases, not only does bandwidth and phase delay improve, but
command following and attitude hold performance improve as well. Though the
improvements are likely related, because the LPV controller is a single controller
with a scheduled gain component it can be inferred that the LPV control problem
likely found an optimum control solution for the higher speed cases and extended its
robustness to include lower airspeed trim conditions. The closed-loop system at
lower airspeeds is stable and performance is adequate but a comparison of figure 12
and figure 13 shows the difference in command following and steady-state error
between the low airspeed, high angle of attack trim case and the high airspeed, low
angle of attack case.
39
~45
D)
5.40
|35
ra
30
4
5
Time (sec)
6
10
FIGURE 12. Linear (blue) and ideal rate (dashed) response to pulse input, 110 kt.
jo
~10
10
Time (sec)
FIGURE 13. Linear (blue) and ideal rate (dashed) response to pulse input, 400 kt.
Nonlinear Simulation Results
The model anomalies seen for both the classical and FL, control designs seem
to be amplified when using the LPV controller. Figure 14 shows the nonlinear
40
simulation results for the previously documented 120 kt. trim case. Recall that
previous simulations have shown that rate command and attitude response match
closely for the linear and nonlinear simulations, but the angle of attack response to
the pitch rate input diverged from the linear result during the nonlinear simulation.
For the LPV-controlled case, an even more peculiar observation is made that not only
is there a steady-state error in pitch rate command, but there is also a sudden "bump"
in pitch rate roughly 5 seconds into the attitude hold portion of the pulse input. This
causes pitch attitude to wash out at a faster rate than the linear model which also
possesses a steady-state rate error.
_40
B35
fi 30
JZ
~25
~40
2,35
•5 30
(O
25
4
5
Time (sec)
6
9
10
FIGURE 14. Linear (blue) and nonlinear (red) response to pulse input, 120 kt.
The cause of this behavior is unknown, though it is believed that the
aerodynamic lookup tables used in the nonlinear simulation contain discontinuities or
anomalies for certain flight conditions and trim speeds. This was documented during
41
the extraction of the linear models for analysis. Trim cases of 120 kt. and those
between 160-200 kt. were identified as problem cases where confidence was not as
high as it was for other cases due to odd behavior in both linear model extraction and
nonlinear desktop simulation.
Notes and Observations
Though bandwidth is comparable to the H^ controller, phase delay is
significantly improved with the LPV controller. This is especially true for higher
airspeeds.
The steady-state error in pitch rate seen with the Hoo control at high speeds is
present over the entire flight envelope with LPV control. This causes a slight roll off
in pitch attitude which may be an issue during piloted simulation.
The model anomalies seen previously are seemingly amplified by the LPV
controller for certain flight conditions. These flight conditions will likely be avoided
as test points in the simulation portion of the project.
The LPV controller works very well for higher airspeeds, but lacks
performance for airspeeds lower than 200 kt. This is in contrast to the Hoo controller,
which performed well for higher airspeeds and adequately for lower airspeed trim
conditions.
42
CHAPTER 6
COMPARISON OF CONTROL DESIGNS
Simulation Considerations
Of the many differences in the three control design methodologies to be
compared, there exist advantages and disadvantages for each in regard to piloted
simulation. First and foremost is the implementation advantage of the classical
controller over the two robust controllers. The classical controller is first order with a
single scheduled gain and can be easily implemented in a simulator. The FL,
controller is of higher order and does not guarantee stability between interpolation
points, thus negating an advantage it holds over the classical design. In addition, the
Hoo and LPV controllers are of higher order and much more computationally
intensive. Though the LPV controller is based on H*, design formulations, it is at a
disadvantage when compared to the Hoo controller because it sacrifices performance at
the edges of the flight envelope to include a stability guarantee for all flight
conditions considered. The advantage of stability is important, but the performance
penalties will likely result in poor pilot ratings.
Performance Metrics
Performance metrics are designed to be measures of system performance
which predict closed-loop flying qualities under piloted control. Listed in [16] are six
43
different criteria for the prediction of pilot opinion, but because the simulation will
consist of up-and-away tasks that will likely require high pilot gain and tight closedloop control, the bandwidth criterion is the primary figure of merit in the comparison
of the closed-loop system characteristics. Because bandwidth is a measure of
tracking precision and disturbance rejection [17], it is expected for the planned tasks
that if a difference was noticeable to a pilot between the control design
methodologies it would likely manifest as a bandwidth or phase delay difference.
Other factors play largely into pilot ratings, such as percent overshoot and attitude
quickness, but a comparison of bandwidth for the three closed-loop systems in
addition to a qualitative analysis of the desktop simulation results is an adequate
means of predicting flying qualities ratings for the piloted simulation.
Table 7 and figure 15 give a summary of the closed-loop performance metrics
for the three designs at four flight conditions of interest. In the figure, the data in
table 7 is overlaid onto the bandwidth/phase delay criterion described in [16]. The
colored regions represent the level 1 (blue), 2 (magenta), and 3 (red) handling
qualities regions. Some trends which were observed in the data:
1. Bandwidth increases and phase delay decreases with increasing airspeed.
2. Bandwidth varies minimally across the flight envelope for all three
designs.
3. Phase delay is almost invariant for the PID controller and is significantly
lower than either of the Hoc, and LPV controllers.
44
4. Bandwidth and phase delay for the FL controller are considered deficient
for all flight speeds when compared to the other designs.
TABLE 7. Closed-Loop Performance Metrics for Various Designs
Classical
VTRUE
(kt)
H«,
LPV
coBw(r/s)
Tp(sec)
coBw(r/s)
xp(sec)
coB\v (r/s)
tp(sec)
110
3.78
0.0192
2.39
0.190
3.00
0.137
160
3.89
0.0191
2.83
0.155
3.29
0.093
250
3.99
0.0190
3.05
0.126
3.42
0.045
400
4.18
0.0190
3.17
0.098
3.46
0.038
2
4
Bandwidth [rad/sec]
a) Classical
2
4
Bandwidth trad/sec]
b) Hx
2
4
Bandwidth trad/sec]
c) LPV
FIGURE 15. Closed-loop bandwidth and phase delay for various flight controllers.
Linear Simulation Comparison
Overshoot
In general, the classical controller exhibited much more overshoot than either
of the two robust control methods. This is to be expected as both robust controllers
45
rely on Hoc control methodology which minimizes resonant peaks and thus damps out
as much overshoot as possible. This provides a predictable response which is
desirable for most control problems. However, for the current application in which
the controller is meant to stabilize and control a highly agile fighter-type aircraft,
damping out overshoot comes with the penalty of lower bandwidth and a much
slower response. Having an acceptable amount of bandwidth can be seen as
beneficial in figure 16, as it induces lead which hastens the rate response and
provides a faster attitude change with minimal overshoot for the response using the
classical controller shown in red. The robust controllers, shown in blue and magenta,
show a response which mimics the ideal response defined in the control problem set
up but contains a much larger amount of phase delay and an attitude response that
lags behind that of the classical controller, which is allowed to have some overshoot
through a lightly damped short period mode that is dominant in the pitch axis.
0
1
2
3
4
5
6
Time (sec)
7
8
9
FIGURE 16. Comparison of linear responses to pulse input, 110 kt.
46
10
Angle of Attack
As evidenced in figure 16, there is a major difference in how the robust
controllers hold angle of attack. The classical controller tends to allow angle of
attack to respond freely as it concentrates more on negating pitch rate error and thus
holding Euler attitude. The MIMO nature of the robust controllers includes a velocity
hold loop which increases throttle input to hold attitude but in doing so causes angle
of attack to return to near its steady-state value. The addition of a velocity hold loop
to the classical controller shows a decrease in the disagreement between the two
design methods in angle of attack response.
Steady-State Error
The final observation made in the qualitative analysis of the responses
generated by the linear simulation is that of a steady-state pitch rate error. The robust
controller formulation used to develop the Hx and LPV controllers allows for a small
amount of steady-state error, which in the case of attitude command-type systems is
acceptable and has little effect on pilot ratings, but for the rate command system
currently implemented the associated pitch attitude roll off may result in degraded
flying qualities.
The robust controllers interpret the steady-state error as a command and thus
tend to drive it rather than attenuate it. For the high airspeed trim cases the angle of
attack responses for all three controllers are almost identical. This is because the
changes in airspeed induced by the pitch rate command are small compared to those
47
at lower airspeeds with larger trim angles of attack and thus the differences in the
MIMO and SISO systems are not observed.
Nonlinear Simulation Comparison
The focus of the nonlinear simulation analysis to this point has been the
verification of agreement between the linear and nonlinear responses in the time
domain. The current qualitative analysis focuses more on differences between the
three control designs, as this is the final analytical check of predicted responses in the
simulator. Figure 17 shows the nonlinear response to a pulse input for the 180 kt.
trim case. The classical and H*, controllers handle the input well and provide an
adequate attitude response. The LPV controller, however, undershoots the return to 0
rate command and holds a steady positive pitch rate input which causes attitude to
ramp. The linear response does not predict this behavior, implying that the problem
is either in the way the nonlinear plant handles the control inputs or in the online
formulation of the controller for these flight conditions. Though this problem falls in
line with other model anomalies previously documented, the currently observed
divergence is the most severe and without explanation.
Because the nonlinear model could not be verified as correct for all flight
conditions, it was determined that linear models would be best suited for piloted
simulation.
48
~20
•E 10
4
5
6
7
8
9
10
Time (sec)
FIGURE 17. Comparison of nonlinear responses to pulse input, 180 kt.
Notes and Observations
Though bandwidth and phase delay are within a reasonable range for all three
controllers, the classical controller exhibits a substantial bandwidth and phase delay
advantage over the two robust control designs.
