Time Delay Systems Part 1 pot

Chapter 1Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Chapter 8 Chapter 9 Preface IX Introduction to Stability of Quasipolynomials 1 Lúcia Cossi Stability of Linear Contin

TIME DELAY SYSTEMS

Edited by Draguti n Debeljković

Time-Delay Systems

Edited by Dragutin Debeljković

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Chapter 1

Chapter 2

Chapter 3

Chapter 4

Chapter 5

Chapter 6

Chapter 7

Chapter 8

Chapter 9

Preface IX Introduction to Stability of Quasipolynomials 1

Lúcia Cossi

Stability of Linear Continuous Singular and Discrete Descriptor Systems over Infinite and Finite Time Interval 15

Dragutin Lj Debeljković and Tamara Nestorović

Stability of Linear Continuous Singular and Discrete Descriptor Time Delayed Systems 31

Dragutin Lj Debeljković and Tamara Nestorović

Exponential Stability of Uncertain Switched System with Time-Varying Delay 75

Eakkapong Duangdai and Piyapong Niamsup

On Stable Periodic Solutions of One Time Delay System Containing Some Nonideal Relay Nonlinearities 97

Alexander Stepanov

Design of Controllers for Time Delay Systems:

Integrating and Unstable Systems 113

Petr Dostál, František Gazdoš, and Vladimír Bobál

Decentralized Adaptive Stabilization for Large-Scale Systems with Unknown Time-Delay 127

Jing Zhou and Gerhard Nygaard

Resilient Adaptive Control

of Uncertain Time-Delay Systems 143

Hazem N Nounou and Mohamed N Nounou

Sliding Mode Control for a Class

of Multiple Time-Delay Systems 161

Tung-Sheng Chiang and Peter Liu Contents

Recent Progress in Synchronization

of Multiple Time Delay Systems 181

Thang Manh Hoang

T-S Fuzzy H∞ Tracking Control

of Input Delayed Robotic Manipulators 211

Haiping Du and Weihua Li Chapter 10

Chapter 11

The problem of investigation of time delay systems has been explored over many years Time delay is very oft en encountered in various technical systems, such as electric, pneumatic and hydraulic networks, chemical processes, long transmission lines, etc The existence of pure time lag, regardless if it is present in the control or/and the state, may cause undesirable system transient response, or even instability Consequently, the problem of stability analysis for this class of systems has been one of the main in-terests for many researchers In general, the introduction of time delay factors makes the analysis much more complicated

So, the title of the book Time-Delay Systems encompasses broad fi eld of theory and

application of many diff erent control applications applied to diff erent classes of afore-mentioned systems

It must be admitt ed that a strong stress, in this monograph, is put on the historical sig-nifi cance of systems stability and in that sense, problems of asymptotic, exponential, non-Lyapunov and technical stability deserved a great att ention Moreover, an evident

contribution was given with introductory chapter dealing with basic problem of

Quasi-polyinomial stability

Time delay systems can achieve diff erent att ributes Namely, when we speak about sin-gular or descriptor systems, one must have in mind that with some systems we must consider their character of dynamic and static state at the same time Singular systems (also referred to as degenerate, descriptor, generalized, diff erential-algebraic systems

or semi–state) are systems with dynamics, governed by the mixture of algebraic and

- diff erential equations The complex nature of singular systems causes many diffi cul-ties in the analytical and numerical treatment of such systems, particularly when there is a need for their control

It must be emphasized that there are lot of systems that show the phenomena of time

delay and singularity simultaneously, and we call such systems singular diff erential sys-tems with time delay These syssys-tems have many special characteristics If we want to

describe them more exactly, to design them more accurately and to control them more

eff ectively, we must tremendously endeavor to investigate them, but that is

obvious-ly very diffi cult work When we consider time delay systems in general, within the

existing stability criteria, two main ways of approach are adopted Namely, one direc-tion is to contrive the stability condidirec-tion which does not include the informadirec-tion on the delay, and the other is the method which takes it into account The former case is oft en called the delay-independent criteria and generally provides simple algebraic condi-tions In that sense, the question of their stability deserves great att ention

In the second and third chapter authors discuss such systems and some signifi cant

con-sequences, discussing their Lyapunov and non-Lyapunov stability characteristics Exponential stability of uncertain switched systems with time-varying delay and

actu-al problems of stabilization and determining of stability characteristics of steady-state regimes are among the central issues in the control theory Diffi culties can be especially met when dealing with the systems containing nonlinearities which are non-analytic

function of phase with problems that have been treated in two following chapters Some of synthesis problems have been discussed in the following chapters covering

problems such as: static output-feedback stabilization of interval time delay systems, controllers design, decentralized adaptive stabilization for large-scale systems with unknown time-delay and resilient adaptive control of uncertain time-delay systems Finally, actual problems with some practical implementation and dealing with slid-ing mode control, synchronization of multiple time delay systems and T-S fuzzy H∞ tracking control of input delayed robotic manipulators, were presented in last three

chapters, including inevitable application of linear matrix inequalities.

