Graduated Persistent Excitation and Steady State
Margins for Adaptive Systems
By
Himani Jain
B.Tech., Aerospace Engineering
Indian Institute of Technology, M umbai, 2004
Submitted to the Department of Mechanical Engineering in partial
fulfillment of the requirements for the degree of
Masters of Science in Mechanical Engineering
at the
MASSACHUSETTS INSTITUTE of TECHNOLOGY
M» 21 , 2007 t1
U~/1e. d.- Oo'lJ
Copyright 2007 Massachusetts Institute of Technology. All rights reserved.
Signature redacted
Author ...... ........ .. ........ .. ........... .. ............ ..... ... ..... '
~
Department of Mechanical Engineering
/
Certified by................. .... ... ..... .... ............. ..
/ May 20, 2007
.;
Signature redacted .....
-----
----
Dr. Anuradha M. Annaswamy
Senior Research Scientist
.I?~,:"
Accepted by......... ....................... .... ....... ...... .
in..~Mechanical
Engineering
p.
1
Thesis Supervisor
,S ignature redacted ..
Lallit Anand
Professor of Mechanical Engineering
MASSACHUSETfS INSTITUTE
OF TEC,H;\.~()L0(;\(
Chairman, Department Committee on Graduate Students
1
Acknowledgement
May 21, 2007
I am most indebted to my thesis supervisor, Dr. Anuradha Annaswamy, for
giving me the opportunity to work on this project. I would like to thank her for her invaluable guidance, motivation and constant support. Her constant criticisms and reviews
gave me the conceptual clarity. Without her help this would not have been possible.
I also wish to thank Dr. Eugene Lavretsky for his invaluable suggestions throughout the duration of this work.
I would like to thank Mac Schwager, Zac Dydek and Jinho Jang. Our conversations
and work together have greatly influenced this thesis.
Finally I would like to thank all my friends who have helped me in the successful
completion of this thesis.
Himani Jain
2
Graduated Persistent Excitation and Steady State Margins for
Adaptive Systems
by
Himani Jain
Submitted to the Department of Mechanical Engineering on May 21, 2007 in Partial
Fulfillment of the Requirements for the Degree of Masters of Science in Mechanical
Engineering
ABSTRACT
The numerous design tools developed for use with linear controllers, specifically gain and
phase margins, do not apply to nonlinear control architectures such as model reference
adaptive control. The first step for the development of Verification and Validation (V&V)
techniques for this class of nonlinear control systems is presented in this thesis in the
context of controlling uncertain flight vehicle dynamics. Using a Reduced Linear
Asymptotic System (RLAS), which characterizes the asymptotic behavior of an adaptive
system, methods for tuning the free adaptive system parameters such as Lyapunov matrix
P to satisfy the desired performance criteria are presented. Making use of the fact that
the RLAS is a linear time invariant system, optimization procedures based on output
feedback and Linear Matrix Inequalities are proposed.
The concept of Persistent excitation in the context of improving stability and robustness
properties of closed loop adaptive systems is discussed. Graduated Persistent Excitation
(GPE) is introduced as an easy to implement alternative to Persistent excitation. Tools
such as MIMO margins based on the singular values of sensitivity matrix are applied on
RLAS to systematically derive stability margins of an adaptive flight control system.
Additionally, a proof of signal boundedness is presented in the presence of both
structured and unstructured uncertainties. The tools are demonstrated on simulations of
a nonlinear 6 DoF aircraft model.
Thesis Supervisor: Anuradha M. Annaswamy
Title: Senior Research Scientist of Mechanical Engineering
3
Contents
Abstract
3
List of Figures
8
List of Tables
11
1 Introduction
12
1.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
1.2 History of Adaptive Control and Persistent Excitation . . . . . . . . . . . .
13
1.3 Previous research on Verification and Validation Procedures . . . . . . . .
14
1.4 Motivation for Current Study . . . . . . . . . . . . . . . . . . . . . . . . .
15
1.5
Road-Map for in-depth Realization of Verification and Validation Procedures 16
2 Optimization of Free Design Parameters
18
2.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
2.2
Previous Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
2.3
Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
2.4 Optimization of P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
4
Algorithm for Optimization of P . . . . . . . . . . . . . . . . . . . .
26
2.5
Simulation Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
28
2.6
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
2.4.1
3
31
Persistent Excitation
3.1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
Exam ples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35
Preliminary Study
3.1.1
3.2
Uniform asymptotic stability and persistent excitation
. . . . . . . . . . .
39
3.3
Properties of Persistently Exciting functions . . . . . . . . . . . . . . . . .
42
3.3.1
Algebraic Transformations . . . . . . . . . . . . . . . . . . . . . . .
42
3.3.2
Dynamic Transformations
. . . . . . . . . . . . . . . . . . . . . . .
44
. . . . . . . . . . . . . . . . .
45
An example of High performance aircraft dynamics . . . . . . . . .
46
Open-Loop Model . . . . . . . . . . . . . . . . . . . . . . . . . . . .
46
. . . . . . . . . . . . . . . . . . . . . . . . . . .
47
Adaptive Controller . . . . . . . . . . . . . . . . . . . . . . . . . . .
48
Persistent excitation of Input Matrix . . . . . . . . . . . . . . . . .
51
Persistent excitation of regressor vector . . . . . . . . . . . . . . . .
53
Simulation studies
. . . . . . . . . . . . . . . . . . . . . . . . . . .
57
Graduated Persistent Excitation . . . . . . . . . . . . . . . . . . . . . . . .
59
Effect of the amplitude and time duration of reference signal . . . .
62
3.4
Persistent Excitation and Adaptive Control
3.4.1
Baseline controller
3.4.2
3.4.3
3.5
3.5.1
4
Performance Metrics for an Adaptive System in Steady State
64
4.1
65
Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
4.2
Frequency Response
. . . . . . . . . . . . . . . . .
. . . . . .
66
4.3
Specific Tools for Adaptive Control . . . . . . . . .
. . . . . .
67
4.4
Stability Margins of MIMO Plant . . . . . . . . . .
. . . . . .
69
4.5
4.4.1
Overview of Transfer Functions
. . . . . . .
. . . . . .
70
4.4.2
Multivariable Nyquist Theorem . . . . . . .
. . . . . .
72
4.4.3
Derivation of MIMO Stability Margins based on Singular Values . .
72
4.4.4
Frequency Response of Sensitivity Matrices.
. . . . . .
76
. . . . .
. . . . . .
77
MIMO margins of linear plant . . . . . . . .
. . . . . .
78
Evaluation of a High-performance Aircraft
4.5.1
5 Persistent Excitation in the Presence of Nonlinearities
81
5.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
81
5.2
Nonlinearity and Radial Basis Function . . . . . . . . . . . . . . . . . . . .
83
Zero Approximation Error . . . . . . . . . . . . . . . . . . . . . . .
85
Persistent Excitation and Analysis . . . . . . . . . . . . . . . . . .
87
Simulation Studies . . . . . . . . . . . . . . . . . . . . . . . . . . .
88
Pitch Break Nonlinearity . . . . . . . . . . . . . . . . . . . . . . . .
89
Boundedness Proof . . . . . . . . . . . . . . . . . . . . . . . . . . .
90
Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . .
95
MIMO Margins . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
96
Unmodeled Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
98
5.3.1
Persistent Excitation in the presence of unmodeled dynamics .
99
5.3.2
Simulation Studies
. . . . . . . . . . . . . . . . . . . . . . . .
100
5.2.1
5.2.2
5.3
6
5.3.3
5.4
6
MIM O Margins .............................
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Summary, Conclusion and Future Work
Future Work .....................................
Bibliography
101
103
104
106
106
7
List of Figures
2.1
Hyper-spherical and hyper-rectangular uncertainty models in parameter space
for lift and drag coefficients that allow comparable and different levels of uncertainties, respectively.
2.2
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
A block diagram showing the nominal inner loop and the adaptive outer loop
of the AFCS with multiple parameter uncertainties. . . . . . . . . . . . . .
23
2.3
Lyapunov surface for distinct P's
28
2.4
Roll rate of JDAM lateral dynamics as a function of P
. . . . . . . . . . .
29
2.5
Vertical acceleration of JDAM longitudinal dynamics as a function of P . .
29
2.6
Closed loop eigenvalues at steady state for distinct Ps . . . . . . . . . . . .
30
3.1
Example 1: Parameter error for (a) r(t) = 1 and (b) r(t) = sin(t)
. . . . .
37
3.2
Example 1: Parameter error for (c) r(t) = e 0 2t sin(t) . . . . . . . . . . . . .
38
3.3
Example 1: Norm of parameter error as a function of state error for (b)
. . . . . . . . . . . . . . . . . . . . . . .
r(t) = sin(t) and (c) r(t) = eO 2t sin(t) . . . . . . . . . . . . . . . . . . . . .
3.4
38
Example 1: Convergence of parameter error vector orthogonal to the regressor vector
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
39
3.5
Overall control structure for the model adaptive control system. The baseline
controller is augmented by the adaptive controller. . . . . . . . . . . . . . .
51
3.6
Eigenvalues of closed loop system . . . . . . . . . . . . . . . . . . . . . . .
58
3.7
Norm of the Parameter error matrix . . . . . . . . . . . . . . . . . . . . . .
59
3.8 Graduated Persistently Exciting inputs . . . . . . . . . . . . . . . . . . . .
62
3.9 Norm of Parameter error matrix for Graduated Persistently Exciting inputs
63
4.1
Bode plot of input u1 with respect to output yi of reference model (solid line),
nominal (dashed line) and various adaptive systems for A = diag(O.25 0.25 0.25) 68
4.2
Bode plot of input ul with respect to output y2 of reference model (solid line),
nominal (dashed line) and various adaptive systems for A = diag(0.25 0.25 0.25) 69
4.3
MIMO system at two distinct break points . . . . . . .
70
4.4
MIMO system with feedback gain . . . . . . . . . . . .
73
4.5
MIMO system with gain and phase uncertainties . . . .
73
4.6
Shapes of Frequency responses of Sensitivity matrices .
77
4.7
Angle of attack tracking performance of nominal and adaptive controller
after 75% loss in elevator and rudder effectiveness . . .
78
5.1
Approximation of fictitious nonlinearity . . . . . . . . .
89
5.2
Pitch Break Nonlinearity vs. Angle of Attack
. . . . .
90
5.3
Graphical representation of trajectory bounds . . . . .
95
5.4
Tracking of a command in the presence of pitch break nonlinearity
. . . .
96
5.5
Pitch break nonlinearity approximation as a function of time . . . . . . . .
97
5.6
Block diagram with additive unmodeled dynamics . . . . . . . . . . . . . .
99
9
. . . . . . . .
5.7
Angle of attack tracking with additive unmodeled dynamics
5.8
Elevator Command for nominal and adaptive controller in the presence of
additive unmodeled dynamics
. . . . . . . . . . . . . . . . . . . . . . . . .
10
101
102
List of Tables
2.1
Eigenvalues of distinct Ps for short period dynamics of the aircraft . . . . .
28
4.1
Simulation Parameter Values . . . . . . . . . . . . . . . . . . . . . . . . . .
78
4.2
Margins based on Singular Values of Error Sensitivity . . . . . . . . . . . .
79
4.3 Margins based on Singular Values of Complementary Sensitivity . . . . . .
80
5.1
Margins based on singular values of error and complementary sensitivity in
the presence of fictitious nonlinearity . . . . . . . . . . . . . . . . . . . . .
88
5.2 Margins based on the Singular Values of Error Sensitivity in the presence of
nonlinearities
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
98
5.3 Margins based on the Singular Values of Complementary Sensitivity in the
presence of nonlinearities . . . . . . . . . . . . . . . . . . . . . . . . . . . .
98
5.4 Margins based on the Singular Values of Error Sensitivity matrix in the
presence of unmodeled dynamics
. . . . . . . . . . . . . . . . . . . . . . . 102
5.5 Margins based on the Singular Values of Complementary Sensitivity in the
presence of unmodeled dynamics . . . . . . . . . . . . . . . . . . . . . . . . 103
11
Chapter 1
Introduction
1.1
Introduction
Adaptive control of aircraft systems promises improvements in stability and robustness in
the presence of parametric uncertainties and is an important element in the design mission of high-performance, safety-critical flight vehicle systems. Early attempts at adaptive
flight control used controllers with unproven stability properties, sometimes with disastrous
consequences. As a result, much of the theoretical work up to the present time has been
rightly focused on stability of adaptive architectures. Currently, there exists an assortment
of stable adaptive control strategies, as well as techniques for preserving stability in the
presence of unknown, bounded disturbances [1], [2]. Three common adaptive architectures
have been investigated exhaustively in conjunction with aircraft control: (i) Direct adaptive control, (ii) indirect adaptive control Ref. [31, and, (iii) neural network based adaptive
control [41,[5]. Perhaps the most promising and elegant architecture is direct adaptive control, in which control parameters are adapted based on some performance error. Stability
12
properties of this architecture are well-known and were explored in Refs. 6-9. In addition,
methods such as Training Signal Hedging (TSH) have been developed to overcome the
real-world problem of saturating actuators 10,11.
1.2
History of Adaptive Control and Persistent Excitation
Adaptive systems have been extensively studied over past years and their stability properties have been derived in the presence of bounded unknown disturbances and unmodeled
dynamics. In 1980 the global stability of the adaptive-control problem was established
for the ideal case (when no disturbances are present) in Narendra et al. (1980), Morse
(1980), Narendra and Lin (1980) and Goodwin et al. (1980). With some prior knowledge
regarding the unknown plant transfer function, it was shown that the adaptive law results
in bounded signals and that the error between plant and reference model outputs tends to
zero asymptotically. It was further shown that if some of the internal signals are persistently exciting, the control parameter vector will converge to the desired value. This was
the first application of the concept of persistent excitation to adaptive-control problems.
A detailed tutorial on persistent excitation in adaptive control is given in Ref. [11].
Following the resolution of the stability problem in the ideal case, attention was focused on robustness questions. These included the effect of external disturbances, variations
in plant parameters and the use of reduced-order reference models (and hence low-order
controllers) on the behavior of the overall system, in all of which persistent excitation was
seen to play an important part. It is now recognized that robustness can be achieved either
by suitably modifying the adaptive law or by increasing the degree of persistent excitation
13
of the reference input.
1.3
Previous research on Verification and Validation Procedures
It is known that the adaptive systems with a modified adaptive law and rich enough input
signals, have good robustness properties. However, it should be noted that an analytical
tool to quantify these robustness properties into more conventional metrics like stability/robustness margins and acceptable transient behavior is absent. This presents the chief
practical obstacle to transitioning adaptive flight controllers into aerospace applications.
This is not a trivial task because the dynamics of an adaptive system in closed loop are
nonlinear. Such concerns can be grouped under the umbrella of Verification and Validation
(V&V) and are obviously of paramount importance in application to aircraft and other
safety critical systems. However these concerns have received curiously little attention in
the adaptive control literature. Researchers have generally relied on extensive simulation
and trial and error to produce adaptive control systems with suitable transient properties.
