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Direct robust adaptive control

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Title:
Direct robust adaptive control
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Reid, Gary Wayne
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English
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vi, 159 leaves : illustrations ; 29 cm

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Adaptive control systems ( lcsh )
Adaptive control systems ( fast )
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bibliography ( marcgt )
theses ( marcgt )
non-fiction ( marcgt )

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Includes bibliographical references.
General Note:
Submitted in partial fulfillment of the requirements for the degree, Master of Science, Department of Electrical Engineering.
Statement of Responsibility:
by Gary Wayne Reid.

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Source Institution:
University of Colorado Denver
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Auraria Library
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ocm28863939
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LD1190.E54 1993m .R44 ( lcc )

Full Text
] DIRECT ROBUST ADAPTIVE CONTROL
by
Gary Wayne Reid
B.S., University of Colorado at Denver, 1986
A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial fulfillment
of the requirements for the degree of
Master of Science
1993


This thesis for the Master of Science
degree by
Gary Wayne Reid
has been approved for the
Department of Electrical Engineering
by
Miloje S. Radenkovic
/
Jan T. Bialasiewicz
of '3_t?3
Date
Tamal Bose


Reid, Gary Wayne (M.S., Electrical Engineering)
Direct Robust Adaptive Control
Thesis directed by Assistant Professor Miloje S. Radenkovic
ABSTRACT
An adaptive control algorithm known as the normalized gradient
algorithm will be analyzed. The mathematical properties as well as the
derivation of the normalized gradient algorithm are discussed. A unique
approach to robustness to unmodelled dynamics and time-variations is also
proposed.
Computer simulations using the normalized gradient and the least
squares algorithms are used to substantiate theory. The simulations show the
performance of the algorithms for some simple systems. A simplified practical
example is also included.
The form and content of this abstract are approved. I recommend its
publication.
Signed:
m
Miloje S. Radenkovic


ACKNOWLEDGMENTS
I would like to thank the Martin Marietta Corporation for sponsoring my
graduate work at the University of Colorado through their program for study
under company auspices.
Dr. Miloje Radenkovic, my thesis advisor, provided me with inspiration
as well as help and guidance throughout my thesis research.
An acknowledgment should also go out to all my family and friends for
their support and patience.
For her assistance during the publication phase of this thesis, I would
like to thank Rebecca Wrinkle for her typing and editorial support.
IV


CONTENTS
CHAPTER
1. INTRODUCTON.............................................1
1.1 Overview........................................ 1
1.2 Thesis Outline.................................. 4
1.3 Notation and Terminology......................... 5
2. ROBUST ADAPTIVE CONTROL OF
TIME-INVARIANT SYSTEMS..................................7
2.1 An Introduction to Robust
Adaptive Control.............................. 7
2.2 Deterministic Adaptive Control....................9
2.3 Global Stability Analysis and Mathematical
Formalization of the Self-Stabilizing
Mechanism in Robust Adaptive Control.............21
2.4 Physical Explanation of the Self-Stabilization
Mechanism and Robustness of the Parameter
Estimates........................................29
2.5 Simulation Examples..............................34
3. ROBUST ADAPTIVE CONTROL FOR
TIME-VARYING SYSTEMS................................. 62
3.1 An Introduction to Time-Varying Adaptive
Control..........................................62


3.2 The Time-Vaiying Plant Model....................64
3.3 T ime-V ary mg Parameter Estimation.............71
3.4 Certainty Equivalence Control Law...............74
3.5 Simulation Examples.............................80
4. CONCLUSIONS.........................................Ill
APPENDIX
A. MATLAB Input Files for Time-Invariant
Simulation Examples.............................113
B. MATLAB Input Files for Time-Varying
Simulation Examples............................ 129
REFERENCES........................................... 155
VI


CHAPTER 1
INTRODUCTION
1.1 Overview
Adaptive control is necessary in many applications, because the
parameters in a system model undergo significant changes or cannot be
measured with sufficient accuracy, rendering the application of classical
feedback either unreliable or impossible. Reasons for this include the effects
of changes in altitude or air speed of high performance aircraft, the inability to
determine system response at high frequencies, or the use of low-dimensional
approximations for complex high-dimensional systems. In some systems, the
suppression of nonlinear or elastic components may lead to simplified models
that exhibit large errors in certain ranges of conditions. Adaptive control
attempts to address those problems by changing the parameters of the feedback.
Historically, early adaptive algorithms employed heuristic models in
classical control such as the Ziegler-Nichols rules and elementary experiments
on the system to tune the parameters of a standard three term controller [6]. In
the mid-seventies, some important advances were made, along two main
approaches.
The first consists of combining an identification scheme to track the
changing values of system parameters and a control scheme designed as if the


parameters were known and time-invariant. The difficulty with this approach,
called "self-tuning," is in the simultaneous presence of identification and
control in the same feedback loop. It may be impossible to identify system
parameters while changing the parameters of the controller. However, it was
shown in 1973 that the optimal control can often be obtained in the limit, even
if the identification is not accurate. This was followed by a considerable
number of applications that confirmed the validity of this approach. Today,
there are several adaptive controllers commercially produced in the United
States and in Europe, especially in Sweden, which use modifications of
"minimum variance" control.
The second approach consists of adjusting the parameters of the
controller so as to make the behavior of the controlled system match closely
that of a chosen reference model. This is called "model reference adaptive
control." The system is assumed to be deterministic. Stability analysis
methods such as those based on the use of Lyapunov functions are a primary
tool for showing the stability of the adaptive control system.
Self-tuning and model reference adaptive control apply primarily to
linear systems. For nonlinear systems specialized adaptive control algorithms
exist. A prime example is adaptive control of manipulator robots, where the
parameters change with mass, shape, and orientation of objects being handled.
An important property of such systems is that the unknown parameters enter
linearly in the nonlinear system equations. By using an approach that
resembles self-tuning, stable controllers have been designed.
Another major breakthrough, which occured in the late seventies, was
when a number of researchers established bounded input bounded state (BIBS)
2


stability of adaptive controllers applied to linear-time-invariant (LTI) systems
[10, 11, 13, 31], However, later it was shown that this theory was inadequate
since it could be shown that the algorithms would fail if applied to systems
having small unmodeled dynamics [36], This problem has been the subject of
intense study and there have been several publications applicable to these type
of situations [15, 17, 18, 21, 25], It also has been established that persistency
of excitation (PE) of the system states lead to exponential stability and that
gives robustness to unmodeled dynamics and time-variations [1, 3, 4, 28].
These results are subject to the constraint that we can persistantly excite all of
the dynamics associated with our system. This can be difficult task in the
presence of potentially destabilizing feedback as it occurs in adaptive control.
It has since been proven that the persistency of excitation constraint in the
presence of time-variations and unmodeled dynamics is not necessary subject
to a stability assumption [24].
It has also been shown that direct adaptive control algorithms using the
certainty equivalence principle can be used to stabilize a linear system with
unknown parameters when the plant can be modeled exactly, there is no
mismatch in the relative degree, the sign and a lower bound for the high
frequency gain is known and a stable invertibility condition is satisfied [13,
32]. Certainty equivalence adaptive controllers estimate the controller
parameters or the process parameters in real-time and then use these estimates
as if they are equal to the true one. The uncertainties of the estimates are not
considered in this type of control.
Certainty equivalent controllers remain the mainstream in adaptive
control due largely to the literature from the 1950's and 1960's on deterministic
3


adaptive control which led to certainty equivalence control as the norm for
adaptive control [7]. To show stability for the linear time-invariant certainty
equivalence controller with no unmodelled dynamics is trivial. However, when
modifications are made to the certainty equivalence controller due to the
addition of unmodelled dynamics and time-variations stability can become an
issue.
The need to apply adaptive methods to high performance systems has
continually been a source for investigation of alternative, more powerful
adaptive control strategies. The goals of adaptive control are still far from
being realized and there is a widespread agreement that the field is still very
much in its infancy. Much of the literature consists of the design and analysis
of particular algorithms. The notion of robustness, together with the now
stated goals of adaptively controlling distributed parameter systems such as
combustion and flow control, will undoubtedly broaden the field of adaptive
control.
This thesis presents an adaptive control algorithm which is robust in the
sense that it is able to maintain exponential stability in the presence of
unmodelled dynamics and time-variations. The algorithm is unique in the
sense that it uses a new approach for the algorithm's normalization sequence.
1.2 Thesis Outline
Chapter 2 begins with an introduction to robust adaptive control. A
brief history of robust adaptive control is given as well as the effect
4


unmodelled dynamics can have on an adaptive system. Next, the derivation for
the normalized gradient algorithm is presented.
The mathematical formalization of the self-stabilization phenoma and a
global stability analysis are then shown. To illustrate these concepts, some
simulation experiments are presented to conclude Chapter 2.
An introduction to time-varying adaptive systems is at the beginning of
Chapter 3. Some models are then developed for time-varying linear systems.
A few examples of time-varying models are given. The operation of the
parameter estimator i$ then explained. The following section shows how
certainty equivalent control can be applied to time-varying systems. Some
simulations of time-varying adaptive systems are shown to conclude this
chapter.
In Chapter 4, the conclusions from Chapter 2 and Chapter 3 are
summarized. Comparisons between different estimation techniques are
discussed and the effects of unmodelled dynamics and time-variations on these
adaptive algorithms are all covered in this final chapter.
where T is the set of positive integers, while R+ is the set of nonnegative real
numbers. When nx(t) is uniformly bounded over allt > o, x is said to be
inl^.
1.3 Notation and Terminology
For a discrete time function x: T->R+, we define the following
seminorm
(1.1)
5


H will denote the space of transfer functions T (z) which are analytic
and bounded outside and on the unit circle in the z plane, s x is the operator
defined by
SxT(z) = t(X1/2z) (1.2)
for a fixed parameter A, o < A < L s XH ~ is the space of transfer function
T (z) such that SxT(z) e H . In other words T (z) e s XH if T (z) is
analytic and bounded outside and on the circle |z | = A1/2 in the z plane. For
T (z) e H the H norm is defined by
|(r(z)||H-: = max(T(z)|. (1.3)
|Z| = 1
Likewise, the norm of the s XH space is defined by
l^(z)|li-:= XT(z)\\h. = max^(A1/2z)|. (1.4)
This norm is induced by the 1 ^ norm of the input and output signals of r (z).
When performing majorizations, in order to account for initial
conditions we will use nonnegative functions
£i(t) = cjF, 0 < cL < oo, 0 < p < l. (1.5)
6


CHAPTER 2
ROBUST ADAPTIVE CONTROL OF TIME-INVARIANT SYSTEMS
2.1 An Introduction to Robust Adaptive Control
In the seventies, significant progress was made in adaptive control
theory, under the assumption that physical systems are described precisely by
linear system models [14,22]. In these works, it was assumed that the system
parameters are unknown and that the relative degree and an upper bound on the
order of the system are known. At the beginning of the eighties, a disturbing
fact was discovered, namely, that an adaptive controller designed for the case
of a perfect system model in the presence of external disturbances or small
modelling errors, can become unstable [10,35].
In order to guarantee stability, a variety of modifications of the
algorithms originally designed for the ideal case have been proposed, such as cr
and £i modification, relative dead-zone, projection of the parameter estimates
and the like [15, 17, 21, 25, 27, 29, 30]. A remarkable unification of the
existing robust deterministic adaptive control methods is developed in [16].
This paper is where most of the significant works in the area of deterministic
adaptive control are cited. In addition to this work other authors have proven
that the projection of the parameter estimates is, in fact, sufficient to guarantee
stability in the presence of small unmodelled dynamics [26, 40],


