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Adaptive control with aerospace applications
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Adaptive control with aerospace applications
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Content
ADAPTIVE CONTROL WITH AEROSPACE APPLICATIONS
by
Ross Gadient
A Dissertation Presented to the
FACULTY OF THE USC GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(ELECTRICAL ENGINEERING)
May 2013
Copyright 2013 Ross Gadient
ii
Acknowledgments
First I would like to thank my advisor Dr. Petros Ioannou, who has been generous
with his time, his guidance, and his wisdom. I am grateful for our many discussions
which led to the exploration of topics contained within. His optimism and enthusiasm
helped fuel my research efforts even at times when hope seemed lost. I consider myself
fortunate to have had an advisor who provided such expertise coupled with sincere
interest and care.
The members of my research committee deserve thanks for providing insight and
constructive criticism, along with enriching my knowledge in the classroom. Professor
Michael Safonov, Professor Henryk Flashner, and Professor Edmond Jonckheere all
contributed in significant ways to the construction of my dissertation and the direction of
my research.
I would like to acknowledge those who originally stoked my interest in the area of
control theory during my days at the University of Illinois. Professor Daniel Metz,
Professor Carolyn Beck, and Professor Juraj Medanic all played pivotal roles in my
introduction to control theory and subsequent continuing study. Special considerations go
to Professor Francesco Bullo, who helped to broaden my research horizons while serving
as my M.S. thesis advisor.
I am indebted to professional colleagues and friends who have helped me better
understand both control theory and application, and I hope my research has positively
contributed to these areas. I would like to express my gratitude to Irene Gregory,
Professor Gary Balas, Professor Anuradha Annaswamy, Professor Naira Hovakimyan,
iii
Brian Whitehead, Travis Gibson, Jason Levin, Ryan Ratliff, Joseph Brinker, and many
others for their contributions.
Special thanks go to Eugene Lavretsky and Kevin Wise for their encouragement, for
their continuing demonstration of excellence in research, application, and teaching of
control theory, and for their friendship.
Most of all, I would like to thank my family: My grandparents Margaret & John and
Bette & Richard for instilling an early love of learning, the denizens of Onanguisse for
sharing positive energy, my brother John for providing inspiration and countless laughs,
my parents Denise and Jeff for their unconditional love and support, and my wife Kat for
her grace, her patience, and her love.
iv
Table of Contents
Acknowledgments............................................................................................................... ii
List of Figures ..................................................................................................................... v
Abstract ............................................................................................................................. vii
Introduction ......................................................................................................................... 1
Chapter 1: Model Following Using Dynamic Inversion.................................................... 7
Chapter 1.1: Baseline Dynamic Inversion Design ....................................................... 8
Chapter 1.2: Aircraft Longitudinal Design ................................................................ 11
Chapter 1.3: Aircraft Directional Design ................................................................... 14
Chapter 2: Model Following Using Dynamic Inversion Controller with State Limiting 18
Chapter 2.1: State Limiter – Motivating Example ..................................................... 20
Chapter 2.2: State Limiter Application to Aircraft Dynamics ................................... 36
Chapter 2.3: Application to X-48B Aircraft Simulation ............................................ 38
Chapter 2.4: Flight Test of the X-48B AOA and AOS Limiting System .................. 49
Chapter 3: Dynamic Inversion with Adaptive Augmentation ......................................... 62
Chapter 3.1: PID Control for Cascaded Systems with Uncertain Dynamics ............. 63
Chapter 3.2: Parameter Adaptation and Closed-Loop System Dynamics ................. 67
Chapter 3.3: Design Example: AOA Tracking .......................................................... 71
Chapter 4: Dynamic Inversion with State Limiting and Adaptive Augmentation ........... 81
Chapter 4.1: Baseline+Adaptive Control+State Limiting for Uncertain Dynamics .. 83
Chapter 4.2: Design Example: AOA Tracking .......................................................... 89
Chapter 5: Adaptive Design with Improved Performance under Input Time-Delays ... 100
Chapter 5.1: Problem Definition and Baseline Optimal Control Design ................. 103
Chapter 5.2: Adaptive Augmentation via Classical MRAC Architecture ............... 110
Chapter 5.3: Alternative Adaptive Augmentation via B-SPM Model ..................... 114
Chapter 5.4: Design Example: AOA Tracking ........................................................ 118
Chapter 6: Concluding Remarks and Suggestions for Future Work .............................. 137
Bibliography ................................................................................................................... 142
v
List of Figures
Figure 1: Limiter Onset Modulation Function 22
Figure 2: Step Response Comparison – Nominal vs. Limited System 24
Figure 3: AOA Limiter Performance in Windup Turn 40
Figure 4: Elevator Response for AOA Windup Turn 42
Figure 5: Reference Model Parameters for AOA Windup Turn 43
Figure 6: AOS Limiter Performance, Step Input 45
Figure 7: Rudder Response for AOA Step Command 46
Figure 8: Reference Model Parameters for AOS Step Command 48
Figure 9: X-48B BWB Configuration 49
Figure 10: X-48B Positive AOA Envelope Limits 51
Figure 11: Windup/Wind-down Turn V-n Diagram 52
Figure 12: Representative Windup/Wind-down Turn V-n Diagram 53
Figure 13: Representative Windup/Wind-down Turn Airspeed/AOA Plot 54
Figure 14: AOA Limiter Results, Slat Retracted/Aft CG 56
Figure 15: AOA Limiter Results, Slat Extended/Fwd CG 57
Figure 16: Representative Sideslip Limiter Maneuvers 59
Figure 17: Sideslip Limiter Results, Slat Ret/Aft CG 60
Figure 18: Sideslip Limiter Results, Slat Ext/Aft CG 61
Figure 19: Representation of the Sets
r b B R
SS 70
Figure 20: Baseline Closed-Loop Response to AOA Doublet without Uncertainties 74
Figure 21: Total Matched Uncertainty vs. AOA at q=0 76
Figure 22: Baseline Closed-Loop Response in Presence of Matched Uncertainties 77
Figure 23: Baseline+Adaptive Closed-Loop Response in Presence of Uncertainties 79
Figure 24: Adaptive Parameter Norm vs. Time 80
Figure 25: Baseline Closed-Loop Response to AOA Doublet without Uncertainties 92
Figure 26: Closed-Loop Response to AOA Doublet: No Uncertainties, with Limiter 93
Figure 27: AOA Doublet Response: No Uncertainties, with Limiter (Zoom) 94
Figure 28: AOA Doublet Response: Uncertainties with Adaptive+Limiter 96
Figure 29: AOA Doublet Response: Uncertainties with Adaptive+Limiter (Zoom) 97
Figure 30: Adaptive Parameter Norm vs. Time 98
Figure 31: Time Response of Pure Time-Delay and Padé Approximations 106
Figure 32: Frequency of Pure Time-Delay and Padé Approximations 107
Figure 33: Baseline Closed-Loop Response to AOA Doublet without Uncertainties 120
Figure 34: Baseline Closed-Loop Response in Presence of Matched Uncertainties 122
Figure 35: Baseline+MRAC Closed-Loop Response in Presence of Uncertainties 124
Figure 36: MRAC Adaptive Parameter Norms vs. Time 125
Figure 37: Baseline+MRAC Closed-Loop Response with High Adaptive Gains 126
Figure 38: Baseline+B-SPM Normalized Response in Presence of Uncertainties 128
Figure 39: Baseline+B-SPM Adaptive Parameter Norms vs. Time 129
Figure 40: Comparison of Adaptive Closed-Loop Responses: with Uncertainties 130
vi
Figure 41: Comparison of Control Input Power Spectral Densities 131
Figure 42: Baseline Response: Input Time-Delay of 935ms 133
Figure 43: Baseline + B-SPM Normalized Response: Input Time-Delay of 935ms 135
vii
Abstract
Robust and adaptive control techniques have a rich history of theoretical development
with successful application. Despite the accomplishments made, attempts to combine the
best elements of each approach into robust adaptive systems has proven challenging,
particularly in the area of application to real world aerospace systems. In this research,
we investigate design methods for general classes of systems that may be applied to
representative aerospace dynamics. By combining robust baseline control design with
augmentation designs, our work aims to leverage the advantages of each approach.
This research contributes the development of robust model-based control design for
two classes of dynamics: 2
nd
order cascaded systems, and a more general MIMO
framework. We present a theoretically justified method for state limiting via
augmentation of a robust baseline control design. Through the development of adaptive
augmentation designs, we are able to retain system performance in the presence of
uncertainties. We include an extension that combines robust baseline design with both
state limiting and adaptive augmentations. In addition we develop an adaptive
augmentation design approach for a class of dynamic input uncertainties. We present
formal stability proofs and analyses for all proposed designs in the research.
Throughout the work, we present real world aerospace applications using relevant
flight dynamics and flight test results. We derive robust baseline control designs with
application to both piloted and unpiloted aerospace system. Using our developed
methods, we add a flight envelope protecting state limiting augmentation for piloted
aircraft applications and demonstrate the efficacy of our approach via both simulation and
viii
flight test. We illustrate our adaptive augmentation designs via application to relevant
fixed-wing aircraft dynamics. Both a piloted example combining the state limiting and
adaptive augmentation approaches, and an unpiloted example with adaptive augmentation
show the ability of our approach to retain desired performance in the presence of relevant
system uncertainty. Finally, we present alternative adaptive augmentation design
developed to mitigate time delays at the system input and which demonstrates significant
improvement over an existing widely used adaptive augmentation approach when applied
to fixed wing aircraft dynamics.
1
Introduction
Demands on safety and performance have necessitated the development of
increasingly sophisticated flight control systems. The design of autopilots for high-
performance aircraft was one of the primary motivations for active research in adaptive
control in the 1950s. In comparison to gain scheduling, adaptive control has a learning
capability which identifies changes in the parameters of the flight dynamics by
processing the input/output data and adjusts its parameters accordingly. Adaptive designs
may offer benefits over fixed-gain counterparts such as improved performance, increased
robustness to uncertainties, decreased design cycle time, and lower cost. These benefits
are a byproduct of their ability to adjust control parameters as a function of online
measurements. Early attempts at adaptive flight control used controllers with unproven
stability properties, which sometimes led to disastrous consequences [5]. Since that time,
significant research in the field of robust and adaptive control has allowed the design of
stable adaptive systems. Various adaptive control methods have been developed for
controlling dynamic systems with parametric and dynamic uncertainties [6]-[17]. These
techniques have been extended and applied to the problem of aircraft control. Recently, a
dynamic inversion baseline controller on the X-36 tailless fighter was augmented by an
adaptive neural network flight control system [18]. In addition, adaptive augmentation
has been used to augment baseline LQR control design in production joint direct attack
munitions (JDAM) unpowered weapons [19]. In this thesis, we take a similar adaptive
augmentation approach to a robust baseline controller and apply to dynamics
representative of an aircraft in all three axes.
2
This thesis considers two model following control system architectures designed to
provide robust baseline with adaptive tracking performance, applied to relevant flight
dynamics. First, given second-order dynamical systems in cascaded form, we design a
baseline model-reference dynamic inversion based tracking control law in both explicit
and implicit forms. Interest in considering this particular class of systems stems from
flight control related applications, where inner-loop controllers for fixed-wing aircraft are
often designed based on simplified models [1]-[4]. These models represent the decoupled
fast responses in aircraft pitch, roll and yaw axes and are given in such cascaded form.
The benefit of such a baseline control design is two-fold. Such a design allows for
favorable closed-loop system characteristics in terms of traditional robustness measures
and disturbance rejection. In addition, the choice of dynamic inversion architecture
allows the designer to satisfy both aircraft performance and handling qualities
requirements in a convenient manner. The latter is of significant importance in the case of
piloted vehicles, where well-known robust control techniques such as LQR may be used
to provide good margins and robustness but may not provide insight into handling
qualities.
An additional challenge addressed is the need to keep the system dynamics in a
prescribed region of the corresponding state space. This region is often referred to as the
“operational flight envelope”. For example, an Angle-of-Attack (AOA) command
tracking controller must include an AOA protection system (often called the “AOA
Limiter”), whose purpose is to maintain the aircraft AOA within a pre-specified range,
outside of which a loss of control is expected. Vehicles with control augmentation
3
systems rely on state limiters to prevent the exceeding of predetermined state limits.
These limiters typically activate once predefined envelope boundaries are exceeded and
introduce sharp changes to control characteristics once active. The resulting
nonlinearities make analysis using conventional control system methods difficult or
invalid. Current state-of-the-art in limiter design consists of ad-hoc limiting schemes
developed specifically for the system under consideration. As a result, such methods
require extensive design and test iterations specific to each individual application. Often,
these designs do not provide stability / performance guarantees. The proposed method
introduces a formal yet numerically efficient limiting technique which gradually modifies
the expected behavior of the system dynamics when user-specified criteria are
approached / exceeded. In essence, such a controller has to blend two subsystems, the
tracker and the limiter, with seamless transitions between the two controllers while
preserving closed-loop stability at all times. In this thesis we present a state limiting
augmentation design via formal methods that keeps the system state in a prescribed
region of the state space while maintaining closed-loop stability.
Another challenge addressed in this thesis is the use of adaptive augmentation to
allow the control system to maintain desired performance in the presence of system
uncertainties. Aircraft operate over a wide range of speeds and altitudes, and their
dynamics are nonlinear and often time-varying. As an aircraft moves through different
flight conditions, the linear design model at an operating point changes. A controller
designed for each operating point can be scheduled to be switched on when the operating
point is reached. Between operating points interpolation or other techniques may be used
4
to modify or mix the controllers of the neighboring operating points. This approach is
referred to as gain scheduling, and does not necessarily possess guarantees of closed-loop
stability when the system undergoes unexpected changes in its dynamics and
environment. We propose the use of adaptive augmentation to provide closed-loop
stability guarantees and improved performance in the presence of system uncertainties.
A resulting development derived in this thesis is the synthesis of a robust dynamic
inversion baseline controller with both state limiting and adaptive augmentation devices.
This design allows for insight into flying qualities design for baseline control of an aerial
vehicle, state limiting to keep the vehicle within a prescribed region of the system state
space, and adaptive augmentation to mitigate matched uncertainties. The combination of
all three elements into an aggregate design yields a flight control design that is both
theoretically justified as well as relevant to real-world flight control problems where
concerns such as permissible flight envelope restrictions are important to maintain safe
piloted flight.
In addition to dynamic inverse based control for a class of cascaded 2
nd
-order
systems, we also develop robust baseline control for aerial vehicles using LQR-optimal
approach. Again using an augmentation approach, we apply classical MRAC adaptive
augmentation to representative flight dynamics to mitigate matched uncertainties. We
advance existing results by formulating the problem to specifically address and mitigate
input time-delay, which is of crucial importance in aerospace systems due to such factors
as sensor latency and actuator performance uncertainty. After reformulating the classical
MRAC architecture to address matched uncertainties and input time-delay, we apply and
5
extend the B-SPM method [7], [28] to representative aerospace systems and illustrate the
significant improvement in system performance under uncertainty and time delay. In all
chapters, relevant real-world aerospace systems and simulations are employed to
illustrate and validate the theoretical advances derived.
The thesis is organized as follows. Chapter 1 derives baseline model following
control using dynamic inversion, with application to aircraft longitudinal and directional
dynamics. In Chapter 2, we derive state limiting modification to the model following
dynamic inversion controller formed in Chapter 1. The efficacy of the state limiting
component is illustrated via high-fidelity piloted simulation and subsequent flight testing
on the X-48B Blended-Wing-Body (BWB) vehicle. In Chapter 3 we derive adaptive
augmentation and apply the results to flight dynamics corresponding to the longitudinal
axis of a fixed-wing aircraft. We present and discuss simulation data to substantiate our
claims that the adaptive augmentation is able to restore system tracking performance in
the presence of matched uncertainties.
Chapter 4 combines the advances developed in the first three chapters to form a
theoretically justified control design consisting of robust dynamic inversion baseline
control, adaptive augmentation to mitigate matched uncertainties, and state limiting
augmentation to keep the system within a prescribed region in the system state space. We
present simulation results using representative aerospace data to show the efficacy of the
design in keeping an aircraft in its permissible flight envelope while restoring tracking
performance in the presence of matched uncertainties. In Chapter 5, we derive robust
LQR-optimal baseline control law and apply classical MRAC-based architecture
6
extended to explicitly address time-delay at the system input. We then apply and extend
the B-SPM adaptive approach to the same problem and show significantly improved
performance in the presence of input time-delay while restoring tracking performance in
the presence of matched uncertainties. Finally, Chapter 6 presents topics for further
investigation.
7
Chapter 1: Model Following Using Dynamic Inversion
This chapter considers a class of model following control systems designed to provide
tracking performance. Given second-order dynamical systems in cascaded form, we
design a baseline model-reference dynamic inversion based tracking control law in both
explicit and implicit forms. Interest in considering this particular class of systems stems
from flight control related applications, where inner-loop controllers for fixed-wing
aircraft are often designed based on simplified models [1]-[4]. These models represent
the decoupled fast responses in pitch, roll and yaw axes and are given in cascaded form.