The need for completely new (V&V) techniques is expanded on in Refs. [12] and
[13], and some necessary features of a successful (V&V) procedure are laid out in Ref.
[14].
Some specific techniques have been proposed for neural network based controllers.
For example, the method in Ref. [15] relies on bounding neural network outputs using
Lipschitz conditions imposed on the chosen set of basis functions, and a second method
employs Support Vector Machines (SVM) to determine if a neural network will produce an
output that is out of specification [16]. These methods are specific to neural network based
adaptive control systems, and it is difficult to envision their use in an industry setting due
14
to their complicated and theoretical nature.
1.4
Motivation for Current Study
Current (V&V) techniques rely on the fact that the underlying control system is linear (at
least locally), which makes them inadequate for adaptive flight control systems which are
intentionally nonlinear. This drawback has severely limited the widespread commercial use
of adaptive flight controllers.
Recently, we have begun the derivation of theoretically verifiable (V&V) techniques
for adaptive systems [3].
It was shown that the closed loop adaptive system converges
asymptotically to a linear time invariant (LTI) system called the Reduced Linear Asymptotic System (RLAS). In this thesis the RLAS, which contains information about the plant
uncertainty, is used as a tool to analyze the steady-state behavior and disturbance rejection properties of the adaptive system. At the same time, this tool also provides practical
guidelines for tuning adaptive controllers to satisfy predetermined performance criteria. In
Ref. [3], a method for aforementioned task was given, using a combination of Lyapunov
theory, asymptotic analysis, and linear systems theory which is made more formal in this
paper.
The contribution of this thesis is to introduce a set of tools based on Lyapunov
theory, asymptotic analysis, and linear systems theory for analyzing the steady state behavior and disturbance rejection properties of adaptive systems. At the same time, the
tools provide practical guidelines for tuning adaptive controllers to satisfy predetermined
performance criteria. The focus of this thesis is limited to an Adaptive Flight Control
15
System (AFCS) which employs model reference adaptive control using state feedback for a
multi input plant which consists of a nominal inner loop and an adaptive outer loop.
1.5
Road-Map for in-depth Realization of Verification and Validation
Procedures
With the brief introduction to the history and evolution of adaptive control, and the need
for a tool for the verification and validation purposes in this first chapter, a detailed examination of the most important aspects of realization of verification and validation tools can
be discussed in the subsequent parts of this thesis. In Chapter 2, optimization of adaptive
control design parameter P to achieve specified performance goals is proposed using the
RLAS, output feedback and Linear Matrix Inequalities (LMIs). Chapter 3, gives a background of persistent excitation and its algebraic and dynamic transformation properties.
Conditions on persistent excitation of a multi input multi output (MIMO) aircraft system are then derived using these properties and are demonstrated in simulation. The idea
of 'Graduated Persistent Excitation' (GPE) is introduced in the later part of the chapter. Chapter 4 starts off with an exhaustive study of available metrics to measure the
performance of MIMO linear plant. MIMO stability margins based on singular value of
sensitivity matrices are introduced and their viability as a metric is successfully demonstrated. In Chapter 5, the MIMO margins of a MIMO aircraft system in the presence of
nonlinearities and unmodeled dynamics with graduated persistent excitation are discussed.
It was shown that the aircraft systems with carefully designed GPE inputs have robustness
16
and disturbance rejection properties close to the reference model. Chapter 6 finally summarizes the contributions of this thesis. Future courses of action in terms of realization of
(V&V) procedures of adaptive flight control systems are recommended.
17
Chapter 2
Optimization of Free Design Parameters
2.1
Introduction
In model reference adaptive control (MRAC), the usual objective is to drive the output
of a partially known plant to asymptotically track the output of a prespecified and stable
reference model for all reference model inputs. The reference model inputs typically belong to the set of all piecewise continuous and bounded functions and the reference model
is designed such as it embodies the desired behavior of controlled plant. Though output
matching of the plant and the reference model and boundedness of all closed loop signals
are the most commonly used criteria to characterize the performance of model reference
adaptive control schemes in the absence of persistently exciting signals, the other important objectives are to achieve good transient behavior, desired steady state response and
robust stability margins for the controlled plant, all of which can be clubbed into adaptive
performance improvement requirements. While output matching has been successfully attempted in past years for a wide variety of plants by designing various adaptive controllers,
18
there have not been clear and concise guidelines to achieve aforementioned performance
goals.
Adaptive control systems often have design parameters which are not completely
specified by adaptive stability conditions alone. Hence, it is natural to consider systematic
methods for choosing these parameters based on performance related criteria. There are
two free parameters in the typical model reference adaptive system, the positive adaptive
gains, IF, and, P, the unique symmetric positive definite solution of the algebraic Lyapunov
equation A'P+ PAm = -Q, with
Q>
0. This chapter will concentrate on taking a further
step to intelligently choose parameter P to achieve specified performance goals.
Section 2 delves into previous research and lays out the need of intelligently choosing
free design parameters to improve transient and steady state performance. Section 3 introduces the problem statement and formulates reduced linear asymptotic system (RLAS).
RLAS is used as a tool in Section 4 for optimization of P based on output feedback control and Linear Matrix Inequalities and provides an algorithm for optimization. Section 5
presents some simulation studies to demonstrate the effect of P matrix obtained by using
optimization algorithm on the performance of adaptive controller. Section 6 concludes the
chapter with key results and observations.
2.2
Previous Research
In [17], Miller et al proved existence results to show that in principle one can achieve
an arbitrarily good transient and steady state response under relatively weak plant assumptions. In [18], Bayard used averaging methods to analyze and optimize the transient
19
response associated with the direct adaptive control of an oscillatory second order minimum
phase system. A certain approximation to the error function was performed and an optimal closed form solution of integral adaptive gain weighting was achieved to optimize the
transient performance. But for implementation, it requires the knowledge of certain plant
parameters which might not be available. In [19], Ydstie investigated the effect of changing adaptation gains on the performance of the adaptive system and concluded that I,,
performance deteriorates while the
12
performance improves as the local rate of adaptation
is decreased. In [20], Datta et al proposed a modified MRAC scheme in which a certain
design parameter can be chosen to improve the transient behavior and smaller possible
bursts at steady state.
Most of the pervious research concentrated on choosing a different adaptive law or
a distinct control law to achieve better performance in transience and steady state. But in
this chapter, we concentrate on choosing the free parameters of already established adaptive
and control laws to improve the performance.
2.3
Problem Statement
The problem under consideration is the control of an uncertain, states accessible plant of
the form:
y = WP(s, 6)u
(2.1)
20
where 6 is a parameter vector that is subject to uncertainty. This uncertainty may be
due to anomalies such as control failures, battle damage, or aircraft reconfigurations. We
use adaptive control design based on augmentation architecture. This is based on the
assumption that the unknown parameter 0 belongs to the set, H(90 , A), where 90 is the
nominal value of 0 which is known, H represents the uncertainty region that may either
be hyper-spherical or hyper-rectangular (see Fig. 2.1), and the uncertainty radius A is
proportional to the uncertainty. While hyper-spherical sets allow handling parameters with
PARAMITERM
SPACE
PARAMETER SPACE
CD0 .
C LC~a
.
.
.
.
.
CL
Figure 2.1: Hyper-sphericaland hyper-rectangularuncertainty models in parameterspace for lift
and drag coefficients that allow comparable and different levels of uncertainties, respectively.
comparable levels of uncertainty, hyper-rectangular sets enable handling of parameters with
different levels of uncertainty. Based on the nominal value, one can then design a nominal
controller. In order to cope with the uncertainty, this nominal controller is augmented with
an adaptive controller. The details of this augmentation architecture are as follows:
We can rewrite the system in Eq. (2.1) in state-space form as:
i = Apx + bpu + d,
(2.2)
21
where Ap E RI", b, E R", and d, E R are unknown. The plant parameters Ap and bp
are therefore functions of the nominal parameter 0 and the uncertainty H. The desired
behavior is for the plant to follow a known reference model given by:
im = Amxm + bm 5c,
(2.3)
where Am is Hurwitz and J, is a reference input. Let the control input u =
6
nom
+
6
ad,
be
given by the nominal control law:
6nom =
(2.4)
kjx,
kd6o +
and the adaptive control law,
6
(2.5)
ad = OTU,
where, w
=
[x
6C
1]TJ
= [eT
8
06
d],
and Ox E R
,65 E R, and Od E R are
the control gains. The control gains are adjusted according to the adaptation law:
(2.6)
= -FwbimPe
where, e = x,-
m
is the system tracking error, P is the unique symmetric positive definite
solution of the algebraic Lyapunov equation AT P + PAm = -Q, with
Q>
0. Also, in
Eq. (2.6), r > 0 is a positive definite symmetric matrix of adaptation rates. A block
diagram of the adaptive augmented system is shown in Fig. 2.2 below. Assuming that
22
Desired
5
Figure 2.2: A block diagram showing the nominal inner loop and the adaptive outer loop of the
AEFCS with multiple parameter uncertainties.
there exist ideal gains 9*, 9) > 0, and 9*, which satisfy the so-called "matching conditions",
dcA
b,*T=Am - A,,
b,9j
=
and
bin,
b,6*, = -dr,
(2.7)
the error dynamics can be written compactly as:
Am
AbmWT]
-TwbiP
where, A =~
J
0
9=
and
:2
6 -9
[( 92 - *T)
9
3d -].l
Equation (2.8), represents the error dynamics of the closed loop adaptive system.
Notice that the error dynamics are nonlinear and time varying due to the presence of the
linear regressor vector w. It was shown
it, in Ref. [3] that for a constant reference input,
the error dynamics in Eq. (2.8) converge to the dynamics,
S
A.+A
6
bmc
Tb
e
(2.9)
23
where,
lI
=
wica and -y is defined to be the known scalar WicPome. Equation (2.9) is
denoted as the Reduced Linear Asymptotic System (RLAS). The RLAS is a simple, linear,
compact approximation to the nonlinear dynamics of the closed loop adaptive system.
This adaptive system has been shown to be stable, with the stability properties described
in Theorem 1 below.
Theorem 1
The error dynamics in (11) have the following properties:
i) The plant state x is bounded.
ii) The controller gains are bounded.
iii) lime.. 0 e = 0.
Proof of Theorem 1
Consider the Lyapunov function candidate V = eTPe+#TA6 I-1#. Taking time derivatives
along the system trajectories gives V = -eTQe < 0. This implies that V is bounded,
and hence e and 9 are bounded. Since Am is stable and 6 , is bounded, xm is bounded.
This, in turn, implies that x and 0 are bounded, proving i) and ii). Now, x bounded and
6c bounded imply that w is bounded; and Am stable, e bounded, and d bounded imply
that 6 is bounded. This implies that V is bounded. Therefore, by Barbalat's lemma [1],
limt,.. V = 0, which directly implies iii) [2].
Since the behavior of nonlinear adaptive system can be practically characterized
by RLAS, hence, if the RLAS is optimized, then the behavior of adaptive system is also
optimized in steady state. Also, the RLAS equation contains the two free adaptive system
24
parameters which can be chosen at will, given some particular conditions are satisfied.
Therefore, RLAS will be used as a tool for optimization.
2.4 Optimization of P
We try to use the concept of static projective control described below, to optimize P using
RLAS. For the following state space representation of any system,
t = Ax + Bu
y = Cx
(2.10)
if x is accessible then the LQR performance optimization problem is:
Optimize J =
J(xTQx
+ ru2 )dt
(2.11)
and to this end, the typical state feedback control law u = -Kx is sufficient. If the full
state feedback is not available due to the unavailability of all of the states, Medanic [21]
proposed static projective control for performance optimization where the feedback control
law is given by:
u = -K.y
Vn: (A - BK)V =VnA
K, = -KVr(CV)-
1
(2.12)
Vn = {Vr, Vnr}
where Vr denotes the modes we wish to retain in the closed loop for performance requirement. We use this concept for the optimization of Lyapunov matrix P in the following
25
sub-section.
Algorithm for Optimization of P
2.4.1
To this end, we rewrite RLAS Eq. (2.9) as:
XR =
(2.13)
AXR + bu
where,
XR
=
,A=
L a
[
L
7bb]c=
0
0
[jbu=
-gbiP
(2.14)
1L1
A method for the calculation of the optimal P was developed using optimal output
feedback techniques [21],[23] and iteratively solving a set of Linear Matrix Inequalities
(LMIs) with MATLAB's LMI solver [24].
The optimization procedure is comprised of the following iterative process:
Step 1 : Choose A6 = A*, 9.c = 9*c where A* and *c are such that the spectrum, -(Am +
A*bm9*cc) is closest to the imaginary axis.
Step 2: Find KLQ such that u = -KLQxR minimizes the cost function given by
J=
(2.15)
(XT QXR + ru2 )dt
Step 3 : Choose V, the set of all eigenvectors of F that we wish to retain, where
F =A -bKLQ
(2.16)
26
Step 4: Calculate K± where,
K 1 = KLQVr(CVr)~',
Cn_1 xn = [In_1xn_1
V: (A - BK)V =VA
(2.17)
0]
Vn = {V, Vn_.}
Step 5: Assuming that only e. is accessible; the optimal control law is given by
u = -7bTPea
which implies that K 1 = ybTP and KI
(2.18)
bP
T= (Note: -y is assumed to be known).
=
Now the problem is to find a P given a K*. Since KI
=
bnP, it is not automatically
guaranteed that every KI calculated using step 1 to 4 will lead to a bTP where P is the
solution of Lyapunov equation AT P + PAm < 0. The next step focuses on this aspect.
Step 6 : Find a P subjected to the following Linear Matrix Inequalities (LMI's):
1.
P=pT>O,
2. AiP + PAm < 0,
3. Pbm= K_*T
This can be carried out using an LMI solver in MATLAB.
Step 7 : Simulate the adaptive system with new P found in step 5 and repeat steps 2 to
5 till the solution converges. Convergence process usually takes 2-3 iterations.
It follows from Kalman-Yakubovich lemma [1] that the solution of Step 5 exists only
if T(s) is SPR [22]. SPR property of T(s) depends upon selection of KLQ, hence a careful
selection is necessary.
27
2.5
Simulation Studies
A series of simulation studies was done on short period aircraft dynamics to investigate
the effect of P matrix on the performance of the aircraft using the optimization algorithm
described in previous section. The Fig 2.5 below shows the Lyapunov surface for distinct
P's achieved during the iterations as described in the algorithm.
15
PI
10
5
0
-5
-10
-15
-15
-10
0
-5
5
10
15
Figure 2.3: Lyapunov surface for distinct P's
Table 2.1 shows the eigenvalues of P matrices obtained during the iterations.
Table 2.1: Eigenvalues of distinct Ps for short period dynamics of the aircraft
Eig(P)
P
P1 (old)
P2
P3
0.1299, 1.0110
0.0429, 0.9372
0.0458, 1.0479
P 4 (new)
0.0516, 1.0060
The algorithm in section 2.4 was applied to the lateral and longitudinal dynamics of
28
Boeing's Joint-Direct Attack Munition (JDAM) with the AFeS. In Fig. 2.4, the roll rate
for a wide variety of locally optimal P is plotted and Fig. 2.5 shows the vertical acceleration
for a variety of distinct P's having distinct eigenvectors. Fig. 2.6 shows the closed loop
eigenvalues in steady state of longitudinal dynamics for distinct P's. In these simulations
the adaptive gains
r
are chosen such that rllb~PII is equal across all P, thus ensuring a
meaningful comparison.