A unified approach to robust deterministic and stochastic adaptive
control is presented where the algorithms with the projection of the parameter
estimates are considered in [34]. It is shown that small algorithm gains may
produce unacceptably large input and output signals in the adaptive loop. In
the same paper, it is proved that the admissible unmodelled dynamics do not
depend on the algorithm gain and it is the same as in nonadaptive control.
The purpose of this chapter is to show that adaptive control is globally
stable and robust with respect to unmodelled dynamics and external
disturbances without introducing a or ei modifications, dead-zone algorithms
or projection techniques. The above-mentioned modifications usually result in
increased algorithm complexity and require a' priority information related to
the parameters of the nominal system model. The normalized gradient
algorithm will be analyzed. This analysis uses a similar approach proposed by
[41]. Global stability and robustness of the considered algorithm is obtained
under the assumption that the reference signal has a sufficiently large level of
excitation compared with the intensities of the unmodelled dynamics and
external disturbances. Then it is proved that during the adaption process there
are time internals in which persistent excitation (PE) conditions are satisfied in
the adaptive loop. This implies that in these time intervals parameter estimates
can drift only in a certain set defined by the level of excitation of the reference
signal and the intensities of the unmodelled dynamics and external
disturbances. It is shown that in the time intervals where the PE conditions in
the adaptive loop can not be characterized, the tracking error is large enough so
that the corresponding Lyapunov function decreases, implying boundedness of
the parameter estimates. The fact that the parameter estimates are bounded is
8


enough to establish global stability of the considered algorithm. Parameter
estimation error and the intensity of admissible unmodelled dynamics and
external disturbances are specified in terms of the H~ norms of the
corresponding transfer functions and the level of excitation of the reference
signal. The results established in this chapter represent mathematical
formalization of the self-stabilization mechanism which is inherent to the
considered adaptive controller. Note that the problem of robust identification
of open-loop unstable systems is still unresolved until now. The presented
results can be considered as a contribution in this direction.
2.2 Deterministic Adaptive Control
Consider the following discrete time single input single output (SISO)
system with unmodelled dynamics
A(g _1)y(t + l) =
(2.1)
= B (g _1)[l + A2(g ) + A(g _1)A2(g _1^i (t) +. co(t + 1)
where {y(0}, MO} and {to(0}are output, input and disturbance sequences,
respectively, while q~l represents the unit delay operator. The polynomials
A(g-1) and B(g) describe the nominal system model and are given by
A(g _1) = 1 + a# _1+. .+amg
B(g_1) = Jb. + (jb. o). (2.2)
In equation (2.1) A/(g-1), / = 1,2, denote multiplicative and additive system
perturbations. The transfer functions A(.(z_1), / = 1,2, are causal, and is
stable.
9


The system in equation (2.1) can be stabilized by designing the adaptive
controller so that for a given reference signal y*(0, the following criterion is
minimized
J = [y(0 y'mf. (2.3)
Concerning disturbance oi(t) and reference signal y*(0, it is assumed:
|u)(0| < ka < oo and |y*(0| < ml < ~ (2.4)
for all t > 1. Note that the system model in equation (2.1) can be written in
the form
+.1) = 6o \ho I Po I
where
mT = [y\t + 1); y(t), , y(t nA + l);
;u(t 1 ),-,u(f nB) ] (2.7)
e(t + 1) = y(t + 1) y\t + 1) (2.8)
and
7,(0 = A0(q-l)u(t) + 0)(t + 1) (2.9)
where
Ao(q_1) = S(q_1)Ai(^_1) + (2.10)
From equation (2.5) it is obvious that when 7j(0 = 0 the control law optimal
in the sense of equation (2.3) is given by
(sgn b0)u(t) = -dl(j>(t) (2.11)
10


assuming a minimum phase nominal system model. Since 80 is unknown, the
adaptive control law is
(sgn b0)u(t) = -8(0T where d(t) is an estimate of 80. From equations (2.5) and (2.12) it follows that the closed-loop adaptive system
is given by e(t + 1) = -|£>o|z(0 + 7,(0 (2.13)
where z(0 = ktfm (2.14)
and 8(0 = 6(0 80. (2.15)
From equation (2.12) it is not difficult to obtain B(q~l)u(0 q[A(q~l) l]y(0 = y*(t + 1) |to|z(0 (2.16)
Combining equations (2.8), (2.13) and (2.16) yields B {q _1)u (t) = A(q _1)[-^0|z (t) + y *(t + 1)] + +[A(q~1) ljr^t) (2.17)
Substituting u(t) from equation (2.17) into equation (2.9), it can be shown
{B(q~l) A0(q-1)[4(cT1) lfly.tf) =
= -A0(q-')A(q-l)[\b0\z(t) y*(t + 1)] + B(q~1)co(t + 1)
(2.18)
Concerning system model in equation (2.1) it is assumed:
(ax) A lower bound A0,0 < A0 < 1, for which zeros of S(z_1) and poles of
the transfer functions
11


D^z'1) = A0(z -1)a(z -1) / {b^"1) Ao^-1)^"1) l]}
and
D (z-1) = ___________S(Z ____________
l } e(z-) a0(z-)K--) i]'
II2
zjt satisfy \z.\ < A0 is known.
Based on the available prior information related to the nominal system model,
the following assumption can be made:
(Aj) The sign of the high frequency gain b0 and an upper bound £>, max of
|ri0|, are known. Without loss of generality it is assumed that b0 > 0
and max > 0.
For the estimation of 0O, the following algorithm is proposed:
0(t + l) = 0(t) + 0(t )[y (t + l) y *(t + l)],
l (219)
, 0 < a < -------.
b0, max
The algorithm gain sequence is given by
r{t) = n0 + n/t)2, 0 < n0 < ~ (2.20)
where
n0(t)2 = An,(t l)2 + ||0(t)||2 (2.21)
and. A is a fixed number satisfying
Aj + A < A < 1, 0 < Aj A (2.22)
where A0 is defined by assumption ). Note that sequence r(t) is similar to
the one used by Praly [33]. The following H~ norms can now be defined:
Ajzt = IIA{z) l||A
S(z)|-' * || B(z) ||.
C* =
Cr =
A0(z)A(z)
B(z) A0(z)[A(z) 1]
12


=
B(z)
B(z) A0(z)[A{z) 1] H.
Throughout all constants are positive, unless otherwise noted.
The following lemma will be useful for future reference.
Lemma 2.1: Let the assumption (/\) hold. Then,
1) nrl(t) < Cy\b,\nz(t) + nv(t) + ^(t)
where
nv(t) = Crny*(t + 1) + canafc + l)
(2.23)
(2.24)
(2.25)
while constants Cr and Ca are defined by equation (2.23). The norms
nyl(t), nz(t )ny*(t) and nm(t) are given by equation (1.1) when
x(t) = yl(t),x(t) = z(t), x(t) = y*(0 and x(t) = co(t), respectively.
2) n,(t) < C\b0\nz(t 1) +
+ 1) + C02nffl(t) + (2-26)
where nAt) is defined by equation (2.21) and the
constants = 1, 2 are given by
'*i
C6, 2. + 1, C^ -
= C,[l + (l + C)C, + CABJ, Cf, =
= c,c.( 1 + CA)
where
(2.27)
/ n
y/2
V = 0
= £ A-' ,n = nax{^, ns}
(2.28)
while H~ -norms CA., Cy and Cm are defined by equation
(2.23).
3) r(t) < max{c0nz(t l)2;kg + £3(t)} (2.29)
where
13


C8 = 4CIM and kB = 4
Cmi + C*2k ./2
--------------- + n
(2.30)
(1 X)
where the constants ka, m{ and n0 are given by equations (2.4) and
(2.20) respectively.
Proof: Statement (1) of the lemma follows from equation (2.18). The second
statement of the lemma can now be proved. From equations (2.7) and (2.21) it
follows that
nf(t) < C\ny{t) + /7u(f)] + nyi,(t + 1) + §19(0 (2.31)
where Cx is given by equation (2.28). Note that from equations (2.8), (2.13)
and (2.24), the following can be obtained
nr(0 < (l + CY)[\b0\nz(t 1) + ny*(0] +
+Cana(t) + <§20 (0-
Similarly, from equations (2.17) and (2.24) it is derived that
nu(t) < (c^ + Cjfr)[\b0\nz(t) + ny*(t + 1)] +
-KjfvnJt + 1) + £2i(t).
Substituting equations (2.32) and (2.33) into (2.31), the second statement of the
lemma follows directly. Note that equation (2.26) implies
nXJ2 + nAV ^ 2 max {C01|fc>o|nz(f 1);
' 7T~T^ (c*>' + c*-) + n;/! + (Z34>
(.1 A)
(2.32)
(2.33)
From the above relation, statement (3) of the lemma directly follows. Thus, the
lemma is proved.
Note that by using equations (2.8), (2.13) and (2.17), the measurement
vector can be written in the form
0(0 = 0*(O + M0 + M0 + *ri(0 (2-35)
where
14


(2.36)
0*(t)r = [y *(t + i), y (t), , y *(t nA + 1);
; ^ y *(fc)' ,7 y *(t - + i)]
Jt)T = [o, o(t), , (0(t nA + l);
; - w(t), , 6)(t nB + 1)1
B B J
z(t)T = [o,-|b0|z(f 1), -\b0\z(t nA)j
~^\b0\z(t l),... ,^|b0|z(f nB)]
0rl(t) = [o, yft i), , n(t ba);
; - 7i(t 1), - n(t Jia)].
B B
(2.37)
(2.38)
(2.39)
In the above relations, for the sake of simplicity, q 1 is omitted in all operators.
Regarding reference signal y*(0. the transfer function of the nominal
system model 8(z) / A{z), the size of admissible unmodelled dynamics Cy
and the intensity of unstructured disturbances, the following assumptions are
introduced:
(a3) For all sufficiently large N,
2 > e^I, el > o
t=l
where A is defined by (2.22) and i = 1 j (1
> 0, H = a |b0|
+ (crm1 + (l + c.)k.) / (1 A)1/2]} > 0
E'?> 2 r (2.40)
2V>-Ier + (2.41)
where
15


X
2
r
max-
16[l -ji + /iCy]2
Pi
(2.42)
where p1 and n are defined in assumption (A,), while
X. = [m,C, + k.C.] / |b|(l A),/2 (2.43)
and n., i = 1,2 in equation (2.41) are given by
", = i A-1 + C^i X-,n2 = £ A-' + C2i A". (2.44)
/=1 1=1 ;=1 /=1
Constants cA,cAB,cr and Ca are defined by equation (2.23). In equation
(2.41), e* represents the level of excitation of the reference signal y*(t) and it
is specified by assumption (A)-
Now, the following two results can be established:
Lemma 2.2: Let the assumptions (A) (A) hold. Then on the subsequence
{/Vp}, where
nz{Npf < Z2 +Z4(Np) (2.45)
for sufficiently large Np, the following holds C N )
AJ £ ^"V(O0(Or 2 el p > 0, 0 < p0,* (2.46)
where nz(t) is given by equation (1.1) when x(t) = z(t), Amir{-}
denotes minimal eigenvalue of the corresponding matrix, while Xr and e* are
given by equations (2.42) and (2.41), respectively.
Proof: From (2.38) and (2.45) it is not difficult to obtain
^ XNp t[q20z(t)]2 < njb.2^ + |5(t)
t=i 7
(2.47)
where nt is given by equation (2.44), and rj is any nonzero vector satisfying
16


