Haddad W.M. Nonlinear Dynamical Systems and Control: A Lyapunov-Based Approach

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INTRODUCTION 5

and chaos, which can make general nonlinear system analysis and control

notoriously diﬃcult. Lyapunov’s results provide a powerful framework

for analyzing the stability of nonlinear dynamical systems. Lyapun ov-

based methods have also been used by control system designers to obtain

stabilizing feedback controllers for nonlinear systems. In particular, for

smooth feedback, Lyapunov-based methods were inspired by Jurdj evic and

Quinn [224], wh o give suﬃcient conditions for smooth stabilization based

on the ability to construct a Lyapunov function for the closed-loop system.

More recently, Artstein [13] introduced the notion of a control Lyapunov

function whose existence guarantees a feedback control law wh ich globally

stabilizes a nonlinear dynamical system. Even though for certain classes of

nonlinear dynamical systems a universal construction of a feedback stabilizer

can be obtained using control Lyapunov functions [13, 406], there does not

exist a uniﬁed p rocedure for ﬁnding a Lyapunov function that will stabilize

the closed-loop system for general n onlinear systems. In light of this,

advances in Lyapunov-based methods have been developed for analysis and

control design for numerous classes of nonlinear dynamical systems. As a

consequence, Lyapunov’s direct method has become one of the cornerstones

of systems and control theory.

The main objective of this book is to present necessary m athematical

tools for stability an alysis and control design of nonlinear systems, with an

emphasis on Lyapunov-based methods. The main contents of the book are

as follows. In Chapter 2, we provide a systematic development of nonlinear

ordinary diﬀerential equations, which is central to the study of nonlinear

dynamical system theory. Speciﬁcally, we develop qualitative solutions prop-

erties, existence of s olutions, uniqueness of solutions, continuity of solutions,

and continuous dependence of solutions on system initial conditions for

nonlinear dynamical systems.

In Chapter 3, we develop stability theory for non linear dynamical

systems. Speciﬁcally, Lyapunov stability theorems are developed for

time-invariant nonlinear dynamical systems. Furthermore, invariant set

stability theorems, converse Lyapunov theorems, and Lyapunov instability

theorems are also considered. Finally, we present several systematic

approaches for constructing Lyapunov functions as well as stability of linear

systems and Lyapunov’s linearization method. Chapter 4 provides an

advanced treatment of stability theory including partial stability, stability

theory for time-varying systems, L agrange stability, boundedness, ultimate

boundedness, input-to-state stability, ﬁnite-time stability, semistability, and

stability theorems via vector Lyapunov functions. In addition, Lyapunov

and asymp totic stability of sets as well as stability of periodic orbits are also

systematically addressed. In particular, local and global stability theorems

are given using lower semicontinuous Lyapunov functions. Furthermore,

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6 CHAPTER 1

generalized invariant set theorems are derived wherein system traj ectories

converge to a union of largest invariant sets contained on the boundary of

the intersections over ﬁnite intervals of the closure of generalized Lyapunov

level surfaces. These resu lts provide transparent generalizations to s tandard

Lyapunov and invariant set theorems.

In Chapter 5, us ing generalized notions of system energy storage and

external energy supply, we present a systematic treatment of dissipativity

theory [456]. Dissipativity theory provides a fundamental framework

for the analysis and design of nonlinear dynamical systems using an

input, state, and output system description based on system-energy-related

considerations. As a direct application of dissipativity theory, absolute

stability theory involving the stability of a feedback system whose forward

path contains a dynamic linear time-invariant system and whose feedback

path contains a memoryless (possibly time-varying) nonlinearity is also

addressed. The Aizerman conjecture and the Lur´e problem, as well as the

circle and Popov criteria, are extensively developed.

Using the concepts of diss ipativity theory, in C hapter 6 we present

feedback interconnection stability results for nonlinear dynamical systems.

General stability criteria are given for Lyapunov, asymptotic, and exponen-

tial stability of feedback dynamical systems. Using quadratic supply rates

corresponding to net system power and weighted inp ut-output energy, we

specialize these results to the classical positivity and small gain theorems.

In addition, notions of a control Lyapunov function, feedb ack linearization,

zero dynamics, minimum-ph ase systems, and stability margins for nonlinear

feedback systems are also introduced. Finally, to address optimality issues

within nonlinear control-system design we consider an optimal control

problem in which a performance function is minimized over all possible

closed-loop system trajectories. The value of the performance function is

given by a solution to the Hamilton-Jacobi-Bellman equation. In Chapter

7, we provide a brief treatment of input-output stability and dissipativity

theory. In particular, we introdu ce input-output system models as well as L

stability. In addition, we develop connections between dissipativity theory

of inpu t, state, and outpu t systems, and input-output dissipativity theory.