The dynamic inversion approach is particularly useful in flight control applications
where the designer may explicitly include vehicle handling qualities requirements in the
selection of reference model desired damping and frequency to meet handling qualities
requirements. After deriving dynamic inversion tracking control for the general case, we
include specific design for aircraft longitudinal and directional axes. The results of the
chapter contribute robust baseline flight control architecture in which both aircraft
performance and handling qualities requirements may be achieved in a convenient
manner. The dynamic inversion design presented will serve as robust baseline control
architecture which to be further improved in the following chapters via adaptive
augmentation and state limiting augmentation components.
8
Chapter 1.1: Baseline Dynamic Inversion Design
This section formulates the system dynamics, poses the control problem, and derives
baseline Dynamic Inversion (DI) Proportional + Integral + Derivative (PID) control
architecture for flight dynamics.
Consider 2
nd
order dynamical systems in the cascaded form:
0
1 1 1 1 2
0
2 2 1 2 2
,
,,
cmd
x F x z B x
x F x x z x
(1.1)
where
12
T
x x x is the system state vector, z is the known bounded external signal,
12
00
, FF are known state-dependent functions,
1
B is a known nonzero constant, and
2
cmd
x
is the system control input. The control goal is to design the control input
2
cmd
x so that the
system 1
st
state component
1
x tracks any given bounded time-varying command
1
cmd
xt ,
while keeping all the signals in the closed-loop system bounded, uniformly in time.
This approach will employ a model reference based design framework. The reference
model is driven by a bounded possibly time-varying reference command
1
cmd
x , and is
chosen to be 2
nd
order, with desired damping ratio and natural frequency :
2
2
1 1 1 1 1 1 22
2
2
m cmd m m cmd m
x x x x x x
ss
(1.2)
We note that in aerospace applications the desired damping ratio and natural frequency
are selected by the designer to achieve performance and, if applicable, handling qualities
for piloted aircraft.
Differentiating the first state component in (1.2) yields:
9
00
11
1 1 1 2
1
FF
x x z B x
xz
(1.3)
Substituting the first equation in (1.1) into (1.3) results in:
12
00
00 11
1 1 1 2 1 2 1 2
1
, , ,
cmd
f x x z z
FF
x F B x z B F B x
xz
(1.4)
or, equivalently:
1 1 2 1 2
, , ,
cmd
x f x x z z B x (1.5)
One may immediately note that in (1.5) the function
12
, , , f x x z z and the constant
1
B are known. Dynamic Inversion (DI) based Proportional + Integral + Derivative (PID)
control is introduced in the form:
1 11
2 1 1 1 1 1 1
ˆ
m
cmd m m m
DPI
xx
x B x f K x x K x x K
s
(1.6)
Because system state derivative is often not available as a measurement, the quantity:
0
1 1 1 1 2 1 1
ˆ
ˆ , , 0 0 x F x z B x x x (1.7)
in (1.6) is estimated / predicted 1
st
state derivative. Let
1 1 1
m
e x x be the reference
model tracking error signal. Substituting (1.6) into (1.5) results in the closed-loop
tracking error dynamics:
1
1 1 1
0
D P I
e
e K e K e K
s
(1.8)
Note that the error dynamics characteristic polynomial is given by:
32
0
D P I
K K K (1.9)
10
As a result, exponential stability of the error dynamics in (1.8) can be achieved with
any appropriate PID gains selected via Routh tables or other methods. If we define
desired error characteristic polynomial as:
22
20 k (1.10)
where , are the desired damping ratio and natural frequency while 0 k is a fixed
positive constant, we may expand (1.10) and compare to (1.9) to yield PID feedback
gains:
2
2
2
2
D
P
I
Kk
Kk
Kk
(1.11)
Using PID gains (1.11), closed-loop tracking error dynamics (1.8) become globally
exponentially stable. This implies that system state component
1
xt tracks reference
signal
1
m
xt exponentially fast.
Under control (1.6), the closed-loop dynamics of the second state component in (1.1)
become:
0
2 2 1 2 2
01 1
2 1 2 1 1 1 1
00
10 1 1 1
1 1 1 1 2 1 1
1
,,
,,
cmd
m
DPI
m
DPI
x F x x z x
e
F x x z B x f K e K e K
s
F F e
B x F B x z K e K e K
x z s
(1.12)
Assuming zero initial conditions, error dynamics (1.8) and the PID gains (1.11) yields:
2
1
2 2 2 2
11
22
cmd
m
x
s s s k x s k s s x
(1.13)
11
Hence, the closed-loop transfer function from the model reference external command
1
cmd
x to the system state
1
x has the desired 2
nd
order form:
2
1
22
1
2
cmd
x
x s s
(1.14)
In order to show that the control in (1.6) solves the tracking problem, note that since
the selected PID gains yield asymptotically stable error dynamics, the error signal
1
et
and its time derivative
1
et are bounded. At the same time, the model reference signal
1
m
xt and its time derivative
1
m
xt are bounded by design. Consequently, the system
state components
1
xt and
1
xt are bounded. Using the first equation in system
dynamics (1.1) implies that
2
x is bounded, and therefore together with (1.8) the tracking
problem is solved in the sense that all signals are bounded and the tracking error
converges to zero exponentially fast.
Chapter 1.2: Aircraft Longitudinal Design
Assuming small angle-of-sideslip (AOS) 0 , negligible lift due to elevator
0 L
, and negligible thrust effects, fixed-wing aircraft short-period dynamics can be
written in the form of (1.1) [1], [4]:
grav
q IC cmd
L Q q
q M M q M q
(1.15)
12
where is aircraft angle-of-attack (AOA), q is the angular pitch rate, L
is the known
lift curve slope,
grav
Q is the known gravity term, M
is the known static stability
(pitching moment),
q
M is the known constant pitch damping,
IC
M is the known pitching
moment increment due to inertial cross-coupling effects, and
cmd
q is the commanded
pitch acceleration (control input).
The AOA reference model dynamics is chosen in the form of (1.2):
2
2
22
2
2
m cmd m m cmd m
ss
(1.16)
To perform the design, one needs to match the required components against the
corresponding AOA/pitch rate dynamics terms. Comparing (1.15) against generic
cascaded dynamics (1.1) yields:
1 2 2
1 1 2
, , ,
, 1,
T
cmd
grav IC cmd
grav q IC
x x q z Q M x q
F L Q B F M M q M
(1.17)
In this case, the baseline PID feedback gains are:
1
1
2
1
2
2
D
P
I
I
K K k
K K k
K K k
(1.18)
where the integrator pole
1
k is chosen as:
1
kL
(1.19)
Based on the AOA equation in (1.15) and the reference dynamics (1.16), define
reference model pitch rate signal:
13
m m m grav
q L Q
(1.20)
Substituting (1.17)-(1.20) into DI PID control (1.6) yields explicit model following pitch
acceleration command:
2
2
m
cmd m q IC m
qq
q q M M q M q q
s
(1.21)
Alternatively, if it is desired to generate control (1.6) without explicit implementation
of reference model dynamics (1.16), an implicit model following pitch acceleration
command may be formed via (1.21), (1.20), and (1.16) as:
2
2
cmd q IC cmd grav grav
sL
q M M q M q Q Q
s
(1.22)
We verify that control input (1.22) results in desired closed-loop dynamics for system
(1.15) as follows. Differentiating the first state equation in (1.15) and substituting the
second state equation yields:
grav q IC cmd
L Q M M q M q
(1.23)
Employing control signal (1.22) in (1.23) gives closed-loop dynamics:
2
2
cmd grav
sL
s s L q Q
s
(1.24)
Noting from the first state equation in system dynamics (1.15) that:
grav
q s L Q
(1.25)
and substituting into (1.24) yields:
2
2
cmd
s s L s L
s
(1.26)
14
Cancelling terms and reducing (1.26) gives:
2
2
2
cmd
s
s
(1.27)
Finally, differentiating (1.27) produces the desired 2
nd
order closed-loop response for the
vehicle angle-of-attack dynamics:
2
22
2
cmd
ss
(1.28)
Based on system dynamics (1.15), the resulting closed-loop response for the pitch rate
dynamics to an angle-of-attack command is:
2
22
2
cmd
sL
q
ss
(1.29)
Note that in this case, if 0 L
then the pitch rate dynamics would be non-minimum
phase, but the pitch rate dynamics would remain bounded in time.
Chapter 1.3: Aircraft Directional Design
Assuming small angles, fixed-wing aircraft AOS / yaw rate dynamics in stability axis
can be written in the form of (1.1) [1], [4]:
grav
r p IC cmd
Y R r
r N N r N p N r
(1.30)
where is aircraft AOS, r is the angular yaw rate, p is angular roll rate, Y
is the
sideforce slope due to AOS,
grav
R is the known yaw rate component due to gravity,
15
, , ,
r p IC
N N N N
represent dimensional yawing moment terms, and
cmd
r is the
commanded yaw acceleration (control input).
The AOS reference model dynamics is chosen in the form of (1.2):
2
2
22
2
2
m cmd m m cmd m
ss
(1.31)
In this case, comparing (1.30) against generic cascaded dynamics (1.1) yields:
1 2 2
1 1 2
, , ,
, 1,
T
cmd
grav IC cmd
grav r p IC
x x r z R N x r
F Y R B F N N r N p N
(1.32)
Similar to AOA case, the baseline PID feedback gains are:
1
1
2
1
2
2
D
P
I
I
K K k
K K k
K K k
(1.33)
where the integrator pole
1
k is chosen as:
1
kY
(1.34)
Based on the AOS equation in (1.30) and the reference dynamics (1.31), define
reference model yaw rate signal:
m m grav m
r Y R
(1.35)
Substituting (1.32)-(1.35) into DI PID control (1.6) yields explicit model following
yaw acceleration command:
2
2
m
cmd m p r IC m
rr
r r N N p N r N r r
s
(1.36)
16
Alternatively, if it is desired to generate control (1.6) without explicit implementation of
reference model dynamics (1.31), an implicit model following yaw acceleration
command may be formed via (1.36), (1.35), and (1.31) as:
2
2
cmd p r IC cmd grav grav
sY
r N N p N r N r R R
s
(1.37)
Similar to pitch dynamics, we verify that control input (1.37) results in desired
closed-loop dynamics for system (1.30) as follows. Differentiating the first state equation
in (1.30) and substituting the second state equation yields:
grav r p IC cmd
Y R N N r N p N r
(1.38)
Employing control signal (1.37) in (1.38) gives closed-loop dynamics:
2
2
cmd grav
sY
s s Y r R
s
(1.39)
Noting from the first state equation in system dynamics (1.30) that:
grav
r s Y R
(1.40)
and substituting into (1.39) yields:
2
2
cmd
s s Y s Y
s
(1.41)
Cancelling terms and reducing (1.41) gives:
2
2
2
cmd
s
s
(1.42)
Finally, differentiating (1.42) produces the desired 2
nd
order closed-loop response for the
vehicle angle-of-sideslip dynamics:
17
2
22
2
cmd
ss
(1.43)
Based on system dynamics (1.30), the resulting closed-loop response for the yaw rate
dynamics to an angle-of-sideslip command is:
2
22
2
cmd
sY
r
ss
(1.44)
Note that in this case, if 0 Y
then the yaw rate dynamics would be non-minimum
phase, but the yaw rate dynamics would remain bounded in time.
18
Chapter 2: Model Following Using Dynamic Inversion
Controller with State Limiting
While the controller developed in Chapter 1 was shown to solve the tracking problem,
in practical application there are additional requirements that must be considered during
the design process. Vehicles with control augmentation systems rely on state limiters to
prevent the exceeding of predetermined state limits. These limiters typically activate once
predefined envelope boundaries are exceeded and introduce sharp changes to control
characteristics once active. The resulting nonlinearities make analysis using conventional
control system methods difficult or invalid. Current state-of-the-art in limiter design
consists of ad-hoc limiting schemes developed specifically for the system under
consideration. As a result, such methods require extensive design and test iterations
specific to each individual application. Often, these designs do not provide stability /
performance guarantees.
An aircraft flight controller must keep the vehicle in a pre-specified region of the
corresponding state. This region is referred to as the operational flight envelope. For
example, an AOA protection system (often called AOA Limiter) is typically employed to
maintain the aircraft AOA within an allowable range outside of which loss of control is
expected. The control challenge in such cases is to blend two subsystems, the tracking
controller and the limiter, with seamless transition between the two controllers while
preserving closed-loop stability at all times. We solve this problem by introducing state
limiting augmentation to the baseline dynamic inversion controllers introduced in
Chapter 1.
19
The proposed method introduces a limiting technique which gradually modifies the
expected behavior of the system dynamics when user-specified criteria are approached /
exceeded. Given a baseline model following dynamic inversion control law designed to
meet robustness and performance requirements throughout a flight envelope, a state-
limiting augmentation component protects the system trajectories from leaving an
allowable subset in the system state space. The proposed design is applied to the X-48B
blended-wing-body (BWB) aircraft in both the longitudinal and lateral/directional axes
with both piloted simulation and flight test results included. The results of this chapter
contribute theoretically justified state limiting to a robust baseline control design,
allowing for aircraft performance as well as protecting the system from exiting a pre-
specified region of the state space. The limiter design will be further improved in
subsequent chapters by including adaptive augmentation to mitigate matched
uncertainties.
20
Chapter 2.1: State Limiter – Motivating Example
In order to illustrate the main idea of the design, first consider scalar dynamics:
xu (2.1)
where x is the system state and u is the control input. The control goal is once again to
choose u such that x tracks the state of a user-defined reference model
m
x whose
dynamics are:
2
2
m m cmd m
x x x x (2.2)
where and represent the desired damping ratio and natural frequency, respectively.
Note that we are dealing with a simplified version of the problem presented in Chapter 1.
Furthermore assume that it is required that the system state
1
x does not exceed pre-
specified limits:
lim 1 lim
x x t x (2.3)
In order to ensure feasibility of a control solution, we assume that the commanded
signal
cmd
x satisfies the same limits, and for clarity we assume that the state derivative x
is available on-line. For simplicity, limits (2.3) are defined as symmetric but we note that,
without loss of generality, this need not be the case. Choosing control solution as:
2
2
cmd
u x x x (2.4)
yields closed-loop system:
2
2
cmd
x x x x (2.5)
Let
21
m
e x x (2.6)
denote the tracking error. Subtracting the reference dynamics from the system dynamics
gives:
2
2 e e e (2.7)
This relation implies global exponential stability of the origin. Consequently, the
system state tracks the state of the reference model exponentially fast starting from any
initial conditions. However, the desired state limits (2.3) are not guaranteed.
We propose a formal design modification to enforce the state limits. The main idea is
to gradually change the reference model if the system state approaches any of its limits.
The modified reference dynamics in turn alter the control input to the system in such
regions. Resulting control commands maximize vehicle state-limited performance within
existing control law architecture. Specifically in this case, the proposed state limiter logic
introduces state-dependent damping ratio and natural frequency. Consider modified
damping ratio and natural frequency:
x
x
xx
xx
(2.8)
In (2.8), x is the modulation function described in [20] and based on work in
[21], and x is the aggregate state vector containing the degrees of freedom of the system.
This is a continuous state-dependent map which allows smooth transition of the desired
damping ratio and natural frequency. The modulation function allows the designer to
define the region sufficiently close to the state limits for the modified reference model
parameters (2.8) to become active. This ensures that for all nominal system conditions
22
(away from state limits), the desired baseline control system behavior and system
dynamics are achieved. Construction of the modulation function is presented below.
Suppose that
0
x is the center point of the sphere
0 R
x x R . Also suppose
that the set
R
represents the allowable state domain. Let 0 be a small positive
constant and define
0 R
x x R
. The multidimensional modulation function
is defined as:
0,
0 1,
1,
R
RR
R
x
x x x
x
(2.9)
Formally, the modulation function can be written as:
0 lim
max 0,min 1,1
x x x
x
(2.10)
where
lim
x is a vector whose components represent the limit value for each system degree
of freedom, and the limiter remains inactive in nominal conditions until system state
approaches defined limits within user-defined tolerance . The limiter is then gradually
applied as the system state approaches the state limits, avoiding undesirable behavior.
Graphical representation of the modulation function is shown in Figure 1.
0
xx
1
0
lim
x
x
lim
x
Figure 1: Limiter Onset Modulation Function
23
Including modification (2.8) results in modified reference dynamics:
2
2
m x x m x cmd m
x x x x x x x (2.11)
Note that in (2.8) the quantities and are user-defined incremental damping
ratio and natural frequency tables, respectively. These two tables are chosen to meet the
desired tracking performance and system robustness requirements. For example, the
reference model can be modified such that its state has no overshoot while tracking a
step-input command. Hence, if the latter is within the limits then the system state and the
reference model state will remain within the desired limits as well.
To illustrate the limiter concept, consider system (2.1) and reference model (2.2) with
baseline parameters 0.3 and 4 rad/sec. Obviously, in this case the baseline step
response will contain significant overshoot since the system is underdamped. Select
limiter parameters:
4.5, 4.5, 0.1 (2.12)
Parameters (2.12) indicate that once the system response comes within 10% of the
defined state limit, the damping and natural frequency will increase and reach values of
4.8 and 8.5 rad/sec once the limiter is fully active, i.e. the system reaches the
limit. Consider a step command with magnitude 2, where the state limit for system state
1
2 x . The step response for the baseline and state limited systems is shown in Figure 2.
The response shows that the limiter is able to arrest the system response and keep the
state response within the specified limit. The modulation parameter x is inactive until
the state response enters the modulation region defined by , at which point it
24
increments from zero to one as the state approaches the limit. The modified system
damping ratio and frequency stay at their respective baseline design values until the
system near the state limit, at which point both evolve to their modified values defined by
increments (2.12).