\\t]\\ = 1. Similarly, from equation (2.40) and statement (1) of the Lemma 2.1,
it can be obtained
2^C[lJr*,1(t)f.|i:,+
+[crm, + CJ(.} / (1 - (2.48)
where n2 and are given by equations (2.44) and (2.42), respectively. From
equations (2.4) and (2.37) it follows that
) (2.49)
t= 1
Relations (2.47) (2.49) yield
|>Vw]2*
<1{n11/2K|Xr-Hn2V2[|b.^rSr +
+(0,111! + (l + c,)k,) / (i A)1/2}2 + {8 where 0, (f) is given by
0i(t) = 02(t) + 0ri(t) + 0o>(t).
(2.51)
From equation (2.35) it follows that
[V0(O]2 > ^ [vTf(t)]2 [ifhitjf (2.52)
where 0^0 is defined by equation (2.51), and 7/ is a nonzero vector satisfying
[|7j|| = 1. From relations (2.50) and (2.52) and by assumption (/\j), it can be
shown that
*i |^/f,'V(tW)T| 2 - Up) (2.53)
17


where e*2 is defined by equation (2.41). Using the fact that for sufficiently
large Np, £g(A/p) < p0,0 < p0 e*, the statement of the lemma directly
follows from relation (2.53). Thus die lemma is proved.
Note that assumption (A,) imposes a condition on the frequency content
of the reference signal y*(t). Specifically, assumption (A3) will be satisfied if
y*(f) has a spectral distribution function which is nonzero at nA + ng + 1
points (or more) and if the transfer function of the nominal system model
B(z) / A(z) is irreducible. If y*(t) does not satisfy this condition, assumption
(A,) can be achieved by superimposing on the reference signal y*(t), an
external signal with at least nA + ng + 1 different spectral lines. It is also
clear that the PE condition defined by assumption (Aj is not stronger than
similar conditions introduced in deterministic adaptive control [39]. From
equation (2.41) it is obvious that on the subsequence where equation (2.45)
holds, the PE condition is satisfied in the adaptive loop, provided intensities of
unmodelled dynamics Cr and external disturbances ka are small enough
compared with the level of excitation e* of the reference signal y*(0-
Lemma 2.3: Let the assumptions (A,) (Aj) hold. Then on the subsequence
{A/p} where

(2.54)
the following holds
2
0(np + i)-0o|| < d0 + £9(t)
(2.55)
where
+SM1 + Cr) Z,
P*f
+
18


(C,m, + CJk) / (1 A)17* ]}
(*: p.)2
where
c* = C^oft-1 xr +(c*1m1 + c02kffl) / (1 A)
1/2
(2.56)
(2.57)
where the constants C*y C^, i = 1,2 are defined by equation (2.27). In
equation (2.56) e* is given by equation (2.41) and 0 < p0 e* .
Proof: From equation (2.13) and statement (1) of Lemma 2.1, the following is
derived
ne(t) < |b0|(l + CY)n2(t) +
(2.58)
+ Crny*(t + 1) + C^n^it + 1) + £10(0
where ng(t) is given by equation (1.1), when x(t) = e(t + 1).
From equations (2.54) and (2.58) it can be shown that
ne(N p)2 [h|(i +Cr) Xy +
(2.59)
+ (crm1 + CJc,) / (1 A)1/2]2 + £t).
Similarly, from statement (2) of Lemma 2.1, it is concluded that
nt{Np) £ C, + |12(f) (2.60)
where C0 is given by equation (2.57). From equations (2.8), (2.15) and (2.19)
it follows that_
6(t + l)rp(0_1 = A6(0rp(t l)"1 +
+ zmtf + 1 e(t + 1)
r(t)
where pity1 is given by
pity1 = apit I)-1 + it)T, pioy1 = p0i, p0 > o
(2.61)
(2.62)
19


and 0(f) is defined by equation (2.15). It is not difficult to see that relation
(2.61) implies
e(w + i)r = )p,p(n) + }p(n) +
+ aff; XN-' S^£L eff + i)
.t=i
Vf=i
r(t)
p{N).
(2.63)
Note that
f A- e(t + 1)
^ r(t)
^ ne(NX(N)
and
X AN-fz(f)0(f)r
r=i
< nz(N)n,(N)
(2.64)
(2.65)
where n0(/V) is defined by equation (2.21). From equation (2.63), by using
relations (2.64) and (2.65) it is shown
that||0(N + i)| < ||p(W)||{po^jr||0(l)| + [nz(N) + ane(W)]n^(w)} (2.66)
Note that by Lemma 2.2, for sufficiently large p, on the sequence [Np] it is
shown
(2.67)
where e* is defined by equation (2.41) and 0 < pQ e*. The statement of
the lemma directly follows from equation (2.66) by using relations (2.54),
(2.59), (2.60), and (2.67). Thus the lemma is proved.
Note that Lemmas 2.2 and 2.3 essentially state that the parameter
estimates are bounded on the subsequence where the PE condition is satisfied
in the adaptive loop. From relation (2.55), it is obvious that on this
subsequence, parameter estimates can drift around 0O only in the certain set
defined by the constant d0 given by equation (2.56).
20


2.3 Global Stability Analysis and Mathematical Formalization of the Self-
Stabilization Mechanism in Robust Adaptive Control
In order to start this analysis the Lyapunov type difference inequalities which
describe the behavior of the parameter estimation error for the considered
adaptive control scheme are needed. From equations (2.19), (2.8) and (2.14) it
follows:
V (t +1) < V (t) 2/if 1 1 +
l 2) r(t)
| 2/1(1 /x)|z(t)|.|y(t)| + n2r(t)2
r(t)
(2.68)
where z(f) is given by equation (2.14), while
V(t) = 0(f) ,\i = a\b0\ and y(f) = y,(f) /
(2.69)
where 0(f) is defined by equation (2.15). Note that fj. < 1, since from
equation (2.19) a < 1 / £>0,max Next it will be shown that global stability of
the proposed adaptive control algorithm can be established by considering the
behavior of the following "Lyapunov" function
S(t + 1) = V(t + 1) + Wit + 1}
r(f)
(2.70)
where
W(t + 1) = X
y=i
1 T + (1 <)Cr + T
ZU)2 -
-2(1 -/i)|z(j)| |/(j)| -/xy(j)2}
It turns out that the behavior of the sequence W(t) is crucial for the
convergence of the recursive scheme (2.68) and consequently for the 12
stability of the sequence z(f). As will be seen in the subsequent analysis, in the
time intervals where w (t) < 0, the stability of the sequencez(f) follows
21


trivially; however, in these time-intervals the Lyapunov function S(t) may be
diverging, thus giving rise to bursting of the sequence z(t).
As a consequence of this phenomenon, die function w (t) becomes positive,
forcing the function S(t) to converge, thus stabilizingz(f).
Global stability, robustness and performance of the considered adaptive
algorithm will be formulated in the following theorem:
Theorem 2.1: (Mathematical formalization of the self-stabilization
mechanism). Let the assumptions (/\) (A,) introduced in Section 2.2 hold.
Then the algorithm in equations (2.12), (2.19) (2.21) provides
1) nz(t)2 < VD +5(f) (2.72)
where Z2 =
= max-
Y2
/

>e
Q+d-lFe
I/JPI
(2.73)
(l "t" d0)Cg jlPX
where Zr is given by equation (2.28), px and n are defined by
assumption (Aj, Ce and ke are given by equation (2.30), and da is
defined by equation (2.56).
2) t + i) y"U + i)f s
i=i
S {|i.|[l + cr] + Ivf + {(<* (2-74)
where ZD is given by equation (2.73), while Zy is defined by equation
(2.43).
3) n,(t)< X-htl ID + ^ ) Jgm' + ) (2.75)
where C* and CÂ¥, i = 1,2 are given by equation (2.27).
4) |9(o e0f < d0 + 4u(t)
22


Proof: The statements of the theorem will be derived by considering the
behavior of the function W(t) given by equation (2.71) and its effects on the
convergence of the recursive scheme in equation (2.68). The following
sequences can be defined tk and ak, k > 1 as follows
1-*! < ^ < r2 < <7, << rk < ok < rk+1-, (2.76)
so that
W(t + 1) < 0 for t g Qk and W(t + 1) > 0 for t e Tk, (2.77)
where the time intervals TK and Qk are defined by
Qk = [xk> Gk)> Tk [k' rk+l)'k 1- (2.78)
\fW(2) > 0, set r, = 0, <7, = 1 and Qk is defined for k > 2. If
W(t + 1) > 0 for all t > 1, define = 1 and t2 = -h. In the case when
W(t + 1) < 0 for all t > 1, then set r1 = l and cj1 = +.
The proof of this theorem will be handled in the following way: regarding the
sequences rk and ok defined by equations (2.76) and (2.77) three possible
cases will be considered:
1) for all finite k, xk < ~ and ok <
2) there exists a finite kQ > 0, such that
Tk < and 3) there exits a finite k{ > 1, such that -
ak]_ < and t^+i = +oo. (2.80)
The last two cases are trivial and are analyzed at the end of the proof of
this theorem. The focus will be on case (1), the case when the function
W(t + 1) changes its sign during the operation of the adaptive system, as
defined by equations (2.77) and (2.78). The main idea in proving global
stability in this case is the following: in the time intervals Qk where the
23


function W(t + 1) is nonpositive, the stability of the adaptive system follows
trivially from the definition of W(t + 1) given by equation (2.71). In the time
intervals Tk when W(t + 1) > 0, it will be shown that the Lyapunov function
s(t) given by equation (2.70) is decreasing. This fact will imply global
stability of the adaptive algorithm.
First consider the time intervals Qk. Since W(t + 1) < 0 for f e Qk,
from equation (2.71) and statement (1) of the Lemma 2.1, it can be shown
[i ^ + (l n)cr + ^-c2jnz(t)2 < 2(1 p)nz(t)
2
[crnz(t) + nv(t) + + p[crnz(t) + nv(t) + ^(t)] .
(2.81)
After simple manipulations, it follows from equation (2.81) that for t e Qk
Pinz(t)2 < 2[l H + //Cy]nz(t)[jiv(t) + ^(t)] +
2 (2.82)
+ p[nv(t) + ^(t)]
where p, is defined by assumption (A,). The above relation implies
p1nz(t)2 < 2 max{2[l p + pc r]nz(t )[nv(t) + ^(t)];
; p[nv(t) + l^t)]2} (2.83)
for t e Qk. From equations (2.83) and (2.25) it is obtained that
nz(tf z rY t S Qk (2.84)
where Xr is defined by equation (2.42).
Next, analyze the time intervals Tk, k > 1, where the function
W(t + 1) > 0. Using the definition of W(t + 1) (equation (2.71)), it is shown
in equation (2.68) that
s(t (2.85)
24


where S(t) is defined by equation (2.70) and p, is given by assumption (A4).
After summation from t = ok + 1 to N < rA+1, it is obtained from equation
(2.85)
S(N + 1) < S(cr, + 1) /ipL £
f=t+i r(t)
(2.86)
Note that from equation (2.71) it follows that
H K + 1) = ^ 1 f + (! - + f Cr2]z((TJ2 -
-2p(i p]p{ok)\. |y(cjfc)| P2r(^)2 + to [k)- (2-87)
Since by definition equation (2.77), W(ak) < 0, equations (2.68), (2.70) and
(2.87) yield
SK +!) s v(a* ) m

