in (1.2.10) to obtain
¨
y = − 2 ˙ y − y − θ∗ 1 ˙ y − θ∗ 2 y + u
(1.2.24)
which may be rewritten as
y = θ∗ 1 ˙ yf + θ∗ 2 yf + uf
(1.2.25)
where
1
1
1
˙ yf = −
˙ y, y
y, u
u
(1.2.26)
s 2 + 2 s + 1
f = − s 2 + 2 s + 1
f = s 2 + 2 s + 1
are signals that can be generated by filtering.
If we now replace θ∗ 1 and θ∗ 2 with their estimates θ 1 and θ 2 in equation
(1.2.25), we will obtain,
ˆ
y = θ 1 ˙ yf + θ 2 yf + uf
(1.2.27)
where ˆ
y is the estimate of y based on the estimate θ 1 and θ 2 of θ∗ 1 and θ∗ 2.
The error
ε 1 = y − ˆ y = y − θ 1 ˙ yf − θ 2 yf − uf
(1.2.28)
is, therefore, a measure of the discrepancy between θ 1 , θ 2 and θ∗ 1 , θ∗ 2, respec-
tively. We refer to it as the estimation error. The estimates θ 1 and θ 2 can
now be adjusted in a direction that minimizes a certain cost criterion that
involves ε 1. A simple such criterion is
ε 2
1
J( θ
1
1 , θ 2) =
= ( y − θ
2
2
1 ˙
yf − θ 2 yf − uf )2
(1.2.29)
which is to be minimized with respect to θ 1 , θ 2. It is clear that J( θ 1 , θ 2) is a
convex function of θ 1 , θ 2 and, therefore, the minimum is given by ∇J = 0.
1.3. A BRIEF HISTORY
23
If we now use the gradient method to minimize J( θ 1 , θ 2), we obtain the
adaptive laws
˙
∂J
∂J
θ 1 = −γ 1
= γ
= γ
∂θ
1 ε 1 ˙
yf , ˙ θ 2 = −γ 2
2 ε 1 yf
(1.2.30)
1
∂θ 2
where γ 1 , γ 2 > 0 are the adaptive gains and ε 1 , ˙ yf , yf are all implementable signals.
Instead of (1.2.29), one may use a different cost criterion for ε 1 and a
different minimization method leading to a wide class of adaptive laws. In
Chapters 4 to 9 we will examine the stability properties of a wide class
of adaptive control schemes that are based on the use of estimation error
criteria, and gradient and least-squares type of optimization techniques.
1.3
A Brief History
Research in adaptive control has a long history of intense activities that
involved debates about the precise definition of adaptive control, examples
of instabilities, stability and robustness proofs, and applications.
Starting in the early 1950s, the design of autopilots for high-performance
aircraft motivated an intense research activity in adaptive control. High-
performance aircraft undergo drastic changes in their dynamics when they fly
from one operating point to another that cannot be handled by constant-gain
feedback control. A sophisticated controller, such as an adaptive controller,
that could learn and accommodate changes in the aircraft dynamics was
needed. Model reference adaptive control was suggested by Whitaker et
al. in [184, 235] to solve the autopilot control problem. The sensitivity
method and the MIT rule was used to design the adaptive laws of the various
proposed adaptive control schemes. An adaptive pole placement scheme
based on the optimal linear quadratic problem was suggested by Kalman in
[96].
The work on adaptive flight control was characterized by “a lot of en-
thusiasm, bad hardware and non-existing theory” [11]. The lack of stability
proofs and the lack of understanding of the properties of the proposed adap-
tive control schemes coupled with a disaster in a flight test [219] caused the
interest in adaptive control to diminish.
24
CHAPTER 1. INTRODUCTION
The 1960s became the most important period for the development of
control theory and adaptive control in particular. State space techniques
and stability theory based on Lyapunov were introduced. Developments
in dynamic programming [19, 20], dual control [53] and stochastic control
in general, and in system identification and parameter estimation [13, 229]
played a crucial role in the reformulation and redesign of adaptive control.
By 1966 Parks and others found a way of redesigning the MIT rule-based
adaptive laws used in the MRAC schemes of the 1950s by applying the
Lyapunov design approach. Their work, even though applicable to a special
class of LTI plants, set the stage for further rigorous stability proofs in
adaptive control for more general classes of plant models.