The classical controller exhibits more command overshoot but its advantage
in rise time is likely to offset the overshoot's negative effects on pilot ratings.
The linear simulation results show comparable responses at higher airspeeds,
with the classical controller showing an advantage in rise time and attitude hold
characteristics for lower airspeeds.
The nonlinear model continues to display odd behavior for certain flight
conditions. There is currently no logical explanation for the anomalies. For this
reason, the flight test plan will focus on linear models for flight conditions which are
viewed with greater confidence.
49
CHAPTER 7
SIMULATION TEST PLAN
Test Objectives
Ground based simulation is a valuable tool in the assessment of how changes
in flight control system parameters affect closed-loop vehicle performance.
Simulation test data can often be used to more accurately predict vehicle performance
during flight and is often regarded as an intermediate step between traditional desktop
analysis and flight test. Incorporating a pilot into the closed-loop system and
performing specific tasks meant to measure and expose any deficiencies in flying
qualities can validate or discredit any claimed improvements in system performance
under more realistic conditions and append the quantitative analysis results with
qualitative pilot ratings and comments.
So far, the current work has involved a design study of three different types of
controllers. Conclusions have been drawn based on the results of a nonlinear desktop
simulation and the quantitative analysis of selected performance metrics and the
analytical results have shown that despite having more overshoot than the robust
control designs, the classical controller shows significant advantages in ease of
implementation and computational intensity as well as bandwidth and phase delay.
These are valid conclusions made from a comprehensive study of several flight
50
control designs, but a validation of these conclusions through piloted simulation is
required for the analysis to be considered complete.
The primary objectives of the piloted simulation study in the current work are:
1. Investigate the differences in closed-loop bandwidth and phase delay for
all controller designs.
2. Attempt to expose any stability deficiencies present in the defined flight
envelope.
3. Evaluate tracking performance for various trim conditions and airspeeds.
4. Track pilot workload and input spectra to provide an additional
performance assessment metric.
An analysis of the simulation test data will provide a well-rounded handling
qualities assessment of the three flight control designs in question. This assessment
will append the desktop analysis and provide a qualitative assessment in a real-world
operational environment of the various controllers.
Test Plan
A full description of the STI flight simulator is given in Appendix A.
Flight Conditions
The flight envelope defined for this study focuses on low speed, high angle of
attack trim conditions for a 20,500 lb. aircraft with a longitudinal center of gravity
location of 30% of mean aerodynamic chord (MAC). For completeness, moderate
speed trim conditions are included to investigate the handling qualities at more
conventional operating points.
51
Because it is both unreasonable and unnecessary to perform a simulation run
for each flight condition in the defined envelope, the simulation test runs shall instead
focus on a few distinct trim cases which together will adequately represent the entire
envelope. A very low speed, high angle of attack case will be examined as the
elevator deflection required for trim is large and encountering surface saturations is
likely. This is a case of interest as encountering rate or travel limits often affects task
performance and may even lead to pilot-induced oscillations (PIO).
Because of concerns over lack of confidence in the model, only flight
conditions which have shown predictable behavior in the desktop analysis will be
selected for the simulation. Shown in table 8, the selected trim conditions span the
entire range of trim airspeeds and include two intermediate flight conditions which
fill in the remainder of the envelope. Tasks will be flown for each controller at each
flight condition in order to collect enough data for a detailed analysis.
TABLE 8. Trim Conditions for Simulation
VTRUE (kt)
a(deg)
5e (deg)
110
33.87
-10.49
160
16.12
-4.23
250
6.09
-2.95
400
1.50
-1.75
52
Sum-of-Sines Tracking Task
The STI flight simulator was designed to simulate aircraft in up-and-away
flight. Because the current work has focused on low to moderately-low airspeeds and
ease of implementation is of importance, the evaluation will focus on a sum-of-sines
(SOS) tracking task in the longitudinal axis only for the selected flight conditions.
This will provide adequate simulation test data to perform a handling qualities
assessment across the entire flight envelope for each controller and investigate pilot
behavior while keeping required simulation time to a minimum. This is important
due to the nature of the project as both simulator and pilot evaluation time is donated
to the investigator by the parties involved.
The sum-of-sines task is a valuable and robust form of computer-generated
command that can provide valuable data used to extract information about the pilotvehicle interaction [26]. Though these measures of performance should not be
interpreted as a direct predictor of handling qualities in actual flight, they can provide
an accurate monitor of pilot workload and ensure that workload is satisfactory or
desirable.
The display for this maneuver will be a head-up display (HUD) as shown in
figure 51 of Appendix A. It is important to note that the signal displayed to the pilot
is the tracking error, not the command signal. In pitch, this means that display = Qe-9,
where 0C is the command from the sum of sine waves and 6\s airplane attitude,
making the task compensatory so that the pilot compensates for changes and does not
pursue an attitude command.
53
The form of the SOS command is as follows:
* c = £4sin(fl>,f + 4)
(7.1)
i=i
Based on previous experience with SOS tracking, some general guidelines for
the selection of the forcing functions include [26]:
1. Use of several discrete frequencies. At least six sine wave signals should
provide a broad range of commands. More offer improvements but are not necessary.
2. For pitch, the frequencies of the sine waves (<y,) should vary from around
0.2 to 0.4 rad/s up to 8 to 10 rad/s. Magnitude of the sine waves (A,) should be
reduced above about 1.5 to 2 rad/s, as needed to provide a bandwidth of roughly 1.5
rad/s.
3. Concentrate the sine waves below the bandwidth frequency of the total
signal. If only six discrete frequencies are used, one or two should be dedicated to
higher frequencies (but with reduced magnitudes as noted above). This provides a
high-frequency "gust-like" component that helps make the command look more
random to the pilot. If more than six sine waves are used, more high-frequency
components can be included.
4. Overall root mean square (RMS) magnitudes should be around 2.5°
attitude in pitch for simulated up-and-away operations.
5. Start some of the sine waves with negative amplitudes (or with nonzero
initial phase angles), such that the initial displayed error is not always large and in the
positive direction. Because one pilot is to fly the task many times, the signal can be
54
made to look random by adding an arbitrary phase shift ($•) to some of the sine waves
and varying this phase shift from run to run.
6. The entire run should be around one minute in length. Anything longer
may be fatiguing to the pilot; a shorter run may not provide sufficient exposure to the
airplane, even if the task is immediately repeated.
7. If pilot/vehicle data are to be extracted, care should be taken to assure that
the entire run of scored data includes at least one full cycle of each sine wave and that
there is an integer number of all sine waves.
8. A warm-up period at the start should be added. This period is several
seconds of non-scoring time during which the command is ramped in, allowing the
pilot to get into the control loop before formal data taking begins.
Recommended forcing functions are given in table 9, taken from reference
[18]. Figure 18 shows a time history of the pitch command signal for reference. The
task time for both command sets is 74.25 seconds, consisting of 10 seconds of warmup (non-scored time), 63 seconds of tracking for scoring, and 1.25 seconds of cooldown (non-scored time). All sine waves complete an integer number of cycles in 63
seconds, as listed in the table. Sine wave frequency, cot (in rad/sec), is computed
from the scoring time, ts, and number of cycles of each sine wave, Nt, as follows:
55
TABLE 9. SOS Command Signal for Pitch
Pitch Attitude
Sine Wave No.
At (deg)
No. Cycles
OJ\ (rad/s)
1
-1
2
0.19947
2
1
5
0.49867
3
1
9
0.8976
4
0.5
14
1.39626
5
-0.2
24
2.39359
6
0.2
42
4.18879
7
-0.08
90
8.97597
-
I
-
A
\
J11
M
1/\i
u
\
; W|
!
;
0
\ /
10
i
11
1
i
\
i
\
'1
A
i
f
k
M
v i
A
1
;
\
1
20
30
40
50
Time (sec)
FIGURE 18. Example pitch axis SOS command signal.
56
60
70
80
In the flight research conducted in [18], the pitch SOS command shown in
table 9 and figure 18 generated load factors between 0.5-1.6g. Load factors higher
than this value would require very large pilot inputs, while lower load factors may not
provide adequate information about the handling qualities of the aircraft.
The objectives of the SOS tracking task in this research are:
1. Evaluate handling qualities in a tight, closed-loop tracking task.
2. Evaluate feel system and control sensitivity characteristics.
3. Identify bobble or PIO tendencies.
Evaluation pilots will aggressively track the displayed signal and attempt to
keep errors within the specified tolerances. Desired and adequate performance limits
are expressed in terms of milliradians (mils) of tracking error on the HUD. The
desired performance criteria is ±10 mils in pitch 50% of the simulation time and the
adequate performance criterion ±20 mils in pitch 50% of the simulation time. These
requirements are based on [18] but adjusted to be double as the work documented in
[26] showed these limits to be too strict for pilots to meet.
Controller Assessment
Evaluation Procedure
The piloted simulation will serve as a handling qualities investigation of the
performance of three independently-designed controllers. The procedures for the test
will be as follows:
57
1. Evaluation pilots will be given a chance to familiarize with the simulator
through several practice runs. Each flight condition will be presented to the pilot for
familiarization with the task and vehicle behavior at that airspeed.
2. Formal evaluations will begin with the classical controller for the high
speed (400 kt.) trim case. The pilot will be asked to perform the evaluation task as
many times as necessary for this controller and flight condition before providing pilot
comments and ratings.
3. Cooper-Harper handling qualities ratings [19] and PIO tendency ratings
[19] [20] will be collected using the rating scales shown in Appendix B. Evaluation
pilots will be strongly encouraged to talk their way through the rating scale decision
trees as a means of extracting additional commentary.
4. Upon completion of the evaluation task and pilot ratings for the classically
designed controller, the scheduled Ho, controller will be flown and evaluated,
followed by the LPV controller for the 400 kt. trim case.