20.---- - r - - - - r - - . - - - - , - - - , - - - . - - - r - - - - , - - - , j - - Prof
.
~
p:;
.
.
:
.
.
.
.
: ----- Pp I
01---:--,·
-5
--------r----
-100'------":10---'--- 30'----'-40--5:'-0--'60':-----L:70--:80::'::----:'90':--~100
Tune, sec
Figure 2.4: Roll rate of JDAM lateral dynamics as a function of P
0.2 ........
: • r. j e:
. ~'!i"" :
0.1
· l,·· t\··
w.I
.1
:~
.
:.
'
-0.1
-0.2
............ ·.. i·· .. ........... : ..........
,........
. .................... ;., .... 0.. ' .... ' ... : ................ . , ·.,.........
. .... ; ...... .. .. .. . ..
"1
-0.3
-0.4 "--_-'--_-'--_-'--_--'-_--'-_----'-_----'-_ _' - - - _ . l . . . - - - - - '
80
90
100
10
20
30
40
50
60
70
o
Figure 2.5: Vertical acceleration of JDAM longitudinal dynamics as a function of P
29
3
i
o
0
0
CX;..
-'1
-z
0
~3
-2
-1.8
-1 .6
-1.4
-1 .2
-1
Re().,)
-0.8
-0,6
-0.4
-0.2
0
Figure 2.6: Closed loop eigenvalues at steady state for distinct Ps
2.6
Conclusion
From figure 2.4, 2.5 and 2.6, it was found that large changes in the eigenvectors of P have
little or no effect on the steady state response of the closed loop system. Additionally, it can
be seen from Figure 2.3 that the result of the optimization was always a locally optimal P in
the neighborhood of the initial guess and the Lyapunov surface doesn't change sufficiently
enough during the iterations of optimization of P. Hence, any reasonable choice for P
can be made and focus can be given to the optimization of the adaptive gains, Ref. [27]
addresses this issue.
It can also be noticed from the eigenvalues shown in Table 2.1 that the P becomes
semi-definite by the end of the iteration, which is not desired as it slows the adaptation and
results in a deteriorated performance. As shown in the optimization algorithm, the SPR
property of T(s) depends upon selection of K LQ , hence a careful selection is necessary.
30
Chapter 3
Persistent Excitation
As discussed in Chapter 2 the closed loop adaptive system including the states of the
adaptive controller, is nonlinear and time varying. Hence, the state vector of the entire
adaptive system is composed of the state variables of the plant on hand and the adjustable
parameters of the adaptive controller on the other. The stability of such a system depends
upon the state error vector e and the parameter error vector . It was proven using
Lyapunov analysis that the state error e goes to zero asymptotically and all the signals in
the system remain bounded for the appropriate choice of the adaptive law. However, these
conditions are not sufficient for the performance and robustness requirements.
We also showed that the adaptive system converges to a linear time invariant system
called RLAS if the command signals are constants. As can be seen from the representation
of RLAS in equation (2.9), it contains the unknown parameter error 9.c which in turn determines the performance and robustness properties. Since O9c is a function of the reference
input r, adaptive gains F, Lyapunov matrix P, initial conditions xO, unknown uncertainty
A6 along with the presence of noise and unmodeled dynamics; it is virtually impossible to
31
estimate 0,c beforehand. As can be seen from equation (2.9) as 0.c -+ 0, the closed loop
adaptive system transfer matrix A,, = Am + A6bmc -+ Am.
Hence, in this chapter we
will discuss about conditions that make the parameter error, 0, go to zero which usually
depends upon the properties of certain signals in the system. Known as Persistent Excitation, it is defined as a condition on the integral of a vector function which assures the
convergence of parameter vector 0 to 0* in control problems.
Since, the reference input in the control problems is a scalar signal that can be
chosen by the designer, relating the conditions of persistent excitation of internal signals
of the system to equivalent conditions on this scalar signal is essential. To this end certain properties of algebraic and dynamic transformation of persistently exciting signals are
studied. The study of dynamics transformations of persistently exciting signals reveals that
the frequency content of the reference scalar signal determines the persistent excitation of
the vector signal obtained by a dynamic transformation. In the following sections we will
find the number of frequencies that are required to guarantee the persistent excitation of
the vector signal for a MIMO plant.
However, for systems with a large number of states, the number of frequencies required in reference signal for the persistent excitation of internal signals of the system,
would be significantly large too, which is usually not preferred due to the interference of
higher frequencies with the unmodeled or structural dynamics of the system. Therefore,
in the later sections of this chapter, the focus shifts to study the parameter convergence in
adaptive control problems as a function of a different class of reference inputs, which are
theoretically not persistently exciting in finite time but still enable us to achieve considerable improvement in the performance of adaptive systems. The property of such inputs is
32
denoted as GraduatedPersistent Excitation. In the subsequent section, we will use simulation studies as a tool to identify the nature of convergence of d for such reference inputs.
In Section 3.1, evolution of the need of persistent excitation is studied and a simple example is shown to introduce persistent excitation. Close relation between uniform
asymptotic stability and persistent excitation is studied in Section 3.2. Section 3.3 lists out
some important algebraic and dynamics transformation properties of persistently exciting
input signals. Section 3.4 introduces the design and evaluation of a high performance aircraft (X-15) and conditions for which the input signal is persistently exciting, are derived.
Simulation studies are performed to demonstrate the effect of persistent excitation on X15 linear model. In Section 3.5, the concept of graduated persistent excitation (GPE) is
introduced and a control group of GPE inputs is formed to assess the effect of graduated
persistent excitation on the performance of the aircraft.
3.1
Preliminary Study
This section forms the basis for the introduction of persistent excitation. As shown in
Chapter 2 in equation (2.8), the system error dynamics can be written as:
AbmwT
6
Am
0
-PwbimP
e
(3.1)
=
.
0
J
which can be compactly written as:
.t
Am
9-C(t)
B(t)
x(32
(3.2)
y
33
It was proved that the state error e goes to zero asymptotically. But the parameter error 9
does not converge to zero in most of the cases until certain conditions on the reference input
signal are satisfied. There are two reasons for this: firstly, the only goal of the adaptive
control law design is to drive the system to the trajectories where e goes to zero and these
trajectories may not be, and in most cases are not, the same trajectories where 0 also goes
to zero. For the state error e as given by the equation
e
(3.3)
= Ame + Abmw T
to go to zero asymptotically, parameter error 9 does not necessarily go to zero. What
usually happens in most of the adaptive systems is that the parameter error converges
to a hyper plane which is orthogonal to the regressor vector w. Hence, the product wTd
converges to zero resulting in the state error dynamics 6 = Ame, where Am is Hurwitz.
Secondly, as the state error goes to zero, it results in even slower movement of adaptive
parameter error towards zero because the adaptive parameters are adjusted according to
9
=
-PwbmPe
(3.4)
which is proportional to the state error. This is the biggest conflict faced in the case of
adaptive control while simultaneously identifying the unknown parameters because of the
nature of the adaptive law. The faster we try to get zero tracking error, the slower will be
the identification of the unknown parameters. Hence, though equation (3.1) is uniformly
stable as shown in chapter 2, uniform asymptotic stability can not be proved as 9 can not
34
be guaranteed to converge to zero. The following subsection shows some simple examples
which introduce the notion of PE and uniform asymptotic stability.
3.1.1
Examples
We will use a simple example of adaptively controlling a dynamical system and use it
to demonstrate the effect of persistent excitation of input signal r on the convergence of
adaptive parameters.
Example 1. A plant with an input-output pair u(.), xp(.) is described by the scalar differential equation
4(t) = apxp(t) + kpu(t)
(3.5)
where ap and kp are plant parameters which are constant and unknown, though the
sign of kp is assumed to be known. The plant state x, is desired to follow the reference
trajectory xm which is described by the following differential equation
±m(t)
= amxm(t) + kmr(t)
(3.6)
where am < 0 and am,km are known constants and r is a piecewise continuous bounded
function of time. The following adaptive laws are chosen to adjust the adaptive parameters
35
which guarantee that e
=
xp(t) - xm(t) -+ 0 as t -+ oo.
O(t)
=
-sgn(kp)e(t)x,(t)
k(t)
=
-sgn(kp)e(t)r(t)
(3.7)
The control input u(t) to the plant is given by the following equation:
u(t) = 0(t)xp(t) + k(t)r(t)
(3.8)
The ideal values of adaptive parameters which guarantee the closed loop transfer function
of plant with the controller is same as the transfer function of reference model, are given
by:
*=
m -a
k* =km
(3.9)
The above system is simulated with am
-1,
ap = 5, km = 1.5, k = 5. The figures
below show the trajectories of state error e and parameter error
#1 = 9 -
0*,
02
= k - k*
for inputs (a) r(t) = 1, (b) r(t) = sin(t), and, (c) r(t) = eo 2t sin(t). As can be noticed
from figure 3.1 and 3.2 that the parameter errors
#1
and
#2
do not converge to zero for
r(t) = 1 and r(t) = eo 2t sin(t) while they converge to zero for r(t) = sin(wt). Figure 3.3
shows the trajectories of norm of state error e and parameter error
#
for input signals (b)
and (c). It can be seen that the parameter error decreases when state error increases and
moves away from zero. This is a very important observation and implies that the reference
input signals should be such that it moves state error away from zero for certain time in
36
(a)
1
1.5
3
1
2
0.5 -
0 .5
-. -.
-.-
.-.
.-.-
-
-
-
--
0
ai)
0
0
-
- -
-1
-2
0
.-
-0.5 -
-0.5
20
10
30
-1
0
10
20
30
0
10
t
t
20
30
t
(b)
3
1.5
1
0 .5
-
.- .
C
-.-C'J
0D
0
--.
. -. -.
-
0
-
-
-0.5
-
-1
-20
20
10
30
-1
-.
.. -.
-.
.
-0.5
-
0
10
t
20
30
-1
0
10
t
Figure 3.1: Example 1: Parameter errorfor (a) r(t) = 1 and (b) r(t)
20
30
t
=
sin(t)
every time interval. Because of this property, parameter convergence is achieved for input
(b) while since input (c), which itself goes to zero after a time duration, is incapable of
providing parameter convergence.
This implies that certain reference input signals have certain properties which result
into the convergence of parameter error. These properties include that the state error e
should not go to zero before the parameter error < converges and the input signal should
persist until the parameters converge to their ideal values.
It can also be seen from Fig. 3.4 that for the cases where the parameter error does
37
2.5
(C)
1.5
2
-0.1
1
-0.2
-
-0.3
1.5
-. . . 0.5
-...
-...
1
-
0.5
-.
-0.4
c'J
g
a)
0
-0.5
-
0
0
-
-
-0.6
-0.5
-0.5
-1
-
10
20
-0.7
-1
30
0
10
20
-0.8
30
-
0
10
t
20
30
t
Figure 3.2: Example 1: Parametererrorfor (c) r(t) = eO.2 t sin(t)
(b)
1.4
1.2
-... .
-
1
0.8
0.6
.....
-- - -- - . -- .- .
-.-.--.-.-
-.-.-.-.-.-- ... ....-.-
.- . -. .- .- . ... .. . . . .-. --. --. - --.-.- --.
E 0.4 -.
- -. -.
-.
. . .. .
0
-.-.
.
0.5
u0
1.4-
----
0.8
-.
- - - -- -
. ..
-..
..- . ... . .- -. -.. . . -..-.. -.
-
-.
... ..
.... . . . .. . . .. . .
.. .
-.-.
1
-- ---.
.-.-
1.5
-.. -..... ----... -.. -)
2
-. -
2.5
.- -..-.-.-.-.-.-
E
0
0.6
nA.0
-.
L-
......
-. .--- - -..--...- - -
... .. -... -.... . .. .. - ...-. . - -.. .-.. .-.. -..-...--..
-..
0.5
norm e
2
1.5
2.5
Figure 3.3: Example 1: Norm of parametererror as a function of state errorfor (b) r(t) = sin(t)
and (c) r(t) = eO.2t sin(t)
not converge to zero, the parameter error vector
#
= [01
#2]T
converges to a plane which is
orthogonal to regressor vector w = [X r]T, thereby the product
when the parameter error vector
as e ->
0 as t
->
#
#TW
-+ 0 as t -- oo. Hence,
is nonzero, it always aligns itself such that
oo.
38
#TW
=
0 So
0.5
--.
---.
--- - ---------. --.--.-.-.
----. ---------------.-.
....
-.-.
--..
---...
0 .---. .. -.
-1.5
..-. .
..-.
..-..
. ..
-----------. ------.
----------....
-------
--
. -.--.-
... .. . . .. -..-.-.--. -. - -- -.-.-...-.--. ..
- -..---. ..
--- -- -.
- 0.5 - ..
-
- - --..---
..
-----...
..
-...
-2
-0
s
10
15
02
Figure 3.4: Example 1: Convergence of parameter errorvector orthogonal to the regressorvector
3.2
Uniform asymptotic stability and persistent excitation
The notion of uniform asymptotic stability in the equations of the form (2.8) have very close
ties with the concept of Persistent Excitation. Hence we first state the uniform asymptotic
stability conditions for the most frequently encountered equation in adaptive control.
As seen in the above example, the uniform asymptotic stability of origin of equation
(3.1) is a function of the properties of reference input signal. The following theorem lists
out the necessary conditions on these input signals for the uniform asymptotic stability of
equation (3.1).
Theorem 1. The origin of Eq. (3.1) is uniformly asymptotically stable if, and only if,
positive constants T,50 and co exist with a t2
39
E [t, t + To] such that for any unit vector
W E Rn
1
t 2 +6 0 W T (T)wdT
> co
(3.10)
Vt > to
For the proof of theorem 1, please refer to [25]. It is shown in the proof that if w
satisfies the condition in Eq. (3.10), then e has to assume a large value at some instant in
every interval [t, t + T]. Since V < -eQTe, this implies that V decreases over every interval
of length To which assures uniform asymptotic stability.
The above definition of uniform asymptotic stability is almost synonymous to the
notion of persistent excitation in a sense that PE determines a class of signals for which
the origins of two linear differential equations are u.a.s. This class of signals is adequate to
prove the u.a.s. of most of the systems and we will restrict our discussion to these particular
systems.
From Theorem 1, the definition of Persistent excitation follows directly.
Definition 1. The set of all functions u : R+
R' that satisfies condition 3.10 over a
period To for all t > to is persistently exciting and is denoted by
Q(n,toToT.
The subscripts n, to, and To in the definition refer to the dimension of the space,
the initial time and the interval over which the function u is persistently exciting and Eo
is defined as the degree of persistent excitation. The condition 3.10 can be interpreted
as a condition on the energy of w in all n directions. The condition can also be written
equivalently in matrix form as:
o
-f 2 +6 u(T)uTr()dr > aol
TO
t2
Vt > to
40
(3.11)
for positive constants to, To and ao. Although the matrix u(r)uT(r) is singular for each r,
condition 3.11 requires that u(r) varies in such a way with time that the integral of the
matrix u(r)uT(r) is uniformly positive definite over any time interval [t, t + To).