The advances in stability theory and the progress in control theory in
the 1960s improved the understanding of adaptive control and contributed
to a strong renewed interest in the field in the 1970s. On the other hand,
the simultaneous development and progress in computers and electronics
that made the implementation of complex controllers, such as the adaptive
ones, feasible contributed to an increased interest in applications of adaptive
control. The 1970s witnessed several breakthrough results in the design
of adaptive control. MRAC schemes using the Lyapunov design approach
were designed and analyzed in [48, 153, 174]. The concepts of positivity
and hyperstability were used in [123] to develop a wide class of MRAC
schemes with well-established stability properties. At the same time parallel
efforts for discrete-time plants in a deterministic and stochastic environment
produced several classes of adaptive control schemes with rigorous stability
proofs [72, 73]. The excitement of the 1970s and the development of a wide
class of adaptive control schemes with well established stability properties
was accompanied by several successful applications [80, 176, 230].
The successes of the 1970s, however, were soon followed by controversies
over the practicality of adaptive control. As early as 1979 it was pointed
out that the adaptive schemes of the 1970s could easily go unstable in the
presence of small disturbances [48]. The nonrobust behavior of adaptive
control became very controversial in the early 1980s when more examples of
instabilities were published demonstrating lack of robustness in the presence
of unmodeled dynamics or bounded disturbances [85, 197]. This stimulated
many researchers, whose objective was to understand the mechanisms of
instabilities and find ways to counteract them. By the mid 1980s, several
1.3. A BRIEF HISTORY
25
new redesigns and modifications were proposed and analyzed, leading to a
body of work known as robust adaptive control. An adaptive controller is
defined to be robust if it guarantees signal boundedness in the presence of
“reasonable” classes of unmodeled dynamics and bounded disturbances as
well as performance error bounds that are of the order of the modeling error.
The work on robust adaptive control continued throughout the 1980s
and involved the understanding of the various robustness modifications and
their unification under a more general framework [48, 87, 84].
The solution of the robustness problem in adaptive control led to the
solution of the long-standing problem of controlling a linear plant whose
parameters are unknown and changing with time. By the end of the 1980s
several breakthrough results were published in the area of adaptive control
for linear time-varying plants [226].
The focus of adaptive control research in the late 1980s to early 1990s
was on performance properties and on extending the results of the 1980s to
certain classes of nonlinear plants with unknown parameters. These efforts
led to new classes of adaptive schemes, motivated from nonlinear system
theory [98, 99] as well as to adaptive control schemes with improved transient
and steady-state performance [39, 211].
Adaptive control has a rich literature full with different techniques for
design, analysis, performance, and applications. Several survey papers [56,
183], and books and monographs [3, 15, 23, 29, 48, 55, 61, 73, 77, 80, 85,
94, 105, 123, 144, 169, 172, 201, 226, 229, 230] have already been published.
Despite the vast literature on the subject, there is still a general feeling that
adaptive control is a collection of unrelated technical tools and tricks. The
purpose of this book is to unify the various approaches and explain them in
a systematic and tutorial manner.
Chapter 2
Models for Dynamic Systems
2.1
Introduction
In this chapter, we give a brief account of various models and parameteriza-
tions of LTI systems. Emphasis is on those ideas that are useful in studying
the parameter identification and adaptive control problems considered in
subsequent chapters.
We begin by giving a summary of some canonical state space models for
LTI systems and of their characteristics. Next we study I/O descriptions
for the same class of systems by using transfer functions and differential
operators. We express transfer functions as ratios of two polynomials and
present some of the basic properties of polynomials that are useful for control
design and system modeling.
systems that we express in a form in which parameters, such as coeffi-
cients of polynomials in the transfer function description, are separated from
signals formed by filtering the system inputs and outputs. These paramet-
ric models and their properties are crucial in parameter identification and
adaptive control problems to be studied in subsequent chapters.
The intention of this chapter is not to give a complete picture of all
aspects of LTI system modeling and representation, but rather to present a
summary of those ideas that are used in subsequent chapters. For further
discussion on the topic of modeling and properties of linear systems, we
refer the reader to several standard books on the subject starting with the
elementary ones [25, 41, 44, 57, 121, 180] and moving to the more advanced
26
2.2. STATE-SPACE MODELS
27
ones [30, 42, 95, 198, 237, 238].