5. The evaluation procedure described above will be repeated for 250, 160,
and 110 kt. trim cases.
6. A detailed run log will be recorded along with all pilot comments and
ratings. Time history data will be saved in MATLAB format for future analysis.
Pilot Debrief Questionnaire
A pilot debrief questionnaire will be used to facilitate the collection of
qualitative pilot comments regarding the performance of the various control designs.
The debrief questionnaire is given in Appendix C.
58
CHAPTER 8
FLIGHT TEST RESULTS AND ANALYSIS
Overview
This section presents the analysis for the simulation test. The test was
performed between 23 July 2009 and 5 August 2009 on the STI fixed base simulator
in Hawthorne, CA. Two pilots participated in the test and gave handling qualities
(HQ) and pilot-induced oscillation (PIO) tendency ratings in addition to qualitative
comments regarding each configuration flown. A questionnaire was also presented
after each evaluation run to gather pilot opinion data regarding the performance and
controllability of each controller at each flight condition.
A background on the analysis methods utilized in the handling qualities study
will first be presented followed by comments and observations of the technique and
rating philosophies employed by the two pilots and their impact on the results of the
study. For the duration of the analysis, the data for both pilots is reported together
but their assessment of the controllers is treated as separate as the focus of the study
was not to compare pilots, but controllers. The assessments are compared at the
conclusion of the analysis to determine a cohesive conclusion regarding each
controller relative to the simulation as a whole.
59
The test analysis portion of the report begins with a summary of the HQ and
PIO ratings for each pilot at each flight condition and a listing of relevant pilot
comments. A tracking performance summary is also included. Questionnaire
answers for each pilot are graphically presented and commentary is provided on the
effect of pilot technique on the qualitative assessment of each controller. Time
histories are then examined and scalograms are used to compare pilot and vehicle
behavior with different controllers and correlations between pilot behavior, tracking
performance, and HQ/PIO ratings are observed. An analysis of the pilot-vehicle
system utilizing describing functions follows to provide further insight into pilot
behavior. The performance and pilot ratings of the aggressive tracking runs
evaluated by pilot 1 are then discussed and the effects of very aggressive pilot
technique are examined. Finally, conclusions are drawn and observations are made
which consider the performance of each controller relative to the others for both
pilots to give a final assessment of the goodness of each design.
The comparisons are made between the three controllers at each distinct flight
condition. As such, all plots and charts are presented for a single flight condition as
part of a family of four. Comparisons between flight conditions for the same or
different controllers are not a part of this investigation. The results of this study will
allow us to draw conclusions as to which controller provides good performance,
robust stability, and desirable vehicle behavior at each distinct flight condition in
terms of pilot opinion.
60
The findings of the simulation study are contrasted throughout with the
predicted handling qualities that resulted from the desktop analysis. Correlations and
anomalies are observed and explained and final conclusions drawn.
Analysis Methods
Wavelet Transform Analysis
The content of this section, adopted from [27], describes the background and
application of wavelet transform analysis techniques.
Fourier transforms have been the primary method of estimating frequency
responses from time history data for many years [21]. Wavelet transforms can be
used for a similar purpose with a particular application to time-varying systems.
Continuous wavelet transforms decompose the time function into basis functions
which translate in time. The basis function length and magnitude scale with
frequency according to the function:
Wt(o>,u)=£j{t)comf[c)(t-uj\dt
(8.1)
The transform W^(co,u) is interpreted as the frequency content of the input
f(t) in the neighborhood of the point (o),u) in the frequency-time plane. The
subscript <j> indicates that the transform is defined for the wavelet function (/>{t), or
"mother" wavelet. All of the basis functions <y12^* [«(/-w)] are derived from ${t)
where <f is the complex conjugate of the wavelet function. The multiplication by
co112 is included to ensure that all basis functions are of the same norm. The mother
wavelet can be real or complex, and in many wavelet applications it is defined as a
61
real function. But, for system identification where phase information is important,
complex exponentials are a natural choice and wavelets of the following form are
used:
t(t)=g{ty
(8.2)
The use of wavelet transforms in system identification and as an analysis tool
in detection of loss of control is documented in more detail in [22]. As an example,
consider the simple system g(s) shown in figure 19 and compute wavelet transforms
input and output time series u and y, respectively. Wu and W are the
corresponding input and output wavelet scalograms. The dynamics of the system can
be estimated using:
Sestet) = J
(8.3)
Wu{(o,t)
«(0
y(t)
g{s)
Wy{0),t)
Wu(a>,t)
FIGURE 19. Transfer function estimation using wavelet transforms.
Ideal conditions can be achieved in practice using sum-of-sine inputs, where
all of the input power is centered at distinct frequencies. Results using this technique
were reported in [23]. Changes in the system, particularly changes that affect
stability, can be estimated and used to analyze pilot and vehicle behavior. With
windowed Fourier transform methods the time window is inversely proportional to
62
the lowest frequency, T = 1/' fmin, while for wavelet transform methods the time
window varies with frequency according to the relation T = \l f for each of the
frequencies at which the transform is estimated. As illustrated in figure 20, the time
window is smaller at higher frequencies and is able to respond more rapidly to
changes at these frequencies.
Windowed Fourier Transform
Wavelet Transform
time
time
FIGURE 20. Comparison of frame sizes for windowed Fourier and wavelet
transforms.
The different frame sizes for the wavelet transform allow for transient
analysis, while the windowed Fourier transform is an average response for that frame.
To further illustrate these differences, the windowed Fourier transform and wavelet
transform of a simple time series input consisting of two sinusoids is shown in figure
21. Note that the power spectra of the windowed Fourier transform shows the two
distinct peaks associated with the input sinusoids. In the top left of figure 21b the
mother wavelet is the shifted Morlet [22]. The Morlet is probably the most popular
of the continuous wavelet transforms and is defined by setting the window function
g(t) to a Gaussian envelope. To make this wavelet causal, the wavelet function is
63
shifted to the right and the tails of the distribution are truncated. A Fourier transform
of the shifted Morlet with a center frequency of 1 rad/s is shown in the top right of
figure 21b. The wavelet scalogram, on the other hand, not only shows the peaks in
power, but also shows when in time the sinusoids occurred. It is this characteristic
that makes wavelets a powerful tool for detecting changes in time varying systems,
and it is this capability that will be exploited for identifying key components of the
PIO signature from the simulation test data set.
Shifted M&Un (1 rad/Mc scale)
y(l) - test signal (1 and 3 cadfcec sines)
TTTHnnnHo i
111! i
D
10
20
» «
£0
GO
~-3
-2
-1
0
R*0fecc
1
2
3
M l signal (1 and 3 r»d/eac sines)
£ffl iWi / V"7" ^mHHKT
30
!
25
20
i
:
i
SsJ
.
* . 6
10
5
>-.,An/J.
0.1
0-3
IIVMUUMM
L.
.
a) Windowed Fourier transform
b) Wavelet transform
FIGURE 21. Transforms of an example time series.
Outside of an experimental setup the conditions used for the estimation are
typically not ideal and some type of smoothing technique must be used. The proper
way to pose the estimation problem is to determine the variance of the estimate; an
estimate with a lower variance uses a smoothed estimate of the cross-spectral density
64
divided by a smoothed estimate of the input power spectral density (PSD). An
estimated function can be given by
where the tildes indicate smoothing, shown as an average over a frequency range, 5
is the smoothed cross scalogram between u, y and frequency co, and Suu is the
smoothed scalogram of the input u and frequency co. The smoothed cross
scalogram can be computed using:
~sa,P K*»0=—— £. F° M V Mda)
which is used to define the smoothing method, whereas Fa is the unsmoothed
scalogram and F*p is its conjugate of the unsmoothed scalogram.
This is the wavelet-based estimation technique used to analyze the simulator
test data. The tradeoff between the smoothing of data and the detection of changes is
always present as the more the signal is smoothed, the longer it will take to identify a
change in the system.
Compensatory Control Describing Functions
In this section the pilot-vehicle system measures for the selected control tasks
are described. MATLAB-based analysis algorithms that compute the pilot-vehicle
system metrics have been created and used for the analysis. The content of this
section is adopted from [24].
65
(8 5)
-
The crossover model is applied as a model used for assessment of pilots
performing precision tracking tasks. The crossover model is valid for single-loop
compensatory control, applicable to the current research which utilizes a precision
tracking task. A block diagram for the compensatory control scenario is shown in
figure 22, where Yp is the pilot describing function and Yc is the controlled element.
Here, the pilot controls the system output, m, in response to the displayed pilotvehicle system error, e.
System
Input +
FIGURE 22. Compensatory control scenario.
In short, the crossover model states that the pilot adjusts his/her characteristics
such that the pilot-vehicle system can be represented by the following open-loop
transfer characteristics:
Yp(je>)Yc{j") =
(o,e
)®
m
e
(8.6)
where coc is the crossover frequency, and re is the effective system time delay. The
key variables, <ycand re, are functions of the controlled element dynamics (airplane
model), mission task variables, and environment (system delays, field-of-view, etc.).
The crossover frequency is defined as the frequency on a Bode plot at which the
66
pilot-vehicle system open-loop describing function amplitude ratio crosses the 0 dB
line. It has been demonstrated through extensive research that those controlled
elements that are most "k/s-like" in the region of crossover require the least
compensation by the pilot. In turn, as pilot compensation increases, the achieved
crossover frequency decreases. The effective system time delay is a function of
fundamental pilot latencies, the high frequency flight control system and aircraft
dynamics (e.g., actuators, structural filters, structural modes, etc.), and added
incremental time delays due to pilot compensation. Once again, the more "k/s-like"
the controlled element is in the region of crossover, the less pilot compensation will
be required and the smaller the effective time delay. When little or no compensation
is required by the pilot and the higher frequency dynamics are negligible, the
effective time delay will consist solely of the delay in the pilot's response.