It is clear from the definition that if u is persistently exciting at time to, then it is
to. Also, if u E Q(n,to,To), then u is persistently
also persistently exciting for any time tj
exciting for any interval of length T1 > To.
To illustrate the above definitions of persistent excitation, some simple examples of
signals are presented.
Example 2. For u : R+ - 91
1. If u(t) = c, where c is a nonzero constant then u(t) E Q(1,to,To) for any finite To > 0.
Also, if u(t) = a sin wt for nonzero a, then u(t) E Q(1,to,To) and To = 27r/w.
2. If (i) u(t) -+ 0 as t
->
oo, (ii) u E V', or (iii) u E
L2,
then u(t) 0 Q(1,t0 ,T0 ) for any
finite To > 0.
Example 3. For u: R+ ,2
1. If u(t) = [sinwt, coswt] then u(t) E
Q(2,to,To),
where To = 21r/w.
2. If u(t) = zi, a constant vector in -R2 , then u(t) 0
Q(2,t,T 0 ).
But if u(t) alternatively
assumes values between z 1 and z 2 for finite periods T and T2 , where z1 and z 2 are
independent vectors in R 2, then u E (2,0,T 0).
3. If u(t) has two spectral lines at frequencies v1 and v2, then u(t) EQ(2,toTo)
41
3.3
Properties of Persistently Exciting functions
In this section we list some simple properties pertaining to algebraic and dynamic transformations of persistently exciting signals which will be useful in finding persistent excitation
conditions for a MIMO plant in the next section.
3.3.1
Algebraic Transformations
Lemma 1. Let u E QNn,to,To) and M be an (m x n) constant matrix, m < n.
Then
Mu E G(,,to,To) if, and only if, M is of full rank.
This lemma establishes that the PE of a vector is invariant under any nonsingular
time-invariant transformation. This is a very important property which we will frequently
use in the next section.
Lemma 2. Let u : R+ -
R", then u E Q(n,to,T) if, and only if, aTu E Q(1,to,T) for every
constant nonzero vector a E Rn.
The proofs of these Lemmas directly follow from Definition 1.
Definition 2. A bounded function u : R+ --+ R is said to be persistently exciting in R'
for r < n, if there exists an (r x n) matrix P such that Pu E Q(r,to,To) and r is the largest
integer for which this holds. The set of all such functions is denoted by
(t
Lemma 3. Let M be a constant matrix. Then the following hold:
1. If u E 00toT,
then Mu E Qi0
2. If u E
then Mu E QrLto T0 ) if M is an (m x n) matrix of rank r 2.
i(n,to,T0 )
if M is square and nonsingular.
42
.
3. If u E
then Mu E
(,,
3oT)
if M is an (m x n) matrix of rank r 2 where
r3 = dim[R(M) n R(P)] and R(A) denotes the range space of matrix A.
Definition 2 and Lemma 3 concern functions that are persistently exciting in a
subspace of Rn.
Lemma 4. If u : R+ -+ Rn, and u is not persistently exciting in any subspace of R", then
u E VI.
Lemma 5.
E
£2,
1. If u : R
then u
-+
f 2 (n,to,To)
2. If u1 E Q(n,to,To),U2 : R+
some t1
>
3. If U1, U2 E
R,
and any component of u(t) -+ 0 as t -+ oo, E
£1,
or
for any T.
-,
Rn and u 2 -+ 0 as t
-
oo, then ui + U2 E
£(n,ti,To)
for
to. The same result also holds if u 2 E L' or £2.
£(n,to,To),
then ui
+
EU2 E £(n,to,To)
for some sufficiently small E E R.
Lemma 5 indicates that if a signal is persistently exciting then the addition of
another signal that is small in some sense, does not affect its persistent excitation.
The following examples list out persistent excitation properties of a special class of
functions which are of theoretical and practical interest in the study of persistent excitation.
Example 4.
1. If si(t)
=
_1 (ai sin wit),
wj, then si E Q(1,tO,TO) for some finite
T > 0 if ai # 0 for some i.
2. if s(.) E R with elements si = aj sin wit, a, # 0, and wi are distinct, then s(.) E
Q(n,to,To) -
43
Proofs of all of the above lemmas are straightforward and follow directly from the
definitions, hence they are avoided. For reference please refer to [1]. It can be seen from the
above lemmas, that if u E
Q(n,to,To)
then a linear algebraic transformation can only result in
a function y E Q(m,to,To) where m < n. The linear dynamic transformation discussed in the
next subsection, can generate a vector that is persistently exciting in dimensions greater
than n from a function u E Q(n,to,To).
3.3.2
Dynamic Transformations
Consider the LTI dynamical system as described by the following differential equation:
(3.12)
. = Ax + Br
where x E Rn, r E W"and A, B are constant matrices with A being asymptotically stable.
The properties of such dynamic transformations of persistently exciting functions
are best described by the following lemmas:
Lemma 6. A necessary condition for x to belong to Q(n,to,To) is that (A, B) be controllable.
Lemma 7. Let y(t) = W(s)u(t), where u, y : R+ -+ Rn. Then the following hold:
1. If W(s) = 1/q(s), where q(s) is Hurwitz, then u E Q(n,to,To) => y E
(nto,To).
2. If W(s) is a proper transferfunction with poles and zeroes in the open left half plane,
then u E Q(n,to,To) => y E
£(n,ti,T1 )
for some t1
to and T 1 : To.
Theorem 2. Let ±1 = A 1 x 1 + B 1r and t2 = A 2x 2 + B2 r be two dynamical systems such
that (Ai, B2 ) is controllable, and A, are n x n asymptotically stable matrices for i = 1,2.
44
Then x, E
£(n,ti,Ti)
if
X2
E
£(n,t2 ,T 2 )
for some ti and t 2 > to, where r(t) is defined for
t > to.
Also, it is known that the number of frequencies that u should contain, in general,
is proportional to the number of unknown plant parameters to be estimated for a linear
plant. Based on this observation, a new definition, related to PE property of input signals
is given by [26]:
Definition 3. A signal u : R
-+
R is called sufficiently rich of order n if it consists of
at least 1 distinct frequencies.
For example, the input u
and w2
:/ wk
=
1 A.
sin wit, where m > n/2, A = 0 are constants
for i =A k is sufficiently rich of order n.
Therefore, roughly speaking, u should have at least one distinct frequency component for each two unknown parameters.
3.4 Persistent Excitation and Adaptive Control
In the previous sections we introduced the notion of persistent excitation and its algebraic
and dynamics transformation properties. Since, the reference input in the control problems
is a scalar signal that can be chosen by the designer, relating the conditions of persistent
excitation of internal signals of the system to equivalent conditions on this scalar signal
is essential. This can be done using the transformation properties of persistently exciting
signals as stated above. To illustrate this, an example of linear model of high-performance
aircraft is shown in this section and condition on the reference signal for persistent excitation
45
of the plant and thereby convergence of parameter error to zero, is derived using the
equivalent conditions on internal control signals.
3.4.1
An example of High performance aircraft dynamics
This section will build up an augmented adaptive controller for a linear model of X-15
obtained by linearizing the nonlinear model [28] at a specific trimming point. It should
be noted that the design method is not restricted to this particular model and that it is
applicable to any plant dynamics that is taken of the following form.
Open-Loop Model
The following nonlinear flight dynamics
X = fP(X, U)
(3.13)
is linearized at the trim point V = 1929.7 fps, h = 60000 ft,T = 7062(lb),6e = -7.3151 deg
#
and a = 5.4643 deg. The fast states xp = (a
ing control input 6 = (,
6a,
p
q
r)T
and the correspond-
6,) can be extracted from the full state vector X and
the full control vector U. This leads to the plant dynamics:
(3.14)
4 = Apxp + Bp6 + dp
where, x, consists of angle of attack a, angle of sideslip 3, body roll rate p, body pitch rate
q, and body yaw rate r. The vector of controls 6 consists of conventional control surface
46
commands and dp E R is the trim disturbance.
Baseline controller
To overcome the drift in lateral dynamics due to the trim disturbance, an integral controller
was added in the roll rate p and the combined yaw rate/sideslip angle term r - 0,
xc
=
[q,
(3.15)
r -3]
PI
The decision to combine r and 3 was made to reduce the number of states by exploiting
their strong coupling. We can write the dynamics of these integral controller states as:
e =
Acxe + Bexp
(3.16)
+ B 2 cu
where u is the vector of inner loop commands,
U = (a'"c
ocmd
Pcmd
(3.17)
r'cd )
The nominal baseline LQ controller is then designed in the standard form as:
6nom =
(3.18)
K~x
where x = [xp
x,] and K, denotes the nominal feedback gain matrix designed for the
dynamics given by Equations (3.14) and (3.16) around the trim point and minimized the
47
cost function:
J =j
(xTQirx
+ 6oTRiqr
(3.19)
nom)
Adaptive Controller
The main problem that needs to be addressed is the accommodation of uncertainties that
occur due to actuator anomalies and unmodeled dynamics. These uncertainties are represented by a combination of two features, one that include a parametric uncertainty matrix
A that represents loss of control effectiveness and the other one that includes an unknown
state dependent vector Ko(xp).
Both of these effects are incorporated in the linearized
dynamics (3.14) as:
z,
=
(3.20)
Apxp + BpA (6 + Ko(xp)) + dp
This leads to an augmented plant dynamics as:
ke
[
:1
Bc Ac
[1+
XC
p]A (6+Ko(xP)) +
0
0[ U +
(3.21)
B2c
or equivalently,
. = Ax + B 1 A (6 + Ko(xp)) + B 2 u + d
(3.22)
The overall dynamics given by equation (3.22) is used for the design of adaptive controller.
48
In order to ensure safe adaptation, a target dynamics is specified for the adaptive
controller using a reference model. Reference model is designed using the plant dynamics
in the absence of uncertainties or unmodeled dynamics and the baseline nominal controller.
This implies that the basic goal of adaptive controller is to achieve the same performance
and response that would have been obtained had there been no failures or unknown dynamics present. Thus, the dynamics of reference model can be written as:
Xm = (A
+ B1K)xm±+B 2U= Axm + B 2U
(3.23)
where the matrix Am is assumed to be Hurwitz.
Using Equations (3.22) and (3.23), an adaptive control input is added to the baseline
controller as:
6
ad
=
E(t)x +
where, 7 = [E
Od(t)
=
(3.24)
8)TW
G] are adaptive parameters that are adjusted according to the adap-
tive law given in Eq. (3.25) and w = [x
1 ]T
is the regressor vector.
In this chapter we will deal with only the first form of uncertainty i.e. A and assume
Ko (xp) = 0. The boundedness proof and the analysis in the presence of KO (xp) is discussed
in the next chapter. For the current purpose, the adaptive law is given by [1]:
O=-
'weTPB1
(3.25)
This adaptive law provides the boundedness of all the signals and the convergence of the
49
plant states to the reference model states. The stability can be proved along the same lines
as in Chapter 2 by choosing an appropriate Lyapunov function.
The adaptive gains F are selected according to the following empirical formula [27],
which arises from the inspection of the structure of the adaptive law:
_
diag(v)
(3.26)
Tminpkcmax
where,
1. v E !R" is a vector given by the sum of the columns of G* where E* corresponds
to the uncertainty A for which the plant has the most unstable eigenvalues.
The
components of G* are given by:
0*
(3.27)
-KxX-
=
[1
*
1
1
T
2. Tmin is the smallest time constant of the reference model.
3. p is the norm of BTP.
4.
6
cmax
is the maximum amplitude of the reference input signal.
5. FO is a small positive definite diagonal matrix which ensures that F is positive definite.
Thus, the full control input can be given by:
6 = 6nom +
6
(3.28)
ad
50
And the overall control architecture can be seen in Fig. 3.5. The block diagram of the full
adaptive system is similar to the one shown in Fig. 2.3.
r_0 Adaptive
6ad
0 Control
no "
Aircraft
Baseline
Integral
Model
Control
Control
----------------.----------------lateral states
fast states
Figure 3.5: Overall control structure for the model adaptive control system. The baseline controller is augmented by the adaptive controller.
3.4.2
Persistent excitation of Input Matrix
The error equation for the aircraft dynamics of the high performance aircraft discussed in
previous section, can be compactly written as:
[]
A -B(t)
x
(3.29)
LYJ
L C
0
JLYJ
For the uniform asymptotic stability of the equation (3.29), i.e. for the parameter error
matrix to go to zero, the input matrix B(t) should satisfy the following persistency of
excitation condition [25]:
Lemma 8. There exists positive numbers To,E 0 and 60 such that given t1 > 0 and a unit
51
vector w E !R", there is a t 2 E [t 1 , t1 + To] such that
J2/
2+60
BT (r)wdr I >0
(3.30)
For the system described in previous section we have:
B(t) = [bA 1WT b2 A2WT bX 3WT]
(3.31)
where B, = [bi b2 b3} is n x m constant matrix, A = diag(Al A, A,) is m x m unknown
constant diagonal matrix with positive nonzero elements and w is the state regressor vector
composed of the system's states. Hence, the condition 3.30 is equivalent to the following:
(3.32)
t2+o
->
I Eo
/2+60
t2
b3 wT]wdr
[biJlwT
2 wTw 2 b3 wTw 3ldT
[biwT b2bwT
Eo
(3.33)
where w is a unit vector of dimensions nm and w = [w, w 2 w3]T. Assuming the following:
Assumption 1.
1. b1 , b2 , b3 are linearly independent vectors.
2. w is persistently exciting vector in n dimensions.
52
Assumption (1) holds true if there is no redundancy in the control allocation matrix.
->
[b1wTw 1 b2 wTW
2+
t2
2
J
1lb1
b3 wTw 3 |]dT
WTwldr
J
- lb
2
t2
WTW 2 dTI -
t2
lb 3
J
(3.34)
WTW 3 dTI
> ||bilai - lb2Ia2 - Ib3 la3l
where, a,
=
5
ft2+ 0
wiTwdw
21
,
a2
=
ft2+0
2
WTW2dr 11and a 3
=
It2+O
2
wT
dT.
d
If w is persistently exciting in 8-dimensions, then max(ai, a2 , a 3 ) = 0. Also, since
bl, b2 , and, b3 are linearly independent, it results into:
lbi jai - lb2 Ia 2
=I
=>
[biwTW1
lbi ai -
b2a2
b2 w Tw 2
-
-
b3 1a3 $ 0
1b
31a
3|
(3.35)
EO
b3 wTW 3]dr
> 60
Hence, if the assumptions 1 are satisfied, then the persistent excitation of equation (3.29) is
guaranteed. The following subsection discusses the sufficient conditions on reference input
signals to satisfy the (2) assumption regarding persistent excitation of the regressor vector.
Persistent excitation of regressor vector
Since the adaptive system is nonlinear and time-varying, it is not possible to determine
conditions under which the regressor vector w is persistently exciting.