2.2
State-Space Models
2.2.1
General Description
Many systems are described by a set of differential equations of the form
˙ x( t) = f ( x( t) , u( t) , t) ,
x( t 0) = x 0
y( t) = g( x( t) , u( t) , t)
(2.2.1)
where
t
is the time variable
x( t) is an n-dimensional vector with real elements that denotes the state
of the system
u( t) is an r-dimensional vector with real elements that denotes the input
variable or control input of the system
y( t) is an l-dimensional vector with real elements that denotes the output
variables that can be measured
f, g
are real vector valued functions
n
is the dimension of the state x called the order of the system
x( t 0) denotes the value of x( t) at the initial time t = t 0 ≥ 0
When f, g are linear functions of x, u, (2.2.1) takes the form
˙ x = A( t) x + B( t) u,
x( t 0) = x 0
y = C ( t) x + D( t) u
(2.2.2)
where A( t) ∈ Rn×n, B( t) ∈ Rn×r, C( t) ∈ Rn×l, and D( t) ∈ Rl×r are matrices with time-varying elements. If in addition to being linear, f, g do not
depend on time t, we have
˙ x = Ax + Bu,
x( t 0) = x 0
y = C x + Du
(2.2.3)
where A, B, C, and D are matrices of the same dimension as in (2.2.2) but
with constant elements.
28
CHAPTER 2. MODELS FOR DYNAMIC SYSTEMS
We refer to (2.2.2) as the finite-dimensional linear time-varying (LTV)
system and to (2.2.3) as the finite dimensional LTI system.
The solution x( t) , y( t) of (2.2.2) is given by
t
x( t) = Φ( t, t 0) x( t 0) +
Φ( t, τ ) B( τ ) u( τ ) dτ
t 0
y( t) = C ( t) x( t) + D( t) u( t)
(2.2.4)
where Φ( t, t 0) is the state transition matrix defined as a matrix that satisfies
the linear homogeneous matrix equation
∂Φ( t, t 0) = A( t)Φ( t,t
∂t
0) ,
Φ( t 0 , t 0) = I
For the LTI system (2.2.3), Φ( t, t 0) depends only on the difference t−t 0, i.e.,
Φ( t, t 0) = Φ( t − t 0) = eA( t−t 0)
and the solution x( t) , y( t) of (2.2.3) is given by
t
x( t) = eA( t−t 0) x 0 +
eA( t−τ) Bu( τ ) dτ
t 0
y( t) = C x( t) + Du( t)
(2.2.5)
where eAt can be identified to be
eAt = L− 1[( sI − A) − 1]
where L− 1 denotes the inverse Laplace transform and s is the Laplace vari-
able.
Usually the matrix D in (2.2.2), (2.2.3) is zero, because in most physical
systems there is no direct path of nonzero gain between the inputs and
outputs.
In this book, we are concerned mainly with LTI, SISO systems with
D = 0 . In some chapters and sections, we will also briefly discuss systems of
the form (2.2.2) and (2.2.3).
2.2. STATE-SPACE MODELS
29
2.2.2
Canonical State-Space Forms
Let us consider the SISO, LTI system
˙ x = Ax + Bu,
x( t 0) = x 0
y = C x
(2.2.6)
where x ∈ Rn. The controllability matrix Pc of (2.2.6) is defined by
Pc = [ B, AB, . . . , An− 1 B]
A necessary and sufficient condition for the system (2.2.6) to be completely
controllable is that Pc is nonsingular. If (2.2.6) is completely controllable,
the linear transformation
xc = P − 1
c
x
(2.2.7)
transforms (2.2.6) into the controllability canonical form
0 0 · · · 0
−a 0
1
1 0 · · · 0
−a 1
0
˙ x
0 1 · · · 0
−a
0
c
=
2
xc +
u
(2.2.8)
.
.
.
.
..
. .
..
..
0 0 · · · 1 −an− 1
0
y = Cc xc
where the ai’s are the coefficients of the characteristic equation of A, i.e.,
det( sI − A) = sn + an− 1 sn− 1 + · · · + a 0 and Cc = C Pc.
If instead of (2.2.7), we use the transformation
xc = M− 1 P − 1
c
x
(2.2.9)
where
1 an− 1 · · · a 2
a 1
0
1
· · · a 3
a 2
M = .
.
.
.
.
..
..
. . ..
..
0
0
· · ·
1
an− 1
0
0
· · ·
0
1
30
CHAPTER 2. MODELS FOR DYNAMIC SYSTEMS
we obtain the following controller canonical form
−an− 1 −an− 2 · · · −a 1 −a 0
1
1
0
· · ·
0
0
0
˙ x
0
1
· · ·
0
0
0
c
=
xc +
u (2.2.10)
.
.
.
.
.
..
..
. .
..
..
0
0
· · ·
1
0
0
y = C 0 xc
where C 0 = C PcM. By rearranging the elements of the state vector xc,
(2.2.10) may be written in the following form that often appears in books
on linear system theory
0
1
0
· · ·
0
0
0
0
1