Substituting equation (2.88) into equation (2.86), it is derived that
S(n + 1) < tr(crA) pPl £
(2.88)
(2.89)
t=cM for N e Tk. Using the fact that in the time intervals Qk relation (2.54) holds
(see equation (2.84)), by Lemma 2.3, it can be concluded that for sufficiently
large k,
V(t) < d0 + |9(t), t g [rk + 1, ak] (2.90)
where V(t) is defined by equation (2.69) and d0 is given equation (2.56).
Consequently, from equations (2.89) and (2.90) it follows that
(2-91)
t = where
ho = (<* + l,(0)r(f). (2.92)
Note that statement (3) of the Lemma 2.1 implies
ht) max{(l + d0)Cgnz(t 1)2; 25


where
9($ = K + ^)[ke + £3(t)]- (2.94)
Next it will be shown that relation (2.91) together with relation (2.93) are
crucial for establishing the stability in the time intervals7^, k > 1.
Specifically, from equation (2.91) it follows that the magnitudes of z(t) for
t e Tk depend on the values of the sequence r(f). On the other hand, by
relation (2.93) r(t) can be order 0(0 or(l + d0)Cgnz(t l)2 The next step
is to partition the time interval Tk in the following way: let the integers
pik e Tk and ik g Tk be defined as follows:
Pck ^ Ilk k ^ lk <" Pik ^ ^~(i+l)k <'""" (2-95)
so that for r(t) given by relation (2.93), the following holds
r(0 < (1 + d0)Cgnz(t l)2 for t g Lik, i > 1, k > 1 (2.96)
send
r(t) < 0(0 for f g D!k,i > 1, k > 1 (2.97)
where the time intervals Ljk and Djk are defined by
AiJt 2 and Dk [I k, P ^ (2.98)
It is obvious that lik < rk+1 and plk < rk+l for / > 0 If
(1 + d0)Cgnz( g(<7k), then pok = (1 + d0)Cgn((Th l)2 < g(ak), then set pok = 0,1^ = ak and the
intervals Lik are defined for / > 2 .In the case when
(1 + d0)Cgnz(t l)2 < 0(0 for all t g Tk, then define pok = o, 1^ = ak
and Plk = rk+l. If (1 + d0)Cgnz(t l)2 > git) for all t g Tk then set
pok = ak and 1& = rk+1. Note that from equations (2.93) and (2.97) it can
be shown that
nz(t l)2 < 0(0 / (1 + d0)Cg for t e Dik, i > 1, k > I (2.99)
26


(2.100)
Relations (2.91), (2.93) and (2.97) imply that for t e Dik,
z(02 < ~r{t) < git).
Since nz(t)2 = z(t)2 + Anz(f l)2, relations (2.94), (2.99) and (2.100) yield
nz(t)2 ^

(2.101)
(l + dopg nPi_
Next, consider the time intervals Lik c Tk for / > 1, k > 1 From equations
(2.92) and (2.96) we can obtain
R* = I
zy
t-T1 nJt 1)'
1 + d o
m
Ce,N e L
From the above relation it follows that
^ nz(t)2 Anz(t l)2 _ -y,
= nz(t l)2 2-
(2.102)
A
-------------0 1
An_(t l) A^(t-i)2
t_P(i-We z
- A = A X {l9 nr(^2 log Anz(t- l)2} =
-P(i-I)k
(2.103)
= ^ log n?(W)rr + p(/-d*]log T
nz[P(l-i)l< ~ ^
Relations (2.102) and (2.103) imply that
i+d0 c
nz{N)2 < eAw *nz(p(M)* 1 )\/V e L*. (2.104)
Since 1 e D(._iy( it can be shown that from relations (2.101) and
(2.104) for t e Ljk,i > 2,
nz(tf <
(2.105)
(l + d0]c, fip1
1+dOr
B**1 + 5i£).
Next, evaluate nz(0 in the time intervals L,fc Jc > 1. If pok = a,,, it can be
derived from equations (2.84) and (2.104)
nzfc)2 Ir2 eAwl + e > 1.
(2.106)
27


In the case pok = 0 and 1^ = ok, time intervals Lik are defined for / > 2.
2
In equation (2.103), the case nz(N)2 < nz(p(/_I)k l) is trivial, and it is
covered by relation (2.104).
Finally, consider the two cases defined by relations (2.79) and (2.80).
In the case defined by relation (2.79), similarly as in the derivation of relation
(2.84), it can be shown that from equation (2.83) that relation (2.84) is valid for
all t > rko. In the case when ak < and xk+ 1 = , for some
it can be concluded that from relation (2.77) that W(t + 1) > 0
t > <7ki and consequently it is derived from relation (2.85) that

z(t)2

< .
for all
(2.107)
From equations (2.29) and (2.107) it is not difficult to obtain that
lim nz(t)2 = O (2.108)
Thus, relations (2.84), (2.101), (2.105), (2.106) and (2.108) establish the first
statement of the theorem. The second statement of the theorem follows from
relations (2.58) and (2.72). Statement (3) of the theorem follows from relations
(2.26) and (2.72). Finally, the last statement of the theorem follows from
equation (2.70) and relations (2.89) and (2.90). Thus the theorem is proved.
Remark 1: (Continuity property of the results established in Theorem 2.1).
From equations (2.41) (2.43) it follows that
lim Xy = O, lim Xr = O, and limX / 2 (2.109)
Cy ^ 0 C. ^ 0 CL "4 0
where from equations (2.56) and (2.73) it is shown that
lim do = 0 and limXD = 0 (2.110)
c-*o c-*o D
Consequently, from relations (2.72), (2.74), (2.109), (2.110) and statement (4)
of the Theorem 2.1, it is derived that
28


limnz(t)2 = 0, lim £ A£-J'[y(j + l) y *(j + l)]2 = 0 (2.111)
rf-.b ctT) j=i
and
lim||0(t) 0o | = 0.
(2.112)
This means that the tracking and parameter estimation errors are continuous
with respect to unmodelled dynamics and external disturbances in the sense
that when unmodelled dynamics and external disturbances tend to zero,
tracking and parameter estimation errors also tend to zero. In the literature on
robust adaptive control, this continuity property usually is established for the
mean-square tracking error, and not for y(t) y *(f).
2.4 Physical Explanation of the Self-Stabilization Mechanism and
Robustness of the Parameter Estimates
The results presented in the previous section show that the presence of
unmodelled dynamics and external disturbances, the adaptive control algorithm
possesses not only self-tuning but also a self-stabilization property. The latter
means the following: whenever, as a consequence of the incorrect parameter
estimates, the adaptive system becomes unstable, the adaptive algorithm will
stabilize itself by generating correct parameter estimates. From the proof of
Theorem 2.1 it is obvious that during its operation, the adaptive controller
passes through two phases characterized by the time intervals Qk and
Tk defined by equations (2.77) and (2.78). In the time intervals Qk, function
W(t + 1) < 0, which implies the stability of the input and output signals for
t e Qk. From equations (2.68) and (2.70) it is clear that in these time-intervals
29


no characterization of the function S(t + 1) can be made As a consequence of
this, the controller parameters may escape from the set of stabilizing controllers
and the adaptive system will become unstable. Accordingly, the time intervals
Qk correspond to the drift phase of the adaptive algorithm. As time
progresses, the norm nz(t) becomes larger than nr(t) and the function
W(t + 1) given by equation (2.71) becomes positive. From equations (2.77)
and (2.78) it is obvius that these periods of operation of the adaptive system
correspond to the time intervals Tk, k > 1. From relation (2.85) it follows
that in these time intervals the Lyapunov function S(t) decreases.
Consequently, the time intervals Tk correspond to the self-stabilization phase of
the adaptive system.
Therefore, during the adaptation process, there are two consecutive
phases: drift phases (time intervals Qk), and self-stabilization phases or phases
of bursting (time intervals Tk). From the proof of Theorem 2.1 it is clear that
bursting phenomena are possible in the time intervals Llk c Tk, where Ljk is
defined by relations (2.96) and (2.98). From equations (2.102) and (2.103) it is
not difficult to see that
which means that bursting periods are finite.
The emphasis on, generally, boundedness of the parameter
estimates is the crucial fact for establishing global stability and robustness of
the adaptive controllers. Bounded parameter estimates imply that the signals in
the adaptive loop can not grow faster than exponentially. In robust adaptive
control design, boundedness of the parameter estimates usually is achieved by
A
30


using <7-modification, dead-zone or projection techniques techniques. The
results presented here show that parameter estimates can be bounded
automatically by the algorithm, if the reference signal y*(f) is persistently
exciting (assumption (AJ). From the proof of Theorem 2.1 it is clear that in
the time intervals Qk where the function W(t + 1) < 0, relation (2.84) is valid.
Then by Lemma 2.2, it follows that for t e Qk, the PE condition is satisfied in
the adaptive loop. As a consequence of this, Lemma 2.3 states that parameter
estimates are bounded for all t e [rk + 1, allows relation (2.91) to be obtained from relation (2.89). On the other hand,
relation (2.91) implies that signals in the adaptive loop can not grow faster than
exponentially, which is obvious from equation (2.94). Note that the PE
condition in the adaptive loop does not have to be satisfied for all t > 1. It is
enough for relation (2.109) to be valid only in the time intervals Qk where
function W(t + 1) is nonpositive. It has been shown that, in the case of
parameter projection, the assumptions (A3) and (As) do not need to be
satisfied [34]. It is obvious that in this case there is no restriction on the
intensity of the external disturbance ka and the admissible unmodelled
dynamics Cr is defined by assumption (A4). It is not difficult to see that
assumption (A4) is equivalent to the condition Cy < 1. Obviously adaptive
control with parameter projection in the estimation algorithm can tolerate larger
unmodelled dynamics since we do not require assumptions (Aj and (A5). In
this case parameter d in equation (2.73) represents the radius of a compact
convex set 0 which contains 0O and where 0(f) is projected orthogonally onto
0 [34]. If there is not enough prior information about 0O, the designer should
31


choose a large do. This means that the upper bound established in equation
(2.73) will be large.
The estimation algorithm without projection (equation (2.19)), requires
assumptions (A,) and (A,), and in this case cf given by equation (2.56) can be
small. Specifically, when y*(t) has a large enough level of excitation e*, then
e* given by equation (2.41) will be large, implying a small d i.e., smaller
drift of the parameter estimates in the drift phase. At the same time, smaller do
implies a small upper bound in equation (2.73). In this context there is a need
to mention that in the self-stabilization phase, for fast adaptation to take place,
it is needed for the adaptation gain fi = a\bQ | to be large ("ultra fast
adaptation" or "chaotic stabilization") [23]. In the time intervals Qk parameter
drift should be minimized, which implies that slow adaptation is preferable. At
the same time, from equation (2.73) it is seen that
lim / = + oo
^~D , implying that for veiy small algorithm gain, performances of
/i 0
the adaptive system are practically unacceptable. It seems that this problem
can be resolved by selecting a signal dependent algorithm gain.
Note that the fourth statement of the Theorem 2.1 characterizes the
parameter estimation error and this result can be considered as a possible
solution of the problem of robust identification in the adaptive loop. To the
author's knowledge, this is the first result related to the problem of the robust
identification of an open-loop unstable plant. However, similar results were
obtained for open-loop stable systems [20, 42].
Remark 2: Assumptions (Aj, (A,) and (As) introduced in Section 2.2 will
now be elaborated upon. In order to get better insight into this condition,
32