However, since
u = wm+e, and limt,, e(t) = 0, from the results of Lemma 5 in section 2.2.1, we conclude
53
that w E
Q(n,tiTo)
if Wm E Q(n,tTo) for some ti
>
to. This also follows from the Theorem
2 as both A,, and Am are asymptotically stable and (A,,, Bp), (Am, Bn) are controllable.
Hence, uniform asymptotic stability of the adaptive loop is assured if the output vector of
stable linear time invariant system (reference model) is persistently exciting. Wm can be
written as :
(3.36)
WM(S) = M(s)r(s)
where M(s)
=
(sI - Am)- 1 Bm is a n x m transfer matrix of transfer functions and r(s) is
the m x 1 reference input. By expanding M(s), equation (3.36) can be written as:
Miiri(s) + M12 r 2 (s) + M 13r 3 (s) + M 14 r 4 (s)
WM(S)
=
(337)
M 21ri(s) + M 22r 2 (s) + M 23rs(s) + M 24 r4 (s)
Mniri(s) + Mn2r 2 (s) + Mn3r 3 (s) + Mn4 r 4 (s)
911(s)
() r1
Ti(S)+
1(s)
q1 (S)
_____~rl(S)
P92(8)
2
r2 (s) +
01(8)
23() r3 (s)+
92(s)
+ Eaa(8)
9 2(8)
r3(s)+
r(s) +
pn3
9~3(9)
r3 (S) +
14 (s)
r4(s)
4(s)
Pq4(8))(r 4
(3.38)
(S)
(3.39)
54
P11(s)
Wn(S)
P21(s)
1
qi(s)
P13(s)
P12 (s)
P22(s)
1
ri(s) +
r2 (s)+
q2(s)
Pni(S)
1
P23(s)
r 3 (s) +
q3 (s)
Pn3(s)
Pn2 (s)
P14(S)
+
P 24 (s)
1
(3.40)
r4 (s)
q4 (s)
Pn4 (s)
1
1
S
ri(s)
S
qi(s)
5n-1
1
r2 (s)
S
r3 (s)
q3 (S)+
q2 (s)
S n-1
Sn-1
1
S
r4 (s)
(3.41)
q4 (s)
n-.
Where Mi contains the coefficients of polynomials p32(s) where
i
=
1, 2,..., n and
i = 1,2, 3, 4. Since qi (s) are Hurwitz polynomial, it follows from Lemma 7 that persistent
55
excitation of wm in equation (3.37) is equivalent to persistent excitation of:
1
Wma(S)
=
1
ss
ri(s) + M 2
M
1
s
r2 (s)+ M 3
n-1
sn-1i
r3(S) +
n-1
1
s
+ M4
r 4 (s)
(3.42)
sn-1
From the Lemma 3, it is clear that if rankMi = mi < n and ri is persistently exciting in
dimensions ni, i.e., ri E
"tT
ni < n, then Miri is persistently exciting in at most ni
dimensions. For example, if ri is a sinusoid, then Mir, is persistently exciting in at most 2
dimension if Mi is of full rank.
However, the advantage of having multi input can be exploited in a way that regressor vector can be persistently exciting even if not all of the inputs have full degree of
persistent excitation. In fact, the following condition will guarantee the persistent excitation of regressor vector in n dimensions:
Claim 1. wn, is persistently exciting in n dimensions if rank(M1P1 + M 2P2 + M3 P3 +
M 4 P 4 ) = n.
where P is such that ri = Pigi E
2
'
rank(P) = ni and gi E
Q(n,to,To)
The proof of Claim 1 follows directly from the lemmas 3. The conditions on the
56
reference input signal which satisfy the claim 1, can be easily found using an offline trial
and error test on the rank of M1 P1 +M 2 P2 +M 3P3 +M 4P4 for different input signals because
only the knowledge of A,, is required to calculate the rank of the matrix. For the system
described in the previous section, the rank of the matrix MIPi + M 2P2 + M3 P3 + M 4 P4
is equal to n if the reference input r consists two distinct frequencies in a command and
one distinct frequency in each of the ,, p and r commands. This guarantees the persistent
excitation of the regressor vector of section 2. It should be noted that the persistent
excitation is independent of the frequency value as long as the frequencies are distinct and
well seperated.
It is important to note that the aim is to find the least number of frequencies that are
required for the persistent excitation of the regressor vector w. w will have a higher degree
of persistent excitation and probably in higher dimensions for more number of distinct
frequencies in the reference input signal. According to the sufficient richness definition 3,
w would be persistently exciting for a stable proper transfer matrix M(s).
3.4.3 Simulation studies
The system as described in equation (3.22) is simulated with the reference signal having 5
distinct frequencies, two in alpha command and one each in other three command signals.
Figure 3.6 shows the eigenvalues of the reference model as denoted by (i), (ii), (iii), (iv)
and the eigenvalues of the closed loop plant with nominal controller alone as denoted by
1, 2, 3,4. It can be seen that in the absence of any adaptive controller, the closed loop
eigenvalues which determine the performance of the closed loop system, axe far from their
57
ideal values. While the eigenvalues of the closed loop adaptive system (RLAS) coincide with
the eigenvalues of the reference model because of the persistent excitation of the reference
signal.
Eigen values of Closed loop system
Ref
PE
Nominal
U
o
0
2
........................
. . . . . . . .. . . . .
......
2 (iv)
......................
-
1 -
.....
04
.
...
...
............ ..........
.0i ... . .
..
E
-1 -
. . . . . . . ... . . . . . . . . . . . . ... . . . . . . . .
-
-2
........................
-3
........................
-4
-7
-6
-5
0
-3i
-4
-2
-1
0
real(?)
Figure 3.6: Eigenvalues of closed loop system
Similarly, the norm of the parameter error matrix decreases as the simulation time
is increased as more information content contained in the reference signal is passed onto the
time.
system. But as can be seen from the figure 3.7, the norm does not go to zero in finite
change
This is a very common behavior of adaptive control systems because the rate of
goes to
of parameter error matrix is directly proportional to the state error vector which
matrix
zero as the control objective is achieved. Therefore, though the parameter error
theoretically goes to zero as t ->
oo, the rate at which the adaptive parameters converge to
58
their true values decreases over time and it is difficult to compare different reference inputs
on their extent of persistent excitation based on this metric. This issue will be addressed
in the next chapter.
220
160 .
. . .%.. . . .. . . . .
1 4 0 - -.----- ----. --- ---- --- -.
12
--- --.
--- --- --.------ ---- ----------------------.-- ----
---- ---- - -. -.-- -- ---.-.----- ---.-..------.-. -.---.
---.--- -
-.--..
-.
-.-- . ..
- --...
-- .
-...
---. --------- -- - - ..
----...
...
-------.
6 0 - ------ --....
20
0
100
200
300
6W
400 0oo
700
-- - - -- - - -- - - -
8W
9W
1C00
t
Figure 3.7: Norm of the Parametererror matrix
3.5 Graduated Persistent Excitation
As noticed in previous section, 5 distinct frequencies are required in the reference signal for
the persistent excitation of the error dynamics given by Eq. (3.22). This would guarantee
the asymptotic convergence of the parameter error to zero and the closed loop matrix would
converge to the reference model matrix resulting in the same robustness and performance
properties as of reference model. However, the persistent excitation condition requires the
59
aforementioned sinusoidal inputs to be present all the time, making it difficult to follow the
desired trajectory as aircraft control systems are designed to follow the reference signals.
Presence of a number of frequencies can also cause interference with unmodeled dynamics
endangering the structure of the aircraft. Therefore, it is not feasible to use persistently
exciting inputs in the aircraft control systems. The next question arises is how to design
inputs which slowly move the plant towards persistent excitation while following the desired
trajectory and without exciting unmodeled dynamics. This question is addressed in this
section and we call such a property of reference inputs as 'GraduatedPersistentExcitation'
and the class of such input would be denoted as (GPE) inputs.
In [29], it was shown that the input time history has a significant impact on the
achievable accuracy for the model parameter estimates computed from the measured data.
The choice of input implicitly includes the length of the maneuver. In [30] Morelli details
certain input forms which have advantages of easy implementation in flight and simple
design based on current estimates of modal frequencies and steady state gain. These input
forms were seen to be very effective for parameter estimation. This observation forms the
basis of designing 'Graduated Persistently Exciting' inputs which are of the similar form
as discussed by Morelli.
Figure 3.8 shows the conceptualized inputs. These include a step input superimposed with (a) a doublet, (b) a 3-2-1-1 , and, (c) a sinusoidal signal. The design procedure
for 3-2-1-1 input is:
1. Match the frequency of the 2 pulse to the current estimate of natural frequency of
dominant oscillatory mode.
60
2. Scale the 3 and 1 pulse widths in proportion to the 2 pulse.
Here, '3' pulse is the 1st step, '2' pulse is the 2nd step and '1' pulse is the third
and fourth steps in the 3-2-1-1 input as seen in Fig. 3.8(b). The frequency of the doublet
and for the sinusoidal is chosen so as to match the estimate of the natural frequency for
the dominant oscillatory mode. Similarly, the frequency of the 2 pulse for 3-2-1-1 signal
is matched with the natural frequency for the dominant oscillatory mode and the 3 and 1
pulse widths are scaled in proportion to the 2 pulse. The amplitudes are chosen so that the
output amplitudes do not exceed values that would invalidate the assumed model structure
using the current model estimate. Also, it is made sure that the root mean square (rms)
value of amplitude of each input is same as the amplitude has an effect on the parameter
estimates.
For the plant described in previous section, the natural frequency of the dominant
mode is 0.6 Hz based on the estimates. Hence the time duration for the doublet is 1.2 s.
The plant given by equation (3.22) is simulated along with the controller given by (2.24-25)
and (2.28) using these three GPE inputs as shown in Figure 3.8 and the norm of parameter
error matrices are compared. As can be seen from Figure 3.9, the norm of the parameter
error for each of these three signals is higher than the norm of parameter error for the
PE input as shown in Figure 3.7. Nonetheless, GPE inputs fare very well and are a step
behind PE inputs with respect to the convergence of parameter error to zero. Also, their
feasibility for implementation gives plenty of incentive to look more into the properties of
GPEs and their effect on the performance of closed loop adaptive systems.
61
(a): doublet
5!
0
5
10
15
2:
-2
(b) 3-2-1 -1
30
3s
40
30
3s
40
(c): sinusoid
5!
40
5
10
15
20
t (88c)
25
Figure 3.8: Graduated Persistently Exciting inputs
3.5.1
Effect of the amplitude and time duration of reference signal
The amplitude and time duration of the reference input has considerable effect on the
parameter estimates achieved, thereby affecting the performance of the closed loop adaptive
system. It is very clear from the figure 3.9 that the longer the input lasts, the closer the
parameter estimates move to their ideal values as more information content in the signal
is passed onto the system, though the rate of convergence gets slower.
Similarly a higher amplitude of input signal results in a better performance until
the output amplitudes do not exceed values that invalidate the assumed model structure
and thereafter the performance deteriorates.
Having demonstrated the advantages and feasibility of GPE inputs and being equipped
62
(a): doublet
200
...... -..
.................- .............. .
2
E
..... -------------.-.
0 150 - ---- -------.--
--.--
z
1001
------------.-
-.-- -----.----.-.-.--
100
50
.
1j50
(b): 3-2-1-1
250
200
0
b
z
..
.......
...............
.- .--------- .---- . --- . ....- ..-------- ---------------..
- ---- ---. ...---- ---
50
0
100
0
15
(C) : sinusold
~200
............- ...............
.................
...
E
0 150 -----------------------------.
z
innl(
0
-------..-...------------.-----...----.-------.------------.----50
time (Sec)
100
150
Figure 3.9: Norm of Parametererrormatrix for GraduatedPersistently Exciting inputs
with the basic understandings of the parameters that effect their performance, we address
the issue of finding a common tool which can access the performance of adaptive systems
in the next chapter.
63
Chapter 4
Performance Metrics for an Adaptive System in
Steady State
As discussed earlier in chapter 2, any adaptive system is essentially represented by a set of
nonlinear and time varying differential equations consisting of some unknown parameters.
Also, we know that this closed loop adaptive system converges to a linear time invariant
system (RLAS) in the steady state. In this chapter we identify some tools which can be used
to measure the effectiveness of an adaptive controller and to ensure that the RLAS, which
the adaptive system converges to, satisfies the pre-specified performance requirements and
has good stability margins.
Section 4.1 provides the overview of the problem statement. Section 4.2 and 4.3
discuss some of the potential tools specific to adaptive control that can be used to measure
the effectiveness of an adaptive controller. MIMO margins based on the singular values
are introduced as an effective tool for this purpose in Section 4.4. Subsequent subsections
highlight the multivariable nyquist theorem, derivation of MIMO margins and desired fre64
quency response of sensitivity functions. An example of a high performance aircraft is given
in Section 4.5 and simulation studies are used to calculate MIMO margins of various closed
loop adaptive systems in steady state (RLAS).
4.1
Problem Statement
In chapter 2, we showed that the RLAS contains some unknown adaptive parameters
such as OE
that determine the performance properties of the adaptive system and are a
function of the reference input. We also discussed in chapter 3 that in the presence of
the persistently exciting reference input the control parameter error Exc converges to zero
and hence the closed loop adaptive system converges to the reference model. Due to the
infeasibility of implementation of persistently exciting inputs in an aircraft system to carry
out pre-specified maneuvers, we introduced the concept of graduated persistent excitation
(GPE).
It was also observed in Chapter 3 that though GPE inputs result in an improved
performance over nominal controller, they do not provide convergence of 0.c to zero in
finite time. Hence, the closed loop transfer matrix at steady state GRLAS(S) is different
than the reference model transfer matrix Gm (s). Here steady state is defined at t = T,
where T denotes the time instant when the adaptive control parameters stop changing.
In general, a plant dynamics with adaptive controller can be written as:
= Ax + B1 A(e
T x-
Kx)+ B2 u+ d
(4.1)
65
In steady state,
. = Ax + B1A(8)x - Kx) + B 2u + d
(4.2)
where Ec = limtT 8 and Ec = Ec - 8*. Hence the closed loop transfer matrix GRLAS(S)
is given by
GRLAS()
=
sI - Ac)-'Bi
(4.3)
Act = A + B1A(T - K);
(4.4)
where
GRLAS(S)
Oc
=
ff(8Oc)
(4.5)
f2(u)
(4.6)
Hence, for every reference signal ui, there is a different GRLASi(S), which is different
from Gm(s) = (sI-Am)-'Bm. Therefore, to quantify the properties of closed loop adaptive
system with respect to the reference model, there is a need of a tool which can measure
the closeness of matrix GRLAS(S) to Gm(s). The following sections focus on this aspect.
4.2
Frequency Response
Bode plot is a very common tool used in SISO linear systems theory to analyze closed loop
system properties such as stability margins, disturbance rejection, DC gain and stability.