Pilot-Vehicle System Metrics
The key pilot-vehicle system parameters (<ycand te) underlie all of the related
performance metrics. The crossover frequency, for example, represents the closedloop system's bandwidth. It will be reflected by the closed-loop pilot-vehicle system
error and output power spectral densities (PSD). If closed-loop damping of the pilotvehicle system is small, thus indicating small phase margin, a>cWi\\ be close to the
peak frequency. The magnitude peak in the PSD will be at the crossover frequency if
the pilot-vehicle system is neutrally stable. The crossover frequency will not
correspond to a peak in the PSD for open-loop forced oscillations such as stick
pumping.
67
The closed-loop characteristics of the crossover model allow the system phase
margin to be defined in terms of the key model parameters as shown in the following
equation:
0M=^-Te<»c
(8-7)
When mission tasks are characterized by known forcing functions and
measured error signals (such as a sum-of-sines tracking task), key parameters such as
crossover frequency are measured directly from the pilot-vehicle system frequency
response Bode plot and do not require estimation [25]. Equation (8.7) is used as an
approximation for effective time delay that becomes more accurate as the actual pilotvehicle system performance approaches the ideal crossover model. Crossover
frequency and phase margin are measured directly.
Pilot Technique and Rating Philosophy
Two pilots participated in the gathering of test data. Pilot 1 is a handling
qualities engineer and has extensive experience conducting flight tests and simulation
studies. Pilot 2 holds a Ph.D. in Aerospace Engineering and is an active Navy reserve
pilot. The two pilots had distinct differences in technique as well as rating
philosophy which impacted study results. These are discussed in detail in the
following sections.
Pilot Technique
Pilot 1 approached sum-of-sines tracking as a task in which error was to be
minimized, while pilot 2 sought only to keep pitch position error within desired and
adequate error limits. As a result, pilot 1 was much more aggressive in his technique
68
and applied much more energy into his input, as shown in figure 23. In this figure,
pilot stick input for the entire task is shown accompanied by a time-varying
scalogram showing the input energy for a 30-second slice of each run. In the
scalograms shown in figure 23b and figure 23d, the red line denotes the most recent
estimate while the blue lines get increasingly light as time marches backward.
The most apparent distinction is the magnitude in which pilot 1 controls in
comparison to pilot 2. Pilot 1 is much more aggressive with his inputs, indicating an
attempt at tighter tracking and reflecting the mentality that the goal of the task is to
zero out any position errors. Pilot 2 uses much less input and changes input direction
much less abruptly, indicating his tendency to keep error within performance limits
and not to negate it completely.
An examination of the scalograms gives additional insight into pilot technique
and behavior in performance of the sum-of-sines task. Pilot 2 concentrates his inputs
in the 0.5-3 rad/s frequency range with a magnitude peak occurring at 1 rad/s. Pilot 1
has a similar magnitude peak at about 1.5 rad/s, but also has a much larger magnitude
peak at 8 rad/s. This correlates with the observations made previously in reference to
the time history data that pilot 1 is controlling much more aggressively with both
higher amplitude and frequency than pilot 2.
This difference in pilot technique is likely to become a factor in controller
assessment as one of the main observations from the desktop analysis was that
controller performance is very similar for higher airspeeds and lower airspeed flight
conditions in which elevator saturations were not encountered. Aggressive pilot
69
technique is likely to uncover these nonlinearities in the system and expose the pilot
to any weaknesses in controller robustness while passive pilot technique is not. It is
possible that the more passive pilot 2 may never encounter any elevator saturation
limits due to his lack of command energy and ratings for these flight conditions will
not reflect controller robustness to nonlinear phenomena. This does not invalidate the
ratings or test data, but rather helps form a conclusion on the effects of nonlinearities
on the performance of the controllers relative to each other.
16
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FIGURE 23. Time histories and scalograms for stick input, PID controller at 250 kt.
70
Rating Philosophy
As explained previously, pilot 1 is a handling qualities engineer wellexperienced in flight test and handling qualities research. He is very familiar with the
Cooper-Harper handling qualities ratings scale as well as the PIO tendency scale as
he has worked extensively with both throughout his career. In his pre-test briefing,
pilot 1 stated that the best rating any configuration can receive for the sum-of-sines
task is an HQR of 3 due to the inherent nature of the task. Detailed examination of
the Cooper-Harper handling qualities ratings shows that for an HQR of 3, minimal
pilot compensation is required to perform the given task as desired. An HQR of 2
states that pilot compensation is not a factor in achieving desired performance, which
for a compensatory task is contradictory. Pilot compensation will always be required
as the task is tailored for that purpose, so pilot compensation will always be a factor
even in a perfectly-handling aircraft. In this sense, an HQR of 3 is the handling
qualities rating a perfectly-performing system should receive under the CooperHarper rating system for the compensatory task under consideration.
Pilot 2 disagreed with the rating philosophy expressed by pilot 1. Pilot 2 was
of the opinion that a perfectly-performing aircraft should always receive a handling
qualities rating of 1 regardless of the task being performed. For this reason, pilot 2
rated well-performing controller-vehicle configurations Is and 2s while pilot 1 rated
well-performing configurations a 3.
This would have been an issue in a study attempting to solidify the opinions
of all pilots tested, but in the current research it is only pilot opinion and rating of
71
each controller relative to the others at each flight condition that is of interest. Pilot
opinion is subjective and though any scientific experiment must provide adequate
variance to arrive at a valid conclusion, the purpose of the current work is to compare
the performance and robustness of the three controllers against each other. Pilot
ratings will still be considered, but only in reference to handling qualities levels and
not absolute ratings when compared for the two pilots. Absolute ratings will still be
used to compare controllers, but for each pilot separately.
Handling Qualities Ratings and Comments
The pilot ratings for each of the 4 flight conditions considered as part of this
study are given in figure 24 to figure 27 and relevant pilot comments are shown in
table 10. Pilot 2 performed the tasks and gave ratings without offering comments, so
only comments given by pilot 1 are shown. The configurations were flown "blind"
and not in any particular order to keep the pilot from rating based on direct
comparison with the previous run.
Pilot Comments
The table lists the pilot comments beginning with the highest airspeed flight
condition flown. HQ and PIO ratings are included in the table for reference. Based
on pilot comments alone, it would appear that pilot 1 had difficulty distinguishing
between controllers at higher airspeeds but noticed significant differences at lower
airspeeds where elevator saturation is an issue for more aggressive pilot technique.
This corresponds to the handling qualities and PIO ratings given by pilot 1 for the
various configurations.
72
Somewhat surprisingly, the 250 kt. case was rated poorly in comparison to the
160 kt. case which possessed less bandwidth and more phase delay. As expected, the
110 kt. case received the worst ratings from pilot 1 as hitting the elevator position
limits affected his ability to perform the task. The exception for this flight condition
was the LPV controller, which received a level 1 handling qualities rating and
showed no tendency to induce pitch bobble or PIO. This controller showed
comparable performance at higher airspeeds but for those flight conditions in which
the other controllers showed unsatisfactory characteristics, it provided adequate
stability and performance and no tendency to induce undesirable motion.
In contrast, the Hoc controller received lower ratings than the other controllers
for almost all of the flight conditions considered. In the desktop analysis, a large
phase delay and low bandwidth predicted degraded handling qualities, and this would
appear to be the case when analyzing the pilot ratings and comments given in table
10.
Handling Qualities Ratings
Pitch tracking performance is presented as percent time versus HQR. For
desired performance, the pilot needed to track the pitch error bar to within 10
milliradians for 50% of the task time, while adequate performance was to track the
pitch error bar to within 20 milliradians for 50% of the time.
73
TABLE 10. Pilot Comments for Pitch Sum-of-Sines Tracking Evaluations
Configuration
HQR/PIOR
PTF> 400
1 /1
^ t a r t s e e m S a bit of a bobble or overshoot when I tighten
the loop. Predictable response. This is good.
LPV 400
3/1
Best I've flown so far. Don't see the bobble on target for
purely compensatory tracking. Very nice recovery to big
error.
Hoo400
3/2
No tendency to bobble or PIO, easy to control, the times I
over controlled I was able to bring it right back.
PID 250
4/2
Sluggish initial response. Difficulty getting error
corrected. Seemed like it took a lot of input to get it going.
LPV 250
3/1
Easy to track, pretty responsive and predictable, no
obvious overshoot or big delays. Could have used a little
more control power at times.
Hoo250
5/2
A little bit of a bobble when you put large inputs in.
Predictability and linearity are not that great.
PID 160
3/1
When the error bar moves I can respond, no real tendency
to overshoot or bobble. Good flying aircraft.
LPV 160
3/1
Nothing really objectionable when I went tighter for
aggressive tracking. One of the better ones I've flown.
HQO 160
3/1
Good response, a little bit of a standoff but I'm getting
desired performance. A little bit of a reversal when I put a
quick input in. It isn't perfect, but it isn't awful.
PID 110
5/3
A lot of bobble on target. Not a level 1 aircraft. I don't
like the bobble it has. But, if I stay on target and don't
have any big inputs it's not bad.
T PV 110
V1
* c a n P u t ^ m des>red eas ''y- Very easy to track. No
obvious adverse characteristics at all.
rM
Very reluctant to move. Unpredictable. Initial response is
not very predictable.