0 0.5 1 1.5 2 2.5 3
0
2
4
x
x
cmd
limit area
x (no limit)
x (limit)
0 0.5 1 1.5 2 2.5 3
0
0.5
1
(x)
0 0.5 1 1.5 2 2.5 3
0
5
x
0 0.5 1 1.5 2 2.5 3
0
5
10
x
time, sec
Figure 2: Step Response Comparison – Nominal vs. Limited System
25
In order to investigate the efficacy of the limiter design, we must consider both the
closed-loop stability of the system as well as determine whether the limit design truly
guarantees that the system state remains within the imposed limit for all time. To simplify
the analysis we define the system in state space form:
12
2
xx
xu
(2.13)
and apply control:
2
21
2
x x x cmd
u x x x x x x (2.14)
yielding closed-loop dynamics:
11
22
22
0 1 0
2
cmd
x x x x
xx
x
x x x x xx
(2.15)
Similarly we may define reference dynamics (2.11) in state space form:
11
22
22
0 1 0
2
mm
cmd
x x x x mm
xx
x
x x x x xx
(2.16)
Based on relations (2.15) and (2.16) with error signal definition (2.6) we can now
derive the closed-loop error dynamics in state space form:
2
01
2
x x x
E E A
ee
x x x ee
(2.17)
alternately expressed as:
E A x E (2.18)
26
It is sufficient to analyze system (2.17) to show closed-loop stability of the system.
Based on definition (2.8) and limiter modulation function (2.10) we find three regions of
interest:
Case 1:
0 lim
x x x . In this region, the modulation function is inactive, and the
error dynamics may be expressed in terms of the nominal damping and natural
frequency . Error dynamics (2.18) are no longer a function of x and instead satisfy
autonomous linear time invariant (LTI) relationship:
2
01
2
EE
(2.19)
The closed-loop poles of (2.19) are given by:
2
12
,1 (2.20)
Because the nominal damping and natural frequency are selected by the designer,
they may be chosen to ensure that error dynamics (2.19) are exponentially stable.
Case 2:
0 lim
x x x . In this region, the modulation function is completely active,
and the error dynamics may be expressed in terms of the full incremented damping
and natural frequency . Error dynamics (2.18) are no longer a function of
x and instead satisfy autonomous linear time invariant (LTI) relationship:
2
01
2
EE
(2.21)
The closed-loop poles of (2.21) are:
2
12
,1 (2.22)
27
Once again, since the nominal damping and natural frequency is selected by the designer
and so are the incremental damping and natural frequency tables, they may be chosen to
ensure that the error dynamics (2.21) are exponentially stable. We note that although the
design goal of the limiter is to ensure that the system dynamics never enter a region
where
0 lim
x x x , in practical application external disturbances may drive the system
into this region. One example of this behavior is a gust acting on an aircraft. This analysis
indicates that even should a system exceed its limit, the error dynamics remain
exponentially stable.
Case 3:
lim 0 lim
x x x x . In this region, the modulation function is active and
likely changing due to the evolution of the system states. The error dynamics become:
2
01
2
x x x
EE
x x x
(2.23)
We immediately note that, as shown in the analyses of cases 1 and 2, the designer may
choose the nominal damping and natural frequency as well as the incremental tables for
each parameter, so that the error dynamics (2.23) could be made exponentially stable for
any value of the modulation parameter . However, since dynamics (2.23) now represent
a time-varying system, one cannot determine the closed-loop stability by simply
inspecting its eigenvalues. In addition, since dynamics (2.23) are not autonomous as
posed, stability results such as LaSalle‟s Theorem do not apply and we must use an
alternate approach to show stability.
We demonstrate three stability results for dynamics (2.23). Since each of the results is
a sufficient condition for stability, they may be conservative in some cases.
28
Method 1: Vanishing Perturbation
Re-write error dynamics (2.23):
0
, E A E g t E (2.24)
where
0
A represents Hurwitz LTI component of error dynamics, while , g t E
represents time varying component due to the activity of the modulation function and
satisfies ,0 0 gt . Since the modulation function operates between the values of zero
and one, we know that
2 2
, g t E E for all 0 t and all
n
E where is a
nonnegative constant. Let 0
T
QQ and solve the Lyapunov equation:
00
T
P A A P Q (2.25)
for P . The quadratic Lyapunov function
T
V E E P E will then satisfy:
22
min max
22
2
0 min
2
max
2 2 2
2
2
2 2 2
T
T
P E V E P E
V
A E E Q E Q E
E
V
E P P E P E
E
(2.26)
The derivative of VE along the trajectories of the perturbed system (2.24) satisfies:
22
min max
22
2 V E Q x P x (2.27)
Hence, the origin is globally exponentially stable if
min max
/2 QP . This ratio is
maximized with the choice QI , and therefore the origin is globally exponentially
stable if:
29
max
1
2 P
(2.28)
We note that (2.28) places a bound on the size of the perturbation permissible while
retaining stability. In this case, it is a restriction on the amount of change permitted in the
damping and frequency from nominal values. The result is likely to be extremely
conservative because the vanishing perturbation analysis is a worst case analysis.
Method 2: Input-to-State Stability
Consider error dynamics (2.23) as:
,, E f t E u (2.29)
where input , u t g t E is a piecewise continuous, bounded function of t for all
0 t . If we again model the system as in (2.24), we see that the unforced system:
0
E A E (2.30)
has a globally uniformly asymptotically stable equilibrium point at the origin. Due to the
modulation function, we may model the external bounded input as:
10
, g t E t A A E (2.31)
where
1
A corresponds to the fully active modulation system (2.21) and
0
A corresponds to
the nominal system (2.19). We can write the system dynamics solution as:
00 0
0
0
,
t
A t t At
t
E t e E t e g E d
(2.32)
and use the bound:
0 0 0
A t t t t
e k e
(2.33)
30
to estimate the solution by:
0
0
0
,
t
tt t
t
E t k e E t k e g E d
(2.34)
Using relation (2.31), we may reduce (2.34):
0
0
0
0
0
0 1 0
,
t
tt t
t
t
tt t
t
E t k e E t k e g E d
k e E t k e A A E d
(2.35)
Noting the modulation parameter satisfies 01 t and that quantity
10
AA now
represents a constant as written in (2.35), we may use the Bellman-Gromwell Lemma to
bound error dynamics as:
10
0
k A A t
E t k E t e
(2.36)
Therefore, for stability we require that:
10
AA
k
(2.37)
Relationship (2.37) gives a bound on the maximum singular value of the matrix
10
AA
in relation to the slowest eigenvalue of the unperturbed system and corresponding
eigenvectors. Again, this result will be conservative but provides another sufficient
stability result.
Method 3: Second Method of Lyapunov
Rewrite error dynamics (2.23) as an unforced second-order differential equation as a
function of time:
0 E f t E g t E (2.38)
where:
31
2
2
xx
x
f t t t
g t t
(2.39)
Because of the relation (2.8), it is apparent that:
min max
min max
0
0
f f t f
g g t g
(2.40)
From dynamics (2.38) or (2.23), we have:
12
2 1 2
ee
e g t e f t e
(2.41)
Following the analysis presented in [22], define Lyapunov function:
22
12
1
, V t E e e
gt
(2.42)
Differentiating (2.42) along the system trajectories yields:
2 22
1 1 2
2
2 2
2 1
,2
2
Kt
ee d
V t E e e e
g t dt g t
g t f t g t
e
gt
(2.43)
Using results from [22], if:
2
2 g t f t g t
gt
(2.44)
and
max
f t f the solution is asymptotically stable.
32
Using (2.44), we may now investigate what the requirement of 0 Kt indicates for
error dynamics with our specific structure (2.39). Using (2.44) while letting 1 , for
stability we require:
2 g t f t g t (2.45)
Substituting structure (2.39) into (2.45) yields stability requirement:
2
2
x x x
t t t (2.46)
We immediately note that in the case where damping and frequency are constant and
satisfy bounds (2.40) that (2.46) is always satisfied, as should be expected. Rearranging
(2.46) yields:
2
2
x
x
x
t
t
t
(2.47)
Integrating both sides of (2.47):
2
00
2
tt
x
x
x
dd
(2.48)
Perform change of variables:
0 0 ,
y dy d
y y t t
(2.49)
Using new variable of integration, (2.48) becomes:
2
00
2
t
t
dy
d
y
(2.50)
Performing integration on the left hand side of (2.50) yields:
33
0 0
1
2
t
t
d
y
(2.51)
Rearranging (2.51) yields condition for stability:
0
0
1 2 0
t
t
d
(2.52)
We now note that it is not necessary to perform the integration in (2.52), because due
to conditions (2.39) and (2.40), we have that:
0
1 2 0 1, 0
t
dt
(2.53)
As a result, stability requirement (2.52) can always be satisfied by using (2.40) and
selecting:
min min
0 g (2.54)
We must now show that the state limiter solution guarantees that the system does not
exceed its pre-specified state limit in addition to providing stable tracking response. First,
we return to Lyapunov function (2.42) which as shown via (2.54) confirms the stability of
error dynamics (2.38) in the region where the modulation function is active and time-
varying. Since Lyapunov analysis only provides sufficient conditions for stability, we
begin by ensuring that Lyapunov function (2.42) is also valid in the cases when the
modulation function is inactive. In these cases, the limiter is either completely active or
inactive, and stability has already been established by virtue of the fact that all
eigenvalues have negative real components. In the time-invariant case, we have:
34
22
12
1
V E e e
g
(2.55)
In (2.55) we have either
2
g or
2
g based on (2.19) and (2.21), but in
either case g is strictly positive. Differentiating (2.55) along the system trajectories
yields:
22
11
2
2
2
2
2
0
ee
V E e e
g
f
e
g
(2.56)
Now, using LaSalle‟s method, from (2.56) we have:
2
22
2
, : 0 0 0
f
S e e V E e e
g
(2.57)
Substituting
2
0 e into error dynamics (2.41) yields:
1 1 1
00 e e t e (2.58)
Additionally,
2 2 1 1
0 0 0 0 e e g e e (2.59)
This indicates that the origin is the only point in the set S where 0 V , which indicates
that the error dynamics are asymptotically stable. Therefore, Lyapunov function (2.55) is
sufficient for use in cases when the modulation function is not varying.
Using error definition (2.6), suppose that system dynamics satisfy:
ek (2.60)
for some 0 k . In terms of system state, we may rewrite (2.60) as:
35
max
m
x k x (2.61)
where
max
m
x is user-defined maximum permissible value of the reference model position
state, which must be less than the absolute system position limit
lim
x . Redefine:
max
m
kx (2.62)
for some 0 . Substituting (2.62) into (2.61) yields:
lim
max
1
m
x
xx (2.63)
Therefore, if error dynamics satisfy (2.60) system position will stay within the limit as
shown in (2.63). Solving (2.63) for , we get:
max
lim
max
m
m
xx
x
(2.64)
Substituting (2.64) into (2.62) yields:
max
lim m
k x x (2.65)
Returning to Lyapunov function (2.42) along with error constraint (2.60), we have:
max
2
2
max
,,
1
,
k g t e
V e e k e
gt
(2.66)
where
max
e is the maximum error rate achievable/allowable for specific system dynamics.
Lyapunov function (2.66) will form an elliptical level set with the specific shape of the
ellipse defined by user-selected value for
max
e . Because of results proved in [22], the
system trajectories will always be negative definite and therefore dynamics will evolve
36
toward the origin. In other words, once system dynamics begin in a Lyapunov level set
the dynamics must evolve in the negative (shrinking) direction. Therefore, we may
formally state:
max lim
0 , , , , 0 e k V e e k g t e x t x t (2.67)
Recalling that
max
m
x is the user-defined maximum permissible value of the reference
model, we see that (2.65) provides a tuning knob by which the designer may define the
maximum permissible error for which the system dynamics are guaranteed to remain
within system limits.
Chapter 2.2: State Limiter Application to Aircraft Dynamics
In order to apply the limiter augmentation to the aircraft pitch dynamics described in
Chapter 1.2, we note the implicit pitch acceleration command (1.22) may alternatively be
expressed as:
2
2
cmd q IC cmd cmd
grav grav
L
q M M q M
s
q Q Q
(2.68)
Including limiter modification (2.8) to (2.68) yields:
2
2
cmd q IC cmd cmd
grav grav
L
q M M q M x
s
x x q Q Q
(2.69)
Pitch acceleration command (2.69) now takes the form:
bl
cmd cmd cmd
q q x q (2.70)
37
where
bl
cmd
q is baseline pitch acceleration command (2.68), and
2
2
2
cmd cmd cmd
grav
L
q
s
qQ
(2.71)
is the state limiting augmentation component.
Similarly, returning to the aircraft yaw dynamics described in Chapter 1.3, we note
the implicit yaw acceleration command (1.37) may be formed without explicit integral as:
2
2
cmd p r IC cmd grav
r N N p N r N Y R
(2.72)
Or, alternatively:
2
2
cmd p r IC cmd grav
r N N p N r N Y R
(2.73)
Including limiter modification (2.8) to (2.73) yields:
2
2
cmd p r IC cmd
grav
r N N p N r N x
x x Y R
(2.74)
Yaw acceleration command (2.74) now takes the form:
bl
cmd cmd cmd
r r x r (2.75)
where
bl
cmd
r is baseline yaw acceleration command (2.73), and
2
22
cmd cmd
r (2.76)
is the state limiting augmentation component.
We immediately note from pitch acceleration command (2.71), yaw acceleration
command (2.75), and construction of the modulation function (2.9) that when the system
38
is in nominal (away from state limits) region the baseline tracking control law provides
the entire command. As the system approaches state limits within user-defined tolerance
, the state limiting augmentation component gradually activates and modifies the
reference model dynamics. This in turn modifies the closed-loop system dynamics, and
based on user-defined incremental damping ratio and natural frequencies may be used to
modify system behavior near state limits. For example, the user may define the
incremental damping ratio such that in regions near the state limits the closed-loop
behavior becomes overdamped, thus preventing overshoot of state limits.
Chapter 2.3: Application to X-48B Aircraft Simulation
The X-48B is an 8.5% scale version of a full-scale blended-wing-body (BWB)
aircraft designed to investigate the stability and control (S&C) characteristics of this
aircraft configuration. The flight test vehicle has successfully completed more than 75
flights at NASA Dryden since July 2007. Information on the vehicle and flight test results
may be found in [23]-[25]. As shown in [24], the latter portion of the first phase of flight
testing was high risk departure limiter assaults intended to investigate the ability of the
aircraft to prevent entry into uncontrolled flight regimes. The AOA and AOS limiters
presented in previous sections were specifically designed to address assault maneuvers
where representative reference model dynamics would experience overshoot into
undesirable regions outside of the flight envelope. We include simulation results in this
chapter but note that the results presented were subsequently taken to flight testing, where
a total of four successful limiter assault flights took place. Details of the flight test
39
approach along with presentation and discussion of the results may be found in the
following section and in more detail in [26].
The baseline longitudinal control system for the X-48B is defined most simply by
(2.68), with some additional components such as thrust and control allocation effects
neglected for the purposes of this work. The X-48B baseline control law originally
contained an AOA limiter that was satisfactory for large portions of the flight regime but
was found to insufficiently arrest AOA during aggressive assault maneuvers. As a result,
limiter augmentation component was included and pitch acceleration command modified
to (2.70). High-fidelity nonlinear piloted simulation results for slats retracted windup turn
maneuver comparing the original limiter with the proposed limiter augmentation are
shown in Figure 3.
40
Time (sec)
(deg)
AOA Response, Wind-Up Turn
Command
Baseline + Aug.
Baseline
Figure 3: AOA Limiter Performance in Windup Turn
The results show that the limiter augmentation is able to arrest the AOA as the
response approaches the command, which in this case is equal to the limit. The original
limiter is not able to arrest the response and the vehicle would enter an undesirable flight
regime. It should be noted that there is some minor overshoot in the AOA response as the
41
windup turn maneuver is continued past the initial transient. In this specific case, this is
due to a modification of the limiter presented above. The portion of (2.71) due to the
AOA error is only active once the AOA limit is exceeded in implementation – this was
done to avoid the alpha rate decreasing prior to the limit being exceeded. The X-48B
program was willing to allow a small overshoot with graceful return to the limit in
exchange for improved pitch tracking (handling qualities) approaching the limit. We note
that the augmentation could provide strict adherence to the AOA limit if so desired, as
demonstrated in the lateral/directional axes.
The elevator surface position and rate responses to the wind-up turn maneuver are
shown in Figure 4. The elevator responses show that the addition of limiter augmentation
does not adversely affect the surface position or rate, both of which remain within
desirable ranges with smooth dynamics. Per design, the surface response remains dictated
completely by the baseline pitch acceleration command when the modulation function
x is not active (zero).
42
e
Elevator Activity
Baseline + Aug.
Baseline
e
dot
Baseline + Aug.
Baseline
Time (sec)
Baseline + Aug.
Figure 4: Elevator Response for AOA Windup Turn
Finally, the reference model parameters for the AOA maneuver are shown in Figure
5. Per design, the reference damping and natural frequency remain static when the
modulation function x is not active. As the modulation function activates, the
43
damping of the reference model increases, causing the closed-loop system to move
toward overdamped characteristics and preventing overshoot of the state limits.