assume that equation (2.1) the multiplicative system perturbation A, (q_1) = 0.
Then the system model in equation (2.1) can be written in the form
co(t + 1)
where P(q_1) denotes the nominal system model given by
P(q-') = S(q-') / 4r)
and A2(q-1) represents the modelling error defined by
A2(q~l) = P(q~l) Pi(q~l)
where P(z-1) is the transfer function characterizing the real physical system
and is of higher order than the nominal system transfer function Pn (z-1). Since
the considered adaptive controller minimizes the criterion (2.3), it is obvious
that the algorithm will attempt to determine a P*(q-1) so that the truncation
modelling error A2(q_1) is minimized. The transfer function P*(q-1) is
referred to as the "centered" nominal system model. There is also an obvious
connection with the tuned parameters where local stability was demonstrated
using the notion of persistent excitation and global properties were established
using the notion of a positive real transfer function [19]. Essentially,
conditions (A), (A ) and (A;) require that for a specified order of the nominal
system model, there exists a P*(q-1) with stable 0*(q_1), so that P*(q~l)
matches P(q_1) as closely as possible. The required degree of matching is
defined by assumptions (A) and (A) and it depends on the frequency
content of the reference signal y*(0. Consequently, among many P(q-1)'s
the algorithm will find such a P*(q-1) so that assumptions (Aj, (Aj and (A)
are satisfied and tracking error 7je(0 is minimized. Obviously, according to the
assumption (A) this P*(q~l) will be irreducible. Note that by assumption
(A), the nominal system model should be stably invertible. The system itself
y(t + 1) = Pn(q~l)u(jt) + A 2(q_1)u(0 +
A(q-
33


need not be minimum phase. Elements which have an unstable inverse may be
"hidden" in the unmodelled term.
2.5 Simulation Examples
In this section, the performance of the previously proposed algorithms
will be verified by some simulation experiments. At the same time, it will be
shown that the developed theory matches the phenomena observed in the
simulations very well. Note that in all the following figures T represents the
number of iterations. To determine the actual time, the number of iterations, T,
must be multiplied by the sampling period.
Simulation 2.1: The system model is represented by
(l L6g_1 + L55g _2)y(t) = (l + 0.5qr-1)u(t 1) +
+(l :L6g_1 + L55g2)A(g_1)u(t 1) (2.113)
where
A(g -1) = 0.04 -3-g 1 (2.114)
1 + 035g 1
The normalized gradient algorithm given in equations (2.19) (2.21)
will be used in this example. The values for the parameters in this algorithm
are set as follows: a = 0.03, A = as5 and y*(t) = 100. In order to
satisfy condition (a3), a white noise sequence is superimposed on the reference
signal y *(t). The mean value and variance of the noise signal are zero and
0.01, respectively. Figure 2.1 represents the output error
e(t) = y (t) y *(t). It is obvious that bursts exist and that the self-
stabilization mechanism provides a bounded tracking error e (t). At the same
34


time on figure 2.1, it is not difficult to recognize the existence of the time
intervals Qk and Tk defined by equations (2.77) and (2.78).
In Section 2.3, where the self-stabilization mechanism is mathematically
formalized, we concluded that the boundedness of the parameter estimates in
the time intervals t e [xk + l, c*], is crucial for the global stability of the
considered adaptive controller. In these time intervals, the parameter estimates
drift has been restricted to a certain set by providing persistent excitation in the
adaptive loop. Figure 2.2 depicts the Amil[p(t )-1}, wherep(t )_1 is defined by
equation (2.62). Observe the obvious "cause and effect" connection between
figures 2.1 and 2.2. Specifically, the time intervals^* and Tk are also
recognizable on figure 2.2. It looks like fort e Qk, Amir[p(t )-1} is zero, as
a consequence of the computer scaling and appearance of the large bursts for
t e [800, 18oo]. In fact AjnijjpCt)-1} > o.i987for allt > 0. It is not
difficult to see that the time intervals Tk, where all signals in the adaptive loop
are large Amil{p(t )-1}, exhibit bursts. Therefore, the considered adaptive
system possesses not only self-stabilization, but also self-excitation property.
35


Time T
Figure 2.1. Simulation 2.1: normalized gradient algorithm with gain of 0.04 on
unmodelled dynamics.
Time history of output error e(t).
Figure 2.2. Simulation 2.1: normalized gradient algorithm with gain of 0.04 on
unmodelled dynamics.
Time history of the minimal eigenvalue of p(t)l.
36


Figures 2.3 2.6 show the parameter estimates as a function of time. These are
the coefficients of the polynomials, A(q _1) and B (q -1), which describe the
nominal system model and are given by equation (2.2). From these figures, it
is obvious that the parameter estimates do not converge in this example. The
actual parameter values for the nominal system are:
Jb0 = LO, b1 = 0.5, a1 = -L6 anda2 = L55. In order to achieve
convergence of the parameter estimates, a lower gain on the unmodelled
dynamics would have to be used and therefore decreasing their effect on the
overall system.
Figure 2.3. Simulation 2.1: normalized gradient algorithm with gain of 0.04 on
unmodelled dynamics.
Time history of parameter estimate bo-
37


Figure 2.4. Simulation 2.1: normalized gradient algorithm with gain of 0.04 on
unmodelled dynamics.
Time history of parameter estimate bj.
Figure 2.5. Simulation 2.1: normalized gradient algorithm with gain of 0.04 on
unmodelled dynamics.
Time history of parameter estimate aj.
38


Figure 2.6. Simulation 2.1: normalized gradient algorithm with gain of 0.04 on
unmodelled dynamics.
Time history of parameter estimate a2-
Figures 2.7 2.10 show the effect of decreasing the gain on the unmodelled
dynamics from 0.04 down to 0.01. By decreasing the effect of the unmodelled
dynamics on the overall system, the parameters converge. Note that the
parameters do converge but not to their actual values. This is due to the effect
of unmodelled dynamics on the system. Also, if the gain on the unmodelled
dynamics were set to zero then the parameters would converge to their actual
values.
39


Figure 2.7 Simulation 2.1: normalized gradient algorithm with gain of 0.01 on
unmodelled dynamics.
Time history of parameter estimate bo*
Figure 2.8. Simulation 2.1: normalized gradient algorithm with gain of 0.01 on
unmodelled dynamics.
Time history of parameter estimate bj.
40


Figure 2.9. Simulation 2.1: normalized gradient algorithm with gain of 0.01 on
unmodelled dynamics.
Time history of parameter estimate aj.
Time T
Figure 2.10. Simulation 2.1: normalized gradient algorithm with gain of 0.01
on unmodelled dynamics.
Time history of parameter estimate a.2-
41


Simulation 2.2: This is a practical example of a very much simplified airframe
using a single feedback loop autopilot. The unmodelled dynamics represent a
structural bending mode of the airframe. Note that in this case the dynamics of
the airframe are held constant whereas in actuality, they do vary as a function
of flight time. The system model is represented by
(l 2.2g -1 + 23 Iq ~2 + 039q ~3)y(t) = (L6 0.6g_1)u(t 1)
-K1 2.2g'1 + 2.5 lg-2 + 0.9 9q _3)A(g _1)u (t 1) (2.115)
where
A(g_1) = 0.70
________1 + Leg'1 Q.63g~2_________
(l L6g_1 + 0.63g'2)(l + 0-86g_1)
(2.116)
The normalized gradient algorithm given in equations (2.19) (2.21)
will be used in this example. The values for the parameters in this algorithm
are set as follows: a- = 1.2, X = 0.8 o and y*(t) = 50. In order to satisfy
condition (a3), a white noise sequence is superimposed on the reference
signal y *(t). The mean value and variance of the noise signal are zero and
0.01, respectively. Figure 2.11 represents the output error
e (t) = y (t) y *(t). From this figure, it is obvious that the adaptive
controller is working well. The output error is reduced to zero after a few
hundred iterations and no periods of instability (i.e., burstings) are observed.
The burstings still occur but are of such magnitude that they are negligible and
have very little effect on our system. This is contrary to Simulation 2.1 where
the unmodelled dynamics were significant enough to cause periods of
instability and burstings were observed.
42


4000
2000-
v
-2000 -
-4000
0
500 1000
Time T
1500
Figure 2.11. Simulation 2.2: normalized gradient algorithm with gain of 0.70
on unmodelled dynamics.
Time history of the output error e(t).
The effect of the unmodelled dynamics can still be observed in the parameter
estimates. Figures 2.12 2.16 show the parameter estimates as a function of
time. From these figures, it can be seen that the parameter estimates converge
but not to their actual values. This convergence of the parameter estimates to
their nonactual value is a consequence of the unmodelled dynamics.
43


Figure 2.12. Simulation 2.2: normalized gradient algorithm with gain of 0.70
on unmodelled dynamics.
Time history of parameter estimate bo-
-0.2
-0.4
^ -0.6
4
-1
-1.2
0 500 1000 1500 2000
TimeT





v
Figure 2.13. Simulation 2.2: normalized gradient algorithm with gain of 0.70
on unmodelled dynamics.
Time history of parameter estimate b^.
44


-1.6
-1.8 -

-2
-2.2
-2.4'---------:----------1----------:-----------
0 500 1000 1500 2000
TimeT
Figure 2.14. Simulation 2.2: normalized gradient algorithm with gain of 0.70
on unmodelled dynamics.
Time history of parameter estimate aj.
Figure 2.15. Simulation 2.2: normalized gradient algorithm with gain of 0.70
on unmodelled dynamics.
Time history of parameter estimate &2-
45


Figure 2.16. Simulation 2.2: normalized gradient algorithm with gain of 0.70
on unmodelled dynamics.
Time history of parameter estimate slj.
Reducing the gain on the unmodelled dynamics from 0.70 down to 0.45
helps reduce the parameter estimation error. The parameter estimates for the
case with the reduced gain of 0.45 on the unmodelled dynamics can be seen in
figures 2.17-2.21.
46


2

if



1.2
0 500 1000 ' 1500 2000
TimeT
Figure 2.17. Simulation 2.2: normalized gradient algorithm with gain of 0.45
on unmodelled dynamics.
Time history of parameter estimate b0.
Figure 2.18. Simulation 2.2: normalized gradient algorithm with gain of 0.45
on unmodelled dynamics.
Time history of parameter estimate b^.
47


-1.8
-2.3
2.4------------1--------------------------------------
0 500 1000 1500 2000
Time T
Figure 2.19. Simulation 2.2: normalized gradient algorithm with gain of 0.45
on unmodelled dynamics.
Time history of parameter estimate aj.
Figure 2.20. Simulation 2.2: normalized gradient algorithm with gain of 0.45
on unmodelled dynamics.
Time history of parameter estimate a2-
48


Figure 2.21. Simulation 2.2: normalized gradient algorithm with gain of 0.45
on unmodelled dynamics.
Time history of parameter estimate a3.
Comparing these figures with figures 2.12 2.16, it can be seen that the lower
the gain on the unmodelled dynamics, the better the parameter estimates match
the actual parameters. Figure 2.22 shows the output error, e(t), for this system
with smaller unmodelled dynamics.
49


600
3 200
I
>; o
-400
-200
400
0
500
1000
1500
Time T
Figure 2.22. Simulation 2.2: normalized gradient algorithm with gain of 0.45
on unmodelled dynamics.
Time history of the output error e(t).
Simulation 2.3: The system model is represented by
(l 2.0g -1 + 5.0g _1)y (t) = (l + 0.6g_1)u(t 1)
The least squares algorithm with exponential data weighting will be
used in this example [14]. The following equations characterize the least
squares algorithm with exponential data weighting
6{t + l) = 9{t) + p(t) +(l 2.0g 1 + 5.0g 2)A(g _1)u (t 1)
(2.117)
where,
(2.118)
where
p(t 1 )0(fc)0(fc) p(fc -t) ^ 120)
A(t) + 0(t)rp(t l)0(t)
with
0(o) > o, P(o) > o
50


and
A(t) = A.A(t l) + (l A.) (2.121)
with
l > A. > o.
The exponential weighting function in equation (2.121) is especially
suitable for time-vaiying systems since it gives a larger weight to the most
recent estimate. However, this example involves a time-invariant system so
this function is not necessary and can be set to A(t) = l. The reference
signal, y *, has a value of 50 and a white noise sequence is superimposed on it.
The mean value and variance of the noise signal are zero and 0.10,
respectively. Figure 2.23 represents the output error e (t) = y(t) y *(t).
This figure shows one very large burst around 350 iterations which reduces to
zero very quickly and remains at zero for the duration of the simulation.
x 1035
Figure 2.23. Simulation 2.3: least squares algorithm with gain of 1.97 on
unmodelled dynamics.
Time history of the output error e (t).
51