H
HQO
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b/:S
Comments
74
In figure 24 through figure 27, red markers indicate ratings and performance
for pilot 1 while blue markers indicate ratings and performance for pilot 2. The
legend distinguishes between controllers and pilots through symbol names according
to the following convention: Pl-PID controller/pilot 1, P2-PID controller/pilot 2,
Ll-LPV controller/pilot 1, L2-LPV controller/pilot 2, Hl-Hoo controller/pilot 1, H2Hoo controller/pilot 2. The task performance plots show both desired and adequate
performance, with desired performance shown as a closed symbol and adequate
performance shown as the open version of the same symbol.
It is desirable to have those configurations which achieved desired
performance in the upper left corner of the task performance plots as this corresponds
to level 1 handling qualities. However, because there are many factors other than
performance that come into play, this is not always the case. For the cases considered
in the current study, desired performance was commonly achievable for moderately
deficient configurations through pilot compensation. This leads to the conclusion that
those configurations rated level 2 which still achieved desired performance did so due
to an increase in pilot compensation and workload. This conclusion will be enforced
by the pilot comments and responses to the questionnaire for the given configurations
and further expanded upon through the scalogram and pilot-vehicle describing
function analysis.
Focusing on the 110 kt. flight condition shown in figure 27, pilot 2 rated all of
the configurations as level 1 and had no problems achieving desired performance
with all of the controllers. Pilot 1, however, gave level 2 ratings and had more
75
difficulty achieving desired performance, though he was still able to do so through
pilot compensation. This is evidenced by the observation that HQR and PIOR
dropped to level 2 while performance remained largely the same, indicating that pilot
compensation and increased workload are to blame for the lower HQ ratings. This
hypothesis will also be explored in further detail through the describing function
analysis.
The differences in pilot opinion for this flight condition can once again be
attributed to pilot technique. Whereas pilot 1 routinely encountered the elevator
saturation limits at 110 kt., pilot 2 did not encounter limits during any of the
evaluations. Figure 28 gives pitch attitude time histories for both pilots in their
evaluation of the PID controller at 110 kt. In addition to the actual aircraft attitude,
the command signal is shown for comparison. In the figure, regions of saturation are
shown highlighted in yellow. Figure 28a shows that pilot 1 spends a considerable
amount of task time on the lower saturation limit while figure 28b shows that pilot 2
does not encounter a saturation limit at all during his evaluation. Examination of the
task performance data shows that pilot 2 was able to achieve desired performance for
65% of the task while pilot 1 was only able to achieve desired performance for 57.2%
of the total task time. This data supports the conclusion that encountering the
elevator saturation limit has an effect on task performance and the previously
discussed conclusion that aggressive pilot technique is tied to this phenomenon. This
is also reflected in the ratings given by both pilots in figure 27.
76
Pilot Questionnaire
Following each evaluation run, both pilots were asked to complete a
questionnaire aimed at enhancing the tracking performance data and pilot ratings with
an assessment of specific characteristics of each controller. The questionnaire was
intended to be brief and was developed based on observations made during the
desktop analysis.
The desktop analysis showed that for higher airspeeds, all of the controllers
exhibited similar behavior and had very closely matched performance metrics. This
led to the prediction that for these flight conditions, all 3 controllers would have
similar handling qualities. At lower airspeeds, however, each controller exhibited
very different characteristics and it was determined that additional data and
information would be beneficial to the assessment of the goodness of each controller.
Of particular interest was the 110 kt. case in which elevator saturation was
expected and the robust controllers showed a considerable bandwidth and phase delay
disadvantage. The PID controller had much better short-term response in the desktop
simulation, but had a significant amount of overshoot that the robust controllers did
not. The questionnaire was developed to address these specific areas of interest with
the knowledge that at higher airspeed flight conditions there was likely to be very
little variance between the controllers.
77
Pitch Sum-of-Sines Tracking Task - 400 kt.
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FIGURE 24. Pilot ratings and tracking performance, 400 kt. flight condition.
78
•
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FIGURE 25. Pilot ratings and tracking performance, 250 kt. flight condition.
79
Pitch Sum-of-Sines Tracking Task -160 kt.
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FIGURE 26. Pilot ratings and tracking performance, 160 kt. flight condition.
80
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FIGURE 27. Pilot ratings and tracking performance, 110 kt. flight condition.
81
70
80
a) Pilot 1-Lower Limit: 17.57% of Task Time
b) Pilot 2-No Saturation Encountered
FIGURE 28. Elevator saturation encountered during sum-of-sines tracking, PID
controller at HOkt.
Pilots were asked to give their response to the 4 questions given in Appendix
C based on a 1-5 scale. The first question intended to address the presence of the
saturation nonlinearity and its impact on system predictability. A system which is not
impacted by the elevator position limit will seem linear and predictable to the pilot,
82
while one in which elevator saturation heavily influences vehicle response will be
highly unpredictable. The second question intended to assess the lack of bandwidth
for some of the controllers in terms of pilot opinion of the rapidity of the short term
response. The third question assesses the impact and noticeability of command
overshoot, and the last question provides insight into the aircraft's tendency to induce
pitch bobble, PIO, or undesirable motions.
Figure 29 and figure 30 show the pilot responses to the predictability
questionnaire for all flight conditions. Grouped within each flight condition are the
responses for each controller so that direct comparisons can be made. Pilot 2 saw all
of the configurations as linear and predictable with slightly degraded marks for the
LPV and Ha, controllers at 110 kt. and the PID and Ho, controllers at 250 kt. Pilot 1
had a similar opinion of the predictability of the PID and FL controllers at 250 kt.,
but gave the FL controller especially poor marks at 110 kt. because of its inability to
effectively handle encounters with the elevator position limit. For this controller,
even brief encounters with the elevator position limit resulted in large tracking errors
that were difficult to recover from. Pilot 1 rated the LPV controller to be predictable
at all four flight conditions.
The bandwidth and phase delay deficiency observed during the desktop
analysis for the robust control designs shows itself in the pilot ratings for system
rapidity, given in figure 31 and figure 32. Again the 250 kt. case is less than ideal in
terms of pilot opinion of the rapidity of the vehicle response. The low bandwidth and
large phase delay of the Hoo controller at 110 kt., however, received the lowest marks
83
as it could not keep up with pilot input and also suffered from large departures due to
elevator saturation.
The tendency of the system to induce a pitch bobble or PIO was the final
question included but provided the most interesting results. As would be expected, a
direct correlation between PIO tendency ratings and PIO questionnaire responses
exists. In addition, those cases which were rated poorly for overshoot also rated
poorly in PIO tendency. Pilot 2 did not see any kind of bobble or PIO in his
evaluations as his more passive technique did not excite the elevator enough to
encounter saturation.
The conclusion to be drawn from the questionnaire responses given by both
pilots is that a large amount of overshoot, coupled with a slow response and the
presence of nonlinearities, leads to a PlO-prone aircraft when performing precision
tracking tasks.
Time Histories and Scalograms
Time-varying scalograms can give insight into the behavior of a pilot-vehicle
system as a function of time. Scalograms can also be used to determine pilot and
actuator workload for a given task and can help to explain a drop in ratings when a
corresponding drop in task performance is not observed. For those runs which the
pilot must increase his compensation or when the augmentation system must put forth
extra effort to maintain stability and performance, the corresponding signal energy
will be much larger than for those which require less pilot compensation and control
system workload.
84
Not Predictable
5
Very Predictable
4 0 0 kt.
2 5 0 kt.
160 kt.
110 kt.
Flight Condition
FIGURE 29. Predictability questionnaire responses, pilot 1.
Not Predictable
5
Very Predictable
4 0 0 kt.
250 kt.
160 kt.
Flight Condition
FIGURE 30. Predictability questionnaire responses, pilot 2.
85
110 kt.
Very Sufficient
400 kt.
250 kt.
160 kt.
110 kt.
Flight Condition
FIGURE 31. Rapidity questionnaire responses, pilot 1.
Very Sufficient
400 kt.
250 kt.
160 kt.
Flight Condition
FIGURE 32. Rapidity questionnaire responses, pilot 2.
86
110 kt.
Excessive
Negligible
4 0 0 kt.
250 kt.
160 kt.
110 kt.
Flight Condition
FIGURE 33. Overshoot questionnaire responses, pilot 1.
Negligible
4 0 0 kt.
250 kt.
160 kt.
Flight Condition
FIGURE 34. Overshoot questionnaire responses, pilot 2.
87
110 kt.
Highly Likely
5
No Tendency
4 0 0 kt.
250 kt.
160 kt.
110 kt.
Flight Condition
FIGURE 35. PlO/bobble questionnaire responses, pilot 1.
Highly Likely
5
No Tendency
4 0 0 kt.
250 kt.
160 kt.
Flight Condition
FIGURE 36. PlO/bobble questionnaire responses, pilot 2.
88
110 kt.
In the example shown in figure 37, elevator deflection for the three controllers
at 110 kt. is shown along with a 3-dimensional mesh scalogram for the 3 runs flown
for pilot 1. It can clearly be seen that those controllers which received level 2 ratings
have a considerable amount of additional elevator activity relative to the case which
received level 1 ratings. In addition, examination of the elevator time histories show
several "flat spots" indicating elevator saturation for the level 2 cases whereas the
level 1 case has only brief encounters with the position limit.
Though the correlation between degraded handling qualities and increased
elevator activity is clearly demonstrated in figure 37, other factors are also to blame
for the drop in ratings. The desktop analysis for this flight condition showed a large
amount of phase delay and a deficient bandwidth for the H*, controller, a problem
which the PID controller did not have. The analysis predicted that though the PID
controller possessed a large amount of overshoot relative to the robust controllers,
pilots would prefer this to a lack of short-term response due to a lack of bandwidth
and an excessive amount of phase delay. This is confirmed with this data as despite
the high pilot workload and large amount of elevator energy required to perform the
task, the ratings were higher for the PID controller when compared with the Hoo
controller.