AOA Augmentation Parameters
Baseline + Aug.
Baseline + Aug.
Time (sec)
Baseline + Aug.
Figure 5: Reference Model Parameters for AOA Windup Turn
In the lateral/directional axes, the baseline control system for the X-48B is defined by
(2.72). The original baseline control system did not include an AOS limiter, so the
44
augmentation scheme described in this paper was designed specifically to address the
limiter assault flight test maneuvers. Limiter augmentation component was included and
yaw acceleration command was modified to (2.75). High-fidelity nonlinear piloted
simulation results for slat extended level flight sideslip step input maneuver comparing
the baseline unlimited control law against the proposed limiter augmentation are shown
in Figure 6.
45
Time (sec)
(deg)
AOS Step Response
Command
Baseline + Aug.
Baseline
Figure 6: AOS Limiter Performance, Step Input
The results show that in the lateral/directional axes, where the theory presented above
was implemented without modification, the limiter is able to fully arrest the closed-loop
dynamics and prevent the AOS from exceeding the limit – which is equal to the
command in this case. We again point out that the nominal AOS response remains intact
46
until such time that the AOS is sufficiently close to the limit to activate the limiter
augmentation. This allows the design to satisfy limiter requirements while still meeting
performance and handling qualities requirements. The rudder surface response is shown
in Figure 7.
r
Rudder Activity
Baseline + Aug.
Baseline
r
dot
Baseline + Aug.
Baseline
Time (sec)
Baseline + Aug.
Figure 7: Rudder Response for AOA Step Command
47
Just as in the longitudinal axis, the rudder response remains smooth and within
desirable ranges for surface deflection and rate with the inclusion of the limiter
augmentation component. The acceleration command and surface response is entirely
dictated by the baseline command until the limiter augmentation component becomes
active, which is illustrated in the similar surface response until the modulation function
becomes nonzero. The reference parameters for the AOS step maneuver are shown in
Figure 8.
48
AOS Augmentation Parameters
Baseline + Aug.
Baseline + Aug.
Time (sec)
Baseline + Aug.
Figure 8: Reference Model Parameters for AOS Step Command
Once again, the reference model parameters remain constant until the modulation
function x becomes active, at which point the parameters change due to the limiter
augmentation. The damping ratio increases, corresponding to a smooth transition to an
overdamped system in order to prevent overshoot in the AOS response.
49
Chapter 2.4: Flight Test of the X-48B AOA and AOS Limiting System
The X-48B‟s Blended Wing Body (BWB) aircraft configuration, presented in Figure
9, represents a design departure from the conventional “tube and wing” shape of
traditional transport aircraft. This novel configuration offers the potential for
revolutionary improvement in performance and efficiency over current day airframe
configurations. A blended-wing configuration is characterized by an overall aircraft
design that provides minimal distinction between wings and fuselage and fuselage and
tail. It closely resembles a flying wing configuration, but concentrates more volume in
the center section of the aircraft than a traditional flying wing.
Figure 9: X-48B BWB Configuration
The X-48B Low Speed Vehicle (LSV) is an 8.5% scale version of a full scale blended
wing body aircraft designed to investigate stability and control characteristics of this
aircraft configuration. Early analysis of blended wing-body aerodynamic characteristics
50
identified the potential for sustained spins and nose-up „tumble‟ post-departure modes.
As a result, one of the goals of the X-48B flight test program was to demonstrate AOA
and sideslip limiters that would provide departure resistance and allow aggressive
maneuvering up to
max
L
C and a sideslip limit equivalent to a full-scale normal landing in a
35kt. crosswind. In the final phase of X-48B handling qualities tests the airplane was
taken to its limit of controlled flight. In the third and final phase, “departure limiter
assaults” were performed to challenge the ability of the aircraft‟s flight control system to
prevent entry into uncontrolled flight regimes and to validate the software algorithms
employed in the computerized flight control system to prevent such occurrence. Through
simulation analysis and flight test the full-envelope AOA and sideslip limits that would
meet program goals and prevent departures from controlled flight were established. The
positive-AOA range of these limits is presented in Figure 10.
The X-48B currently uses a modified model-following control scheme with dynamic
inversion control mixing. The longitudinal and directional reference models are standard
2
nd
order equivalent systems with frequency and damping selected for handling qualities
requirements and stability margins. Tuning for handling qualities in general provides
predictable initial response and tracking characteristics; however, if the desired damping
characteristics are less than critically damped ( <1) will result in an overshoot beyond the
desired steady-state response. If this response, for example, was an aggressive pull to the
AOA limit (coincident with
max
L
C ) the ideal system would experience an overshoot of up
to 2-4 deg AOA for X-48 representative reference model dynamics. Similar performance
would be experienced in the directional axis in response to an aggressive pedal input.
51
Flight testing indicated that this would result in AOA excursions into a region
characterized by an abrupt uncontrollable right wing drop, an undesirable flight
characteristic during maneuvers requiring maximum performance and minimum altitude
loss. Sideslip excursions beyond the recommended limit were not intentionally tested.
- deg
- deg
limit variation with
Slat Retracted Limit
Slat Extended Limit
Figure 10: X-48B Positive AOA Envelope Limits
In order to reduce limit overshoots while maintaining excellent handling qualities
away from the limiter, the limiter developed in Section 2.2 was developed and applied to
the X-48B vehicle.
AOA limiter testing was conducted using the standard windup/wind-down turn
method [27]. A windup turn is a constant airspeed, increasing AOA (and thus normal
acceleration) turn at fixed power, where altitude is traded to maintain airspeed. Above
52
corner speed, the lowest airspeed at which the normal acceleration limit can be attained, a
windup turn will increase AOA until the normal acceleration limit is reached; below
corner speed the windup turn increases AOA and normal acceleration until the AOA limit
is reached. Once the limiting condition (AOA or normal acceleration) is reached, the
windup turn transitions to: a wind-down turn, a constant normal acceleration deceleration
to corner speed, and then a constant AOA deceleration below corner speed. Deceleration
rate, typically specified in knots per second, is modulated by increasing or decreasing the
rate of descent of the airplane. This is a high workload closed-loop test; however, test
tolerances are typically very broad, and usable test data can be obtained with wide
variations in airspeed during the windup portion and in deceleration rate during the wind-
down portion. A windup/wind-down turn progression is presented graphically in Figure
11.
WUT
WDT
Airspeed
N
Z
1
0
Nz Limit
AOA Limit
Vc
Figure 11: Windup/Wind-down Turn V-n Diagram
53
To illustrate the interpretation of X-48B flight test data a representative windup turn
(WUT) / Wind-down turn (WDT) maneuver with annotations is presented in Figure 12.
All WUT/WDT maneuvers presented in this section adhere to this example.
Nz
V
c
B
D
F
G
A
C
E
Figure 12: Representative Windup/Wind-down Turn V-n Diagram
The same maneuver, presented in terms of airspeed and AOA with identical phases
annotated, is presented in Figure 13.
54
AOA
cf
V
c
B
D
F
G
A
C
E
Figure 13: Representative Windup/Wind-down Turn Airspeed/AOA Plot
The segments of the maneuver presented in Figure 12 and Figure 13 are as follows:
A. Initial Condition. This corresponds to the start of data collection and is typically just
after a roll-in to a descending turn is initiated but before AOA has increased
significantly above 1g AOA at the test conditions.
B. Windup segment. During this portion of the maneuver altitude is sacrificed to
maintain airspeed as AOA and normal acceleration are increased. The goal is a
near-constant airspeed, increasing normal acceleration turn up to the test normal
acceleration limit
C. Point C corresponds to the limit normal acceleration for the test. At this point the
maneuver is continued; however, airspeed is allowed to decrease so that AOA
will increase.
55
D. Constant Normal Acceleration Wind-down segment: During this portion of the
maneuver the descending turn is shallowed such that airspeed decreases. AOA is
increased to maintain normal acceleration. The goal is a near-constant normal
acceleration, increasing AOA, decreasing airspeed turn up to the test AOA limit.
E. “Corner Speed”. This corresponds to the point at which the airplane achieves the limit
normal acceleration at the limit AOA. Below this speed the airplane is AOA-
limited; above this speed, normal acceleration limited.
F. Constant AOA Wind-down segment: During this portion of the maneuver the
descending turn is continued with decreasing airspeed and normal acceleration,
while normal acceleration decreases.
The maneuver is intended to be flown as a seamless transition from initial condition
through the windup segment and transitioning to the wind-down segments until
termination at some proscribed minimum airspeed. Ideally once at a limiting condition
(segments C-G) the airplane can “ride the limiter” and transition through each maneuver
phase with no pilot compensation to avoid exceeding a limit. Limiter functionality was
evaluated during all phases, and included limiter performance in the presence of high
approach rates, changing limits while limited, and changing flight conditions while
limited. AOA limiter test results for slat retracted/aft CG and slate extended/forward CG
are presented in Figure 14 and Figure 15.
56
AOA
cf
V
c
WUT
g-limited WDT
AOA-limited WDT
Nz
V
c
WUT
g-limited WDT
AOA-limited WDT
Figure 14: AOA Limiter Results, Slat Retracted/Aft CG
57
Nz
V
c
WUT
g-limited WDT
AOA-limited WDT
AOA
cf
V
c
WUT
g-limited WDT
AOA-limited WDT
Figure 15: AOA Limiter Results, Slat Extended/Fwd CG
58
The AOA limiter limited overshoots to less than 2 deg in all cases tested, and
generally less than approximately 1 deg for all configurations and conditions tested,
including up to 5 deg/sec AOA rates and 5 kt/sec deceleration rates. In the slat
extended/forward CG case approaching the limiter from above corner speed resulted in
an approximate 1 deg steady-state error below corner speed (i.e. the airplane settled
approximately 1 deg above the AOA limit rather than at the limit) with minimal
overshoot. Limiter approaches from below corner speed converged to the limit AOA
condition with minimal overshoot. Limiter performance was acceptable in all cases
evaluated, and the limiter demonstrated functionality in the presence of changing limits.
It should be noted that the X-48B g-limiter is actually an AOA limiter, with the AOA
limit estimated as that AOA that will result in the normal acceleration limit at the current
flight condition (also known as the „AOA-for-g limiter‟). The AOA-for-g function did
not account for lift differences with control surface deflection; consequently, the limiter
with trailing edge up elevon during a windup turn typically underestimates g for any
given AOA, resulting in less than the normal acceleration limit when on the g-limit
portion of the V-n diagram. G-limiter performance was noted to improve somewhat
during aft CG testing, when trailing edge up control deflections were reduced. This
phenomenon is characterized by the „g-limited WDT‟ lines in Figure 14 and Figure 15.
The sideslip limiter was tested by abrupt full pedal inputs from a stabilized flight
condition. Buildup was conducted by initially performing these inputs from steady-
heading sideslips, resulting in smaller pedal inputs and sideslip command changes,
eventually progressing to step inputs from zero sideslip, the worst-case condition. Inputs
59
were tested in both directions to establish symmetry of the limiting function. To illustrate
the interpretation of X-48B flight test data representative step pedal input maneuvers with
annotations are presented in Figure 16. All sideslip limiter test maneuvers presented in
this report adhere to this example.
- deg
- deg
limit variation with
Slat Ret Limit
B2
A1
A2
C2
C1
B1
Figure 16: Representative Sideslip Limiter Maneuvers
A. Initial Condition (Sideslip may be nonzero)
B. Sideslip Change due to Step Pedal Input, Near-Constant AOA and Airspeed.
C. Test Sideslip Limit and Maneuver Completion
In this example the maneuver with the smaller sideslip increment (A1-C1) was flown
prior to the larger sideslip input (A2-C2), thus building up in time rate of change of
2 2 1 1 A C A C
2 2 1 1 A C A C
60
sideslip. Sideslip limiter test results for slat retracted/aft CG and slat extended/aft CG are
presented in Figure 17 and Figure 18 respectively.
- deg
- deg
limit variation with
Slat Ret Limit
Step Input from Stabilized Sideslip
Step Input from Stabilized Sideslip
Figure 17: Sideslip Limiter Results, Slat Ret/Aft CG
61
- deg
- deg
limit variation with
Slat Ext Limit
Step Input from Stabilized Sideslip
Figure 18: Sideslip Limiter Results, Slat Ext/Aft CG
Sideslip limiter performance was considered excellent at all conditions tested, with
typically less than 0.5 deg overshoot. The limiter also demonstrated excellent
compensation during sideslip limit changes as AOA varied (Figure 18).
The AOA and sideslip limiter system developed for the X-48B demonstrated
acceptable performance at all conditions tested, including slat extended and retracted,
forward and aft CG, and low and high assault rates. The incorporation of state-dependent
damping improved the limiter performance by decreasing the AOA and/or sideslip
overshoot at high assault rates. The limiter is now a part of the baseline X-48B control
laws and will be used for envelope protection in subsequent flight tests.
62
Chapter 3: Dynamic Inversion with Adaptive Augmentation
The baseline dynamic inversion control solution presented in Chapter 1 assumed
complete knowledge of the system dynamics. In practical applications, the inevitable
presence of system uncertainties requires additional control design. We propose using
baseline dynamic inversion control presented in Chapter 1 along with adaptive
augmentation to provide tracking performance in the presence of system uncertainties.
The adaptive augmentation component provides tracking performance in the presence of
the system uncertainties.
As in the baseline control design, the control architecture including adaptive
augmentation will be developed for a class of second-order systems in the cascaded form.
This particular class of systems is chosen primarily to clarify and expose key features of
the design process. These systems also naturally appear in flight dynamics and control
problems, which constitute the primary focus and motivation for the control
development. One may immediately note that since the design is based on the dynamic
inversion method, the developed controller can be extended to a generic class of feedback
linearizable MIMO systems with cascade-connected dynamics.
The results of this chapter contribute theoretically justified adaptive augmentation
design for such a generic class of feedback linearizable MIMO systems in cascade form.
The design will be further improved in the next chapters by including the state limiter
previously developed to form an aggregate system containing the combined benefits of
each of the design components.
63
Chapter 3.1: PID Control for Cascaded Systems with Uncertain Dynamics
Similar to Chapter 1.1, we consider 2
nd
order dynamical systems in the cascaded
form, but now include system uncertainties:
0
1 1 1 1 2 1 1
0
2 2 1 2 2 2 1 2
,,
, , , ,
cmd
x F x z B x f x z
x F x x z x f x x z
(3.1)
where components
12
, ff are unknown continuously differentiable functions that
represent the system uncertainties. The control goal remains to design the control input
2
cmd
x so that the system 1
st
state component
1
x tracks any given bounded time-varying
command
1
cmd
xt while keeping all the signals in the closed-loop system bounded,
uniformly in time, now in the presence of system uncertainties
12
, ff .
As in (1.5), we differentiate the first system state component to yield:
1 1 2 1 2 1 2
, , , , , ,
cmd
x f x x z z B x d x x z z (3.2)
where
12
, , , d x x z z now represents the unknown system uncertainty. We now introduce
modified Dynamic Inversion (DI) based Proportional + Integral + Derivative (PID)
control is introduced in the form:
1 11
2 1 1 1 1 1 1
ˆ
m
cmd m m m
DPI
xx
x B x f K x x K x x K v
s
(3.3)
where v represents adaptive augmentation component of baseline dynamic inversion
control law first introduced in (1.6). Substituting (3.3) into (3.2) yields the updated
tracking error dynamics:
64
12
1
1 1 1 1
, , ,
D P I D
D x x z z
e
e K e K e K d K f v
s
(3.4)
Via Introduction of aggregate state
12
, , ,
T
x x x z z , (3.4) may be expressed as:
1
1 1 1 D P I
e
e K e K e K D x v
s
(3.5)
Select adaptive augmentation control signal v to dominate the system uncertainties
online:
1 2 1 2
ˆ ˆ
, , , , , ,
T
DD
v D x x z z x x z z (3.6)
We immediately note that an online approximation of the system uncertainty is not
required in the proposed design, only the ability to dominate the system uncertainty. This
is an important facet of the design, since approximation would require persistency of
excitation conditions [6], [28] that are not verifiable in real-time system operation.