Dr Dragutin Lj Debeljković

University of Belgrade Faculty of Mechanical Engineering Department of Conrol Engineering

Serbia

Introduction to Stability of Quasipolynomials

Lúcia Cossi

Departamento de Matemática, Universidade Federal da Paraíba

João Pessoa, PB, Brazil

1 Introduction

In this Chapter we shall consider a generalization of Hermite-Biehler Theorem1 given by Pontryagin in the paper Pontryagin (1955) It should be understood that Pontryagin’s

generalization is a very relevant formal tool for the mathematical analysis of stability of

quasipolynomials Thus, from this point of view, the determination of the zeros of a quasipolynomial by means of Pontryagin’s Theorem can be considered to be a mathematical method for analysis of stabilization of a class of linear time invariant systems with time delay Section 2 contains an overview of the representation of entire functions as an infinite product

by way of Weierstrass’ Theorem—as well as Hadamard’s Theorem Section 3 is devoted to

an exposition to the Theory of Quasipolynomials via Pontryagin’s Theorem in addition to a generalization of Hermite-Biehler Theorem Section 4 deals with applications of Pontryagin’s Theorem to analysis of stabilization for a class of linear time invariant systems with time delays

2 Representation of the entire functions by means of infinite products

In this Section we will present the mathematical background with respect to the Theory

of Complex Analysis and to provide the necessary tools for studying the Hermite-Biehler Theorem and Pontryagin’s Theorems At the first let us introduce the basic definitions and general results used in the representation of the entire functions as infinite products2

2.1 Preliminaries

Definition 1. (Zeros of analytic functions) Let f :Ω−→ C be an analytic function in a region Ω—i.e., a nonempty open connected subset of the complex plane A value α ∈ Ω is called a zero of f with multiplicity (or order) m ≥ 1 if, and only if, there is an analytic function g :Ω−→ C such that

f(z) = (z − α)m g(z), where g(α ) = 0 A zero of order one(m=1)is called a simple zero.

Definition 2. (Isolated singularity) Let f :Ω−→ C be an analytic function in a region Ω A value

β ∈ Ω is called a isolated singularity of f if, and only if, there exists R > 0 such that f is analytic in

{ z ∈C: 0< | z − β | < R } but not in B(β, R ) = { z ∈C:| z − β | < R }

1 See Levin (1964) for an analytical treatment about the Hermite-Biehler Theorem and a generalization of this theorem to arbitrary entire functions in an alternative way of the Pontryagin’s method.

2 See Ahlfors (1953) and Titchmarsh (1939) for a detailed exposition.

1

Definition 3. (Pole) Let Ω be a region A value β ∈ Ω is called a pole of analytic function

f :Ω−→ C if, and only if, β is a isolated singularity of f and lim

z−→β | f(z )| = ∞.

Definition 4. (Pole of order m) Let β ∈ Ω be a pole of analytic function f : Ω −→ C We say that

β is a pole of order m ≥ 1 of f if, and only if, f(z) = A1

z − β+ A2

(z − β)2+ + A m

(z − β)m +g1(z), where g1is analytic in B(β, R)and A1, A2, , A m ∈ C with A m = 0.

Definition 5. (Uniform convergence of infinite products) The infinite product

+∞

∏

n=1(1+f n(z)) = (1+f1(z))(1+f2(z)) (1+f n(z)) (1)

where { f n } n∈IN is a sequence of functions of one variable, real or complex, is said to be uniformly convergent if the sequence of partial product ρ n defined by

ρ n(z) = ∏n

m=1(1+f m(z)) = (1+f1(z))(1+f2(z)) (1+f n(z)) (2)

converges uniformly in a certain region of values of z to a limit which is never zero.

Theorem 1. The infinite product (1) is uniformly convergent in any region where the series

+∞

∑

n=1| f n(z )|

is uniformly convergent.

Definition 6. (Entire function) A function which is analytic in whole complex plane is said to be entire function.