The parameter estimates for this example are shown in figures 2.24 2.27.
The actual parameter values for the nominal system are:
jbg = 1.0, b1 0.6. a = 2.0 and a 2 = 5.0.
Figure 2.24. Simulation 2.3: least squares algorithm with gain of 1.97 on
unmodelled dynamics.
Time history of parameter estimate b0.
52


Figure 2.25. Simulation 2.3: least squares algorithm with gain of 1.97 on
unmodelled dynamics.
Time history of parameter estimate bj.
Figure 2.26. Simulation 2.3: least squares algorithm with gain of 1.97 on
unmodelled dynamics.
Time history of parameter estimate aj.
53


Figure 2.27. Simulation 2.3: least squares algorithm with gain of 1.97 on
unmodelled dynamics.
Time history of parameter estimate a2-
These figures show that the parameter estimates converge to the actual
parameters very rapidly and with a very small parameter estimation error.
Reducing the gain on the unmodelled dynamics does not have a large effect on
the parameter estimate errors. This can be seen in figures 2.28 2.31 where the
gain on the unmodelled dynamics is reduced from 1.97 down to 0.01.
54


1.2
1.15
o 1.1
JD
1.05
0
500 1000
TimeT
1500
Figure 2.28. Simulation 2.3: least squares algorithm with gain of 0.01 on
unmodelled dynamics.
Time history of parameter estimate b0.
1
0.9
0.8
0.7
0.6
0.5
0
500 1000
Time T
1500
Figure 2.29. Simulation 2.3: least squares algorithm with gain of 0.01 on
unmodelled dynamics.
Time history of parameter estimate bj.
55


-1.7
-1.8
_ -1.9
4_i
-2.1 ................j.......-....... .................
_2 2-----------------1----------------1-----------------
' 0 500 1000 1500
Time T
Figure 2.30. Simulation 2.3: least squares algorithm with gain of 0.01 on
unmodelled dynamics.
Time history of parameter estimate aj.
5.2 -----------------:------------------:-----------------
5 r.................i..................!.................
4.8...................:.................\..........;.......
§ 4.6...................i- ...............I.................
cd
4.4...................:.......<.........:..................
4.2 .................:..................|.................
4I------------------:------------------:-----------------
0 500 1000 1500
Time T
Figure 2.31. Simulation 2.3: least squares algorithm with gain of 0.01 on
unmodelled dynamics.
Time history of parameter estimate a2-
56


However, this reduction in gain does cause a large reduction in the transient
behavior of the system. This can be seen by comparing output errors of the
two systems, figures 2.23 and 2.32.
Figure 2.32. Simulation 2.3: least squares algorithm with gain of 0.01 on
unmodelled dynamics.
Time history of the output error e(t).
Simulation 2.4: This simulation example uses the same system model as the
one used in Simulation 2.2 defined by equations (2.115) and (2.116).
However, in this example, the least squares algorithm defmed in equations
(2.119) (2.121) will be used instead of the normalized gradient algorithm.
Again, A(t) defined in equation (2.121) will be set to one since our system is
time-invariant. The reference signal, y *, has a value of 50 and a white noise
sequence is superimposed on it. The mean value and variance of the noise
signal are zero and 0.01, respectively. The gain of the unmodelled dynamics
was chosen to be 0.45, the same as the second part of Simulation 2.2. Figure
2.33 represents the output error e (t).
57


I
Figure 2.33. Simulation 2.4: least squares algorithm with gain of 0.45 on
unmodelled dynamics.
Time history of the output error e(t).
Comparing this figure with figure 2.22 in Simulation 2.2, it can be seen that the
least squares algorithm exhibits a smaller transient than the normalized gradient
algorithm. The parameter estimates can be seen in figures 2.34 2.38.
1.9
1.8
^ 1.7
o
1.6
1.5
1.4
0 500 1000 1500 2000
TimeT
Figure 2.34. Simulation 2.4: least squares algorithm with gain of 0.45 on
unmodelled dynamics.
Time history of parameter estimate bo.
58


'



0.5------------:-----------------------------------
0 500 1000 1500 2000
Time T
Figure 2.35. Simulation 2.4: least squares algorithm with gain of 0.45 on
unmodelled dynamics.
Time history of parameter estimate b^.
Figure 2.36. Simulation 2.4: least squares algorithm with gain of 0.45 on
unmodelled dynamics.
Time history of parameter estimate ai.
59


4



l.

0 500 1000 1500 2000
Time T
Figure 2.37. Simulation 2.4: least squares algorithm with gain of 0.45 on
unmodelled dynamics.
Time history of parameter estimate a2-
Figure 2.38. Simulation 2.4: least squares algorithm with gain of 0.45 on
unmodelled dynamics.
Time history of parameter estimate 0.3.
60


Comparing these figures to the equivalent ones, figures 2.17 2.21 in
Simulation 2.2 using the normalized gradient algorithm it is seen that the least
squares algorithm converges to the parameter estimate quicker than the
normalized gradient algorithm but the normalized gradient algorithm generally
has a smaller parameter estimation error.
61


CHAPTER 3
ROBUST ADAPTIVE CONTROL FOR TIME-VARYING SYSTEMS
3.1 An Introduction to Time Varying Adaptive Control
One of the basic functions adaptive control serves is an approach to deal
with plant uncertainty. The basic idea of adaptive control is to have a
controller which tunes itself to the plant being controlled. A major advantage
that adaptive controllers have over controllers with fixed gains stems from their
ability to adjust their parameters on-line. This property is extremely beneficial
for the control of plants that are slowly time-varying or can be considered as
such. Some of the most interesting results in adaptive control have been done
on the problem of time-varying plants.
The difficulty in applying most of the adaptive control schemes
designed for linear-time-invariant (LTI) plants to linear-time-varying (LTV)
ones is their lack of robustness with respect to perturbations such as bounded
disturbances and unmodelled dynamics. Parameter variations can be shown to
introduce similar types of pertubations and so the stability of most of the
schemes cannot be guaranteed in a time-varying environment. However,
several results have been published on the adaptive control of time-varying
plants.


Many of the difficult issues in adaptive control dealing with time-
vaiying parameters are receiving more and more attention lately. One of these
issues deals with how to choose the control signal for a system with time-
varying parameters. Almost all current adaptive controllers are certainty
equivalent controllers where the parameter estimation is separated from the
control. Certainty equivalent control uses a plug-in control law obtained by
plugging in parameter estimates to the known parameter control specification.
Most of the current approaches make some modification to the certainty
equivalent control law for cases of time-varying parameters [39], The form of
these modifications has been the focal point of much research.
Other difficult issues dealing with time-vaiying parameters deal with the
stability or boundedness properties of the algorithm. The stability properties in
the time-invariant parameter case are fundamental in determining behavior in
the time-varying parameter situation. The common key-point between the
time-invariant case and the time-varying case is the idea that a linear slowly
time-varying system behaves, at each instant, almost like a linear time-invariant
one. The basic idea is as follows. The error system is a nonlinear time-varying
difference equation forced by the velocity parameter. To tolerate this forcing
some sort of exponential stability must be established for the unforced system.
Perhaps the earliest treatment of this type of problem is in a model reference
adaptive control problem [3].
It has been postulated that there is a relation between the speed of
adaption /rand the true speed of change p. At a recent conference, it was
mentioned that it seems necessary that P be known in order that /rbe chosen
[37]. This point seems to be emphasized more in adaptive signal processing.
63


Robustness properties of the adaptive control system is another difficult
issue which must be dealt with when designing a system with time-varying
parameters. A measure of robustness of a system is how well the system
performs in the presence of unmodelled dynamics. Various schemes for
dealing with unmodelled dynamics have been tried. These include Praly's
normalizing device as well as the use of relaxation algorithms (or a-
modification), normalized dead zones and others [38]. There is still doubt
about how various parameters needed by these schemes should be chosen. It
does not appear to be a problem to apply these methods in a deterministic
adaptive control time- varying setting although there is not much detail
available. It is also worth noting that time-varying parameters can not be
treated as unmodelled dynamics and dealt with using such techniques. The
idea of adaptive control is to track time-varying parameters and use that
tracking to obtain superior sensitivity, it would seem counter productive to treat
the time-varying parameters as perturbations to be ignored. This idea fails
anyway because only small amplitude unmodelled dynamics can be handled
whereas large amplitude paramter variations need to be tracked. The point
about parameter variations is that their amplitude can be large but only their
speed has to be small.
3.2 The Time-Varying Plant Model
Continuous systems are treated because this trivially extends the
discrete case (the converse is not true, in general) and the analysis gives
confidence to cases where the algorithms are implemented with fast sampling.
Consider a class of time-varying linear systems modelled as follows:
64


(3.1)
y(t) = x(t)r6(t) + 77(0
where y(f), u(t) denote the system input and output and
Dn_1 1 Dn~lu
x{t)T =
F
y,
7y'
i
FU
(3.2)
F is a Hurwitz polynomial in the differential operator D. In equation (3.1), 0(f)
denotes a vector of time-varying parameters and rj(t) denotes unmodelled
errors.
To illustrate how this model might arise, here are several examples of
time-varying linear systems which belong to the above class.
Example 3.1 (Time -varying Differential Equation): This class of system is
modeled as follows:
A{t, D)y = B(t, D)u (3.3)
where
A(t,D) = Dn + an_1(f)Dn_1+--+&(0 (3.4)
and
B{t,D) = bJt)D+"-+bM). (3.5)
Now, operate on equation (3.3) by 1 / F(D), and commute this
operation with A(t, D) and B(t, D) which yields a model of the form equation
(3.1). Proceed as follows.
Rewriting equation (3.3) gives
Fy = (F A)y + Bu
= (F- A)- F.iy + e- F fu. (3.6)
Operating on equation (3.6) by 1/F then gives
65


(3.7)
y = -(F-A)- F- y + B F-u
y F ' ' F y F F
= (F A)-j:y + B-u
+ j [(F-A)-F-F-{F-A)]-ly
+ L[B.F-F-B]^u (3.8)
= xTe + n, (3.9)
where 0 denotes the vector of parameters in the polynomials (F A) and B,
and where T]l denotes errors which arise due to the commuting of time-varying
operators
= [F A A F)y + [B F F B]u.
(3.10)
It shall be important to subsequent analysis that the swapping errors
above be a small function of past plant inputs and outputs. To examine this
question, rewrite equation (3.10) as
lr
1,1 = F
.1=1
where

(0

1 (i) 1 t -y -B(D).Fi(D).7uj (3.11)
L-i Dn~l+---+ al
denotes f ] a, UJ ; (3.12)
(3.13)
[and similarly for B(D)] and
by expanding the expression in equation (3.11) and recombining terms the
following is obtained
, = i {(-i)
A=1
i Fj(D)
F(D)
(1) 1 (i) 1
B u A y
F F
(3.14)
66