66
Therefore, this section explores the viability of Bode Plots in assessing the closeness of
matrices GRLAS(s) and G,,(s) by comparing the outputs generated by each of them. For
MIMO systems, bode plot of each input/output channel, uj/yi, have to be considered to
analyze aforementioned properties. To this purpose, linear model of a high performance
aircraft was simulated and the bode plots of two input/output channels are plotted for
distinct reference inputs.
Figures 4.1 and 4.2 show that bode plot of angle of attack, yi, with respect to the
elevator command, u 1 , and pitch rate, y2, with respect to the elevator command respectively. It can be noticed that these plots are not a good metric to compare the effect of
different reference inputs as (i) there is a many to one mapping with respect to the reference
input i.e. the change in the bode plot with distinct reference inputs is not easily noticeable
rendering it difficult to compare various reference inputs and (ii) for higher order MIMO
systems, it is tedious to look at multiple bode plots.
4.3 Specific Tools for Adaptive Control
This section focuses on few other tools unique to the adaptive systems that can be used
to measure their performance. It is evident that the closer the estimates of the control
parameters EOc are to their ideal values 8*, the closer the performance of the adaptive
system is to the performance of ideal reference model. Hence, one measure of the closeness
of the adaptive system to the reference model could be the norm of the error between
the estimated value of the control parameters and their ideal values, i.e.
110, -
8*11.
But, in the absence of persistent excitation, distinct uncertainties and various inputs can
67
Bode Diagram
-50-
-200
CO
Ca
0 2- -
-
; i
-
-
-
-180
-270 102
t
0
0
I
a
I
10
10
Frequency (rad/sec)
102
10104
Figure 4.1: Bode plot of input u1 with respect to output yi of reference model (solid line),
nominal (dashed line) and various adaptive systems for A = diag(O.25 0.25 0.25)
lead to the same norm of parameter approximation error although the spatial position of
parameters might be favorable for stability and performance properties for certain input
and uncertainties compared to others. Hence, this measure also results in many to one
mapping and does not quite distinguish between the various spatial positions of control
parameters.
Since there is a direct relationship between the closed loop eigenvalues and the
performance of any linear system, another measure could be how close the eigenvalues of
the closed loop adaptive system A(Ad) are with respect to the eigenvalues of the reference
model A(Am).
This measure also suffers from many to one mapping drawback and does not
serve purpose of comparing different reference inputs (in turn different GRLASi ). Hence in
68
Bode Diagram
-20
-
-
-
-
-
-
--
--30 -40 -
--
-3
e
-
- -
-
--
-60 ---70 -
--
-so10
\01010
-
45 -
9
-...-
-
-45 ---
CL-90 -135
-180
-
---
\
--
-
--
10-2
1
10
10,
Frequency (rad/sec)
Figure 4.2: Bode plot of input u 1 with respect to output Y2 of reference model (solid line),
nominal (dashed line) and various adaptive systems for A = diag(0.25 0.25 0.25)
the next section we introduce MIMO margins which provide a unique scalar that enables
us to distinguish and compare the performances with respect to various reference inputs
and uncertainties.
4.4
Stability Margins of MIMO Plant
As discussed in previous sections, there is a need of a tool which can provide a measure
of the stability and performance properties of an adaptive system in the steady state at
the same time providing distinction among various GPE reference inputs in their degree of
persistent excitation. In this section we discuss MIMO stability margins which could be a
69
potential tool for this purpose.
For SISO systems the gain margin and the phase margin are both well accepted criteria for measuring relative stability. In [32], the multivariable phase and gain margins are
developed by examining the polar decomposition of an uncertainty matrix in the feedback
path. Here, we will use the method called singular value analysis which is based on the
singular values of some important closed loop transfer function matrices between specified
inputs and outputs in the system [33]. It is a direct extension of the concept of SISO stability margins but it allows for simultaneous independent variations, while the SISO analysis
allows only for single-loop variation.
The following subsections give an overview of the important transfer functions,
multivariable Nyquist theorem and the derivation of MIMO stability margins based on the
singular values.
4.4.1
r
Overview of Transfer Functions
e K(s)
U1
G(s)
yieU
r
K(s)
G(s)
U0
Uy
output loop break point
input loop break point
(b)
(a)
Figure 4.3: MIMO system at two distinct break points
The figure above shows a multi-input multi-output (MIMO) system with the unity
feedback gain at two different break points. The returned signal can be written for each
70
case as:
Uo(s)
(4.7)
= - K(s)G(s) Ui(s)
L1(s)
Uo(s)
(4.8)
= - G(s)K(s) Ui(s)
L2 (s)
where u E R"u, y E R"Y,K(s) E Cnu n, G(s) E Cny "n and L(s) is the loop gain matrix.
For SISO systems, loop gain is identical at plant input and plant output but for MIMO
plants, it is distinct at both the loop break points. Therefore, the margins need to be
calculated with respect to both loop gain matrices to identify the worst case margins. Two
important closed loop transfer function matrices are given by:
E(s)
S(s),
E(s) = (I + L(s))-=
R(s)
L~) 1
Y(s)
Y(s) = (I + L(s))-'L(s) = T(s)
R(s)
(4.9)
where S(s) is sensitivity matrix, T(s) is complementary sensitivity matrix, I + L(s) is the
return difference matrix and I + L-'(s) is called the stability robustness matrix .
Sensitivity matrix embodies reference command tracking and disturbance rejection
properties while complementary sensitivity matrix represents robustness properties with
regards to high frequency unmodeled dynamics and sensors noise. Since, in many real situations the reference and the disturbance signals contain mostly low frequencies, margins
based on the singular values of sensitivity matrix are more crucial for lower frequencies in
order to achieve good tracking of the reference signal and good disturbance rejection properties. Similarly, margins based on the singular values of complementary sensitivity matrix
are of more importance at high frequencies for robustness to high frequency unmodeled
71
dynamics and sensors noise. Hence, margins based on both of the sensitivity matrices need
to be considered to calculate worst case margins over all frequencies.
Multivariable Nyquist Theorem
4.4.2
For the calculation of the margins for MIMO plants, first an understanding of multivariable
Nyquist theorem is essential. Please note that the zeroes of return difference matrix are
same as the poles of closed loop transfer function matrix. Hence, instead of looking at
the encirclements of (I + L(s))-L(s) around (-1, J0), let us look at the encirclements of
I+ L(s) around (0, j0).
Theorem 4.1 The control loop transfer function matrix (I + L(s))-L(s) is closed
loop stable iff det ((I+ L(s)))
n times around (0,
0, Vw and furthermore the plot of det ((I+ L(s))) encircles
j0), where n is defined as the number of right half plane poles of the
determinant of the denominator matrix of the rational coprime factorization of the loop
matrix [34],
[35].
This theorem forms the basis of the MIMO stability margins discussed in the next
section.
4.4.3
Derivation of MIMO Stability Margins based on Singular Values
Let us consider the following MIMO system with state feedback:
±=Ax+ Bu
(4.10)
u=-Kx
Assuming that the feedback gain K stabilizes the system shown in the above figure, let us
72
( sI - A)~I
B
K
Figure 4.4: MIMO system uith feedback gain
insert the gain and phase uncertainties of the form A = diag[kiemi] and find out how much
k and /i the system can tolerate before it becomes unstable and this gives us the stability
margins of the above MIMO system. According to theorem 4.1, the distance of the plot
diag(kje o)
-*
(sI - A)~' B
1
K
Figure 4.5: MIMO system with gain and phase uncertainties
of det ((I + L(s))) from (0, jO) is a measure of stability robustness, which is measured by
the 'size' of return difference matrix I + L(s) which in turn is represented by the singular
values of return difference matrix for MIMO systems. Since the nominal system without
the uncertainties is stable, the return difference matrix is nonsingular.
For an unstable
system, the return difference matrix becomes singular. The addition of
A,
changes the
encirclements of det ((I + L(s)A)) around (0, JO) which results in the singularity of the
new return difference matrix (I + L(s)n). Using this, a sufficient test of stability in the
73
presence of gain and phase uncertainty is given by the following inequality:
1 - I)
o(I + L) ;> o(a~
(4.11)
where - represents the minimum singular value of a matrix and - represents the maximum
singular value of a matrix. Singular values of a square matrix A are defined as the square
roots of the eigenvalues of AHA, where AH is the conjugate transpose. Hence minimum
singular value of the return difference matrix must be larger than the uncertainty for the
stability. For details of the stability test, please refer to [351.
Let min,, (I + L(jw)) = a, and using ai(classical gain margin &-'
=
1 -
I) =
,
,
Eq. (4.11), the
and phase margin ,&- = e-A can be written as:
1 - a, 51/k i 1 +a,
<k <
+a,
1
1
(4.12a)
a,
PM = t2arcsin(-)
2
(4.12b)
Hence minimum singular values of return difference matrix measure the margins based on
error sensitivity matrix. A value of a, = 1 results in the best sensitivity based margins i.e,
-6 dB < k2
+oo and PM : ±60'. Similarly, to measure the margins based on comple-
mentary sensitivity matrix, we need to look at the singular values of stability robustness
matrix I + L(s)~1 and the following condition has to be satisfied for the stability in the
74
presence of uncertainties:
j(I + L~)
> F(A
Let min, a (I + L(jw)-)
(4.1)
I)
-
, the classical gain margin and phase margin can be written
=
as:
1-
, 5 k i 1+
#,
(4.2)
PM = ±2 arcsin(3 )
2
#,
= 1 results in the best complementary sensitivity based margins. Hence the MIMO
margins of the closed loop system can be summarized as follows:
mino(I + L(jw))
=
a,
GMI+L
=
I
GMI+L-1
GM
I
,
1 + a, 1 - a,J
[1=
min (I+L(jw)- 1 ) =,3
f,,1+ Oa]
PMI+L = ±2 arcsin(-)
2
PMI+L-1
GMI+L U GMI+L-1
(4.3)
PM
=
±2 arcsin(/)
(4.4)
(4.5)
PM+L U PMI+L-1
(4-6)
and 1 +/,3, nega-
Equations (4.4) and (4.5) define the positive gain margins as 1t
1
tive gain margins as 1y, and 1-0., and, phase margins as ±2 arcsin(RQ) and +2 arcsin(La)
based on error sensitivity and complementary sensitivity matrix respectively.
75
4.4.4
Frequency Response of Sensitivity Matrices
As noted earlier, we would like both a, and
0,
to be large and close to 1 for the best possi-
ble stability margins. But the restriction £(S + T) = 1 presents a control design dilemma.
Thereby, both a, and 0, can not be made close to 1 at the same time. Small minimum
singular value for return difference matrix (inverse of sensitivity matrix) results indicates
poor stability robustness while small minimum singular value for stability robustness matrix (inverse of complementary sensitivity matrix) indicates large peak resonance. Hence,
a control designer's task is to make sensitivity small at low frequencies for command following and to make complementary sensitivity small at high frequencies for robustness to
unmodeled dynamics and sensor noise. Figure 4.6 shows the performance specification for
singular values of Sensitivity and Complementary sensitivity matrix in frequency domain.
It can be seen from fig 4.6 that a good design ensures that the minimum singular
value of sensitivity matrix is small at low frequencies for small tracking error and disturbance rejection. Similarly, the minimum singular value of complementary sensitivity matrix
rolls off at high frequency for robustness to noise and high frequency dynamics.
It is important to note here that the MIMO margins are calculated once the steady
state has reached. Here, we refer steady state as a point when the adaptation to the unknown uncertainty has been completed and the adaptive parameters do not change any
longer. Hence, MIMO margins provide a measure to assess the stability of thus converged
linear system (RLAS) in steady state. By comparing the MIMO margins of RLASs corresponding to these reference inputs, the degree of persistent excitation provided by each of
these reference inputs can be compared, which is done in the following section.
76
U(S)
0dB
(a) Performance specification in frequency domain for Sensitivity
Matrix
!z(T)
0 dB
(b) Performance specification in frequency domain for Complementary Sensitivity Matrix
Figure 4.6: Shapes of Frequency responses of Sensitivity matrices
4.5 Evaluation of a High-performance Aircraft
The modern adaptive controller described in Chapter 3 by Equations (2.24-2.28) is simulated with the X-15 model. The additional parameter values used for these simulations can
be found in Table 4.1. In the nominal case where no failures are present (A = Ix'),
the
nominal controller tracks the reference model well but in the presence of A, the nominal
controller is not able to follow the desired trajectory and hence the adaptive controller steps
77
Table 4.1: Simulation Parameter Values
Qiqr
diag([0 0 100 300 1000 3000 3000 3000])
Q 10 I8x8
Riqr
I3x3
0.3
A
Tmin
A
0.1 13x3
diag(O.25 0.25 0.25)
130
6Cmax
in and restores the tracking as is clear from Fig. 4.7.
U-Un I
I
----
-
Ref
Nom
Nomd Adap
0.04
0.02
Cu
0
7 2 II
CM
-0.02
4
-0.04 -
K
0
50
100
150
200
250
time
300
350
400
450
500
Figure 4.7: Angle of attack tracking performance of nominal and adaptive controller after 75%
loss in elevator and rudder effectiveness
4.5.1
MIMO margins of linear plant
After establishing the effectiveness of modern adaptive controller in maintaining the stability and bounded tracking in the presence of uncertainty, the next step is to assess the
margins of such a system once the adaptation has been completed. Since the linear system
78
to which the nonlinear adaptive system converges to in the steady state, is a function of
the reference input, the margins are calculated for the control group of reference inputs as
elaborated in Chapter 3. Doublet, 3-2-1-1 and sinusoidal are denoted as Grad PE 1, Grad
PE 2 and Grad PE 3 respectively. Tables 4.2 and 4.3 list out the MIMO margins thus
obtained. The comparison of margins of adaptive systems are made with respect to the
margins of the reference model and nominal plant which are not a function of the reference
input.
Table 4.2 shows the margins based on the singular values of Sensitivity matrix while
Table 4.3 shows the margins based on the singular values of complementary sensitivity
matrix. These margins are calculated by using the loop gain matrix L(s) at input break
point. The results are shown for the case when there is 75% loss in the elevator and rudder
effectiveness.
Table 4.2: Margins based on Singular Values of ErrorSensitivity
Plant
Negative Gain Margin
Positive Gain Margin Phase Margin
Reference
-6
00
60
Nominal
Adaptive
PE
Grad PE 1
Grad PE 2
Grad PE 3
-5.7
22.6
55.1
-6
-6
-6
-6
59.5
39.4
46.8
42.8
60
59.3
59.7
59.5
As can be seen from Table 4.2, the loss in control effectiveness approximately results
in a loss of 20% gain margin and 20% phase margin for the nominal plant. When the
reference input is persistently exciting, the adaptive controller restores back the loss in
margins due to the uncertainty. If the reference input is gradually persistently exciting, it
results in slightly improved margins than the nominal controller. Results listed in Table 4.3
79
Table 4.3: Margins based on Singular Values of Complementary Sensitivity
Plant
Negative Gain Margin
Reference
-19.2
Nominal
-13
Adaptive
PE
-19.4
Grad PE 1
-14
Grad PE 2
-14.8
Grad PE 3
-15.2
Positive Gain Margin
5.5
5
Phase Margin
52.9
46.4
5.5
5.1
5.2
5.2
53
48
48.4
48.9
also tell the same story. It can be noticed that MIMO margins offer an excellent tool to
compare distinct reference inputs and various uncertainties that the plant can encounter.