Linear-in-parameters online representation of the uncertain function Dx in (3.5) is
performed on a compact x region , using an N-dimensional regressor vector
N
D
x , with radial basis functions (RBFs) [29]:
ˆ ˆ
T
DD
D x x (3.7)
where
ˆ
N
D
is the vector of online estimated parameters. It is assumed that the
number of RBF components N is large enough, and the components are chosen so that the
uncertainty Dx can be represented within the prescribed tolerance
max
D
on :
*
T
D D D
D x x x (3.8)
65
In (3.8),
*
D
denotes the vector of true unknown constant parameters, and
D
is the
unknown bounded representation error, with a known upper bound:
max
DD
x (3.9)
Subtracting (3.8) from (3.7), the function representation error can be expressed in terms
of the parameter estimation error:
*
ˆ ˆ
D
T
T
D D D D D D D D
e D D
(3.10)
Returning to the tracking error dynamics, substituting (3.6) into (3.5) and using
relation (3.10) yields:
1
1 1 1 D P I D
e
e K e K e K e
s
(3.11)
Using previously derived PID gains (1.11) and substituting into (3.11) yields:
22 1
1 1 1
22
D
e
e k e k e k e
s
(3.12)
Regrouping terms in (3.12) gives:
2 1
1 1 1 1 1
2
D
ke
e k e e k e e e
s
(3.13)
We now introduce the filtered tracking error:
1
1 1 1
f
sk ke
e e e
ss
(3.14)
Using (3.13) and (3.14) yields filtered tracking error dynamics:
2
1 1 1
2
f f f
D
e e e e (3.15)
Filter tracking error dynamics may be expressed in matrix form as:
66
11
2
11
0 1 0
21
ref ff ref
ff
D
ff
B ee A
ee
e
ee
(3.16)
The main idea we wish to exploit now is that by controlling the filtered tracking error,
we will in turn ensure control of the original tracking error, or more specifically:
1 1 1
0 0 0
ff
e e e
(3.17)
We show (3.17) holds as follows. First, if the time derivative of the filtered tracking error
1
f
et is driven to become small then the original tracking error signal
1
et will also
become small. This statement directly follows from the definition (3.14), which could
alternatively be written as:
1 1 1
f
e k e e (3.18)
The explicit solution to (3.18) is:
0
0
1 1 0 1
t
k t t kt f
t
e t e e t e e d
(3.19)
Consequently, if T such that
1 1 0
,
f
e t t t T , then as t we have:
0
0
00
1 1 0 1
11
10
1
t
k t t T kt
tT
k t t T k t t T
e t e e t e d
e e t e
kk
(3.20)
In other words, the absolute value of the original tracking error
1
et approaches
1
k
exponentially fast.
67
Second, if the filtered tracking error
1
f
et is driven to become small then the original
tracking error signal
1
et will also become small. Integrating (3.19) by parts yields:
0
0
0
0 0
0
0
1 1 0 1
1 0 1 1
1 0 1 1 1
t
k t t kt f
t
t t
k t t k t k t ff
t t
t
k t t kt f f f
t
e t e e t e e d
e e t e e k e e d
e e t e t e t k e e d
(3.21)
Consequently, if T such that
1 1 0
,
f
e t t t T , then as t we have:
0
0
00
1 1 0 1 0 1 1
1 0 1 0 1 1 1
12
t
k t t T kt f
tT
k t t T k t t T f
e t e e t T e t T k e d
e e t T e t T e
(3.22)
In other words, if
11
f
et , then as t the following asymptotic relation takes
place:
11
21 e t o (3.23)
Chapter 3.2: Parameter Adaptation and Closed-Loop System Dynamics
Choose a symmetric positive-definite matrix 0 Q and solve the following algebraic
Lyapunov equation:
T
ref ref
P A A P Q (3.24)
Since
ref
A is Hurwitz by design, the Lyapunov equation has unique positive-definite
symmetric solution P , which is used to introduce a Lyapunov function candidate in the
form:
68
1
,
TT
f D f f D D D
V e e Pe
(3.25)
where symmetric positive-definite matrix
D
defines the rate of adaptation.
Differentiating (3.25) along the trajectories of the system (3.16) yields:
1
ˆ
22
T T T
f f f ref D D D D
V e Qe e P B e
(3.26)
Regrouping terms and using (3.10) gives:
1
ˆ
22
T T T T
f f f ref D D D f ref D D
V e Qe e P B e P B
(3.27)
To make the time derivative of V in (3.27) negative outside of a compact , e
subset of choose the following parameter adaptation laws:
ˆˆ
proj ,
T
D D D D f ref
e P B (3.28)
In (3.28) proj denotes the projection operator [30], which forces the adaptive parameters
to evolve in a pre-specified
D
region. Substituting (3.28) into (3.27) yields:
2
TT
f f f ref D
V e Qe e P B (3.29)
Using (3.29), we must now show closed-loop stability and boundedness of the error
dynamics.
Assumption 1: The command
1
cmd
x for reference model (1.2) is chosen such that
12
T
mm
x t x t z t z t (3.30)
forward in time.
We begin by noting that (3.29) implies that
69
2
max
min
2
2
TT
f f f ref D
f f ref D
V e Qe e P B
Q e e P B
(3.31)
where
min
Q is the minimum eigenvalue of Q and
max
max
D x D
x
. We also note
that because of the projection operator the norm of the parameter estimation error will
stay uniformly bounded; that is,
max
t (3.32)
Using (3.31) we can now establish uniform ultimate boundedness (UUB) [8] of the
closed-loop system trajectories. Toward that end, define the following compact subset in
the
f
e region:
max
min
2
ref D
rf
PB
S e r
Q
(3.33)
Also define a minimal level set
T
b f f
e Pe b that contains
r
S . Since
22
min max
T
f f f f
P e e Pe P e (3.34)
then choosing
2
max
b P r (3.35)
implies that for all
f
er ,
2
2
max max
T
f f f
e Pe P e P r b (3.36)
Hence, the set
r
S is contained in the level set
b
.
70
Suppose that all initial values of the filtered tracking error
0 f
et belong to a
compact set
Rf
S e R . Let
T
B f f
e Pe B be the maximal level set which
belongs to
R
S . To maintain closed-loop system stability, a specific relation between the
boundaries for the sets , , ,
b B r
S and
R
S must be imposed. These sets will be used to
prove that the closed-loop system trajectories are UUB. Graphical representation of the
four sets is shown in Figure 19.
Figure 19: Representation of the Sets
r b B R
SS
Choose:
2
min
B P R (3.37)
Then if
T
ff
e Pe B then using (3.34) yields:
2
2
min min
T
f f f
P e e Pe B P R (3.38)
Consequently,
f
eR ; that is, the filtered tracking error is in
R
S . Because of (3.31) and
(3.33), the time derivative V is negative outside of
r
S . Therefore, the filtered tracking
71
error
f
e will enter level set
b
in finite time, and will remain in the set from then on.
Therefore, the closed-loop system trajectories are UUB. Moreover, due to the use of the
projection operator, all the estimated parameters are bounded. Hence, the tracking
problem is solved. In summary, the corresponding total explicit model following control
signal can be written using (3.3), (3.6), and (3.7):
PID controller
11
2 1 1 1 1 1
adaptive augmentation
baseline dynamic inversion controller
ˆ
cmd m m T I
D P D D
K
x B x f x K s K x x B x
s
(3.39)
Chapter 3.3: Design Example: AOA Tracking
This section applies the developed adaptive control methodology to construct an
AOA tracking system for a fixed-wing aircraft, whose short period dynamics including
lift and pitching moment uncertainties can be written as a modified version of (1.15):
,
grav
q IC cmd
L Q q L
q M M q M q M q
(3.40)
where now L is the lift force uncertainty and , Mq represents the pitching
moment uncertainty. The AOA reference model dynamics is chosen in the form of (1.16)
and the corresponding baseline dynamic inversion controller is in the form of (1.21):
2
2
bl m
cmd m q IC m
qq
q q M M q M q q
s
(3.41)
The system dynamics are in the form of (3.1), where:
72
1 2 2
1 1 2
12
, , ,
, 1,
,,
T
cmd
grav IC cmd
grav q IC
x x q z Q M x q
F L Q B F M M q M
f L f M q
(3.42)
Using (3.15), the filtered tracking error signal becomes:
1
11
f m
m
s k s L qq
ee
s s s
(3.43)
Hence, the filtered tracking error vector is:
11
T
T
ff m
fm
qq
e e e q q
s
(3.44)
The regressor vector
D
is chosen to depend on AOA only. Then parameter
adaptation laws are written based on (3.28):
0
ˆˆ
proj ,
1
m
D D D D m
qq
q q P
s
(3.45)
where 0
T
PP is the unique positive definite symmetric solution of the algebraic
Lyapunov equation (3.24) via the Hurwitz reference model matrix
ref
A as specified in
(3.16). Based on (3.7), adaptive pitch acceleration augmentation command becomes:
ˆ
ad T
cmd D D
qt (3.46)
In summary, the total pitch acceleration command consists of the baseline dynamic
inversion command (3.41) and adaptive augmentation (3.46):
bl ad
cmd cmd cmd
q q q (3.47)
The simulation model is chosen to represent longitudinal dynamics of an aerial
vehicle, specifically an F-16 aircraft. Similar to previously derived longitudinal dynamics
73
(1.15), by neglecting the effects of gravity and thrust the short period aircraft dynamics
may be written in matrix form:
1
e
e
e
q
Z
Z
V V
qq
MM M
(3.48)
where is aircraft AOA, q is aircraft pitch rate,
e
is elevator deflection (control
input), V is the trimmed (constant) airspeed, and
,,
e
q
Z Z Z
and
,,
e
q
M M M
are
partial derivatives of the aerodynamic vertical force Z and pitching moment M with
respect to ,,
e
q respectively. Numerical values for the vehicle aerodynamic
derivatives were taken from [4] (Example 5.5-3, Table 3.4-3). These data represent an F-
16 aircraft trimmed at:
2
502ft/sec, alt 0ft, 300lb/ft
c.g. 0.35 , 2.11 deg
T
Vq
c
The resulting open-loop system matrices are:
1.0189 1 0.0022
,
0.8223 1.0774 0.1756
AB
(3.49)
where is in radians, q is in radians/second, and
e
is in degrees.
Baseline flight control is designed for the baseline system without uncertainties
according to (3.41) with selected dynamics 0.6 and 2 rad/sec. The eigenvalues of
the reference dynamics along with their corresponding natural frequencies are shown
below:
74
1 1 1
2 2 2
1.2 0.6 0.6, 2
1.2 0.6 0.6, 2
j
j
(3.50)
The baseline system response to a series of angle-of-attack command doublets is shown
in Figure 20. The response shows that in the absence of uncertainties, the baseline system
tracks the AOA doublet commands while maintaining elevator deflection and rate that are
well within acceptable limits.
0 10 20 30 40 50 60 70 80
-10
-5
0
5
10
, deg.
cmd
ref
0 10 20 30 40 50 60 70 80
-0.4
-0.2
0
0.2
0.4
e
, deg
0 10 20 30 40 50 60 70 80
-10
-5
0
5
10
e
dot
, dps
Time, sec
Figure 20: Baseline Closed-Loop Response to AOA Doublet without Uncertainties
75
Including system pitching moment uncertainty yields updated system dynamics:
0 1.0189 1 0
, 0.8223 1.0774 1
cmd
q
Mq qq
(3.51)
Three types of matched uncertainties are added to the system: 1) linear-in-state
uncertainty
T
pert
Kx , 2) control effectiveness constant uncertainty 0 , and 3) nonlinear-
in-state uncertainty in the form of (3.8). Addition of the uncertainties updates dynamics
(3.51) to:
0 1.0189 1 0
0.8223 1.0774 1
T cmd
pert
q
K x D x qq
(3.52)
Numerical values for the uncertainties were chosen as:
2
2
2
0.411 0.8619 , 0.5, 0.5
c
T
pert
K D x D e
(3.53)
where the center of the Gaussian was set to 2 /180
c
and its width was 0.0233 .
This particular selection of numerical values for
T
pert
Kx and is equivalent to 50%
increase in the static instability M
, 80% decrease in the pitch damping
q
M , and 50%
decrease in the control input effectiveness. These changes imply that the vehicle became
50% more statically unstable, lost 80% of its pitch damping ability, and the aircraft
controllability decreased by 50%. In fact, this combination causes the open-loop system
to become unstable (eigenvalues enter the right-half plane), and in addition there is the
simultaneous injection of a significant nonlinear-in-state uncertainty. Such drastic
changes were motivated by intent to demonstrate the effectiveness of the proposed design
methodology. This particular example was also selected to be similar to previous work
76
presented in [31], [32] so that relevant performance comparisons may be available. The
system total uncertainty versus AOA was calculated at 0 q and is shown in Figure 21.
-5 -4 -3 -2 -1 0 1 2 3 4 5
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
AOA, deg
Figure 21: Total Matched Uncertainty vs. AOA at q=0
77
With only the baseline controller in operation and with the uncertainties included, the
closed-loop system tracking performance degradation can be clearly observed from the
data that are shown in Figure 22. Although the tracking performance is poor, both the
control input and its rate remain small.
0 10 20 30 40 50 60 70 80
-10
-5
0
5
10
, deg.
cmd
ref
0 10 20 30 40 50 60 70 80
-1.5
-1
-0.5
0
0.5
e
, deg
0 10 20 30 40 50 60 70 80
-10
-5
0
5
10
e
dot
, dps
Time, sec
Figure 22: Baseline Closed-Loop Response in Presence of Matched Uncertainties
78
To counteract the effects of the uncertainties, adaptive laws (3.45) were constructed
by solving algebraic Lyapunov equation (3.24) with reference matrix
ref
A as shown in
(3.16). The regressor vector
D
consisted of 11 -dependent and evenly spaced
Gaussians (RBFs). RBF centers were placed at [-10:2:10] degrees of AOA, and all RBF
widths were set to 0.0233 . The rate of adaptation was chosen to be:
100
D
(3.54)
and the total control (elevator deflection) was formed as shown in (3.47). With the
baseline + adaptive augmentation control active, closed-loop system tracking
performance was recovered with acceptable elevator deflection and rates as shown in
Figure 23.
79
0 10 20 30 40 50 60 70 80
-10
-5
0
5
10
, deg.
cmd
ref
0 10 20 30 40 50 60 70 80
-2
-1
0
1
e
, deg
0 10 20 30 40 50 60 70 80
-10
-5
0
5
10
e
dot
, dps
Time, sec
Figure 23: Baseline+Adaptive Closed-Loop Response in Presence of Uncertainties
Finally, the norm of the adaptive parameter is shown in Figure 24. In addition to the
projection operator robustness modification in adaptive laws (3.28) employed to keep the
adaptive parameters within a bounded region, standard dead-zone modification is
80
included to keep the adaptive parameters from evolving when the tracking error is small.
The effect of the dead-zone is seen in the adaptive parameter response, where once the
tracking error becomes small the adaptive parameter stops evolving and remains constant.
0 10 20 30 40 50 60 70 80
0
0.5
1
1.5
2
2.5
Theta hat
Time, sec
Figure 24: Adaptive Parameter Norm vs. Time
81
Chapter 4: Dynamic Inversion with State Limiting and Adaptive
Augmentation
In Chapter 1 we derived a baseline dynamic inversion control solution which assumed
complete knowledge of the system dynamics. The results of the chapter contribute robust
baseline flight control architecture in which both aircraft performance and handling
qualities requirements may be achieved in a convenient manner. While the controller
developed in Chapter 1 was shown to solve the tracking problem, in practical application
there are additional requirements that must be considered during the design process.
In order to control a system while keeping the dynamics within a permissible subset
of the system state space, in Chapter 2 we introduced state limiting augmentation to the
baseline dynamic inversion controllers introduced in Chapter 1. The proposed method
introduces a limiting technique which gradually modifies the expected behavior of the
system dynamics when user-specified criteria are approached / exceeded. Given a
baseline model following dynamic inversion control law designed to meet robustness and
performance requirements throughout a flight envelope, a state-limiting augmentation
component protects the system trajectories from leaving an allowable subset in the
system state space.
In addition, in practical applications the inevitable presence of system uncertainties
requires additional control design. In Chapter 3, we proposed using baseline dynamic
inversion control presented in Chapter 1 along with adaptive augmentation to provide
tracking performance in the presence of system uncertainties. The adaptive augmentation
component provides tracking performance in the presence of the system uncertainties.
82
In this chapter we propose combining the contributions of Chapters 1-3 into an
integrated design. The resulting closed-loop system contains robust baseline control by
design, state limiting to maintain system trajectories within a prescribed region in the
system state space, as well as adaptive augmentation to maintain tracking performance in
the presence of system uncertainties. Formal stability proofs and analyses are included
for the combined design. The results of this chapter contribute a theoretically justified
robust + adaptive design including state limiting for a generic class of feedback
linearizable MIMO systems in cascade form.
83
Chapter 4.1: Baseline+Adaptive Control+State Limiting for Uncertain
Dynamics
Similar to Chapter 3.1, we consider 2
nd
order dynamical systems in the cascaded form
including system uncertainties:
0
1 1 1 1 2 1 1
0
2 2 1 2 2 2 1 2
,,
, , , ,
cmd
x F x z B x f x z
x F x x z x f x x z
(4.1)
We again form baseline dynamic inversion PID control:
1 11
2 1 1 1 1 1 1
ˆ
m
cmd m m m
DPI
xx
x B x f K x x K x x K v
s
(4.2)
where v represents adaptive augmentation component of baseline dynamic inversion
control law first introduced in (1.6). We once again form tracking error dynamics:
1
1 1 1 D P I
e
e K e K e K D x v
s
(4.3)
and select adaptive augmentation control signal v to dominate the system uncertainties
online:
1 2 1 2
ˆ ˆ
, , , , , ,
T
DD
v D x x z z x x z z (4.4)
Again using radial basis functions in the regressor vector and analyses (3.7)-(3.10) we
can express the updated tracking error dynamics:
1
1 1 1 D P I D
e
e K e K e K e
s
(4.5)
We now update baseline PID gains (1.11) to include state limiting component developed
in Chapter 2:
84
2
2
2
2
D x x
P x x x
Ix
K x x k
K x x k x
K x k
(4.6)
Using methodology described in (3.12)-(3.15) we have updated filtered error dynamics
(3.16) that now include both the adaptive augmentation and state limiting design
improvements:
11
2
11
01
0
1 2
ref ff
ref
ff
D
ff
x x x
B ee
A
ee
e
ee x x x
(4.7)
The dependence of
ref
A on system state x is due to the inclusion of state limiter, while
the use of filtered tracking error and parameter estimation error
D
e is due to the inclusion
of adaptive augmentation.