Then, provided the first n derivatives of the time-varying coefficients are
small, it is clear that 77 is a small function of u and y.
Example 3.2 (Time-Varying Observer State-Space Form): Suppose the plant is
described as
1 i3 1 M H O O i 'ton-lit)'
-an_2(t) 0 10 0 bn-2(t)
: : 0 xp + .
: : 1
-a0(t) 0 0
(3.15)
y = [ 1 0 ..... 0] xp. (3.16)
By successive differentiation of equation (3.16), and using equation
(3.15), this is equivalent to
A(D,t)y(t) = B{D,t)u(t) (3.17)
where
A(D, f) = Dn + Dn 1 3n_l+'"+D + 30 (3.18)
and similarly for B. Proceeding as for the previous class,
y = xtQ + i\2 (3.19)
where
=j(F-A-A-F)y + (B-F-F B)u (3.20)
with A and B as in equation (3.4) and equation (3.5). Then rearrange equation
(3.20) as
n2
l(F.A-A .F)i/-I(F.fl-e.F)iU
+ L(A-A)y-(B-B)u
- 7?1 + ' Z*\ Ai y + B U
F i=l\ 7
(3.21)
(3.22)
67


where rjl is as in equation (3.14) and
(A 1 ( d Vf n_! (i) (i) U)
Ai-ril*Jr a^+-+D !+
(i)
(3.23)
(similarly for b i)
Thus, it is clear that provided the first n derivatives of the time-varying
parameters are small, then tj2 is a small function of u and y.
Example 3.3 (General Time-Varying State-Space Equation): Suppose
Dxp = A(t)xp + B(t)u (3.24)
and o II (3.25)
Then define C (0 = C(t), pu = C. (00(f),
Ci(t) = Ci_!(t) + C^fcjAfc) (3.26)
and Pijt = Pi-lfr + P-\k-l (3.27)
for i = l--- ri; k < i
and P, A o.
With the above definitions it can be shown that
D'y = C,(t)xp + X P^D'-'u. (3.28)
#f=i
Now define the observability matrix o(t) as
o(t) = [ca(t)rc1(t)r--c/3_1(t)rf (3.29)
Note that in the case where C(t) and >4(0 are fixed, o (t) is simply the
observability matrix. In order to obtain a time-varying model of the same
dimension as the state-space model, it is assumed that o (t )_1 is bounded and
68


that there exists a k such that ||o (t) 1|| < k for any t. This is called uniform
observability [5].
Then from equation (3.28), and letting (t) = cn(tXKt)-1,
Dny = cn(t)xp + £ PnP1^
i=l
= a(t)o(t)xp + £ pnip1 \i
n-1
i=l
where
= X +£ /?niDi-3u
j=0 i=l
= [ From equation (3.28) the following is obtained
Dny = xfa^y - +£ PnP1^-
j=oV U=1 ^ i=l
(3.30)
(3.31)
(3.32)
Equation (3.32) now has a time-varying differential equation form.
Note, however, that the coefficients in the time-varying differential equation
depend on derivatives, of order up to n of C(t), (n 1) of A(t), and (n 1)
of B(t). Thus, in order to ensure the swapping terms, in a model of the form in
equation (3.1), are small the derivatives of order up to 2n of C(t) and (2n 1)
of A(t) and 8(0 need to be small.
It has been shown that each of the above examples has a representation
of the form in equation (3.1). The time variables will be restricted to be small
in the following assumption.
(a-J : The parameter time variations are uniformly small in the mean in the
sense that there exists > 0, k > 0 such that
ft+T
I 6(c) dc < ejT + k,
for all t, T where ex is sufficiently small.
(3.33)
69


Remark 3.1: The above assumption clearly covers the case of slowly drifting
parameters where
||0(t)|| S Sl. (3.34)
It also covers the case of jump parameters if impulses are allowed in 0(t).
Assumption (aJ places a limit on the size and frequency of jumps.
The following assumption regarding the error term 77 in equation (3.1)
can be made.
(a2): There exists known constants e2, a > 0 such that
|n(t)| < e2 sup {e_ OSTSt
where
0 < v(x) < ||x|| V x (3.36)
Remark 3.2: There are two possible sources of errors contributing to the term
77(f) in equation (3.1); namely undermodelling of the system order, and the
commuting of time-varying operators as illustrated in the discussion of
examples (3.1)-(3.3). Assumption (a2) requires that these errors are a small
function of an exponentially weighted sum of plant inputs and outputs.
The model described in equation (3.1) together with assumptions (ax)
and (a2) is the natural extension of the corresponding time-invariant model
used to describe linear plants having unmodelled dynamics [21, 25],
70


3.3 Time-Varying Parameter Estimation
For simplicity, the first case that will be treated is where 77(f) = 0 and
assumption (a-J holds. It will be indicated how the results can be extended to
the case of unmodelled dynamics (77(f) 0).
In the case of time-varying parameters, the usual parameter estimation
schemes do not necessarily result in bounded parameter estimates. This is a
potential failure mode and thus presumed that sufficient prior knowledge is
available to give an upper bound on the parameter norm. More specifically,
the following assumption is made.
(a3) : There exists a known convex region c c 9?n+J7I+1 such that:
i) 0(f) e C for all f
ii) 3k, such that ||0j 02||2 < k for all 9lf 02 e c.
The following standard parameter estimation algorithm is now
introduced [12, 14]:
and P denotes the projection operator necessary to ensure dec while
retaining the other properties of the parameter estimation [12, 14],
This leads to the following result.
Lemma 3.1: The esimator equations (3.37), (3.38), applied to the system
equation (3.1) with 77(f) = 0, and subject to assumptions (a2) and (a3), has the
following properties for all t, T:
0 < a < 00
(3.37)
where,
e(t) A y(t) -x(tfd(t)
(3.38)
71


e(t) A
... ft+T - 2, K K KtS
11) e(r) dr < e{T + +
* a 2a a
iii) £+T 6(t)t9(t)3t <
e(t)
[l + x(tfx(tj\
1/2
is bounded
kjc
k e-i T + + JrJc
a
(3.39)
(3.40)
(3.41)
where k0, k, and are as defined in assumptions (a^ and (a3) and a is as in
equation (3;37).
Proof:
i) From Equations (3.1),
\xT(e 9'
'3.38), and (3.39), the following is obtained
H (1 + xrx)1/2 [1 + ||xf]l/I k' (3.42)
ii) Let
where v(t) = -9(tf9(t) 2 (3.43)
9 = 6-9. (3.44)
V < -ae 2 9t9 (3.45)
Then
where equality holds for those times when the projection operatorp is invoked.
Using assumptions (ax) and (a3) ii), it is shown that
V < -ae +.
(3-46)
and
ffc+r n ft+r ffc+2* ii- 11
a I e <5r < -I Vdr + I |0|pcdT
< k2 + elkT + kX
Hi) e4 s MIx ~2'2
(l + xTx)
r a'fe
(3.47)
(3.48)
72


The first inequality, is an equality for when the projection is invoked
[12]. Equation (3.41) follows from equation (3.47) and equation (3.48).
Remark 3.3: Note that essentially the same properties as given above hold for
the regularized least-squares parameter estimator provided the algorithm is
modified (e g. by covariance resetting) to ensure Pand P-1 are bounded where
P is the covariance matrix [12].
When 77(f) is nonzero, it is needed to ensure that the gain of parameter
estimator is reduced whenever 77(f) is large. This can be achieved in a number
of ways [17, 21]. By way of illustration, a relative dead zone is used in which
case the parameter estimator becomes
and [<7(0 is defined in equation (3.4)].
The following is an extension of Lemma 3.1.
Corollary 3.1: The estimator equations (3.49)-(3.50), applied to the system
equation (3.1) where 77(f) satisfies assumption (a2), and subject to assumptions
(a2) and (a3), has the following properties for all f, T:
(3.49)
where
(3.50)
i) g----------^Yj2 bounded
(3.51)
(3.52)
(3.53)
Proof:
i) Using equations (3.1), (3.38), (3.44), the following is obtained.
73


(3.54)
e = x T6 + 7J.
Hence, |gf| = max{|e| d,0}
< max||lx-1| ||d| ,0}, using equation (3.5). (3.55)
The result then follows as in Lemma 3.1.
ii) Introducing V as in equation (3.44), it follows that
V < 09 ^ - 6t6 < -ag2 dTd. (3.56)
1 + x x
The result then follows as in Lemma 3.1.
iii) It can be shown that
eTe < a2g2. (3.57)
The result then follows as in Lemma 3.1.
3.4 Certainty Equivalence Control Law
Essentially the same analysis holds for a wide range of control laws
including model references and linear quadratic optimal. To illustrate,
certainty equivalence pole assignment will be used. The control law is
implemented as
£(Fu) = ^Fy*Fy) (3'58)
where y* denotes a bounded reference signal and L, P are of degrees n and
n 1, respectively, and satisfy
al + BP = A* (3.59)
where &*(#) is a polynomial of degree 2n such that dk* / 38 is bounded and
such that A* has a uniform stability margin, i.e., for all 8 e c
RejA^A*^)]} < -a < 0 (3.60)
where A( [] denotes the /th-zero of the polynomial and Re {} denotes real part.
74


A technical difficulty arises in connection with the solution of equation
(3.59). In particular, it is required that
be bounded (where || || denotes
the norm of the vector of polynomial coefficients). This is equivalent to
requiring that the estimated model A, B be uniformly controllable if A* is
fixed, or uniformly stabilizable in the more general framework above.
(a4) : The convex region of assumption (a3) has the further property that for all
6 e C, the solution to equation (3.59) gives bounded L and P .
Remark 3.4: Assumption (a4) is everywhere in the literature on adaptive
control for time-invariant systems [14]. Moreover, it has recently been shown
that an arbitrary large region of the parameter space can be covered by the use
of multiple convex regions. The analysis presented below can be trivially
extended to the case of multiple convex regions [25]. Alternatively, one may
use parameter search techniques [8]. A single convex region is used here since
our emphasis is on the issue of time-varying parameters.
The following theorem established global BIBS stability for the adaptive
control law in the case where 77(f) = 0.
Theorem 3.1: If the adaptive controlTaw, equations (3.58), (3.59), (3.37),
(3.38) is applied to the system (3.1) with 77(f) = 0 and subject to
assumptions (a^, (a3) and (a4), then the closed-loop system is globally BIBS
stable provided in assumption (ax) is sufficiently small.
Proof: Combining equations (3.58) and (3.38) it can be shown that
75