2.2 Factorization of the entire functions

In this subsection, it will be discussed an important problem in theory of entire functions,

namely, the problem of the decomposition of an entire function—under the form of an infinite

product of its zeros—in pursuit of the mathematical basis in order to explain the distribution

of the zeros of quasipolynomials

2.2.1 The problem of factorization of an entire function

Let P(z) =a n z n+ +a1z+a0 be a polynomial of degree n, (a n =0) It follows of the

Fundamental Theorem of Algebra that P(z) can be decomposed as a finite product of the following form: P(z) =a n(z − α1) (z − α n), where theα1,α2, , α nare—not necessarily

distinct—zeros of P(z) If exactly k jof theα jcoincide, then theα j is called a zero of P(z)of

order k j [see Definition (1)] Furthermore, the factorization is uniquely determined except for

the order of the factors Remark that we can also find an equivalent form of a polynomial

function with a finite product of its zeros, more precisely, P(z) =Cz m

N

∏

j=1(1− α z

j), where

C=a n

N

∏

j=1(− α j), m is the multiplicity of the zero at the origin, α j =0(j=1, , N) and

m+N=n.

We can generalize the problem of factorization of the polynomial function for any entire

function expressed likewise as an infinite product of its zeros.

Let’s supposed that

f(z) =z m e g (z)

∞

∏

n=1(1− α z

where g(z)is an entire function Hence, the problem can be established in following way: the representation (3) should be valid if the infinite product converges uniformly on every compact set [see

Definition (5)]

2.2.2 Weierstrass factorization theorem

The problem characterized above was completely resolved by Weierstrass in 1876 As matter

of fact, we have the following definitions and theorems

Definition 7. (Elementary factors) We can to take

E p(z) = (1− z) exp(z+z2

2 + .+z p

p), for all p=1, 2, 3, (5)

These functions are called elementary factors.

Lemma 1. If | z | ≤ 1 , then |1− E p(z )| ≤ | z | p+1, for p=1, 2, 3,

Theorem 2. Let { α n } n∈IN be a sequence of complex numbers such that α n = 0 and lim

n−→+∞ | α n | = ∞.

If { p n } n∈IN is a sequence of nonnegative integers such that

∞

∑

n=1( r

r n)1+pn< ∞, where | α n | = r n, (6)

for every positive r, then the infinite product

f(z) = ∏∞

n=1E p n(α z

define an entire function f which has a zero at each point α n , n ∈IN, and has no other zeros in the complex plane.

Remark 1. The condition (6) is always satisfied if p n=n − 1 Indeed, for every r, it follows that

r n > 2r for all n > n0, since lim

n−→+∞ r n=∞ Therefore, r

r n <1

2for all n > n0, then (6) is valid with respect to 1+p n=n.

Theorem 3. (Weierstrass Factorization Theorem) Let f be an entire function Suppose that f(0) = 0, and let α1, α2, be the zeros of f , listed according to their multiplicities Then there exist an entire function g and a sequence { p n } n∈IN of nonnegative integers, such that

f(z) =e g (z)

∞

∏

n=1E p n(α z

n) =e g (z)

∞

∏

n=1



1− α z

n



e



z

αn+ 1 (z

αn) 2 + + 1

−1(z

αn)n−1



(8) 3

Introduction to Stability of Quasipolynomials

Notice that, by convention, with respect to n=1 the first factor of the infinite product should be

1− α1

1.

Remark 2. If f has a zero of multiplicity m at z=0, the Theorem (3) can be apply to the function

f(z)

z m

Remark 3. The decomposition (8) is not unique.

Remark 4. In the Theorem (3), if the sequence { p n } n∈IN of nonnegative integers is constant, i.e.,

p n=ρ for all n ∈IN, then the following infinite product:

e g (z)∏∞

n=1

E ρ(α z

converges and represents an entire function provided that the series ρ+11

∞

∑

n=1(| α R

n |)ρ+1converges for all R > 0 Suppose that ρ is the smallest integer for which the series ∑∞

n=1

1

| α n | ρ+1 converges In this

case, the expression (9) is denominated the canonical product associated with the sequence { α n } n∈IN

and ρ is the genus of the canonical product3.

With reference to the Remark (4) we can state:

Hadamard Factorization Theorem. If f is an entire function of finite order ϑ, then it admits

a factorization of the following manner: f(z) =z m e g (z)

∞

∏

n=1E p(α z

n), where g(z) is a polynomial function of degree q, and max { p, q } ≤ ϑ.

The first example of infinite product representation was given by Euler in 1748, viz., sin(πz) =πz∏∞

n=1(1− z2

n2), where m=1, p=1, q=0[g(z ) ≡0], andϑ=1

3 Zeros of quasipolynomials due to Pontryagin’s theorem

We know that, under the analytic standpoint and a geometric criterion, results concerning the existence and localization of zeros of entire functions like exponential polynomials have received a considerable attention in the area of research in the automation field In this section the Pontryagin theory is outlined

Consider the linear difference-differential equation of differential order n and difference order

m defined by

n

∑

μ=0

m

∑

ν=0a μν x

3 See Boas (1954) for analysis of the problem about the connection between the growth of an entire function and the distribution of its zeros.