We immediately note that similar to state limiter results in Chapter 2, there are three
cases for which stability of system (4.7) must be investigated:
Case 1:
0 lim
x x x . In this region, the modulation function is inactive, and the
error dynamics may be expressed in terms of the nominal damping and natural
frequency . In this case, the stability results of the LTI system including adaptive
augmentation presented in Chapter 3 directly apply with
ref
A in (4.7) corresponding to
the nominal damping and frequency.
Case 2:
0 lim
x x x . In this region, the modulation function is completely active, and
the error dynamics may be expressed in terms of the full incremented damping
85
and natural frequency . Again, in this case the stability results of the LTI system
including adaptive augmentation presented in Chapter 3 directly apply with
ref
A in (4.7)
corresponding to:
2
01
2
ref
A
(4.8)
Case 3:
lim 0 lim
x x x x . In this region, the modulation function is active and
likely changing due to the evolution of the system states. Without adaptive augmentation
present, we illustrated the stability of this system using Lyapunov function (2.42):
22
12
1
, V t E e e
gt
(4.9)
We now note that Lyapunov function (4.9) may alternatively be expressed as:
,
T
V t E e R t e (4.10)
where:
10
, 0 0 1
0
R t R t t
gt
(4.11)
due to the properties of gt documented in (2.39)-(2.40). Motivated by (4.10) and by
Lyapunov function used in the stability proof for LTI case with adaptive augmentation
(3.25), we propose Lyapunov function:
1
,
TT
f D f f D D D
V e e R t e
(4.12)
86
where Rt is defined in (4.11). Differentiating (4.12) along the system trajectories (4.7)
yields:
21
ˆ
22
TT
f f ref D D D D
V K e e R t B e
(4.13)
where K is defined as in (2.43). We immediately note that based on analysis shown in
(2.44)-(2.54), term
2
f
Ke in (4.13) ensures asymptotic stability in the case without
uncertainties and adaptive augmentation, and therefore the second and third terms in
(4.13) are of interest here. Rearranging terms in (4.13) yields:
21
ˆ
22
T T T
f f ref D D D f ref D D
V K e e R t B e R t B
(4.14)
To make the time derivative of V in (4.14) negative outside of a compact
,
f
e
subset of choose the following parameter adaptation laws:
ˆˆ
proj ,
T
D D D D f ref
e R t B (4.15)
Substituting (4.15) into (4.14) yields:
2
2
T
f f ref D
V K e e R t B (4.16)
Using (4.16), we must now show closed-loop stability and boundedness of the error
dynamics. Similar to analysis presented in Chapter 3:
Assumption 1: The command
1
cmd
x for reference model (1.2) is chosen such that
12
T
mm
x t x t z t z t (4.17)
forward in time.
We begin by noting that (4.16) implies that
87
2
2
max
2
2
T
f f ref D
ref f f ref D
V K t e e R t B
K t B e e R t B
(4.18)
We further note that because previous analysis has shown that term
2
f
Ke ensures
asymptotic stability in the case without uncertainties, we may bound (4.18) further by
considering the case where Kt is minimum and Rt is maximum, corresponding to
the “worst” case from a Lyapunov stability analysis:
2
max
min max
2
ref f f ref D
V K B e e R B (4.19)
We note that the minimum and maximum values of matrices Kt and Rt are known
due to their definitions and the properties of , f t g t given in (2.39). We also note that
because of the projection operator the norm of the parameter estimation error will stay
uniformly bounded; that is,
max
t (4.20)
Similar to Chapter 3, using (4.19) we can now establish uniform ultimate
boundedness (UUB) of the closed-loop system trajectories. Toward that end, define the
following compact subset in the
f
e region:
max
max
min
2
ref D
rf
ref
RB
S e r
KB
(4.21)
Also define a minimal level set
T
b f f
e R t e b that contains
r
S . Since
22
min max
T
f f f f
R e e R t e R e (4.22)
88
then choosing
2
max
b R r (4.23)
implies that for all
f
er ,
2
2
max max
T
f f f
e R t e R e R r b (4.24)
Hence, the set
r
S is contained in the level set
b
. Suppose that all initial values of
the filtered tracking error
0 f
et belong to a compact set
Rf
S e R . Let
T
B f f
e R t e B be the maximal level set which belongs to
R
S . To maintain
closed-loop system stability, a specific relation between the boundaries for the sets
, , ,
b B r
S and
R
S must be imposed. These sets will be used to prove that the closed-
loop system trajectories are UUB. Choose:
2
min
B R R (4.25)
Then if
T
ff
e R t e B then using (4.22) yields:
2
2
min max
T
f f f
R e e R t e B R R (4.26)
Consequently,
f
eR ; that is, the filtered tracking error is in
R
S . Because of (4.19) and
(4.21), the time derivative V is negative outside of
r
S . Therefore, the filtered tracking
error
f
e will enter level set
b
in finite time, and will remain in the set from then on.
Therefore, the closed-loop system trajectories are UUB. Moreover, due to the use of the
projection operator, all the estimated parameters are bounded. Hence, the tracking
problem is solved.
89
Chapter 4.2: Design Example: AOA Tracking
In this section, we apply the combined baseline control + adaptive augmentation
along with state limiting methodology to construct an AOA tracking system for a fixed-
wing aircraft. Similar to the example developed in Chapter 3, we note that fixed-wing
aircraft short period dynamics including lift and pitching moment uncertainties can be
written as a modified version of (1.15):
,
grav
q IC cmd
L Q q L
q M M q M q M q
(4.27)
where L is the lift force uncertainty and , Mq represents the pitching
moment uncertainty. The AOA reference model dynamics is chosen in the form of (1.16)
and the corresponding baseline dynamic inversion controller is in the form of (1.21) but
now including the state limiter improvement:
2
2
bl m
cmd m q IC x x m x
qq
q q M M q M x x q q x
s
(4.28)
The system dynamics are in the form of (3.1), where:
1 2 2
1 1 2
12
, , ,
, 1,
,,
T
cmd
grav IC cmd
grav q IC
x x q z Q M x q
F L Q B F M M q M
f L f M q
(4.29)
Using (3.15), the filtered tracking error signal becomes:
1
11
f m
m
s k s L qq
ee
s s s
(4.30)
90
Hence, the filtered tracking error vector is:
11
T
T
ff m
fm
qq
e e e q q
s
(4.31)
The regressor vector
D
is chosen to depend on AOA only. Based on stability
analysis developed in the previous section, we note that the parameter adaptation laws
have changed due to the presence of state limiting component and are now written based
on (4.15):
0
ˆˆ
proj ,
1
m
D D D D m
qq
q q R t
s
(4.32)
where 0
T
R t R t is defined in (4.11). Similar to Chapter 3, adaptive pitch
acceleration augmentation command becomes:
ˆ
ad T
cmd D D
qt (4.33)
In summary, the total pitch acceleration command consists of the baseline dynamic
inversion command (4.28) and adaptive augmentation (4.33):
bl ad
cmd cmd cmd
q q q (4.34)
The simulation model is again chosen to represent longitudinal dynamics of an F-16
aircraft. Similar to previously derived longitudinal dynamics (1.15), by neglecting the
effects of gravity and thrust the short period aircraft dynamics may be written in matrix
form:
1
e
e
e
q
Z
Z
V V
qq
MM M
(4.35)
91
where is aircraft AOA, q is aircraft pitch rate,
e
is elevator deflection (control
input), V is the trimmed (constant) airspeed, and
,,
e
q
Z Z Z
and
,,
e
q
M M M
are
partial derivatives of the aerodynamic vertical force Z and pitching moment M with
respect to ,,
e
q respectively. The same numerical values for the vehicle aerodynamic
derivatives were employed as in Chapter 3, with resulting open-loop system matrices:
1.0189 1 0.0022
,
0.8223 1.0774 0.1756
AB
(4.36)
where is in radians, q is in radians/second, and
e
is in degrees.
Baseline flight control is designed for the baseline system without uncertainties and
without limiting according to (4.28) with the same selected dynamics of 0.6 and
2 rad/sec as used in previous examples. Just as shown in Chapter 3, the nominal
system response to a series of angle-of-attack command doublets is shown in Figure 25.
The response shows that in the absence of uncertainties, the baseline system tracks the
AOA doublet commands while maintaining elevator deflection and rate that are well
within acceptable limits.
92
0 10 20 30 40 50 60 70 80
-10
-5
0
5
10
, deg.
cmd
ref
0 10 20 30 40 50 60 70 80
-0.4
-0.2
0
0.2
0.4
e
, deg
0 10 20 30 40 50 60 70 80
-10
-5
0
5
10
e
dot
, dps
Time, sec
Figure 25: Baseline Closed-Loop Response to AOA Doublet without Uncertainties
We note that because of the selected damping ratio of 0.6, the response does exhibit
some overshoot. In order to arrest the overshoot, we employ the limiter feature developed
93
in Chapter 2. Without uncertainties and assuming an AOA limit of 5 degrees, 0.1 and
0.5, 0.5 , the response including limiter is shown in Figure 26.
0 10 20 30 40 50 60 70 80
-10
-5
0
5
10
, deg.
cmd
ref
0 10 20 30 40 50 60 70 80
-0.5
0
0.5
e
, deg
0 10 20 30 40 50 60 70 80
-20
-10
0
10
20
e
dot
, dps
Time, sec
Figure 26: Closed-Loop Response to AOA Doublet: No Uncertainties, with Limiter
94
The response shows that both the reference model and corresponding system response
now remain within the system state limits. A detailed plot of the system response
showing the limiter activation is shown in Figure 27.
0 1 2 3 4 5 6 7
0
5
10
, deg.
cmd
limit area
ref
0 1 2 3 4 5 6 7
0
0.5
1
( )
0 1 2 3 4 5 6 7
0.5
1
1.5
x
0 1 2 3 4 5 6 7
2
2.5
x
time, sec
Figure 27: AOA Doublet Response: No Uncertainties, with Limiter (Zoom)
95
In Chapter 3 we showed that in the presence of significant uncertainties, adaptive
augmentation is able to restore tracking performance for the system. We now show the
total baseline + adaptive augmentation + limiter system is able to retain tracking
performance in the presence of uncertainties while simultaneously keeping the system
within the state limits. Consider the same uncertainties as previously investigated, that is:
50% increase in the static instability M
, 80% decrease in the pitch damping
q
M , and
50% decrease in the control input effectiveness. These changes imply that the vehicle
became 50% more statically unstable, lost 80% of its pitch damping ability, and the
aircraft controllability decreased by 50%.
The adaptive law implemented in Chapter 3 must be slightly modified to
accommodate the inclusion of the state limiting feature, and is updated to (4.15). The
limiter parameters remain the same: assume an AOA limit of 5 degrees, 0.1 and
0.5, 0.5 . Although the adaptive law has changed due to the inclusion of Rt
rather than P , we keep the adaptive architecture the same is previous: the regressor
vector
D
consisted of 11 -dependent and evenly spaced Gaussians (RBFs). RBF
centers were placed at [-10:2:10] degrees of AOA, and all RBF widths were set to
0.0233 . The rate of adaptation was chosen to be:
200
D
(4.37)
We increased the adaptation rate to account for the additional deviation from the
reference model that the limiter may induce. The system response is shown in Figure 28:
96
0 10 20 30 40 50 60 70 80
-10
-5
0
5
10
, deg.
cmd
ref
0 10 20 30 40 50 60 70 80
-2
-1
0
1
e
, deg
0 10 20 30 40 50 60 70 80
-20
-10
0
10
20
e
dot
, dps
Time, sec
Figure 28: AOA Doublet Response: Uncertainties with Adaptive+Limiter
The response shows the cumulative benefit of the total design. The baseline design
provides robust control that is able to handle significant uncertainties. In the presence of
extreme uncertainty, the baseline performance degrades but is restored by the inclusion of
97
adaptive augmentation. The limiter modification further adds the ability to keep the
system within state limits while maintaining the benefits of the adaptive augmentation. A
detailed response plot is shown in Figure 29.
0 1 2 3 4 5 6 7
0
5
, deg.
cmd
limit area
ref
0 1 2 3 4 5 6 7
0
0.5
1
( )
0 1 2 3 4 5 6 7
0.5
1
1.5
x
0 1 2 3 4 5 6 7
2
2.5
x
time, sec
Figure 29: AOA Doublet Response: Uncertainties with Adaptive+Limiter (Zoom)
98
The initial deviation of the response from the reference model shown in Figure 29 is
due to the uncertainties, but as shown the adaptive augmentation is able to restore
tracking performance while the limiter keeps the system within limits. The norm of the
adaptive parameter is shown in Figure 30.
0 10 20 30 40 50 60 70 80
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Theta hat
Time, sec
Figure 30: Adaptive Parameter Norm vs. Time
99
In addition to the projection operator robustness modification in adaptive laws (4.32)
employed to keep the adaptive parameters within a bounded region, standard dead-zone
modification is included to keep the adaptive parameters from evolving when the tracking
error is small. The effect of the dead-zone is seen in the adaptive parameter response,
where once the tracking error becomes small the adaptive parameter stops evolving and
remains constant.
100
Chapter 5: Adaptive Design with Improved Performance under Input Time-
Delays
In Chapters 1-4 we derived a model-following control design resulting in a closed-
loop system containing robust baseline control by design, state limiting to maintain
system trajectories within a prescribed region in the system state space, as well as
adaptive augmentation to maintain tracking performance in the presence of system
uncertainties. While the dynamic inversion approach employed for the baseline control
design was derived for general second-order dynamical systems in cascaded form, it is
used extensively in aerospace flight control applications, particularly in the area of
piloted flight vehicles. The popularity of this design approach is largely due to the ability
of the designer to easily define target closed-loop natural frequency and damping ratios
that satisfy flying qualities requirements for piloted aircraft. These requirements can be
found in the FAA Federal Air Regulations (FAR) Part 25: Airworthiness Standards for
Piloted Aircraft and MIL-F-8785C (Military Specification for Flying Qualities of Piloted
Airplanes) for commercial and military piloted aircraft, respectively.
Another popular control architecture that is used heavily in aerospace applications is
optimal control. Application of optimal control methods, specifically Linear Quadratic
Regulator (LQR) techniques, relies on the inherent robustness properties provided by
LQR-optimal controllers. It is well known that proper selection of LQR design
parameters will achieve 6 dB gain margin and at least 60 degrees of phase margin at the
system control input break points. This is of significant importance for aerospace
applications, where uncertainties including latency and actuator dynamics can reduce
101
system robustness at the system control input. Therefore, the inherent robustness
properties of LQR make this technique very attractive for baseline control design in
aerospace applications, particularly in cases such as unmanned vehicles and munitions,
where uncertainties may be more dramatic and design goals often include maximum
performance. While the LQR approach does not offer the immediate insight into closed-
loop frequency and damping that dynamic inversion allows, LQR techniques are
attractive for piloted applications because of the inherent robustness and a good design
will be tailored to also meeting flying qualities requirements.
It has been shown that LQR optimal controllers can tolerate classes of uncertainties
that may exist in the system control channels, also called “matched” uncertainties since
they appear where control inputs exist in the system dynamics. In the presence of such
uncertainties, baseline closed-loop system performance will degrade. While the inherent
robustness properties of LQR are attractive, it is worth noting that these controllers are
designed to be robust to the entire class of matched uncertainties and therefore may
become overly conservative.
In this chapter, we use the LQR-optimal technique to provide a robust baseline
control design. Similar to the philosophy presented in previous chapters, we again use
adaptive augmentation to provide tracking performance in the presence of system
uncertainties. The now-classical model reference adaptive control (MRAC) design
approach is applied and modified to specifically address systems with matched
uncertainties and input time-delay. We then propose an alternative design approach to
address the uncertainties and input time-delay while improving performance under input
102
time delay. Using aerospace simulations consistent with previous chapters, we illustrate
that the design approach is able to restore baseline closed-loop performance in the
presence of both matched uncertainty and input time-delay, while showing improved
performance in the presence of input time-delay compared with classical MRAC design.
103
Chapter 5.1: Problem Definition and Baseline Optimal Control Design
Similar to Section 3.1 we consider dynamical systems including system uncertainties,
but without restricting the problem formulation to 2
nd
order cascaded form systems.
Instead consider a general class of MIMO uncertain systems:
p
TT
p p p p m m u u d d p
s
dx
x A x B I s u x
(5.1)
where
p
n
x is the system state vector and
m
u is the control input. The term
T
p d d p
d x x (5.2)
represents linear-in-parameters state-dependent matched uncertainty, where
Nm
d
is
a matrix of unknown constant parameters, and
N
dp
x is the known N-dimensional
regressor vector whose components are Lipschitz continuous functions of
p
x . The term
T
uu
ss (5.3)
represents linear-in-parameters control multiplicative dynamic uncertainties, where
u
Nm
u
is a matrix of unknown constant parameters, and
u
Nm
u
s
is the known
u
Nm -dimensional regressor matrix whose components are strictly proper stable
transfer functions. In order to shorten notation, we are going to mix time and frequency
domain variables in one and the same equation, such as in (5.1). This simply allows us to
define a time-domain function whose Laplace transform equals (5.3). Furthermore, there
104
is no loss of generality to assume that components of the regressor are strictly proper. In
fact, suppose
u
s is only proper. Then we can write it as:
0 uu
ss (5.4)
where
0
is a constant matrix and
u
s is a strictly proper transfer function matrix.