Also, this tool is clearly an extension of SISO systems and provides a single scalar number
which can be used for assessment of performance.
As is clearly evident, the nominal controller is very robust and is well equipped to
deal with uncertainties of the type of A. Hence using a modern adaptive controller on the
top of baseline controller seems little advantageous. However, it is essential to note that
these simulation results are for a linear plant with no unmodeled dynamics or unknown
nonlinearity: an impossible scenario for practical systems. The role of modern adaptive
controller and the need of persistent or graduated persistent excitation becomes evident
when we discuss MIMO margins in the presence of nonlinearities and unmodeled dynamics
in the next chapter.
80
Chapter 5
Persistent Excitation in the Presence of
NonIinearities
5.1
Introduction
In last chapter we analyzed persistent excitation properties of distinct forcing functions on
a Linear model of X-15 obtained by linearizing the nonlinear model at a specific trimming
point. The trimmed X-15 was assumed to be linear, finite dimensional plant with unknown
parameters whose input and output could be measured exactly. It was also shown that
an adaptive controller can be implemented such that the output of the plant tracks the
output of the reference model and all the signals in the system remain bounded. However,
in practical applications, no plant is truly linear or finite dimensional. Plant parameters
tend to vary with time, and measurements of system variables are invariably contaminated
with noise. Also, the plant model used for analysis is almost always approximate, thereby
consisting of some unmodeled dynamics.
[36] discusses different type of instability that
81
can occur in adaptive systems. Some examples are nonlinear behavior of pitching moment
of an aircraft at high angle of attacks and onset of unmodeled structural dynamics at high
frequencies. These are many such sources which render the plant to be controlled as a time
varying, nonlinear and uncertain plant. Consequently, it is important from the practical
viewpoint to examine whether the same boundedness and stability properties can be derived
in a realistic environment where only an approximation of the overall plant transfer function
is available and the plant input and output are affected by unknown disturbances. In this
chapter we will address such class of uncertainties and nonlinearities and approaches to
ensure the boundedness of all the signals in the system.
Since, underlying differential equations that describe adaptive systems are nonlinear
and time varying, the addition of external inputs in the form of disturbances or internal
inputs in the form of unmodeled dynamics and nonlinearities, makes the analysis of the
resultant adaptive systems considerably more difficult.
Therefore, new approaches are
needed to establish stability and robustness properties of these systems. Robustness in
this context implies that the adaptive system essentially performs in the same manner
even when external or internal perturbations are present. The approaches used to this
end can be broadly classified into two categories (i) Modifications in the adaptive law, and
(ii) increasing the degree of persistent excitation of the reference input. Modifications in
the adaptive laws include use of dead zone, bound on 0*, the a modification scheme and
the el modification scheme. For details on each of these modified adaptive laws, please
refer to
[1].
Our focus would be on the second category of modification i.e. changing
the persistent excitation properties of the reference input. It should be noted that in the
presence of modeling errors, exact model-plant transfer function matching is no longer
82
possible in general and therefore the control objective of zero tracking error at steady state
for any reference input signal may not be achievable. The best can be hoped, in the nonideal
situation in general, is signal boundedness and small tracking errors that are of the order
of the modeling error at steady state.
Section 5.2 considers the effect of nonlinearity on the stability and robustness properties of adaptive control systems in steady state. A network of radial basis functions is
used to approximate the nonlinearity in aircraft parameters and the stability analysis is
performed for (i) nonlinearity with zero approximation error and (ii) PitchBreak Nonlinearity. Simulations results corroborate the analysis and MIMO margins are calculated. In
Section 5.3, the unmodeled dynamics is added in the plant model and persistent excitation condition is discussed in its presence and MIMO margins are calculated. Section 5.4
concludes with reiterating the observations of previous sections.
5.2
Nonlinearity and Radial Basis Function
To explain the effect of nonlinear behavior of plant parameters at certain flight conditions,
we will look at the nonlinearity, Ko(x,). In the presence of Ko(x,), it is necessary to modify
the adaptive controller to ensure the boundedness of all the signals in the system. To this
end, the uncertainty is approximated using multi-input-multi-output feedforward neural
network with No radial basis function neurons in its inner layer [37], [38]. The network
computes m linear combinations of a suitable chosen set of radial basis functions {qS3 (x,) ,
83
Ref. [39],
ko(x)
ZNO
eo(xp)
$(..
=
(5.1)
(X
9moj(xp)
Ko(xp) - Ko(xp)
=
Ko(xp) - 0'4<(xp)
where co(xp) represents the uncertainty approximation error and AOnl =
(5.2)
OnI -
One
repre-
sents unknown parameter estimation error. The adaptive law for adjustment of parameters
is given by:
(5.3)
$n= F=i<DT(Xp)ePB1
where Pnl is a positive definite diagonal matrix of adaptive gains.
The RBF NN Universal Approximation Theorem [40 states that given an approximation tolerance E* > 0, and a compact set X C R , there must exist an integer No and
a 'true' constant matrix One E RNoxm such that for all xP E X c Rn:
(5.4)
||EO(XP)|| < E*
In other words, given enough neurons, one can approximate a nonlinear function to
within any accuracy on a compact domain. The problem arises when X does not belong to
a compact domain, which would be a case for an unstable, uncertain plant. In the problems
that are addressed in this chapter, the uncontrolled plant is unstable therefore X would
84
not be a bounded vector of system's states and the Eq. (5.4) is not guaranteed to hold for
a given KO while using a finite number of neurons (RBFs).
Having said this, we breakdown our analysis to two cases: when, 1) KO is such that
the uncertainty approximation error is zero, hence boundedness of all signals is guaranteed
and when, 2) KO is such that there will be a nonzero uncertainty approximation error and
closed loop adaptive system can be guaranteed to be stable only upto a certain amplitude
of the nonlinearity.
Let us first discuss scenario (1) with a representation of Ko where Co(xp) = 0,
i.e., the nonlinearity can be estimated accurately by the given number of RBFs and the
approximation error is zero.
5.2.1
Zero Approximation Error
We concoct a fictitious pitch break nonlinearity just to demonstrate the effect on margins of
a nonlinearity which can be estimated accurately. Since, gaussian radial basis functions are
used to estimate the unknown nonlinearity, we use the fictitious pitch break nonlinearity
of the form:
Ko
=
-5
/
0.5 * exp(-
+ exp(-
2
(a - 2)2
2
) + 0.5 * exp(-
) + exp(-
2
(a - (-2))2
2
)+
)
(5.5)
Four gaussian radial basis functions centered at a = -3, -2, 2, 3 degrees as given by:
#j (a) = exp(-
-
2
)
(5.6)
85
where, i = -3, -2, 2,3, are used to estimate KO. The Gaussian width, a, were set to
0.25 which provides for a reasonable overlap between the individual basis functions. The
adaptation rates in Eq. (5.3) were chosen to be:
re, = 1000 eye(4)
(5.7)
Inclusion of nonlinearity changes the closed loop dynamics 3.20 and the controller command
to:
±
Ax + B1A(6 + Ko(xp)) + B 2u
6
6
while the
3
ad +
6
nom
(5.8)
-ko(XP)
"d component of KO was set to zero,
1
" and the
2 nd
components were chosen
to represent the fictitious nonlinearity as given in Eq. (5.5). KO is approximated as given
by Eq. (5.1) using the RBFs as given in Eq. (5.6) and the adaptive parameters On are
updated according to Eq. (5.3).
According to Gorinevsky [41], the approximation error Eo will converge to 0 provided
that the regressor vector sequence 4<(xp) has the persistency of excitation property. [41],
provides formulation and proof of PE conditions on input variables stating that if the input
variables belong to domains around network node centers, they provide PE. Therefore, the
reference input commands are chosen such that they lie in the neighborhood of radial
basis functions centers a = -3, -2,2,3 for a significant duration to provide PE. Thus,
the convergence of approximation error to zero can be guaranteed. Since the radial basis
86
functions <j are linearly independent, the approximation error co can be zero only if $i,
converges to its true value. Therefore, we need to look into the persistent excitation of the
closed loop system given by equation 5.8.
In Chapter 3, we discussed about the persistent excitation conditions for the linear X-15 model. The next question arises is the persistent excitation in the presence of
nonlinearities such as KO which is addressed in the following section.
Persistent Excitation and Analysis
As discussed in Chapter 3, the regressor vector w is persistently exciting for the linearized
X-15 model if the reference input contains two distinct frequencies in a command and one
distinct frequency in each of the P, p and r commands. Similarly, the regressor vector
4(x,) for the estimation of nonlinearity is PE if input belongs to domains around network
)
node centers. In this subsection we delve into the conditions required for the PE of the
augmented regressor vector,
,
when individual regressor vectors x, and
4(x,) is persistently exciting.
According to the definition of Persistent excitation given by Eq. 3.11, the matrix
I(t)JT(t) should be positive definite over any time interval [t, t + T].
[
T
P;=
D(XP)XT
X D(Xp)T
1
P
D(Xp,D(X,)T
(5.9)
Since, 4(x,) and x, are persistently exciting in No and n dimensional space respectively,
therefore, rank(4(xP(t))I(x,(t))T) = No and rank(x,(t)xT(t)) = n over any time interval
87
[t, t + T]. From Eq. 5.9, the rank of QI(t)IIT(t) will be full only if only one of the radial
basis functions
width
#i
is excited at a time which can be achieved by a lower value of gaussian
-. The conditions for the persistent excitation of the augmented regressor vector
can be succinctly stated as:
1. x, is persistently exciting in dimension n.
2. <D(xp) is persistently exciting in dimension No.
3. o is small to provide one at a time excitation of radial basis functions
#i(xp).
Simulation Studies
The plant as given by equation 5.8 with A = diag[O.25 0.25 1], In = 1000eye(No) was simulated with KO given by Eq. 5.5 using PE input. The figure 5.1 shows the approximation
of fictitious nonlinearity as a function of time. It is clear that the nonlinearity is approximated well with the PE input. Also, it can be seen from the following table that the MIMO
margins of closed loop adaptive system is same as the margins of the reference model. This
implies that the persistent excitation of augmented regressor vector is guaranteed as the
conditions 5.2.1 in the previous section are met.
Table 5.1:
Margins based on singular values of error and complementary sensitivity in the
presence of fictitious nonlinearity
Negative Gain Margin
Plant
Error Sensitivity based margins
-6
Reference
PE
Positive Gain Margin
Phase Margin
00
60
00
60
-6
Complementary Sensitivity based margins
Reference
-19.2
5.5
52.9
PE
-19.1
5.5
52.8
88
t =20
t = 200
- - - Kappm
2
0
0
..
-2
...
. _ _ .-. . ...
.- -- --. - ---.-.- --- .-
-2 - -
-
-4
-4
-61
-4
-2
0
2
-4
4
-
-- - - - -- - - -
- ~~~-- -------2
t = 400
-- - - - -
--- - -- --- --.
2
0
4
t = 600
0
U
-1
. ...
..
... .
-2
-2
-----.
------------..------- ------
-3
-4
-2
4
0
2
--V-V---
-----
-4
-6,
4
-2
t =800
0
2
4
t = 950
-1
-1
-2
-2
-3
-3
-4
-4
-5
-2
0
2
-2
0
2
Figure 5.1: Approximation of fictitious nonlinearity
5.2.2 Pitch Break Nonlinearity
Having discussed the simpler case where uncertainty can be estimated accurately, we now
embark onto the more realistic representation of pitch break nonlinearity. According to
Lavretsky et al. [42], a typical representation of pitch break nonlinearity that depends
solely on angle of attack is given in the figure 5.2:
To approximate the 'Pitch Break phenomenon', radial basis functions were chosen in the
form of Gaussians:
O (xp)
3
=
exp (
-
a
(5.10)
89
--------20 --- --
I- -
-
-
-
-
--------
-
-
-
-
-
-
-
-
-
I
-------
a) -40 ---- --- --- ---- ------- J
-
-
--- --- - -
------ L--- ----
-o
0 -60 -------- --------
100
0
2
-
4
-
-- ------ - --- -- -
6
8
10
AOA (deg)
Figure 5.2: Pitch Break Nonlinearity vs. Angle of Attack
AOA break points
were spaced evenly between -5 and 5 degrees and half a
af
S=...21
degree apart from each other. The Gaussian widths were set to - = 0.5. The adaptation
rates were chosen to be F1
=
1000eye(21), where 'eye' denotes identity matrix.
For such a KO, the approximation error eo(xp) can not be guaranteed to be bounded
for an unbounded x,. Hence, the following subsection provides the details of boundedness
proof of all the signals in the system after certain assumptions on the structure of KO.
Boundedness Proof
By subtracting reference model dynamics (3.23) from the equation (5.8), the tracking dynamics can be written in the form:
6 = Ame + B1 A(QT1p(x, u) +
(5.11)
60 (xp))
90
where T =
Ko(xp)
-
,b
OT>D(xP)
denotes the vector of radial basis functions and co(xp) =
is the approximation error.
Assumption 2. Let us assume that 60 can be approximated as:
11
co ||$ P 11X 11",
where m >
(5.12)
1 and [ > 0 small.
Assumption 2 implies:
I6|I1
p|x||"
->60|ol1
=
I||xm + el m
1/|(IIXmIIm + ||elm)
(5.13)
(5.14)
P(Hlle|m + co)
where co is the upper bound on the reference model trajectory.
To establish boundedness of all signals, let us consider the following positive definite
Lyapunov function candidate
V(e, Q) = eTPe + trace(QT F-(A)
(5.15)
where trace is defined as the sum of its diagonal elements and r is a positive definite
diagonal symmetric matrix.
rX
0
F =(5.16)
0
re",
91
For each pi > 0 and some constants co > 0, a > 0, the inequality
V(e, Q) <; c- c 2
defines a closed sphere L(pu, a, c). Computing the time derivative of the Lyapunov function
candidate V along the system trajectories Eq. (5.11) yields:
V(e, Q) = -eTQe + 2eTPB1AEo(xp) + 2trace((
T (F- 1
+ 1p (x, u)eTPB,)A)
(5.17)
Based on Eq. (5.17), adaptive laws can be written as:
O=
J'Proj(Q,- T (x, u)eTPB1)
(5.18)
where Proj(.,.) denotes the projection operator, [ref. anna's book]. It is defined as:
Proj(Q, - T(x, u)eTPBI) = -IQ(x, u)eTPBI -
(1 -
*
Qmax
) 2 f(Q)
(5.19)
where
I fI1IQII
> Q*a
tw)
10
(5.20)
x
otherwise
It ensures that the matrix of adaptive parameters Q does not exceed its pre-specified norm
bound Q*a.