0
D
n-1 y
F
.F
an-l ........ _a bn-1
1 0 0
1 0
~Pn-1 ........ ~P ~bi-l
1
u
1 0
u
0
0
F
F
or
x = Ax + + B2r
(3.61)
(3.62)
where
r
(3.63)
and so in view of assumption (a4) and since y* is bounded, r is bounded.
The stability of equation (3.62) together with other relevant equations
will now be studied. Do this by first establishing exponential stability of the
homogenious part of equation (3.62). Consider the following unforced system:
x = Ax'. (3.64)
Show that this system is exponentially stable, provided ex is sufficiently
small.
For any fixed Q = QT > 0, let T(f) denote the symmetric solution to
AT(t)nt) + mm = -q. (3.65)
The 2n eigenvalues of A(t) are the 2n zeros of A*(t), and in view of
assumption (a4), A(t) is bounded. These conditions are sufficient to establish
76


that; i) T(t) is bounded and positive definite; ii) T(t) 1 is bounded, and iii)
there exists kl such that
< k-
0(t)
(3.66)
Now consider a second auxiliary system
*' = [*" 7 ^
This system is exponentially stable since with
V = x"T Tx"
it can be shown that
V = -x"T Qx"
Returning to equation (3.64), it is found that
x = a i r_1rjac +^ r_1rj* .
In view of the exponential stability of equation (3.68), and using
equation (3.66), the following relation can be shown
||x-'(t)|| < kfa^Wx'iT)! +
(3.67)
(3.68)
(3.69)
(3.70)
'Ki
0(T)
[|x' (t)|(5t.
(3.71)
Squaring both sides and using Schwarz's inequality yields the following
Ik ft S 2*^-*-n||*(T)||2 + df
t( 2k^\
i
0 )

dr
- (3.72)
or
e£lk'(t)||2 Ik'CDir
4

6{ r)

using Gronwall's Lemma and Lemma 3.1 it can be shown that
e \\x'(t )||2 < k^a(P\\x(T)\\2
(3.73)
77


r r k2 ^
* expj/f4al /c^! (f T) + + k0k
(3.74)
From equation (3.74), it is clear that for £( sufficiently small, the system
equation (3.64) is exponentially stable. In particular, if (t, r) is the state
transition matrix corresponding to /4(f), then
|0(t, t)| < *5 expj- ( =
where cri define
a%t-r)
O i =
O, fc4a/c£1
(3.76)
Returning toequation (3.62) shows
t ftT)
|[x(t)|| < ^5|lJClle~ Where the fact that IbJI = |[s2|| = l is used. Squaring both sides of
equation (3.77), using Schwarz's inequality, and since r is bounded equation
(3.77) yields
||x(t)||2 < k6e-** +k7 + fQkQef\2WT (3.78)
rt lfc-T)
- k9 + )0kBeffl Ik(T)||e 2Wx (3.79)
since e2(t) < (l + |x(T)f)e2 and e is bounded.
Then using the Bellman-Gronwall Lemma, it can be shown that
||x-(t)||2 < k9 + kQeait j*{kgeai'e2(x) exp jjk8efyjdvJjr (3.80)
< k9 + K 10J^exp(-ox(t r)) expp^-fijtt r)^Jr (3-81)
where the parameter estimator properties are used, equations (3.39) and (3.40)
[9]. From equation (3.81), it is seen that if
k,k
o, > -J£,, (3.82)
a
78


from the definition of cr,, if
( kAk k'kY1
£l <
a -4 + "8
a*,
(3.83)
2 a
then x(0 is bounded. Boundedness of 0, e, x, and all other variables can be
easily established.
Remark 3.5: The tradeoffs inherent in the choice of a are evident in equation
(3.75). In the estimation algorithm, ainfluences the speed of adaptation, large
a giving more rapid parameter variations (i.e., larger 6) but lower normalized
prediction errors (in the mean square sense).
Finally, turn to the important practical case, where the plant contains
unmodelled dynamics satisfying assumption (A2). The following corollary
covering time-varying parameters as well as unmodelled dynamics is useful in
this case.
Corollary 3.2: The adaptive law equations (3.58), (3.59), (3.49), (3.50) is
applied to the system (3.1) and subject to assumptions (aJ, (a2), (a3), and
(a4). The closed-loop system is globally BIBS stable provided £, in
assumption (Aj and e2 in assumption (a2) are sufficiently small.
Proof: With unmodelled dynamics equation (3.62) still holds. This equation
can be rewritten as
x = Ax + Bxg + £?,(e g) + B2r. (3.84)
As in the proof of Theorem 3.1, exponential stability of the unforced
system equation (3.64) can be established.
The term in B2r is treated by a BIBS type argument as in Theorem 3.1,
the term in Byg is treated using the Bellman-Cronwall Lemma, also as in
Theorem 3.1 (since g is small in the mean square). This leaves the term
B{ (e g) to be treated. This is handled by a small gain argument [9, 25].
79


3.5 Simulation Examples
In this section, the performance of the previously proposed algorithms
will be verified by some time-varying simulation experiments. Again, it will
be shown that the developed theory matches the phenomena observed in the
simulations very well.
The system model used in this section is a similar model to the one used
in Simulation 2.2 of Chapter 2. However, in this section, sinusoidal time-
variations have been added to the coefficients of the A(t, D) term in the system
model, see equation (3.4). In particular, time-variations are added to the a! and
a 2 terms. All the simulations in this section will use the following system
model represented by
(l (2.2 + 0.3 sin(Q)t))g -1 + (2.51 + 0.5 cos(cot))q~2
-03 9q 3)y (t) = (L6 0.6g_1)u(t 1) + (l (2.2
+0.3 sin(iut))y_1 + (2.51 + 0.5 cos (rat))?-2 -
-039q -3)A(g -1)u (t l)
where co is the frequency of the time-variations and
, -i\ = 1 + Leg"1 0.63g ~2
g P (l L6gr_1 + 0.63g-2)(l + O-Segr'1)
where p is the gain of the unmodelled dynamics.
(3.85)
(3.86)
Simulation 3.1: The normalized gradient algorithm with parameter projection
given in equation (3.37) will be used in this example. The values for the
parameters in this algorithm are set as follows: a = L2 and y*(t) = 50. A
white noise sequence is superimposed on the reference signal y *(t). The mean
value and variance of the noise signal are zero and 0.01, respectively. In this
80


example, there are no unmodelled dynamics, so set p= 0.0 in equation (3.86).
The frequency of the time-variations is chosen to be 6.525 x io-4 Hz or
co = 0.004lRad/Sec.
Figure 3.1 represents the output error e(t) = y (t) y *(t). This
figure shows that even in the presence of time-variations the output error is
driven close to zero. There are two separate bursting sequences observed in
this figure; the first at around 550 iterations and the second around 850
iterations.
Figure 3.1. Simulation 3.1: normalized gradient algorithm with p= 0.0 and
o)= 0.0041 Rad/sec.
Time history of parameter output error e (t).
These burstings are similar in nature to those observed in Simulation 2.1, with
the difference being that here our system model has time-variations and in
81


Simulation 2.1 the system had unmodelled dynamics. It is interesting to note
that the self-stabilization mechanism of the normalized gradient algorithm
works in the time-varying case. The parameter estimates are shown in figures
3.2 - 3.6. The actual parameters for the nominal system are:
Jb0 = L6, bi = 0.6,3! = -2.2 0.3 sin(0.0043t),
a2 = 2.51 + 0.5 cos(0.004It) and a3 = -039.
Figure 3.2. Simulation 3.1: normalized gradient algorithm with p= 0.0
and co = 0.0041 Rad/sec.
Time history of parameter estimate bp.
82


Figure 3.3. Simulation 3.1: normalized gradient algorithm with p=0.0
and co = 0.0041 Rad/sec.
Time history of parameter estimate bj.
Figure 3.4. Simulation 3.1: normalized gradient algorithm with p=0.0 and
co = 0.0041 Rad/sec.
Time history of parameter estimate aj.
83


Figure 3.5. Simulation 3.1: normalized gradient algorithm with p =0.0 and
on = 0.0041 Rad/sec.
Time history of parameter estimate a
TimeT
Figure 3.6. Simulation 3.1: normalized gradient algorithm with p =0.0 and
co = 0.0041 Rad/sec.
Time history of parameter estimate a3-
84


From these figures, the time-variations can be observed. It can also be seen in
these figures that there is a very small parameter estimation error in our system
and hence our estimator has excellent performance in this case.
Figure 3.7 is plot of the closed-loop roots on the unit circle for each
iteration in the simulation. From this figure, it can be seen that all of the roots
are within the unit circle (stable) for every iteration.
85


Figure 3.7. Simulation 3.1: normalized gradient algorithm with p-0.0 and
O) = 0.0041 Rad/sec.
Plot of the closed-loop roots of the adaptive system.
86


Simulation 3.2: The normalized gradient algorithm with parameter
projection given in equation (3.37) will again be used in this example. The
nominal system model is defined by equations (3.85) and (3.86). However, in
this example, the amplitude and the frequency of the time-varying parameters
have been increased. The values for the parameters in this algorithm are set as
follows: a = L2 and y *(t) = 50. A white noise sequence is superimposed
on the reference signal y *(t). The mean value and variance of the noise signal
are zero and 0.01, respectively. In this example, there are no unmodelled
dynamics, so set p =0.0 in equation (3.86). All of the previous conditions have
been the same as Simulation 3.1. In this example, the frequency,co, of the
parameter variations will be increased from, co = 0.004lRad/sec to
co = a0205Rad/sec. This increase in frequency is evident from the output
error, e(t), in figure 3.8.
Figure 3.8. Simulation 3.2: normalized gradient algorithm with p=0.0 and
co = 0.0205 Rad/sec.
Time history of the output error e (t).
87


Comparing this figure to figure 3.1 in Simulation 3.1, it can be seen that
increasing the frequency of the parameter variations causes many more bursts
in the output error e (t). The increase in amplitude can also be seen by
comparing the magnitude of the bursts. The parameter estimates can be seen in
figures 3.9-3.13. These figures show the increase in frequency and amplitude
of the time-variations. The actual parameters for the nominal system are:
jb0 = 1.6, = 0.6, ax = -2.2 0.7 sin(0.020Bt),
a2 = 2.51 + L2 cos(0.0205t) and a3 = -039.
Figure 3.9. Simulation 3.2: normalized gradient algorithm with p=0.0 and
to = 0.0205 Rad/sec.
Time history of parameter estimate bo-
88


0
Figure 3.10. Simulation 3.2: normalized gradient algorithm with p =0.0 and
(0 = 0.0205 Rad/sec.
Time history of parameter estimate bj.
Figure 3.11. Simulation 3.2: normalized gradient algorithm with p= 0.0 and
io= 0.0205 Rad/sec.
Time history of parameter estimate aj.
89


Figure 3.12. Simulation 3.2: normalized gradient algorithm with p= 0.0 and
o)= 0.0205 Rad/sec.
Time history of parameter estimate &2-
Figure 3.13. Simulation 3.2: normalized gradient algorithm with p= 0.0 and
(o- 0.0205 Rad/sec.
Time history of parameter estimate a3.
90


Comparing these figures to the parameter estimates in Simulation 3.1, it
can be seen that the increase in frequency and amplitude of the time-variations
have increased the parameter estimation error. As the frequency and/or the
amplitude of the time variations are increased, the estimator has a much more
difficult time tracking the actual parameters. The plot of the closed-loop roots
is shown in figure 3.14. This figure shows that in some instances, some of the
roots go unstable. This was not seen in Simulation 3.1, see figure 3.7. This
increase in frequency and amplitude in the time-variations has a destabilizing
effect on the system.
91


1.5
Real Axis
Figure 3.14. Simulation 3.2: normalized gradient algorithm with p=0.0 and
a = 0.0205 Rad/sec.
Plot of the closed-loop roots of the adaptive system.
92


Simulation 3.3: The normalized gradient algorithm with parameter projection
given in equation (3.37) will again be used in this example. The nominal
system model is defined by equations (3.85) and (3.86). The amplitude and
frequency of the time-variations will be the same as in Simulation 3.2. The
values for the parameters are set as follows: a = L2 and y *(t) = 50.
Again, a white noise sequence is superimposed on the reference signal y *(t).
The mean value and variance of the noise signal are zero and 0.01,
respectively. This example differs from the previous two examples because
this time unmodelled dynamics are included along with the time-variations, so
now set p=0.05 in equation (3.86). Figure 3.15 represents the output error
e(t).
Figure 3.15. Simulation 3.3: normalized gradient algorithm with p=0.05 and
0) = 0.0205 Rad/sec.
Time history of the output error e (t).
It is evident, if a comparison is made between this figure and the previous two
cases output error, figures 3.8 and 3.1, that the unmodelled dynamics are
93


causing more bursts in the output error e (t). Comparing figure 3.15 to figure
3.8, it seems that most of the larger bursts are coming from the time-variations
and most of the smaller burstings are from the unmodelled dynamics. Even
with time variations and unmodelled dynamics, the self-stabilization
mechanism works. In figure 3.15, there are periods where the output error is
reduced very small. The parameter estimates are shown in figures 3.16 -3.20.
Figure 3.16. Simulation 3.3: normalized gradient algorithm with p=0.05 and
(o= 0.0205 Rad/sec.
Time history of parameter estimate bo-
94