A( s)
A( s)
A( s)
which may be written in the compact form
y = k 0( θ∗ φ + z 0)
(2.4.31)
60
CHAPTER 2. MODELS FOR DYNAMIC SYSTEMS
α
( s)
α
( s)
where θ∗ = [ q , p ] , φ =
n− 2
u, n− 1
y
, and z
u. We can
A( s)
A( s)
0 = sn− 1
A( s)
also filter each side of (2.4.30) using an arbitrary stable filter
1
whose
Λ( s)
order nλ satisfies 2 n − m − 1 ≥ nλ ≥ n − 1 to obtain
y = W ( s) k 0( θ∗ φ + z 0)
(2.4.32)
α
( s)
α
( s)
where now φ =
n− 2
u, n− 1
y
, z
u, and W ( s) = Λ( s) is a
Λ( s)
Λ( s)
0 = sn− 1
Λ( s)
A( s)
proper transfer function.
In (2.4.31) and (2.4.32), φ and z 0 may be generated by filtering the input
u and output y of the system. Therefore, if u and y are measurable, then
all signals in (2.4.31) and (2.4.32) can be generated, and the only possible
unknowns are k 0 and θ∗. If k 0 is known, it can be absorbed in the signals φ
and z 0, leading to models that are affine in θ∗ of the form
¯
y = W ( s) θ∗ ¯
φ
(2.4.33)
where ¯
y = y − W ( s) k 0 z, ¯
φ = k 0 φ. If k 0, however, is unknown and is part
of the parameters of interest, then (2.4.31) and (2.4.32) are not affine with
respect to the parameters k 0 and θ∗, but instead, k 0 and θ∗ appear in a
special bilinear form. For this reason, we refer to (2.4.31) and (2.4.32) as
bilinear parametric models to distinguish them from (2.4.7) to (2.4.9) and
(2.4.33), which we refer to as linear parametric or affine parametric models.
The forms of the linear and bilinear parametric models are general enough
to include parameterizations of some systems with dynamics that are not
necessarily linear, as illustrated by the following example.
Example 2.4.3 Let us consider the nonlinear scalar system
˙ x = a 0 f( x, t) + b 0 g( x, t) + c 0 u
(2.4.34)
where a 0 , b 0, and c 0 are constant scalars; f( x, t) and g( x, t) are known nonlinear functions that can be calculated at each time t; and u, x is the input and state of
the system, respectively. We assume that f, g, and u are such that for each initial
condition x(0) = x 0, (2.4.34) has only one solution defined for all t ∈ [0 , ∞). If x and u are measured, (2.4.34) can be expressed in the form of parametric model (2.4.33)
by filtering each side of (2.4.34) with a stable strictly proper transfer function Wf ( s),
i.e.,
z = Wf ( s) θ∗ φ
(2.4.35)
2.5. PROBLEMS
61
where z = sWf ( s) x, θ∗ = [ a 0 , b 0 , c 0] , and φ = [ f( x, t) , g( x, t) , u] . Instead of (2.4.35), we may also write (2.4.34) in the form
˙ x = −amx + amx + θ∗ φ
for some am > 0, or
1
x =
[ a
s + a
mx + θ∗ φ]
m
Then
a
1
z = x −
m
x =
θ∗ φ
(2.4.36)
s + am
s + am
which is in the form of (2.4.35) with Wf ( s) =
1
. We may continue and rewrite
s+ am
(2.4.35) (respectively (2.4.36)) as
z = θ∗ φf ,
φf = Wf ( s) φ
(2.4.37)
which is in the form of (2.4.8).
The nonlinear example demonstrates the fact that the parameter θ∗ ap-
pears linearly in (2.4.35) and (2.4.37) does not mean that the dynamics are
linear.
2.5
Problems
2.1 Verify that x( t) and y( t) given by (2.2.4) satisfy the differential equation (2.2.2).
2.2 Check the controllability and observability of the following systems:
(a)
− 0 . 2
0
1
˙ x =
x +
u
− 1
0 . 8
1
y = [ − 1 , 1] x
(b)
− 1
1
0
0
˙ x = 0
− 1
0 x + 1 u
0
0
− 2
1
y = [1 , 1 , 1] x
62
CHAPTER 2. MODELS FOR DYNAMIC SYSTEMS
(c)
− 5 1
1
˙ x =
x +
u
− 6 0
1
y = [1 , 1] x
2.3 Show that ( A, B) is controllable if and only if the augmented matrix [ sI −A, B]
is of full rank for all s ∈ C.