Time histories and scalograms for the same configuration flown by pilot 2 are
shown in figure 38. The ratings for his runs are much more closely correlated and all
are level 1, and an examination of the elevator energy shows a similar magnitude for
all 3 runs, all of which are much less than the energy shown in figure 37 for pilot 1.
89
The conclusion to be drawn from this observation is that even though the
configurations are the same, the aggressiveness of the pilot is what will expose the
deficiencies in the aircraft. For passive pilot technique with the tracking task used for
this study, all of the given controllers exhibit similar characteristics and thus receive
similar ratings.
If a similar analysis is applied to the 400 kt. case, the conclusion that elevator
activity directly correlates to handling qualities ratings is affirmed. Pilot 1 rated all 3
controllers as level 1, or satisfactory without improvement, for this flight condition.
The elevator activity shown for these is correspondingly low relative to that observed
for the 110 kt. case and all 3 controllers exhibit similar elevator energy for the task.
The passive technique used by pilot 2 did not show any significant differences from
the 110 kt. case when compared with the 400 kt. case other than absolute magnitude
and dominant frequency. For the 400 kt. case, both pilots controlled predominantly
around 1 rad/s while pilot 1 also has significant energy in the elevator signal at about
6 rad/s. This is attributable to pilot technique and the similar behavior exhibited by
the elevator at higher airspeeds corresponds with both pilots giving level 1 handling
qualities ratings for those configurations. The time histories and scalograms for
pilots 1 and 2 are shown in figure 37 through figure 40.
Pilot-Vehicle Describing Functions
With a known input forcing function, the open and closed-loop transfer
functions for the system defined in figure 22 can be directly computed. Thus, the
pilot and pilot-vehicle describing functions can be derived from the tracking task data
90
and are of particular importance as they can be used to directly compare pilot
behavior for the 3 controllers tested and determine if increased pilot compensation or
workload is a factor in the degradation of ratings.
In figure 41 and figure 42, time histories are given for the 4 signals identified
in figure 22: i-6c, e-0e, c-5ion , and m-6. Figure 41 gives the pilot and pilot-vehicle
describing function Bode plots for pilot 2's evaluation of the PID and Ho, controllers
at 250 kt. The pilot-vehicle system metrics are summarized in table 11 and indicate a
higher level of pilot compensation for the Ha, controller. The crossover model states
that the crossover of the pilot-vehicle system (PVS) decreases with increased pilot
compensation, indicated in the data shown in table 11 and shown graphically in figure
41c.
In addition, there is additional effective delay present and a decreased amount
of phase margin, both of which also indicate an increase in pilot compensation for
this configuration. The pilot describing function shown in figure 41d also leads to
this conclusion, as the pilot is controlling with an additional amount of phase lead in
an attempt to recover phase margin and to provide the additional compensation
required for desired task performance. Though both configurations showed
comparable performance, the Hoc controller likely received lower ratings due to the
higher pilot workload required.
91
a) PID control, HQR 5/PIOR 3
b) LPV control, HQR 3/PIOR 1
c) H„o control, HQR 6/PIOR 3
FIGURE 37. Time histories and scalograms for elevator deflection, 110 kt. flight
condition (pilot 1).
92
1
I
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c) Hoo control, HQR 2/PIOR 1
FIGURE 38. Time histories and scalograms for elevator deflection, 110 kt. flight
condition (pilot 2).
93
B r a b Ooflacfon (desrWft;
!•
a) PID control, HQR 3/PIOR 1
1
1
(1
1
1
1
1
10
illP ill
1
1
1
'111 Ji ' iTil '
. i
1
-
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20
b) LPV control, HQR 3/PIOR 1
c) Hoo control, HQR 3/PIOR 2
FIGURE 39. Time histories and scalograms for elevator deflection, 400 kt. flight
condition (pilot 1).
94
Elevator Detection (deg^/ratfs
a) PID control, HQR 2/PIOR 1
1
1
1
1
1
1
i
i
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b) LPV control,
f
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c) H„o control, HQR 1/PIOR 1
FIGURE 40. Time histories and scalograms for elevator deflection, 400 kt. flight
condition (pilot 2).
95
The 110 kt. case shown in figure 42 and summarized in table 12 gives a
similar characterization. Pilot-vehicle system crossover is considerably lower for the
LPV controller, which also possesses a large amount of effective delay when
compared to the PID controller for the same airspeed. Though the pilot describing
function for both controllers is shown to be similar at mid to high frequencies, when
flying the LPV controller at low frequencies the pilot is controlling with much less
lead compensation, likely due to a combination of pilot 2's low frequency control
technique and the substantial amount of compensation required for this control
configuration. With the PID controller, the pilot was able to control with more low
frequency lead compensation, a higher crossover, and less additional delay than the
LPV controller, resulting in better handling qualities ratings.
Both cases show a consistent, low gain pilot who attempts to compensate for a
low bandwidth or lag-ridden controller with additional phase lead and lower
crossover. The evidence presented in figure 41 and figure 42 and table 11 and table
12 clearly shows the reason for the degraded handling qualities ratings given to
configurations that were still able to achieve desired performance. The pitch sum-ofsines tracking task is such that even with a "level 2" aircraft adequate or even desired
performance is achievable through pilot compensation; for those controllers with
handling qualities deficiencies, performance can be sustained through lower
crossover and phase margin to accommodate additional pilot lead compensation.
96
Aggressive Tracking
Pilot 1 participated in additional evaluation runs in which he attempted to
force tracking error to zero through very aggressive technique. This served to not
only provide additional tracking data, but also to allow the pilot to give ratings and
comments for each system as applied to very tight pilot control. Though it would
provide a great amount of insight into the behavior of each system under what can be
considered "high gain" control, this data would not be considered in the evaluation of
each controller as it was not part of the original flight test experiment. Pilot 2 did not
participate in aggressive tracking evaluations.
Pilot Performance and Ratings
Because the pilot was not performing a formal evaluation, achieving desired
performance was secondary to qualitatively assessing controller behavior for very
aggressive pilot technique. This is reflected in table 14, which gives tracking
performance for compensatory and aggressive pilot technique at 250 kt.
Performance was roughly 10% lower for both desired and adequate as the
pilot flew much more aggressively and at times purposely allowed error to build in
order to test the response to very abrupt, high amplitude inputs. The primary goal of
the aggressive tracking runs was to determine the handling qualities and PIO
tendency ratings for the controllers in highly aggressive tracking as they related to the
HQR and PIOR given for purely compensatory tracking. A summary of the ratings
for both types of pilot technique are given in figure 43 and figure 44.
97
n
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20
30
40
60
70
b) Hoo-HQR 3/PIOR 2
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Freq (raoVs)
Freq {rad/s)
c) PVS Describing Function (6/9t)
d) Pilot Describing Function {dior/9e)
FIGURE 41. PVS describing function variation amongst controllers, pilot 2/250 kt.
TABLE 11. Pilot-Vehicle System Metrics, Pilot 2/250 kt.
„ x „
Controller
Crossover
, ,, .
(rad/s)
Effective Delay
, ,
(sec)
Phase Margin
,, . °
(deg)
PID
1.93
0.406
45.14
H«
1.79
0.483
40.55
98
1
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80
a) PID-HQR 2/PIOR 1
2
0
3
0
4
0
TlmafsaC)
5
0
8
0
7
0
8
b) LPV-HQR 3/PIOR 2
Freq (ractfs)
Freq (rad/s)
d) Pilot Describing Function (<W#e)
c) PVS Describing Function (O/0e)
FIGURE 42. PVS describing function variation amongst controllers, pilot 2/110 kt.
TABLE 12. Pilot-Vehicle System Metrics, Pilot 2/110 kt.
Controller
Crossover
(rad/s)
Effective Delay
(sec)
Phase Margin
(deg)
PID
2.05
0.334
50.75
LPV
1.61
0.520
41.96
99
0
For the 250 kt. configurations, pilot 1 did not change the ratings given to the
PID or LPV controllers for compensatory tracking, but did change the Hoc controller
ratings to reflect the additional deficiencies uncovered by aggressive tracking. A
drop of 2 ratings takes the aircraft from level 2 handling qualities into level 3 and is
characterized as a change from moderately objectionable deficiencies which require
considerable pilot compensation for adequate performance to major deficiencies
which make adequate performance not attainable with tolerable pilot compensation.
This is a major drop in handling qualities for aggressive tracking and it should be
noted that aggressive tracking or other high gain tasks are not uncommon for the F-16
aircraft considered in this study.
The 110 kt. case, of interest due to its exposure to elevator saturation, shows a
more profound change in pilot assessment for aggressive control. Both the PID and
Hoo controllers experienced divergent PIOs where the pilot had to completely back out
of the loop to regain control. This is reinforced by the comments made in response to
the aggressive tracking evaluations, presented in table 14. The LPV controller fared
much better, being degraded from level 1 to level 2 but not experiencing any
divergent PIOs for aggressive maneuvers.
Pilot Behavior and Technique
As previously discussed, pilot 1 was inherently more aggressive than pilot 2,
even for the compensatory tracking tasks. The aggressive evaluation runs were made
with pilot 1 attempting to close the loop at very high bandwidth and thus showed high
pilot-vehicle crossover and low phase margin, shown in figure 45. In this example,
100
the pilot pushes crossover to nearly 2 rad/s while sacrificing a substantial amount of
phase delay in order to pursue zero tracking error. A direct comparison with the
compensatory tracking run is shown in figure 46.
An examination of stick input scalogram prior to the divergent PIOs
experienced at 110 kt. also provides a good measure of the level of aggressiveness
shown in these runs. Shown in figure 47 are time histories and input scalograms for
the compensatory tracking run as well as the aggressive tracking run for the Hoo
controller which resulted in a divergent PIO.
The LPV controller performed best for aggressive tracking in the presence of
nonlinearities, suffering degraded handling qualities ratings but maintaining stability.