Due to Eq. (5.18) an upper bound for the time derivative of Lyapunov function
92
can be found as
V(e, n) < -Ile||(Amin(Q)leI - 211PBiIlmax(A)p(IIeI m
+ co))
(5.21)
since lehl(ilehlm + co) < 1(e 2 + e2 m+ cg + 2hlemilco), Eq. (5.21) can be re-written as
222
-Amin(Q)le 2 |I+ ,p|IPBiIImax(A)(e + e
#
+ c + 2||emlico)
-Amin(Q)Ile 2I + pIIPBiImax(A)e 2 (1 + e2 m- 2 + 2|lem 2 lco)
(5.22)
+
+ pilPB1|lmax(A)cs
(5.23)
Inside L(p, a, c), hell, IIfn can grow up to O(p-c). Hence, there exists positive constants
k1 , k2 such that inside L(/, a, c), we have
||ell < k1 ip
,
I|n| <
k2A~"
For all e, f inside L(p, a, c),Eq. (5.23) can be simplified to
V < -
2e2
Se2 (Ain(Q)
-0t,
1-(2m-2)G)
for some positive constant 81. Now for 0 < a <
each p E (0,p*I
Amin(Q) >
-(m-2)a
2
93
+ pIIPBi||max(A)cO2
(5.24)
'-, there exists a /-* > 0 such that for
Hence, for each p E (0, p*] and e,!n inside L(p, a, c), we have
V< -Ai
Q e2 + p|IPB1||lmax( A)c 2
(5.25)
We define the set
2pPB Ilmax(A)
()
|ell <
DO(P) =e,( H
c(|||
OI1 1: ; Qmax
)
A mi()
(5.26)
(.
which for fixed a and for sufficiently small p is inside L(p, a, c) if
(1
<
21 IPBiI:max(A)CO\
\min (Q)
P2
<
max
P3
<
2,31
p= min(PI,
k)
(5.27)
P2, A3 )
Outside Do(p) and inside L(p, a, c), V < 0 and, therefore, V(e, Q) decreases. And
any solution e(t), 0(t) which starts in Do(p) remains inside L(p, a, c). The stability result
thus obtained is semi-global because as p -* 0 the size of L(p, a, co) becomes the whole
space. Hence, for a given ki, k2 > 0 and 31 > 0, the signals are bounded only for p as given
by Eq. (5.27). The graphical representation of all the sets is given in the figure 5.3.
Note: The assumption 5.12 is a stronger condition than the semi-global Lipschitz condition.
Work is under progress to relax this assumption to include broader class of nonlinearities
in the signal boundedness proof.
94
Figure 5.3: Graphicalrepresentationof trajectory bounds
Simulation Results
After proving the boundedness of all signals for nonlinearities of the form 1 1o ||!;
| x
Ii"
and for small /,t the linear X-15 model developed in Chapter 3 was simulated with the
pitch break nonlinearity KO as given by Fig. 5.2. It can be seen from the figure 5.4 that
the nominal controller fails in maintaining the tracking in the presence of KO, while the
augmented controller regains tracking and all the signals in the closed loop stay bounded.
Fig. 5.5 shows the nonlinearity approximation by PE input as a function of time. It can be
noticed that a nonzero approximation error EO exists which can be reduced by having a input
containing more information content in the domain of radial basis function centers. Due to
a finite non-zero approximation error, the PE input for linear system is no longer PE in the
presence of pitch break nonlinearity. The best that one can hope for is to have bounded
signals and as close performance to the reference model as possible. Nonetheless the PE
95
2
-
.
1.5
aad
-am
-
.
m
nom
1
#
0.5
0-0.5::
-2C--1
-2.5 L
10
20
30
40
50
60
70
80
90
100
time (sec)
Figure 5.4: Tracking of a command in the presence of pitch break nonlinearity
input provides a very good performance in the steady state. To quantify the performance
of adaptive system, the following subsection shows the calculated margins using PE and
GPE inputs.
MIMO Margins
Once the stability and tracking is established, the MIMO margins of the augmented adaptive system are calculated for the control group of PE and GPE reference inputs. The
margins based on error sensitivity matrix are listed in the Table 5.2. It can be noticed
that the margins are worse with pitch break nonlinearity compared to their counterparts
for the linear system.
The margins based on complementary sensitivity matrix are listed in the Table 5.3.
96
t=20
t=200
..
--- -..
- 1 0 --.
-0
K pro
.--- .------ ..----. -.
-10
.................................. .
-30 ...
-40
-6
....
-4
-2
0
2
4
- 30
-40-6
6
------
--.
.---
.------ .-.---
...
-: --..--.:- --. ----.
--- -.
--4
-2
t = 400
0
2
-4
-2
0
2
4
-40
6
-6
t= 800
.1 . . .......
0
.....
.... ...
-30
-40
-40
-4
-2
0
4
6
2
4
-4
-2
0
2
---------....
4
6
t= 950
---- ---.
.....
-..
..
-.----.-
.
-30 - ......-..-.-....-- ...-- ..-..-- .--- ... - ..
-6
-.
-10 -- ---....
...
----------------.-.--
.... ------ . .- ..
-------------..
- 20 - - -----
- ------
t =600
- 10 ------- -- ---- --- -:- --- ---
408
-6
.-.-.-----
6
-6
. ...-..---.-..-..--------.-.--..-.--.-.---4
-2
0
2
4
..
6
Figure 5.5: Pitch break nonlinearity approximation as a function of time
It can be noticed that there is virtually no difference between the margins of adaptive
system with pitch break nonlinearity and linear system. Rather, GPE inputs fare better
than PE inputs. This implies that closed loop adaptive systems with GPE reference inputs
are more robust to noises and unmodeled dynamics as compared to PE reference inputs.
It can be attributed to the fact that the GPE input has less frequency content than the
PE input and hence results in less interference with high frequency noise and unmodeled
dynamics.
97
Table 5.2: Margins based on the Singular Values of Error Sensitivity in the presence of nonlinearities
Plant
Negative Gain Margin Positive Gain Margin Phase Margin
Reference
-6
00
60
Adaptive
PE
Grad PE 1
Grad PE 2
Grad PE 3
-6
-6
-6
-6
51.6
44.6
46.0
45.8
59.8
58.8
59.8
59.6
Table 5.3: Margins based on the Singular Values of Complementary Sensitivity in the presence
of nonlinearities
Plant
Negative Gain AMargin Positive Gain Margin Phase Margin
-19.2
52.9
Reference
5.5
Adaptive
PE 1
Grad PE 1
Grad PE 2
Grad PE 3
5.3
-19
-25.2
-25.0
-25.2
5.5
5.78
5.8
5.78
52.7
56.4
57.0
56.4
Unmodeled Dynamics
In this section we discuss the effect of unmodeled dynamics on the robustness properties
of the closed loop adaptive system and how these properties can be improved by the use
of persistent excitation and graduated persistent excitation. Let us consider the following
plant:
y
=
(5.28)
Go(s)u + Aa(s)u
where Go(s), A,(s) are proper and stable, Aa(s) is an additive perturbation of the modeled
part Go(s). We would like to identify the plant by exciting it with input u in the presence
of unmodeled dynamics. Since, Aa(s)u is treated as a disturbance, the input u should be
98
chosen so that at each frequency wi contained in u, we have IGo(ji)I > IAa(jwi)1. Furthermore, u should be rich enough to excite the modeled part of the plant that corresponds to
Go(s) so that y contains sufficient information about the coefficients of Go (s). Since usually
system response at low frequencies is of interest for the analysis, |Go(jw)| >
IAa(jW)
in
the low frequency range. For high frequencies we may have Ia(jw)I of the same order or
higher than IGo(hw)I. Therefore, the richness of the input signal u should be achieved in
the low frequency range. An input signal with theses properties is called dominantly rich
because it excites the dominant (modeled) part of the plant much more than the unmodeled
one [26].
5.3.1
Persistent Excitation in the presence of unmodeled dynamics
High frequency unmodeled dynamics is added in the linear X-15 model to study the persistent and graduated persistent excitation in the presence of unmodeled dynamics. The
block diagram is shown in figure 5.6. The transfer function of unmodeled dynamics
r+
Controller
Aa(S)
Nonlinear
Figure 5.6: Block diagram with additive unmodeled dynamics
is chosen to be stable and proper and can be represented by:
2
Aa(S)
=
2n
s2+ 2(wns + w,
(5.29)
2
99
The variable q > 0 is the small singular perturbation parameter that can be used as a
measure of the separation of the spectrums of the dominant dynamics and the unmodeled
high frequency dynamics. Here, Aa(s) represents the aeroelastic modes which are primarily
fuselage bending modes. The disturbance generated from the unmodeled dynamics Ao(s)
adds the perturbation to the angle of attack a and thereby changes the dynamics of closed
loop adaptive system. The damping, (, should be very small and W" is in the range of
6-10 rad/s for 1st mode [43].
As discussed in previous section, the richness of the PE
and GPE inputs should be in the low frequency range. As a general rule of thumb, input
frequencies wi should be in the range 0 < wi < O(1/I) to avoid excitation of unmodeled
dynamics and small signal to noise ratio. Hence, if the input signal is dominantly rich and
contains frequencies away from the range of unmodeled dynamics, then the PE property
of the regressor vector
Wm
can not be destroyed by the unmodeled dynamics [26].
The next subsection discusses the MIMO margins of adaptive system in steady state
in the presence of unmodeled elastic modes.
5.3.2
Simulation Studies
For the simulations, a value of 1 = 0.3, (
=
0.1 and w,, = 10 rad/s is chosen. It is made sure
that PE and GPE inputs do not contain frequencies in the aforementioned range. In figure
5.7, the angle of attack following is shown in the presence of loss of control effectiveness and
the unmodeled dynamics. As can be seen from the figure 5.7, the nominal controller is not
able to provide stability while the adaptive controller regains the stability and command
following. Figure 5.8 shows the control power used by each of the nominal and adaptive
100
controller with the limits of [-20 20] degrees on the magnitude of elevator command. It can
6
be observed that the adaptive controller adds an extra command
ad
well within actuator
limits to the JnOm to regain the stability and command following. The tracking error e is
of the order of q.
acmd
4-
aad
3-
2
0
4
-
-
nom -
1
0
,
I'
,
/
/
-1.
-2
-3--4-
0
50
100
150
200
250
300
time (sec)
Figure 5.7: Angle of attack tracking with additive unmodeled dynamics
5.3.3
MIMO Margins
This subsection lists out the MIMO margins of closed loop adaptive system using distinct
PE and GPE inputs.
The margins are calculated in the same manner as discussed in
chapter 4. The following tables represent the MIMO margins calculated based on the error
sensitivity and complementary sensitivity matrix. It can be seen that for q < 0.3, the
margins of the adaptive system with unmodeled dynamics in steady state are similar to the
margins of the adaptive system without unmodeled dynamics shown in table 5.2 and 5.3.
101
25-
20-
15-
10-
0-5
-10-
-15 0
50
100
15
200
25
300
time (sec)
Figure 5.8: Elevator Command for nominal and adaptive controller in the presence of additive
unmodeled dynamics
This demonstrates the fact that the PE property of PE and GPE inputs does not change if
the inputs contain frequencies away from high frequency range and are PE in the absence
of unmodeled dynamics.
Table 5.4: Margins based on the Singular Values of Error Sensitivity matrix in the presence of
unmodeled dynamics
Negative Gain Margin Positive Gain Margin Phase Margin
Plant
60
00
-6
Reference
Adaptive
PE
Grad PE 1
Grad PE 2
Grad PE 3
36
45.0
46.0
45.8
-5.9
-6
-6
-6
102
59.0
59.6
59.8
59.6
Table 5.5: Margins based on the Singular Values of Complementary Sensitivity in the presence
of unmodeled dynamics
Plant
Negative Gain Margin Positive Gain Margin Phase Margin
Reference
-19.2
5.5
52.9
-18.8
-23.2
-25.2
-23.5
5.5
5.7
5.85
5.78
52.5
55.3
56.7
55.5
Adaptive
PE
Grad PE 1
Grad PE 2
Grad PE 3
5.4 Conclusion
The effect of nonlinearities in the form of nonlinearity in constant parameters and unmodeled dynamics on the robustness and performance properties of closed loop adaptive was
discussed. Using combination of quantitative and simulation studies it was shown that
although use of GPE reference inputs does not result into identification of unknown plant,
they do provide stability and robustness properties very close to that of the reference model.
This observation coupled with the easy implementation of GPE inputs in flight commands
makes study of GPE inputs very attractive and useful.
103
Chapter 6
Summary, Conclusion and Future Work
This work is best concluded by revisiting the last five chapters in a succinct manner and
extrapolating the thread of thought into work for the future.
" Chapter 1 introduced the concept of adaptive control in flight control applications
besides discussing the absence of verification and validation procedures for adaptive
control based flight controllers and therefore laying out the need for a comprehensive
tool for this purpose.
" In Chapter 2 the RLAS was formulated to provide a compact linear approximation
to nonlinear adaptive systems. It further delved into intelligently choosing adaptive
control parameter - solution of algebraic Lyapunov equation P. Tools such as output feedback control, Linear Matrix Inequality and Lyapunov analysis were used on
RLAS to find an optimal P. The analysis resulted into the conclusion that the effect of eigenvectors of P does not have much effect on the transient or steady state
performance of closed loop adaptive system.
104
"
Chapter 3 presented persistent excitation and its fundamental properties exhaustively and its application in adaptive control. Conditions for the internal signals to
be persistently exciting were derived and verified on a full linear model of high performance aircraft. Graduated persistent excitation was introduced as a feasible and
easy-to-implement alternative for persistent excitation.
" Chapter 4 offered a thorough study in identifying apt metrics for assessing adaptive
controllers performance in steady state. MIMO margins based on singular values of
sensitivity matrices were found to be suitable for this purpose and were demonstrated
on the linear model of X15.
" Chapter 5 finally demonstrated the ability of graduated persistent excitation (GPE)
inputs in achieving performance close to the reference model in the presence nonlinearities and unmodeled dynamics.
The preceding chapters thus clearly demonstrate the feasibility of implementing
GPE inputs and thereby providing performance characteristics close to the reference model.
RLAS is used as a tool for the analysis of robustness margins of adaptive controller in
steady state in the presence of uncertain parameters, nonlinearities and unmodeled dynamics. Thus this thesis takes pride in analyzing some crucial aspects of adaptive control
based flight controllers such as - (i) intelligent choice of adaptive control parameters, (ii)
easy calculation of required properties for persistent excitation of internal signals, (iii) introduction of singular value based MIMO margins which turn out to be an excellent tool to
calculate the robustness properties in steady state, and, (iv) use of GPE inputs which are
easy to implement to achieve good robustness characteristics. All of these can be grouped
105
together as a set of simple and compact tools to assess and quantify the performance of
adaptive controllers in so defined 'steady state'.
Future Work
Several avenues of research remain ongoing.
Some of the potential future deliverables
include:
" Demonstration of the aforementioned tools in a full nonlinear aircraft with sensor and
actuator dynamics included.
" Developing a set of tools to quantify the performance in the transient phase of adaptive controllers.
" Equipping the adaptive controller with an elegant means to deal with actuator saturation both in magnitude and rate and,
" Modifying current control design techniques to suit adaptive controllers will likely aid
the transition of adaptive control technologies into aircraft and other safety critical
applications.
106
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