Substituting (5.4) into (5.1) gives:
0
0
1
00
T
u
T
p p p p m m u u p
TT
p p p m m u u u p
T T T
p p p m m u m m m m u u u p
s
x A x B I s u d x
A x B I s u d x
A x B I I I s u d x
(5.5)
where we assumed that
0
T
m m u
I
is invertible. We assume that
nm
p
B
is
constant and known,
nn
p
A
is constant and unknown, and
mm
is a constant,
diagonal, and unknown matrix with positive diagonal elements corresponding to control
input deficiencies in each input channel.
Motivation for the inclusion of (5.3) in the problem statement comes from the desire
to mitigate time delays in the input channel. The transfer function of an input time delay
is given by
s
eu
(5.6)
This expression can cause difficulties in control design and analysis because (5.6) is not a
rational transfer function. Therefore in practice, Padé approximations [33] are often used
105
to approximate a pure time delay by a proper rational transfer function. Use of Padé or
other rational function approximation (RFA) techniques to model time delay allows for
application of problem formulation (5.1) by noting that
mm
U t I s u t
(5.7)
For example consider a time-delay with 0.01 second corresponding to a one-
sample time-delay in a system with a sample rate of 100Hz, which is a typical
representative rate for modern aerospace control systems. Figure 31 below shows the
time response of the pure time-delay, along with that of two Padé approximations of
order one and four respectively:
106
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Time, sec
Input
u
u
delay
u
1st order Pade
u
4th order Pade
Figure 31: Time Response of Pure Time-Delay and Padé Approximations
In addition, the frequency response is shown below in Figure 32.
107
10
-1
10
0
10
1
10
2
10
3
-1
-0.5
0
0.5
1
Magnitude (dB)
10
-1
10
0
10
1
10
2
10
3
-600
-500
-400
-300
-200
-100
0
Phase (deg)
Frequency, rad/sec
u
delay
u
1st order Pade
u
4th order Pade
Figure 32: Frequency of Pure Time-Delay and Padé Approximations
The time and frequency domain responses show that the Padé approximation is able
to approximate the pure-time delay with increasing fidelity by increasing the order of the
Padé approximation. However, increased order can lead to numerical instability, and
108
therefore in practice the designer must balance accuracy of the approximation against
numerical robustness requirements. In the case of the 1
st
order Padé approximation, the
corresponding transfer function is:
200
200
s
s
(5.8)
Therefore in this example, the dynamic uncertainty in terms of (5.7) would be:
11
400
1
200
I
s
s
(5.9)
Returning to the problem formulation, the control goal remains the same: Design the
control input u such that the system controlled (i.e. regulated) output
m
pp
y C x (5.10)
tracks any given bounded time-varying external command while keeping all the signals in
the closed-loop system bounded, in the presence of the system uncertainties
, , ,
p d u
A .
Define the system tracking error:
y
e t y t r t (5.11)
Augmenting the system dynamics (5.1) with the integrated tracking error
yI y
e e y r (5.12)
yields extended open-loop dynamics:
m m p c
x A x B I s u d x B r
(5.13)
where
T T n
yI p
x e x is the extended system state vector. The system matrices are:
109
0 0
,,
00
pp
m m p m m mm
c
n m p n m p
CI
A B B
A B
(5.14)
with system controlled output
0
p m p
y C x C x
(5.15)
While constant matrix A is assumed to be unknown, we assume that the well-known
model matching conditions [7] are met for the system of interest. That is, given a known
constant matrix
nn
k
A
and the unknown positive-definite diagonal constant matrix
, there must exist a possibly unknown constant gain matrix
d
K such that:
T
kd
A A B K (5.16)
Using assumption (5.16), system dynamics (5.13) may be written as:
T
TTT
k d d u d p c
u
x
x A x B u K x B r
su
(5.17)
As in previous chapters, a baseline control design is performed for the system
dynamics (5.17) in the absence of uncertainties, which corresponds to the situation when
, 0, 0
TT
d d p d
I x K and the time delay 00 s . Using the LQR
servomechanism design technique [34], an optimal linear state feedback control solution
in PI form is derived as:
T
bl x
u K x (5.18)
110
We note that any properly designed state feedback control solution of the form (5.18)
would be acceptable for baseline design. We choose the LQR optimal architecture
because of the robustness properties inherent in the design. Since the adaptive control
signal will be designed as an augmentation, it is desirable to have maximum robustness in
the baseline design to guarantee adequate system performance in the absence of
uncertainties. Substituting baseline control (5.18) into the system dynamics (5.17) in the
case without uncertainties leads to the closed-loop dynamics:
ref
T
k x c
A
x A B K x B r (5.19)
The closed-loop system (5.19) represents the desired closed-loop dynamics in the
absence of uncertainties, and will be used as the reference model for the model reference
adaptive control augmentation scheme.
Chapter 5.2: Adaptive Augmentation via Classical MRAC Architecture
The baseline controller was designed for the system under nominal conditions. If
uncertainties are present, the baseline controller may no longer provide adequate
performance or stability. Therefore, we introduce classical MRAC adaptive augmentation
to cope with the system uncertainties. We use the reference model defined in (5.19) and
synthesize the total control input as the sum of the baseline LQRPI component and
adaptive augmentation (to be constructed):
111
ˆ
ˆ ˆ
ˆ
bl
ad
TT
bl ad x x p
u
u
T
bl
u u u K x K x d x s u
u
(5.20)
In (5.20),
ˆ nm
x
K
is the incremental adaptive feedback gain designed to counteract the
effects of system uncertainties and
d
K ,
ˆ
m
s is the on-line approximator of the
system input time-delay uncertainty approximation, and
ˆ
m
p
dx is the on-line
approximator of the matched system uncertainty
p
dx . On-line approximator
ˆ
s
contains regressor matrix
u
Nm
u
s
whose components are strictly proper transfer
functions with stable poles and incremental adaptive weight
ˆ u
Nm
u
:
ˆˆ
T
uu
ss (5.21)
With the Padé approximations, the regressor matrix
u
s in (5.21) would contain
u
Nm Padé transfer functions which may be a combinations of various orders and time-
delay values selected based on the knowledge of the system characteristics. The on-line
approximator
ˆ
p
dx contains the regressor vector
Nm
p
x
with N radial basis
functions (RBFs) and the adaptive weight matrix
ˆ
Nm
d
:
ˆ ˆ
T
p d d p
d x x (5.22)
Substituting (5.20) into the original system dynamics (5.17) gives:
1
T
TT
ref ad x c
x A x B u I K x B r
(5.23)
112
or equivalently,
T
ref ad c
x A x B u B r (5.24)
with the defined regressor vector:
T
d p u
x x s u (5.25)
and the matrix of unknown / ideal parameters:
1
T
x
T
TT
dx
K
K I K
(5.26)
Substituting adaptive component
ad
u from (5.20) into dynamics (5.24) yields
ˆ
T
ref c
x A x B B r
(5.27)
where is the matrix of parameter estimation errors. We now introduce the state
tracking error
ref
e x x (5.28)
and calculate the tracking error dynamics:
T
ref
e A e B (5.29)
Using Lyapunov design approach, bounded output tracking is achieved through on-
line parameter adaptation laws [34]:
ˆ T
e P B
(5.30)
Or, in terms of the system parameters:
113
ˆ
ˆ
ˆ
d
u
T
xx
T
d d p
T
uu
K x e P B
x e P B
s u e P B
(5.31)
In (5.31), symmetric positive-definite matrices
,,
du
x
represent the adaptation
rates and P is the unique symmetric positive-definite solution of the Lyapunov algebraic
equation:
T
ref ref
P A A P Q (5.32)
for a symmetric positive-definite matrix Q. Extending the design to MIMO systems with
non-parametric uncertainties is straightforward and can be accomplished using well-
known in adaptive control robustness methods. For this control scheme in
implementation, the dead-zone and the Projection Operator robustifications are critical.
We note that design approach similar to that presented here resulting in adaptive laws
(5.31) is well established and has been applied to many aerospace applications in the past
decade [1]-[3], [5], [19], [31], [32], [34]. A notable difference presented here is the
inclusion of term (5.3). The motivation for this term is to address time-delays, but the
analysis can be used for any stable proper dynamic uncertainty meeting the conditions of
(5.3).
Relations (5.20) and (5.30) solve the tracking problem with globally asymptotically
stable closed-loop dynamics for any symmetric positive definite rates of adaptation
.
However, it is well known that if this matrix has large singular values, the system will
114
often contain undesirable transient oscillations. We will show such behavior in the
forthcoming aerospace simulations.
Chapter 5.3: Alternative Adaptive Augmentation via B-SPM Model
In order to explore the performance of adaptive design (5.31) under time-delay, we
use an alternative design architecture developed in [7], [28]. Begin by assuming the ideal
case, that is, when the uncertainties present in the system are exactly known. Substituting
total control signal (5.20) into the system dynamics (5.17) yields:
ˆ ˆˆ
T T T
k x x p c
x A x B K x K x d x s u B r (5.33)
We make the following assumptions:
1. Matrix is invertible
2. Matrix B is full rank
Note: assumption 1 is satisfied by the definition of as a positive definite diagonal
matrix. In order to achieve the desired system response indicated by the reference model
(5.19), setting:
1
ˆ
ˆ ˆ
ˆˆ
T T T
x d x
TT
p d d p d d p
TT
u u u u
K K I K
d x x x
sss
(5.34)
will recover the reference dynamics in the presence of uncertainties. This is due to the
augmentation explicitly cancelling the system uncertainties. The fact that relations (5.34)
may be derived to solve the tracking problem proves the existence of a control solution,
indicating that the problem must be solvable.
115
The assumption that the system uncertainties are exactly known is unrealistic – if it
were true such terms would have been included in the baseline system dynamics and not
treated as uncertainties. We therefore attempt to approximate parameters
ˆ
x
K ,
ˆ
s ,
ˆ
p
dx using methods from [7], [28] and bilinear static parametric model (B-SPM).
Given (5.17), we wish to express the system in the form:
T
Yu (5.35)
by collecting unknown terms on one side and then filtering both sides. First, define the
matrix of unknown parameters:
T T T T
d d u
K
(5.36)
and the known regressor matrix:
T
T T T
du
x s u
(5.37)
The system dynamics (5.17) may now be expressed as:
T
kc
x A x B u B r (5.38)
Dynamics (5.38) make up the model that will be used for estimation. Collecting known
terms on one side, and again assuming that B is full rank yields:
T
kc
B x A x B r u
(5.39)
where B
is the Moore-Penrose inverse. We do not assume availability of the state
derivative as a measurement, and therefore filter each side of (5.39) with a strictly proper
stable transfer function Gs . An additional benefit of the filtering technique is the
ability to shape the measured quantities and to avoid known frequency regions associated
116
with uncertain dynamics, noise, and/or disturbances. Denote filtered quantities with a
subscript, i.e.
f
x G s x (5.40)
Filtering each side of (5.39) results in:
T
f k f c f f f
Y
B x A x B r u
(5.41)
We can now compute
f
x due to relation (5.40), and see that (5.41) is in the desired form
of the B-SPM (5.35). The unknown parameters are on the right-hand side and are
replaced with their respective estimates to form the estimated output:
ˆ ˆ ˆ
T
ff
Yu (5.42)
Using a gradient-based adaptive law [7], [28] we obtain:
ˆ
ˆˆ
T
f
TT
ff
u
(5.43)
where the normalized estimation error is:
2
ˆ
s
YY
m
(5.44)
and the normalizing signal
s
m is designed to bound
f
,
f
u from above. An example of
s
m with this property is [7], [28]:
2
1
TT
s f f f f
m u u (5.45)
The adaptive gains are free design parameters that satisfy 0
T
and 0
T
.
For simplicity we may allow
to take the form:
117
00
00
00
d
u
d
(5.46)
Using (5.46) we may express the parameter update equations (5.43) as:
ˆ
ˆ
ˆ
ˆ ˆ ˆ
df
u
f
T
d d f
T
dd
T
uu
f
T T T
f d f d d p
Kx
su
u K x x
(5.47)
Extending the design to MIMO systems with non-parametric uncertainties is
straightforward and can be accomplished using well-known in adaptive control
robustness methods. For this control scheme in implementation, the dead-zone and the
Projection Operator robustifications are critical. The total control signal (5.20) is derived
in terms of adaptive parameters (5.47):
1
ˆ ˆ ˆ ˆ
bl
ad
T T T T T
bl ad x d x d d p u u
u
u
u u u K x K I K x x s u
(5.48)
This is a dynamic controller and its output, the control signal u , is realizable (i.e.
computable) since
u
s is a strictly proper transfer function. Following analysis
presented in [7], [28] adaptive law (5.47) provides closed-loop stability with all signals in
the closed loop system bounded. In addition, it provides that the regulation error will be
of the order of the modeling error in mean square sense. In this case, the modeling error
does not exist since we have assumed that the dynamic uncertainty may be exactly
matched via (5.3). This implies that 0 e as t , which indicates that the system
118
state trajectory will track the reference model. Consequently, the closed-loop system is
stable, all signals in the closed-loop system are bounded, and the model reference
tracking error asymptotically converges to zero forward in time.
Chapter 5.4: Design Example: AOA Tracking
We demonstrate the efficacy of the adaptive schemes via simulation on a simplified
model which is representative of flight dynamics and which has been used extensively in
published adaptive control literature. The simulation will also be used to show the
improved performance characteristics of the proposed alternative adaptive design
including normalization. As in previous chapters, the simulation model is again chosen to
represent longitudinal dynamics of an F-16 aircraft. Similar to previously derived
longitudinal dynamics (1.15), by neglecting the effects of gravity and thrust the short
period aircraft dynamics may be written in matrix form:
1
e
e
q
e
q
Z Z
Z
V V V
qq
M M M
(5.49)
where is aircraft AOA, q is aircraft pitch rate,
e
is elevator deflection (control
input), V is the trimmed (constant) airspeed, and
,,
e
q
Z Z Z
and
,,
e
q
M M M
are
partial derivatives of the aerodynamic vertical force Z and pitching moment M with
respect to ,,
e
q respectively. The same numerical values for the vehicle aerodynamic
derivatives were employed as in Chapter 3, with resulting open-loop system matrices:
119
1.0189 0.9051 0.0022
,
0.8223 1.0774 0.1756
AB
(5.50)
where is in radians, q is in radians/second, and
e
is in degrees.
Baseline flight control is designed for the baseline system without uncertainties.
Similar to previous chapters, we define the system controlled output to be the vehicle
AOA:
10
C
y
q
(5.51)
Augmenting the system dynamics with the integrated tracking error following the
procedure shown in (5.11)-(5.15) and designing baseline control (5.18) via LQR methods
with weighting matrices:
diag 100 0 0 , 1 QR (5.52)
yields baseline LQR feedback gains:
10 10.8786 6.0589
T
x
K (5.53)
Baseline feedback control gains (5.53) lead to the reference model (5.19) that will be
used in adaptive augmentation designs. The eigenvalues of the reference dynamics along
with their corresponding natural frequencies are shown below:
1 1 1
2 2 2
3 3 3
0.613 0.67 0.675, 0.907
0.613 0.67 0.675, 0.907
1.96 1, 1.96
j
j
(5.54)
The baseline system response to a series of angle-of-attack command doublets is shown
in Figure 33. The response shows that in the absence of uncertainties, the baseline system
120
tracks the AOA doublet commands while maintaining elevator deflection and rate that are
well within acceptable limits.
0 10 20 30 40 50 60 70 80
-10
-5
0
5
10
, deg.
cmd
ref
0 10 20 30 40 50 60 70 80
-1
-0.5
0
0.5
1
e
, deg
0 10 20 30 40 50 60 70 80
-1
0
1
2
e
dot
, dps
Time, sec
Figure 33: Baseline Closed-Loop Response to AOA Doublet without Uncertainties
Including system uncertainties yields dynamics in the form of (5.17):
121
00 0 1 0
0 1 0
0
ee
pert
T
d
ee
k
yI yI
q T
xe
u
K
q
xx
A B
A
ee
ZZ Z
Z
Kd
V V V V
qq
M M M M
1
0
0
c
cmd
B
(5.55)
Similar to Chapter 3, three types of matched uncertainties are added to the system: 1)
linear-in-state uncertainty
T
pert
Kx , 2) control effectiveness constant uncertainty 0 ,
and 3) nonlinear-in-state uncertainty in the form of (5.2). Numerical values for the
uncertainties were chosen as:
2
2
2
4.6839 9.8197 , 0.5, 0.5
c
pert
T
xp
K d x d e
(5.56)
where the center of the Gaussian was set to 2 /180
c
and its width was 0.0233 .
This particular selection of numerical values for
pert
T
x
K and is equivalent to 50%
increase in the static instability M
, 80% decrease in the pitch damping
q
M , and 50%
decrease in the control input effectiveness. These changes imply that the vehicle became
50% more statically unstable, lost 80% of its pitch damping ability, and the aircraft
controllability decreased by 50%. In fact, this combination causes the open-loop system
to become unstable (eigenvalues enter the right-half plane), and in addition there is the
simultaneous injection of a significant nonlinear-in-state uncertainty. Such drastic
changes were motivated by intent to demonstrate the effectiveness of the proposed design
methodology. This particular example was also selected to be similar to previous work
presented in [31], [32] as well as in Chapter 3 so that relevant performance comparisons
may be available.