No divergent PIO was observed and though the aircraft suffered deficiencies for the
aggressive tracking task, they were only moderately objectionable and could be
overcome through pilot compensation. This further asserts its superior robustness
against nonlinearities at low speeds when compared with the other controllers, even
for very aggressive maneuvers.
101
Pitch Sum-of-Sines Tracking Task - 250 kt.
P1 • L1 A H1 • P1* • L1* • H1*
1
2
3
4
5
6
7
8
Handling Qualities Rating
9
10
'Aggressive run ratings
FIGURE 43. Pilot rating comparison for aggressive pilot technique, 250 kt.
Pitch Sum-of-Sines Tracking Task -110 kt.
• P1 • L1 A H1 • P1* • L1* • H1*
6
5
Tendency R
1
4
4
3
Old
•
A
*
i
2
1
i
•
2
3
I
i
4
5
6
7
8
Handling Qualities Rating
i
i
9
10
'Aggressive run ratings
FIGURE 44. Pilot rating comparison for aggressive pilot technique, 110 kt.
102
TABLE 13. Pilot Comments for Pitch Sum-of-Sines Tracking Evaluations
Configuration
HQR/PIOR
PID 400
3/1
Start seeing a bit of a bobble or overshoot when I tighten
the loop. Predictable response. This is good. Overall, a
very good aircraft.
LPV 400
3/1
I'm being quite aggressive but I'm getting good
performance. Reasonably predictable. A little bit of a
slow initial response but once it got started it was nice.
„
Comments
Tiny change with aggressiveness. Almost a
hesitation/standoff when I put an aggressive input in. No
tendency to bobble or PIO, easy to control, the times I over
controlled I was able to bring it right back.
A™
o//}
PTP> 9S0
A/0
T PV O^n
1/1
^ e " behaved aircraft, just at level 1 performance.
Response still predictable and linear, even for large inputs.
Hoo 250
7/2
I can't get the error to zero because I overshoot.
Unpleasant, almost worse than a 5 for aggressive flying.
Not able to track it very well at all.
PID 160
3/1
No sign of any divergent oscillations, good response.
Very predictable, very linear. Nothing like some of the
others we've had that were easily divergent or ugly.
1/1
' c a n P u t '* o n t a r S e t a n d it pretty much stays on target,
some of the others got out of phase with me really fast.
00
T PV 1
TT i^n
fift
1/9
00
PTn 11 n
1 f>/^
T P V 11 n
s/1
Hoo 110
10/5
I can see a small initial delay with large inputs. Causes me
^ t t ' e m o r e hunting for the target for aggressive
tracking. When I push, it isn't enough. I have to put 2
inputs in to get it going.
to nave a
A little over control when I tighten up. A little bit of a
bobble or oscillation on target. No divergent PIO
tendencies like some of the other ones. For this level of
aggressiveness, it isn't level 1.
Large input reversal. Big delay in the initial response.
Easy to get into a PIO. Predictability problem.
Additional delay for sure, a little bit of an overshoot as a
result. Not level 1 for aggressive maneuvers.
Bobble on target as I try to be aggressive. The others
wouldn't do that. It's out of phase with me. It isn't doing
what I want it to do. Big divergent PIO.
103
TABLE 14. Comparison of Task Performance for Aggressive Tracking
riif>
Controller
Task
Adequate (%)
Desired (%)
250
PID
Compensatory
90.97
57.05
250
PID
Aggressive
82.93
52.65
250
LPV
Compensatory
92.36
60.47
250
LPV
Aggressive
84.32
50.72
250
Hoo
Compensatory
83.74
52.61
250
Hoc
Aggressive
69.99
37.06
104
1
1
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1
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1 1
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1
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o PM=S.S6,Te=0.56 . 1
1
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1 1 1 1
m n
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1
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11
~*~^~*
1 1
I 1
1 1
1 1
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r n n r
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r m r
1 1 1 1
i
i x
i
i
i I X
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i
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i P Yj
, , ,- , ,r
FIGURE 45. PVS frequency response for H» controller at 250 kt. (aggressive).
I
I
I I I I I I I
"K^ i
i i i i i i i
t -xil
ir-r-i~r-rr
j__
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i i
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-270
-360
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i
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i i
r?
FIGURE 46. Comparison of compensatory and aggressive pilot technique, Hoo/250
kt.
105
l
J
0
I
1
1
[
1
1
1
10
20
X
1
1
1-
1
L
_1
1
1
1
1
1
1
1
1
1
l_
1
1
I
a) Compensatory Tracking Run
b) Scalogram
c) Aggressive Tracking Run
d) Scalogram
FIGURE 47. Time history and scalogram comparison for compensatory and
aggressive tracking runs.
106
;—-.--/K
¥I0(^
40
1
1
1
i
I
i
L^
r r
:..... ,-U__^_A—-
20
0
-20
-40
—-<---
V.;J._:LJ:.
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i
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I
10
20
30
40
Time (sec)
50
60
70
80
FIGURE 48. Divergent PIO encountered during aggressive tracking, Hoo controller at
HOkt.
The conclusion that can be drawn from this data is an extension of that
reached previously: for compensatory tracking, higher airspeed flight conditions
show no differences in handling qualities between controllers. It is at lower airspeeds
that the controllers show distinguishable characteristics and the more aggressive the
pilot technique, the more likely a pilot will encounter system nonlinearities and
uncover the deficiencies in the system. For very aggressive pilot technique, this
problem presents itself as a complete loss of control such as the one shown in figure
48 for the Hm controller.
107
CHAPTER 9
SUMMARY AND CONCLUSIONS
Project Summary
This project sought to address the question of which control design methods
are best suited to a high performance aircraft application. Traditionally, aircraft have
been augmented with classically-designed controllers which have been exhaustively
studied and analyzed, but in recent decades the emergence of highly capable,
technologically-advanced aircraft has spawned a new area of research that seeks to
apply modern and robust control methods to flight control design. Though these
designs have shown undeniable results in other applications, their application to flight
control has been largely unstudied in the application sense; several paper studies exist
but very few extend the results to include pilot opinion. Qualitative assessment of
these designs is ultimately the best tool for measuring goodness and a comparison
between these new systems and the classically-designed systems they are intending to
replace is the only way to arrive at a definitive answer.
The work presented herein sought to tie all of the important aspects of flight
control design and handling qualities into one cohesive study. The design of a
classical controller was first presented and its robustness and performance were
studied. Next, two types of robust controllers, Hoc and LPV, were designed and
108
analyzed, and the results of the analysis compared using traditional handling qualities
metrics and desktop simulation. A hypothesis was then formed seeking to predict the
results of a piloted simulation study based on the findings of the desktop analysis.
A two-pilot simulation study was conducted using the fixed base simulator at
Systems Technology, Inc. in Hawthorne, CA. The simulator features a high fidelity
graphics rendering, heads up display with task bar and performance indicators, and a
McFadden hydraulic feel system used to simulate the feel system dynamics of a
modern fighter aircraft. Pilots flew 12 total configurations, 4 flight conditions and 3
controllers at each flight condition, and gave handling qualities ratings based on the
Cooper-Harper and PIO tendency rating scales. A questionnaire seeking to address
specific areas of interest was filled out following each evaluation run and the data
tabulated.
The performance, rating, and questionnaire data was then used to conduct a
full handling qualities study comparing the performance and robustness of each
controller at each flight condition. Wavelet techniques were used in addition to pilotvehicle describing function analyses to support the conclusions drawn from the
qualitative pilot ratings, comments, and questionnaire responses. The predictions
made at the conclusion of the desktop analysis were then revisited to provide a
comparison of predicted and rated handling qualities.
The research now concludes with the selection of the controller which is best
suited for application to a modern, fighter-type aircraft.
109
Observations and Conclusions
The results of the piloted simulation study clearly show that for higher
airspeed flight conditions, there is no noticeable difference in the 3 controllers tested.
This is in accordance with the findings of the desktop analysis and simulation, as it
predicted similar flying qualities for the 400 kt. flight condition. It was found in the
preliminary desktop analysis that closed-loop bandwidth and phase delay steadily
degraded with airspeed and trim angle of attack for all controllers, but that the
classically-designed controller held an advantage at very low airspeeds due to its lack
of phase delay and superior bandwidth. It was predicted that pilot opinion would be
favorable for this controller despite a significant amount of overshoot, but this was
refuted in that pilots rated the overshoot unfavorably when asked to specifically
address the issue.
The piloted simulation study proved that in the presence of nonlinearities such
as elevator saturation, the LPV controller was superior to both the classicallydesigned and Hoo controllers. The classical controller tended to have excessive
overshoot and a general lack of predictability and the FL, controller was deficient in
rapidity, leading to pilot overcontrol in an attempt to salvage desired performance.
The LPV controller, however, was well suited for this flight condition and was able to
both avoid encounters with the elevator position limit and reject the negative effects
when saturation was unavoidable. The LPV controller received level 1 ratings for the
compensatory task and was rated highly by both pilots because it allowed them to
stay in the loop more aggressively to ensure desired performance.
110
It was found that the deficiencies seen by the more aggressive pilot 1 were not
an issue with the more passive pilot 2, proving that all of the controllers designed
provided adequate performance and stability for normal tasks and pilot technique. It
was the high gain approach to the task that pilot 1 took that uncovered weaknesses in
the designs at low airspeeds.
For aggressive tracking, the LPV controller again was superior to the other
control designs considered. It was the only controller able to avoid a divergent PIO
and maintain adequate performance for aggressive pilot technique. There were no
changes to the assessment of the controllers at higher airspeeds. When considering
task performance, robustness to nonlinearities, and the ability to accommodate very
aggressive pilot technique, it is concluded that the LPV control design technique is
best suited for the aircraft and task considered in the current work.