2.4 The following state equation describes approximately the motion of a hot air
balloon:
˙ x 1
− 1
0
0
0
τ
x 1
1
1
˙ x
1
2
=
σ
− 1
0
x
+
0
u +
w
τ
2
2
τ 2
˙ x 3
0
1
0
x 3
0
0
y = [0 0 1] x
where x 1: the temperature change of air in the balloon away from the equilib-
rium temperature; x 2: vertical velocity of the balloon; x 3: change in altitude
from equilibrium altitude; u: control input that is proportional to the change
in heat added to air in the balloon; w: vertical wind speed; and σ, τ 1 , τ 2 are
parameters determined by the design of the balloon.
x (temperature)
1
u(hot air)
w (wind)
x3
(a) Let w = 0. Is the system completely controllable? Is it completely
observable?
(b) If it is completely controllable, transform the state-space representation
into the controller canonical form.
(c) If it is completely observable, transform the state-space representation
into the observer canonical form.
2.5. PROBLEMS
63
(d) Assume w = constant. Can the augmented state xa = [ x , w]
be
observed from y?
(e) Assume u = 0. Can the states be controlled by w?
2.5 Derive the following transfer functions for the system described in Problem 2.4:
(a) G 1( s) = Y ( s) when w = 0 and y = x
U ( s)
3.
(b) G 2( s) = Y ( s) when u = 0 and y = x
W ( s)
3.
(c) G 3( s) = Y 1( s) when w = 0 and y
U ( s)
1 = x 1.
(d) G 4( s) = Y 1( s) when u = 0 and y
W ( s)
1 = x 1.
2.6 Let a( s) = ( s + α)3 , b( s) = β, where α, β are constants with β = 0.
(a) Write the Sylvester matrix of a( s) and b( s).
(b) Suppose p 0( s) , l 0( s) is a solution of the polynomial equation
a( s) l( s) + b( s) p( s) = 1
(2.5.1)
Show that ( p 1( s) , l 1( s)) is a solution of (2.5.1) if and only if p 1( s) , l 1( s) can be expressed as
p 1( s) = p 0( s) + r( s) a( s)
l 1( s) = l 0( s) − r( s) b( s)
for any polynomial r( s).
(c) Find the solution of (2.5.1) for which p( s) has the lowest degree and
p( s) /l( s) is a proper rational function.
2.7 Consider the third order plant
y = G( s) u
where
b
G( s) =
2 s 2 + b 1 s + b 0
s 3 + a 2 s 2 + a 1 s + a 0
(a) Write the parametric model of the plant in the form of (2.4.8) or (2.4.13)
when θ∗ = [ b 2 , b 1 , b 0 , a 2 , a 1 , a 0] .
(b) If a 0 , a 1, and a 2 are known, i.e., a 0 = 2 , a 1 = 1, and a 2 = 3, write a parametric model for the plant in terms of θ∗ = [ b 2 , b 1 , b 0] .
(c) If b 0 , b 1, and b 2 are known, i.e., b 0 = 1 , b 1 = b 2 = 0, develop a parametric model in terms of θ∗ = [ a 2 , a 1 , a 0] .
64
CHAPTER 2. MODELS FOR DYNAMIC SYSTEMS
2.8 Consider the spring-mass-dashpot system shown below:
u
M
x
k
f
where k is the spring constant, f the viscous-friction or damping coefficient,
m the mass of the system, u the forcing input, and x the displacement of
the mass M . If we assume a “linear” spring, i.e., the force acting on the
spring is proportional to the displacement, and a friction force proportional
to velocity, i.e., ˙ x, we obtain, using Newton’s law, the differential equation
M ¨
x = u − kx − f ˙ x
that describes the dynamic system.
(a) Give a state-space representation of the system.
(b) Calculate the transfer function that relates x with u.
(c) Obtain a linear parametric model of the form
z = θ∗ φ
where θ∗ = [ M, k, f ] and z, φ are signals that can be generated from
the measurements of u, x without the use of differentiators.