In Chapter 8, we develop a uniﬁed framework to address the problem of

optimal nonlinear analysis and feedback control. Asymptotic stability of the

closed-loop nonlinear system is gu aranteed by means of a Lyapunov function

which can clearly be seen to be the solution to the s teady-state form of the

Hamilton-Jacobi-Bellman equation, and hence guarantees both stability and

optimality. The overall framework provides the foundation for extendin g

linear-quadratic controller synthesis to nonlinear-nonquadr atic problems.

Guaranteed stability margins for nonlinear optimal and inverse optimal

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INTRODUCTION 7

regulators that minimize a nonlinear-nonquadratic performance criterion

are also established. Using the optimal control framework of Chapter 8,

in Chapter 9, we give a uniﬁcation between nonlinear-nonquadratic optimal

control and backstepping control. Backstepping control has received a great

deal of attention in the nonlinear control literature [247, 395]. The popularity

of this control methodology is du e in large part to the fact that it provides a

systematic procedure for ﬁ nding a Lyapunov function for nonlinear closed-

loop cascade systems.

In Chapter 10, we develop an optimality-based framework to address

the problem of nonlinear-nonquadratic control for disturbance rejection of

nonlinear systems with bounded exogenous disturbances. Speciﬁcally, using

dissipativity theory with appropriate storage functions and supply rates

we transform the nonlinear disturbance rejection problem into an optimal

control problem by modifying a nonlinear-nonquadratic cost functional to

account for the exogenous d isturbances. As a consequence, the resulting

solution to the modiﬁed optimal control prob lem guarantees disturbance

rejection for nonlinear systems with bounded input disturbances. Fur-

thermore, it is s hown that the Lyapunov function guaranteeing closed-loop

stability is a solution to the steady-state Hamilton-Jacobi-Isaacs equation

for the controlled system. The overall framework generalizes the Hamilton-

Jacobi-Bellman conditions developed in C hapter 8 to address the design of

optimal controllers for nonlinear systems with exogenous disturbances.

In Chapter 11, we concentrate on developing a uniﬁed framework

to address the problem of optimal nonlinear robust control. As in the

disturbance rejection problem, we transform the given robust control

problem into an optimal control pr oblem by properly modifying the cost

functional to account for the system uncertainty. As a consequence, the

resulting solution to the modiﬁed optimal control problem guarantees robust

stability and performance for a class of nonlinear uncertain systems. In

Chapter 12, we extend the framework developed in Chapter 11 to nonlinear

systems with nonlinear time-invariant real parameter uncertainty. Robust

stability of the closed-loop nonlinear system is guaranteed by means of a

parameter-dependent Lyapunov function composed of a ﬁxed (parameter-

independ ent) and variable (parameter-dependent) part. The ﬁxed part

of the Lyapunov function can be seen to be the solution to the steady-

state Hamilton-Jacobi-Bellman equation for the n omin al system. Finally,

in Chapters 13 and 14 we give a condensed presentation of the continuous-

time analysis and control synthesis results developed in Chapters 2–12 for

discrete-time systems.

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Chapter Two

Dynamical Systems and Diﬀerential

Equations

2.1 Introduction

In this chapter, we provide a thorough treatment of some of the basic

results in nonlinear diﬀerential equations. The study of diﬀerential equations

in dynamical systems and control is of the highest importance since

diﬀerential equations arise in nearly all disciplines of science, engineering,

medicine, economics, biocenology, and demography. I n as much as

diﬀerential equations are central to these ﬁelds, their s tudy has additionally

led to the development of entire ﬁelds of abstract mathematics. In

particular, algebraic topology and group theory were developed to solve

problems in dynamical system theory, wh ich was originally a branch of

diﬀerential equations. Fourier analysis was developed for analyzing the

heat equation, while topology and set theory were essential developments

for the study of convergence problems for diﬀerential equations. And of

course most recently control theory, whose strength has been its eﬀective

use of diﬀerential equations, has signiﬁcantly contributed to the theory

of diﬀerential equations. The study of diﬀerential equations is usually

divided into two p arts; qualitative theory and quantitative theory. In

this chapter, we will focus almost exclusively on qualitative analysis of

diﬀerential equations. The notions of openness, convergence, continuity,

and compactness that we will use throughout this book will refer to the

topology generated on the n-dimensional Eu clidean space R

by the vector

norm k · k.