Ill
APPENDICES
112
APPENDIX A
SIMULATOR DESCRIPTION
113
The material presented in this section has been adapted from [26].
System Hardware
The key hardware system components to the STI pilot-in-the-loop flight
simulator are shown in figure 49.
Pilot
McFadden
Feel System
Simulator
Computer
Projected
Display
FIGURE 49. Pilot-in-the-loop simulator elements.
Simulator Computer and Projected Display
The simulator shown in figure 50 is a Win32 console application designed to
interface with MATLAB for data input and output. A dynamically linked library (dll)
was developed to simulate the linear equations of motion for the F-16 aircraft based
on the model provided by Dr. Eugene A. Morelli which accepts stick displacements
as inputs and sends vehicle rates and attitudes as outputs.
114
FIGURE 50. STI simulator with McFadden control loader and projected display.
The current PC-based simulator features:
1. Linear airframe equations of motion.
2. Software rate limits, surface position limits, and actuator models.
3. Data recording of unlimited output states.
4. Vehicle dynamics that update at 120 Hz. and graphics that update at a
minimum of 30 Hz.
5. Texture-mapped PC graphics with a superimposed head-up display that
supports pitch and roll axis tracking tasks.
The pilot is presented a HUD superimposed on a suitable 3D environment
(see figure 51). At the end of each run, the input vectors and output time responses of
all simulated systems are saved to a MATLAB file for future analysis. The simulator
uses tracking bars on the HUD to support sum-of-sines tracking tasks which provide
the high gain environment that is required to induce unfavorable pilot-vehicle
interactions such as PIO and expose any deficiencies in control system design or
robustness.
115
FIGURE 51. Projected head-up display.
McFadden Control Loader
A McFadden Series 292A 2-axis (pitch and roll) fighter stick control loader
shown in figure 52 has been integrated with the STI PC-based flight simulator. The
McFadden feel system provides proprioceptive cues to the pilot and can be
configured to provide feel system dynamics appropriate for the study. It can provide
a wide range of control force characteristics that are encountered in real flight such as
linear and nonlinear spring gradients, damping, deadband, breakout, coulomb friction,
and deflection limits.
116
FIGURE 52. McFadden series 292A 2-axis fighter stick.
The stick characteristics listed above are set via an electronics control box
(see figure 53a). Descriptions of the many control box options are provided below.
The outputs from the control box to the loader and computer are shown in figure 53b,
while the signal transfer box (from the control box to the computer Analog/Digital
card) and force feedback (on/off) switch are shown in figure 53c and figure 53d,
respectively. Calibration tests to verify the force/volt and position/volt values
provided by McFadden Services were undertaken. The results of these tests are
provided in Appendix A of [26].
117
a) Pitch and roll axis electronics control box
b) Pitch and roll outputs to control loader & computer
c) Signal transfer from control box to A/D card
d) Control loader force feedback switch
FIGURE 53. McFadden electronic control box and related wiring.
Flight Control System
In this research, controller evaluations were conducted in the pitch axis only.
Lateral and directional characteristics were not considered for evaluation.
Feel System
The feel system components are shown in figure 54 and include an optional
time delay designed to degrade pilot-vehicle system performance in order to induce
phenomena that are typically hard to replicate in ground based simulation such as
pilot-induced oscillation (PIO). For the current work, as shown in figure 54, the stick
time delay is set to zero for all flight conditions and configurations.
118
Also shown in table 15 are the feel system characteristics used in the
simulation. These values were chosen to match the F/A-18 ground based simulator at
NASA DFRC using data supplied by NASA, as the McFadden stick was designed to
emulate that of the F/A-l 8 and it is generally regarded by pilots to be favorable. The
F-16 VISTA, also an agile fighter-type aircraft, is unique in that it has a position
sensing center stick rather than the small displacement side stick typical to the
standard F-16. The similarity in aircraft type and presence of the position sensing
center stick allows for the substitution of the F/A-18 stick parameters in light of
hardware availability.
Despite the F-16 stick having a semi-parabolic feel system gradient, we can
use a linear stick gradient and linear control system gearing because the nature of the
sum-of-sines task is such that we are conducting flight experiments about an
operating point. That is, all maneuvers will be contained within a small deflection of
the zero deflection point where stick gradient and control system gearing is
essentially linear.
Stick
Force
F,
Stick
Position
Feel
System
s,
Rate
Command
Time
Delay
s
delay
w
Control
System
Gearing
s
1c
delay
FIGURE 54. Feel system elements.
119
TABLE 15. Baseline Feel System Characteristics
Parameter
Pitch Axis
Spring Gradient (lb/in)
7.00
Damping (lb-sec/in)
0.237
Natural Frequency (rad/sec)
26.46
Damping Ratio
0.448
Inertia (lbsec2/in)
0.010
Breakout (lbs)
2.00
Travel (in)
2.5 Fwd, 5.0 Aft
Time Delay (sec)
0
Control System Gearing (deg/in)
-5.0
Longitudinal Axis
The longitudinal flight control system is a rate command system and consists
of a flight controller, a software rate limiter, a second-order surface actuator model,
software-imposed surface position limit, and the nonlinear aircraft equations of
motion.
The 3 flight controllers were developed independently and are extensively
documented as chapters 4, 5, and 6 of this report. Each was evaluated using the same
aircraft setup as the other two so that direct comparisons of tracking performance and
pilot opinion can be made. In addition to the classically-designed and LPV
120
controllers, a scheduled Hoo controller was also be evaluated as a comparison to LPV
control. This is a result of the desktop analysis which showed that LPV control
suffers moderate to severe performance penalties for flight conditions which the Hoo
controller handles well.
The software rate limit can be arbitrarily set to induce nonlinearities that may
lead to PIO. This capability was built into the STI simulator to allow for
investigations into aircraft loss of control [27], but for the current study the stabilator
rate limit will be set to 65 deg/sec for all flight conditions and simulation runs.
Similarly, the elevator is limited to ±25 degrees of deflection as this is representative
of the F-16 aircraft.
The surface actuator is modeled as a second-order, critically damped linear
transfer function with a 10 Hz. natural frequency. The transfer function is intended to
be representative of the F-16 surface actuator.
Elevator Position Command
y^
5ec
Controller
Software
Rate Limit
Elevator Surface Position
•w
Surface
Actuator
6e
Surface
Position
Limit
i
FIGURE 55. Longitudinal axis control system elements.
121
S
Airplane
g,e,«
TABLE 16. Longitudinal Axis Flight Control System Parameters
FCS Element
Form
Parameter Values
Controller
Varies
N/A
Software Rate Limit
VL = 65 (Baseline)
(deg/sec)
Surface Actuator
col
s + 2£ans + orn
Surface Position Limits
con = 62.8 rad/sec; £ = 0.707
±25 deg
122
APPENDIX B
PILOT RATING SCALES
123
Adequate performance requires extensive
pilot compensation
Adequate performance not attainable with
maximum tolerable pilot compensation.
Controllability not In question,
Considerable pilot compensation Is required
for control
Intense pilot compensation is required to
retain control
Control will be lost during some portion of
required operation
Very objectionable but
tolerable deficiencies
Major deficiencies
Major deficiencies
Major deficiencies
Major deficiencies
no]
PILOT '
RATING
* Definition of required operation involves designation of flight phase and /or
subphases with accompanying conditions.
Adequate performance requires
considerable pilot compensation
Moderately objectionable
deficiencies
Minimal pilot compensation required for
desired performance
Fair - Some mildly
unpleasant deficiencies
Desired performance requires moderate
pilot compensation
Pilot compensation not a factor for
desired performance
Good
Negligible deficiencies
Minor but annoying
deficiencies
Riot compensation not a factor for
desired performance
DEMANDS ON THE PILOT
IN SELECTED TASK OR REQUIRED OPERATION*
Excellent
Highly desirable
AIRCRAFT
CHARACTERISTICS
Cooper-Harper Ref. NASATND-5153
FIGURE 56. Cooper-Harper ratings scale.
Riot decisions
Yea
1
ADEQUACY FOR SELECTED TASK OR
REQUIRED OPERATION*
to
FIGURE 57. PIO tendency scale.
Pilot Attempts
to Enter Control
Loop
Pilot Initiates
Abrupt Maneuvers
or Tight Control
yes
(
2
3
4
Undesirable motions easily induced when pilot initiates abrupt
maneuvers or attempts tight control. These motions can be
prevented or eliminated but only at sacrifice to task performance
or through considerable pilot attention and effort
Ocillations tend to develop when pilot initiates abrupt maneuvers
or attempts tight control. Pilot must reduce gain or abandon task
to recover
Disturbance or normal pilot control may cause divergent oscillation.
Pilot must open control loop by releasing or freezing the stick
Divergent oscillations tend to develop when pilot initiates abrupt
maneuvers or attempts tight control. Pilot must open loop by
releasing or freezing slick
1
Rating
Undesirable motions tend to occur when pilot initiates abrupt
maneuvers or attempts tight control. These motions can be
prevented or eliminated by pilot technique
Description
No tendency for pilot to induce undesirable motions
APPENDIX C
PILOT DEBRIEF QUESTIONNAIRE
126
PILOT QUESTIONNAIRE
Pilot:
Date:
Aircraft response in pitch was essentially linear and predictable:
Very Predictable
1
2
3
4
5
Not Predictable
Rapidity of airplane response in pitch was sufficient:
Very Sufficient
1
2
3
4
5
Not Sufficient
3
4
5
Excessive
4
5
Highly Likely
Overshoot of the pitch response was:
Negligible
1
2
Tendency to induce pitch bobble or PIO:
No Tendency
1
2
3
127
REFERENCES
128
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131
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