2.9 Verify that (2.4.11) and (2.4.12) are nonminimal state-space representations
of the system described by (2.4.1). Show that for the same input u( t), the
output response y( t) is exactly the same for both systems. (Hint: Verify that
C 0 [ adj( sI − Λ c)] B 0 = C [ adj( sI − A)] for some C 0 ∈ Rn, B 0 ∈ Rn×n by using the identity
[adj( sI − A)] = sn− 1 I + sn− 2( A + an− 1 I) + sn− 3( A 2 + an− 1 A + an− 2 I)
+ · · · + ( An− 1 + an− 1 An− 2 + · · · + a 1 I)
and choosing C 0 such that ( C 0 , Λ c) is an observable pair.)
2.10 Write a state-space representation for the following systems:
2.5. PROBLEMS
65
(a) φ = αn− 1( s) u, Λ( s) is monic of order n.
Λ( s)
(b) φ = αn− 1( s) u, Λ
Λ
1( s) is monic of order n − 1.
1( s)
(c) φ = αm( s) u, m ≤ n − 1, Λ
Λ
1( s) is monic of order n − 1.
1( s)
2.11 Show that
α
( sI − Λ
n− 1( s)
c) − 1 l =
Co ( sI − Λ o) − 1
=
Λ( s)
where (Λ c, l) is in the controller form and ( Co, Λ o) is in the observer form.
2.12 Show that there exists constant matrices Qi ∈ R( n− 1) ×( n− 1) such that
α
( sI − Λ
n− 2( s)
0) − 1 di = Qi
,
i = 1 , 2 , . . . , n
Λ( s)
In− 2
where d
1 = −λ; Λ( s) = sn− 1 + λ αn− 2( s) = det( sI − Λ0), Λ0 = −λ
−− ;
0
di = [0 , . . . , 0 , 1 , 0 , . . . , 0] ∈ Rn− 1 whose ( i − 1)th element is equal to 1, and i = 2 , 3 , . . . , n.
Chapter 3
Stability
3.1
Introduction
The concept of stability is concerned with the investigation and characteri-
zation of the behavior of dynamic systems.
Stability plays a crucial role in system theory and control engineer-
ing, and has been investigated extensively in the past century. Some of
the most fundamental concepts of stability were introduced by the Rus-
sian mathematician and engineer Alexandr Lyapunov in [133]. The work of
Lyapunov was extended and brought to the attention of the larger control
engineering and applied mathematics community by LaSalle and Lefschetz
[124, 125, 126], Krasovskii [107], Hahn [78], Massera [139], Malkin [134],
Kalman and Bertram [97], and many others.
In control systems, we are concerned with changing the properties of dy-
namic systems so that they can exhibit acceptable behavior when perturbed
from their operating point by external forces. The purpose of this chapter is
to present some basic definitions and results on stability that are useful for
the design and analysis of control systems. Most of the results presented are
general and can be found in standard textbooks. Others are more specific
and are developed for adaptive systems. The proofs for most of the general
results are omitted, and appropriate references are provided. Those that are
very relevant to the understanding of the material presented in later chapters
are given in detail.
In Section 3.2, we present the definitions and properties of various norms
66
3.2. PRELIMINARIES
67
and functions that are used in the remainder of the book. The concept of I/O
stability and some standard results from functional analysis are presented
in Section 3.3. These include useful results on the I/O properties of linear
systems, the small gain theorem that is widely used in robust control design
and analysis, and the L 2 δ-norm and Bellman-Gronwall (B-G) Lemma that
are important tools in the analysis of adaptive systems. The definitions of
Lyapunov stability and related theorems for linear and nonlinear systems are
presented in Section 3.4. The concept of passivity, in particular of positive
real and strictly positive real transfer functions, and its relation to Lyapunov
stability play an important role in the design of stable adaptive systems.
Section 3.5 contains some basic results on positive real functions, and their
connections to Lyapunov functions and stability that are relevant to adaptive
systems.
In Section 3.6, the focus is on some elementary results and principles that
are used in the design and analysis of LTI feedback systems. We concentrate
on the notion of internal stability that we use to motivate the correct way of
computing the charact