As discus sed in Chapter 1, a nonlinear dynamical system consists of a

set of possible states such that the knowledge of these states at some time

t = t

, together with the knowledge of an external input to the system for

t ≥ t

, completely determines the behavior of the dynamical system at any

time t ≥ t

. Hence, a dynamical system can generate a d iﬀerential equation

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10 CHAPTER 2

of the form

˙x(t) = F (t, x(t), u(t)), x(t

) = x

, t ∈ [t

, t

], (2.1)

where x(t) ∈ D, t ∈ [t

, t

], is the state of the dynamical system, D is an

open subset of R

with 0 ∈ D, u(t) ∈ U ⊆ R

, t ∈ [t

, t

], is the input

or control signal to the system, and F : [t

, t

] × D × U → R

is piecewise

continuous in t and continuous in x and u on [t

, t

] × D × U. The input

or control u is assumed to be a piecewise continuous function of time with

values in U, that is, u : [t

, t

] → U. Here, the time index t runs over the

set R

= [0, ∞), or sometimes the set R = (−∞, ∞) of all reals. Even

though for physical dynamical systems t ∈ R

, on some occasions involving

the mathematical analysis of diﬀerential equations it becomes necessary to

assume that t ∈ R. In studying dynamical systems of the form (2.1), we can

distinguish between system analysis and system synthesis. Speciﬁcally, in

analyzing (2.1) we evaluate the behavior of the system for a speciﬁed ﬁxed

input or control signal. Alternatively, synthesis refers to a design procedure

which yields a control inpu t that achieves a desired behavior of the system

state. In this book we consider both the analysis and control synthesis

problems for nonlinear dynamical systems.

In the remainder of this section we provide several classiﬁcations of

(2.1) along w ith some deﬁnitions used throughout the book. First, we

refer to (2.1) as a nonlinear disturbed or controlled time-varying dynamical

system. Alternatively, in the case where u(t) ≡ 0, we let f (t, x) = F (t, x, 0)

so that (2.1) becomes

˙x(t) = f (t, x(t)), x(t

) = x

, t ∈ [t

, t

], (2.2)

where f : [t

, t

] × D → R

is piecewise continuous in t and continuous in x

on [t

, t

] × D. In this case, we refer to (2.2) as a nonlinear undisturbed or

uncontrolled time-varying dynamical system. Note that if the input u(·) is

a priori uniquely speciﬁed, then deﬁning f(t, x) = F (t, x, u(t)), (2.1) can be

written as (2.2). Hence, in this case, the distinction between a disturbed and

undisturbed s ystem is not precise s ince an undisturbed system can describe

the case where u(t) ≡ 0 or the case when u(t) is speciﬁed over [t

, t

]. Note

that if u : [t

, t

] → U is piecewise continuous, then f(t, x) = F (t, x, u(t))

is also piecewise continuous in t over [t

, t

] for all x ∈ D. The following

deﬁnitions provide several classiﬁcations of the nonlinear dynamical system

(2.2). Analogous classiﬁcations also hold for (2.1).

Deﬁnition 2.1. Consider the nonlinear dynamical system (2.2). If

f(t, x) = f(t

, x) for all (t, x) ∈ [t

, t

] × D, then (2.2) is called a time-

invariant or autonomous dynamical s ystem.

Deﬁnition 2.2. Consider the nonlinear dynamical system (2.2) with

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DYNAMICAL SYSTEMS AND DIFFERENTIAL EQUATIONS 11

f : [t

, ∞) × D → R

. If there exists T > 0 such that f (t, x) = f (t + T, x)

for all (t, x) ∈ [t

, ∞) ×D, then (2.2) is called a periodic dynamical system.

Furth ermore, the minimum time T > 0 for which f(t, x) = f (t + T, x) is

called the period.

Deﬁnition 2.3. Consider the dynamical system (2.2) with D = R

If f(t, x) = A(t)x, where A : [t

, t

] → R

n×n

is piecewise continuous on

, t

] and x ∈ R

, then (2.2) is called a linear time-varying dynamical

system. If, alternatively, for all t ∈ [t

, ∞) there exists T > 0 such that

A(t) = A(t + T ), then (2.2) is called a linear periodic dynamical system.

Finally, if f(t, x) = Ax, where A ∈ R

n×n

and x ∈ R

, then (2.2) is called a

linear autonomous dynamical system.

Next, we introduce the concept of an equilibrium point of a nonlinear

dynamical system.