122
With only the baseline controller in operation and with the uncertainties included, the
closed-loop system tracking performance degradation can be clearly observed from the
data that are shown in Figure 34. Although the tracking performance is poor, both the
control input and its rate remain small.
0 10 20 30 40 50 60 70 80
-10
-5
0
5
10
, deg.
cmd
ref
0 10 20 30 40 50 60 70 80
-4
-2
0
2
4
e
, deg
0 10 20 30 40 50 60 70 80
-2
-1
0
1
2
e
dot
, dps
Time, sec
Figure 34: Baseline Closed-Loop Response in Presence of Matched Uncertainties
123
To counteract the effects of the uncertainties, MRAC adaptive laws (5.31) were
constructed by solving algebraic Lyapunov equation (5.32) with reference matrix
ref
A as
designed via (5.53) and with
diag 0.1 1 800
ref
Q (5.57)
The regressor vector
d
consisted of 11 -dependent and evenly spaced Gaussians
(RBFs). RBF centers were placed at [-10:2:10] degrees of AOA, and all RBF widths were
set to 0.0233 . The regressor matrix
u
s consisted of 11 1
st
-order Padé
approximations of input time-delays with magnitudes of [0:0.02:0.2] seconds. The rates
of adaptation were chosen to be:
diag 1 200 200 , 20, 0.5
du
x
(5.58)
and the total control (elevator deflection) was formed as shown in (5.20). In order to
improve robustness, projection and dead-zone modifications are included in the adaptive
laws. With the baseline + MRAC adaptive augmentation control active, closed-loop
system performance was recovered with acceptable elevator deflection and rates, as
shown in Figure 35.
124
0 10 20 30 40 50 60 70 80
-10
-5
0
5
10
, deg.
cmd
ref
0 10 20 30 40 50 60 70 80
-2
0
2
4
e
, deg
0 10 20 30 40 50 60 70 80
-10
-5
0
5
e
dot
, dps
Time, sec
Figure 35: Baseline+MRAC Closed-Loop Response in Presence of Uncertainties
The norms of the MRAC adaptive parameters are shown in Figure 36. The effect of
the dead-zone is seen in the adaptive parameter response, where once the tracking error
becomes small the adaptive parameters stop evolving and remain constant.
125
0 10 20 30 40 50 60 70 80
0
1
2
3
4
5
K
x
hat
0 10 20 30 40 50 60 70 80
0
0.5
1
1.5
2
2.5
d
hat
0 10 20 30 40 50 60 70 80
0
0.05
0.1
0.15
0.2
u
hat
Time, sec
Figure 36: MRAC Adaptive Parameter Norms vs. Time
As previously discussed, if the adaptive gains are too high in the classical MRAC
design, poor performance characteristics may occur. Increasing the gains to:
diag 1 200 200 , 40, 5, 8
du
d
(5.59)
126
that is, setting gains
d
and
u
to twice and ten times as high as in the original MRAC
design, yields the response shown in Figure 37. While the tracking response appears
acceptable, inspection of the control input positions and rates show unacceptable
characteristics and indicate the stability of the system may be tenuous.
0 10 20 30 40 50 60 70 80
-10
-5
0
5
10
, deg.
cmd
ref
0 10 20 30 40 50 60 70 80
-5
0
5
10
e
, deg
0 10 20 30 40 50 60 70 80
-1000
-500
0
500
e
dot
, dps
Time, sec
Figure 37: Baseline+MRAC Closed-Loop Response with High Adaptive Gains
127
In order to improve the performance of the baseline + adaptive augmentation system,
we next implement the alternative design featuring normalization developed in (5.35)-
(5.48). The rates of adaptation are selected to be those defined in (5.59), where the
selection of
d
is chosen to be similar to
x
in the MRAC case for comparison, while
gains
d
and
u
are twice and ten times as high as the original MRAC case for the
alternative design. We note that this gain set showed poor performance characteristics in
the classical MRAC case. Increasing rates of adaptation leads to oscillations and
instability in general, yet we will show that the normalized design is able to facilitate
larger rates of adaptation while still improving performance by reducing oscillations. The
filter in (5.40) is selected to be a single pole low-pass filter:
20
20
Gs
s
(5.60)
We note that while the Padé approximations may not accurately represent the phase loss
of the pure time-delay (as illustrated in Figure 32), as previously stated the design of filter
(5.60) allows the ability to shape the measured quantities to avoid known frequency
regions associated with uncertain dynamics, noise, and/or disturbances. More rigorous
investigation of the filter design and its relationship to the effects of time-delay remains a
future research area. With the baseline + normalized adaptive augmentation control
active, closed-loop system performance was again recovered with acceptable elevator
deflection and rates, as shown in Figure 38.
128
0 10 20 30 40 50 60 70 80
-10
-5
0
5
10
, deg.
cmd
ref
0 10 20 30 40 50 60 70 80
-2
0
2
4
e
, deg
0 10 20 30 40 50 60 70 80
-4
-2
0
2
4
e
dot
, dps
Time, sec
Figure 38: Baseline+B-SPM Normalized Response in Presence of Uncertainties
The norms of the B-SPM normalized adaptive parameters are shown in Figure 39.
Once again, the effect of the dead-zone is seen in the adaptive parameter response, where
once the tracking error becomes small the adaptive parameters stop evolving and remain
129
constant. The adaptive norms show smoother evolution than the data from the MRAC
design.
0 10 20 30 40 50 60 70 80
0
1
2
K
d
hat
0 10 20 30 40 50 60 70 80
0
1
2
d
hat
0 10 20 30 40 50 60 70 80
0
0.2
0.4
u
hat
0 10 20 30 40 50 60 70 80
0.5
1
1.5
hat
Time, sec
Figure 39: Baseline+B-SPM Adaptive Parameter Norms vs. Time
130
To further demonstrate the advantage of the alternative adaptive design, we compare
the responses of the two designs. As shown in Figure 40, the alternative design shows
smoother elevator transients and does not contain unwanted high frequency oscillations.
0 10 20 30 40 50 60 70 80
-10
-5
0
5
10
, deg.
0 10 20 30 40 50 60 70 80
-2
0
2
4
e
, deg
0 10 20 30 40 50 60 70 80
-10
-5
0
5
e
dot
, dps
Time, sec
cmd
MRAC
Alt
Figure 40: Comparison of Adaptive Closed-Loop Responses: with Uncertainties
131
Another way of quantifying control input activity is via the frequency domain by
plotting the power spectral densities (PSDs) of the elevator inputs. As shown in Figure
41, the alternative adaptive design displays less control input and does not contain the
high frequency control input components present in the MRAC response.
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
-140
-120
-100
-80
-60
-40
-20
0
20
Frequency, Hz
Elev. Pos.(deg
2
/Hz dB)
MRAC
Alternative
Figure 41: Comparison of Control Input Power Spectral Densities
132
In order to more formally quantify the performance improvements provided by the
alternative adaptive design scheme, we investigate the effects of time delays on the
control input signal using each of the adaptive schemes. The input time delay margin of
the baseline LQR system can be obtained through linear analysis. Many attempts have
been made to quantify time delay margins in adaptive systems [35]-[37], but only
recently have theoretical results been established that can determine the time delay
margin of such nonlinear systems [38]. These recent results are only applicable for single
input state feedback MRAC systems, and the vector (MIMO) case remains an open
problem. Therefore in this work the time delay margins of the adaptive controllers will be
found numerically through simulation. We include a realistic 2
nd
order actuator model
with parameters:
60rad/sec, 0.7
act act
(5.61)
Using LTI analysis, the phase margin of the baseline LQR system defined by (5.53)
and actuator (5.61) without uncertainties at the plant input is 66.56 degrees at a crossover
frequency at 1.24 rad/sec corresponding to a time-delay margin of approximately 933ms.
This result is verified as shown in Figure 42, which shows baseline system response
when time-delay is set to 935ms in simulation, with the response just becoming unstable.
133
0 10 20 30 40 50 60 70 80
-10
-5
0
5
10
, deg.
Time-Delay = 0.935
cmd
ref
0 10 20 30 40 50 60 70 80
-2
-1
0
1
2
e
, deg
0 10 20 30 40 50 60 70 80
-4
-2
0
2
4
e
dot
, dps
Time, sec
Figure 42: Baseline Response: Input Time-Delay of 935ms
It is expected that the adaptive controllers will decrease the time-delay margin while
simultaneously adding improved performance under uncertainty. To determine the time-
delay margins of the adaptive designs, the same AOA doublets are used for tracking and
134
the uncertainties are removed from the simulation and the same gains, projection bounds,
and adaptive laws are used. The input time-delay is increased until instability occurs. The
baseline + MRAC design is able to remain stable until a time-delay of 30ms, or only
roughly 3% of the baseline system. This is a significant reduction in system performance
due to the addition of classical MRAC augmentation.
The baseline + B-SPM normalized adaptive augmentation system response with an
input time-delay of 935ms is shown in Figure 43. The response shows the B-SPM
normalized design able to retain acceptable AOA excursions with stability. In effect, the
B-SPM system is able to match the time-delay characteristics of the baseline LQR design,
but with the significantly improved performance in the presence of uncertainties shown in
previously.
135
0 10 20 30 40 50 60 70 80
-10
-5
0
5
10
, deg.
Time-Delay = 0.935
cmd
ref
0 10 20 30 40 50 60 70 80
-2
-1
0
1
2
e
, deg
0 10 20 30 40 50 60 70 80
-4
-2
0
2
4
e
dot
, dps
Time, sec
Figure 43: Baseline + B-SPM Normalized Response: Input Time-Delay of 935ms
136
The time-delay results are summarized below, with the phase margin calculated by
using the baseline closed-loop crossover frequency:
Control Scheme
Time-Delay Margin
(msec)
Corresponding
Phase Margin
(deg)
Baseline LQR 933 66.29
LQR + MRAC 30 2.13
LQR + B-SPM 933 66.29
The same simulation analyses were performed for AOA maneuvers with magnitude of
one degree and generated similar results, which indicates that the adaptive design
approach developed here is scalable.
137
Chapter 6: Concluding Remarks and Suggestions for Future Work
In this dissertation, we focus on the development of robust baseline and adaptive
control augmentation designs for general classes of dynamic systems, with results that
address known challenges in flight dynamics and aerospace applications. Our design
philosophy combines well-known modern control techniques to provide robust baseline
control design with augmentation-based adaptive control and state limiting to retain
system performance in off-nominal and degraded conditions. Throughout, we present
rigorous analysis to theoretically justify the approach while using application to relevant
real world aerospace examples to illustrate the benefit of our developments. We believe
the contributions of this work are the following:
Use of modern control techniques (dynamic inversion, LQR) for baseline design
provides robust system performance under nominal conditions, with well known
LTI analysis tools to quantify robustness and system performance.
Theoretically justified state limiting augmentation combined with robust dynamic
inversion design satisfies performance specifications while keeping the system
within a prescribed state space. In practice such a design allows an aircraft to
meet robustness and performance specifications while staying within its flight
envelope. Application to a flight test program illustrated the efficacy of the
design, while also receiving positive pilot feedback.
Adaptive augmentation combined with robust dynamic inversion provides a
method to retain desired system performance in the presence of parametric
uncertainties. Additional robustness modifications are employed to address non-
138
parametric uncertainties, resulting in a system that provides performance both
under nominal and degraded conditions. Application to representative flight
dynamics illustrated the design is able to retain desired closed-loop flight
dynamics even in the presence of severe uncertainty and system degradation.
Combining state limiting and adaptive augmentation with robust dynamic
inversion effectively combines the benefits of the three design components.
Application to representative flight dynamics illustrated the design is able to
satisfy robustness and performance specifications under nominal conditions, keep
the aircraft within its prescribed flight envelope, and retain desired closed-loop
dynamics in the presence of severe uncertainty and system degradation.
Use of LQR optimal baseline design provides attractive guaranteed stability and
robustness properties. Again, use of classical model reference adaptive control
augmentation provides a method to retain nominal closed-loop performance in the
presence of uncertainty. However, it is known that increased adaptive gains can
lead to oscillation and instability. In this work, we presented an alternative design
approach using bilinear static parameter model adaptive design with modification
to specifically address input time delay. The resulting design is able to retain
desired closed-loop dynamics in the presence of severe uncertainty and system
degradation, with the additional benefit of significantly improved time-delay
margin as illustrated via representative flight dynamics.
139
There are a number of worthwhile potential extensions to the research presented in
this thesis. In Chapter 2 we designed state limiting augmentation for application to 2
nd
order cascaded system dynamics often found in flight dynamics. A future extension of
this work could generalize the state limiting augmentation to a more generic class of
dynamic systems, such as that employed in Chapter 5. Some work has been accomplished
in this regard [39], but more investigation is warranted.
In the B-SPM design presented in Chapter 5, a stable strictly proper filter is used to
filter the system dynamics as part of the adaptive design. Future work should focus on
selection of this filter and the corresponding impact on system performance and time-
delay margin. Some recent work on employing closed-loop reference models [40], [34]
approaches this problem from another direction and shows promising results.
The adaptive augmentation schemes presented are able to recover desired closed-loop
performance, and in the B-SPM design show significantly improved performance under a
class of dynamic input uncertainties. However, in all cases the time-delay margin is
numerically derived without a complete analytic solution. Recent work [38] has provided
analytic derivation of time-delay margin, but the results are only applicable for single
input state feedback MRAC systems. Future work should extend such results to the
MIMO case to facilitate more practical applications.
In a larger sense, a more thorough robustness analysis needs to be developed for the
adaptive augmentation schemes presented in this thesis. Both the classical MRAC design
derived in Chapter 3 and the B-SPM design derived in Chapter 5 contain modifications
designed to improve robustness to non-parametric uncertainties. Chapter 3 includes
140
analysis of the estimation error and corresponding UUB properties of the closed-loop
system, including both static parametric and bounded non-parametric uncertainties, as
seen in the uncertainty definition:
*
T
D D D
D x x x (6.1)
In this case, we derived the ultimate bound and found that it depended on the size of the
estimation error, but did not do additional robustness analyses. The B-SPM design in
Chapter 5 assumes both the static and dynamic parametric uncertainties may be perfectly
matched, without considering non-parametric uncertainties such as estimation error:
TT
p p p p m m u u d d p
x A x B I s u x
(6.2)
Addition of such estimation errors complicates the stability and robustness results,
and while some generalized results exist [7], these results need to be extended and
applied to the aerospace examples presented in this work. Future work should expand the
analysis to first include static and dynamics non-parametric uncertainties:
T
p p p p m m d d p
x A x B I s u x
(6.3)
where s and
T
d d p
x may not be perfectly estimated. Beyond this case, future
work should include unmatched dynamic uncertainties. For example, consider the
dynamics
T
p p p p m m d d p
x A x B I s u x t
(6.4)
where t is a bounded function of time which it is assumed cannot destroy the
controllability of the system. The derivation of robustness properties similar to those
141
known for existing LTI designs (gain and phase margin, Nyquist margins, etc.) will be
crucial for increased certification and implementation of robust adaptive control schemes
in aerospace applications.
142
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Abstract (if available)
Abstract
Robust and adaptive control techniques have a rich history of theoretical development with successful application. Despite the accomplishments made, attempts to combine the best elements of each approach into robust adaptive systems has proven challenging, particularly in the area of application to real world aerospace systems. In this research, we investigate design methods for general classes of systems that may be applied to representative aerospace dynamics. By combining robust baseline control design with augmentation designs, our work aims to leverage the advantages of each approach. ❧ This research contributes the development of robust model-based control design for two classes of dynamics: 2nd order cascaded systems, and a more general MIMO framework. We present a theoretically justified method for state limiting via augmentation of a robust baseline control design. Through the development of adaptive augmentation designs, we are able to retain system performance in the presence of uncertainties. We include an extension that combines robust baseline design with both state limiting and adaptive augmentations. In addition we develop an adaptive augmentation design approach for a class of dynamic input uncertainties. We present formal stability proofs and analyses for all proposed designs in the research. ❧ Throughout the work, we present real world aerospace applications using relevant flight dynamics and flight test results. We derive robust baseline control designs with application to both piloted and unpiloted aerospace system. Using our developed methods, we add a flight envelope protecting state limiting augmentation for piloted aircraft applications and demonstrate the efficacy of our approach via both simulation and flight test. We illustrate our adaptive augmentation designs via application to relevant fixed-wing aircraft dynamics. Both a piloted example combining the state limiting and adaptive augmentation approaches, and an unpiloted example with adaptive augmentation show the ability of our approach to retain desired performance in the presence of relevant system uncertainty. Finally, we present alternative adaptive augmentation design developed to mitigate time delays at the system input and which demonstrates significant improvement over an existing widely used adaptive augmentation approach when applied to fixed wing aircraft dynamics.
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Asset Metadata
Creator
Gadient, Ross
(author)
Core Title
Adaptive control with aerospace applications
School
Viterbi School of Engineering
Degree
Doctor of Philosophy
Degree Program
Electrical Engineering
Publication Date
03/19/2013
Defense Date
02/27/2013
Publisher
University of Southern California
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Tag
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Language
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Ioannou, Petros (
committee chair
), Flashner, Henryk (
committee member
), Lavretsky, Eugene (
committee member
), Safonov, Michael G. (
committee member
)
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gadient@usc.edu,rossgadient@gmail.com
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