Deﬁnition 2.4. Consider the nonlinear dynamical system (2.2) with

f : [t

, ∞) × D → R

. A point x

∈ D is said to be an equilibrium point of

(2.2) at time t

∈ [t

, ∞) if f (t, x

) = 0 for all t ≥ t

Note that in the case of autonomous systems and periodic systems,

∈ R

is an equilibrium point at time t

if and only if x

is an equ ilibrium

point at all times. Furthermore, if x

is an equilibrium point at time t

, then

deﬁning τ

△

= t − t

and τ

△

= t

− t

yields

˙x(τ + t

) = f(τ + t

, x(τ + t

)), x(τ

+ t

) = x

, τ ∈ [τ

, ∞), (2.3)

where ˙x(·) denotes diﬀerentiation with respect to τ, and hence x

is an

equilibrium point of (2.3) at τ = 0 since f (τ + t

, x

) = 0 f or all τ ≥ 0.

Hence, we assume without loss of generality that t

= 0. Furthermore,

if f(·, ·) has at least one equilibrium point x

∈ R

then, without loss of

generality, we can assume that x

= 0 so that f(t, 0) = 0, t ≥ 0. To see this,

deﬁne x

△

= x − x

and note that

˙x

(t) = ˙x(t) = f (t, x(t)) = f (t, x

(t) + x

)

△

= f

(t, x

(t)),

) = x

− x

, t ∈ [t

, ∞). (2.4)

Now, the claim follows by noting that f

(t, 0) = f(t, x

) = 0, t ≥ t

. Similar

observations hold for the forced dynamical system (2.1) by noting that a

point x

∈ R

is an equilibr ium point of (2.1) if and only if there exists

∈ R

such that F (t, x

, u

) = 0 for all t ≥ 0. Finally, we note that if

∈ R

is an equilibrium point of (2.2) and x(t

) = x

, then x(t) = x

t ≥ t

, is a solution (see Section 2.7) to (2.2).

Example 2.1. Consider the nonlinear dynamical system describing the

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12 CHAPTER 2

motion of a simple pendulum with viscous damping given by

˙x

(t) = x

(t), x

(0) = x

, t ≥ 0, (2.5)

˙x

(t) = −cx

(t) −

sin x

(t), x

(0) = x

, (2.6)

where c > 0 is the viscous damping coeﬃcient, g is the acceleration

due to gravity, and l is th e length of the pendulum. Note that (2.5)

and (2.6) have the form of (2.2) with f (t, x) = f (x). Even though the

physical pendulum has two equilibrium points, namely, (0, 0) and (π, 0), the

mathematical model of the pendulum given by (2.5) and (2.6) has countably

inﬁnitely many equilibrium points in R

given by f(x

) = 0. In particular,

, x

) = (nπ, 0), n = 0, ±1, ±2, . . .. △

Example 2.2. There are dynamical systems that have no equilibrium

points. In particular, consider the system

˙x

(t) = α + sin[x

(t) + x

(t)] + x

(t), x

(0) = x

, t ≥ 0, (2.7)

˙x

(t) = α + sin[x

(t) + x

(t)] − x

(t), x

(0) = x

, (2.8)

where α > 1. For this system there does not exist an x

= [x

]

such

that f(x

) = 0. Hence, (2.7) and (2.8) does not possess any equilibrium

points in R

. △

As seen in Examples 2.1 and 2.2, the nonlinear algebraic equation

f(x) = 0 may have multiple solutions or no solutions. However, f(x) = 0

can also have a continuum of solutions. To see this, let f(x) = Ax, where A ∈

n×n

, which corresponds to the dynamics of a linear autonomous system.

Now, Ax = 0 has a unique solution x

= 0 if and only if det A 6= 0. However,

if det A = 0, then x

∈ N(A)

△

= {x ∈ R

: Ax = 0}, which corresponds

to a continuum of equilibr ia in R

. This observation leads to the following

deﬁnition. For this deﬁnition we deﬁne the open ball B

)

△

= {x ∈ R

kx − x

k < ε}, where k ·k denotes a vector norm in R

(see Section 2.2).

Deﬁnition 2. 5. An equilibrium point x

∈ R

of (2.2) is said to be an

isolated equ i lib rium point if there exists ε > 0 such that B

) contains no

equilibrium points other than x

The following proposition provides suﬃcient conditions for the exis-

tence of isolated equilibria of the dyn amical s ystem (2.2). For this result we

assume that f : [t

, t

] × D → R

is continuously diﬀerentiable on D (see

Section 2.3) and we use the notions of vector norms as deﬁned in Section

2.2.

Proposition 2.1. Consider the nonlinear dynamical system (2.2).

Assume f (t, ·) : D → R

is continuously diﬀerentiable on D for every

t ∈ [t

, t

], f(t, x

) = 0, t ∈ [t

, t

], where x

∈ R

, and f(·, x) : [t

, t

] → R

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DYNAMICAL SYSTEMS AND DIFFERENTIAL EQUATIONS 13

is piecewise continuous on [t

, t

] f or every x ∈ D. If for every t ∈ [t

, t

A(t)

△

∂f

∂x

(t, x)



x=x

(2.9)

is nonsingular, then x

is an isolated equilibrium point.

Proof. Let t ∈ [t

, t

]. Since det A(t) 6= 0, there exists ε > 0 such

that kA(t)xk ≥ εkxk, x ∈ D (see Problem 2.1). Next, expandin g f (t, x) via

a Taylor series expansion about the equilibrium point x = x

and using the

fact that f(t, x

) = 0 yields

f(t, x) = f(t, x

) +

∂f

∂x

(t, x

)(x − x

) + O(t, x)

= A(t)(x −x

) + O(t, x). (2.10)

Now, since

∂f

∂x

(t, x) is continuous on D it follows that lim

kx−x

k→0

kO(t,x)k

kx−x

0. Hence, for every ˆε > 0 there exists δ > 0 such that

kO(t, x)k ≤ ˆεkx −x

k, kx − x

k < δ. (2.11)

Setting ˆε = ε/2, using (2.11), and using the fact that kA(t)xk ≥ εkxk,

x ∈ D, it follows that

kf(t, x)k = kA(t)(x − x

) + O(t, x)k

≥ kA(t)(x −x

)k − kO(t, x)k

≥

kx − x

k, kx − x

k < δ, (2.12)

which implies that kf (t, x)k > 0 for all x ∈ B

), x 6= x

Example 2.3. Consider the nonlinear d ynamical system

˙x

(t) = −x

(t) + 2x

(t)x

(t), x

(0) = x

, t ≥ 0, (2.13)

˙x

(t) = −2x

(t)x

(t), x

(0) = x

. (2.14)

Note that with x = [x

, x

]

, f (x) = 0 implies that x

= [x

]

[0 x

]

, where x

∈ R, which shows that every point on the x

axis is an

equilibrium point of (2.13) and (2.14). △

In the remainder of this chapter we address the problems of existence

and uniqueness of solutions to the nonlinear dynamical system (2.2) along

with continuous dependence of solutions on the system initial conditions

and system parameters. First, however, we review some basic deﬁnitions

and results on vector and matrix norms, topology and analysis, and vector

and Banach spaces.

O(t, x) denotes the Landau order, which means that kO(t, x)k/kx − x

k is bounded as kx −

k → 0.

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14 CHAPTER 2

2.2 Vector and Matrix Norms

In this section the concepts of vector and matrix norms are introduced.

Given a number α ∈ R, the absolute value |α| of α denotes the magnitude

of α while discarding the sign of α. For x ∈ R

and A ∈ R

n×m

we can

extend the deﬁnition of absolute value by replacing each component of x

and each entry of A by its absolute value, that is, by deﬁning |x| ∈ R

and

|A| ∈ R

n×m

|x|

(i)

△

= |x

(i)

|, |A|

(i,j)

△

= |A

(i,j)

for all i = 1, . . . , n and j = 1, . . . , m. For many applications it is more useful

to have a scalar measure of the magnitude of x or A. Vector and matrix

norms pr ovide s uch measures.

Deﬁnition 2.6. A vector norm k·k on R

is a function k·k : R

→ R

that satisﬁes the following axioms:

i) kxk ≥ 0, x ∈ R

ii) kxk = 0 if and only if x = 0.

iii) kαxk = |α|kxk, α ∈ R, x ∈ R

iv) kx + yk ≤ kxk + kyk, x, y ∈ R

Condition iv) is known as the triangle inequality. It is important to

note that if there exists α ≥ 0 such that x = αy or y = αx, then iv) holds

as an equ ality. There are many diﬀerent norms. The most useful class of

vector norms are the p-norms (also called H¨older norms).

Proposition 2.2. For 1 ≤ p ≤ ∞, k ·k

deﬁned by

kxk

△

i=1

(i)

1/p

, 1 ≤ p < ∞,

kxk

∞

△

= max

i=1,...,n

(i)

is a vector norm on R

Proof. The only diﬃculty is in proving the triangle inequality, which

in this case is known as Minkowski’s inequality. If kx + yk

= 0, then

obviously kx + yk

≤ kxk

+ kyk

since kxk

and kyk

are nonnegative for

1 ≤ p ≤ ∞. Suppose kx + yk

> 0. The cases p = 1 and p = ∞ are

immediate. Hence, assume 1 < p < ∞ and note that

i=1

(i)

+ y

(i)

≤

i=1

(i)

+ y

(i)

p−1

(i)

| +

i=1

(i)

+ y

(i)

p−1

(i)

|. (2.15)