TIMEDELAY SYSTEMS Edited by Dragu n Debeljković Time-Delay Systems Edited by Dragutin Debeljković Published by InTech Janeza Trdine 9, 51000 Rijeka, Croatia Copyright © 2011 InTech All chapters are Open Access articles distributed under the Creative Commons Non Commercial Share Alike Attribution 3.0 license, which permits to copy, distribute, transmit, and adapt the work in any medium, so long as the original work is properly cited. After this work has been published by InTech, authors have the right to republish it, in whole or part, in any publication of which they are the author, and to make other personal use of the work. Any republication, referencing or personal use of the work must explicitly identify the original source. Statements and opinions expressed in the chapters are these of the individual contributors and not necessarily those of the editors or publisher. No responsibility is accepted for the accuracy of information contained in the published articles. The publisher assumes no responsibility for any damage or injury to persons or property arising out of the use of any materials, instructions, methods or ideas contained in the book. Publishing Process Manager Katarina Lovrecic Technical Editor Teodora Smiljanic Cover Designer Martina Sirotic Image Copyright Mihai Simonia, 2010. Used under license from Shutterstock.com First published February, 2011 Printed in India A free online edition of this book is available at www.intechopen.com Additional hard copies can be obtained from orders@intechweb.org Time-Delay Systems, Edited by Dragutin Debeljković p. cm. ISBN 978-953-307-559-4 free online editions of InTech Books and Journals can be found at www.intechopen.com 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 Contents VI 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 Pref ac e The problem of investigation of time delay systems has been explored over many years. Time delay is very o 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 admi 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 a 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 a 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 systems 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 X Preface existing stability criteria, two main ways of approach are adopted. Namely, one direc- tion is to contrive the stability condition which does not include the information on the delay, and the other is the method which takes it into account. The former case is o en called the delay-independent criteria and generally provides simple algebraic condi- tions. In that sense, the question of their stability deserves great a 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 [...]... analytic function f : Ω −→ C We say that A1 A2 Am + + + + g1 (z), β is a pole of order m ≥ 1 of f if, and only if, f (z) = z−β (z − β)m ( z − β )2 where g1 is analytic in B ( β, R) and A1 , A2 , , Am ∈ C with Am = 0 Definition 5 (Uniform convergence of infinite products) The infinite product +∞ ∏ (1 + f n (z)) = (1 + f 1 (z)) (1 + f 2 (z)) n =1 (1 + f n (z)) (1) where { f n }n∈IN is a sequence of... that f (0) = 0, and let 1 , α2 , be the zeros of f , listed according to their multiplicities Then there exist an entire function g and a sequence { pn }n∈IN of nonnegative integers, such that f (z) = e g( z) ∞ ∏ Ep n =1 n ( ∞ z z ) = e g(z) ∏ 1 − αn αn n =1 e z αn 1 + 1 ( αzn )2 + + n 1 ( αzn ) n 1 2 (8) 4 Time- Delay Systems Notice that, by convention, with respect to n = 1 the first factor of the... take E0 (z) E p (z) = 1 − z, and (4) = z2 zp (1 − z) exp(z + + + ), for all p = 1, 2, 3, 2 p (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 {αn }n∈IN be a sequence of complex numbers such that αn = 0 and If { pn }n∈IN is a sequence of nonnegative integers such that ∞ r ∑ ( r n )1+ p n =1 n lim n−→+ ∞ < ∞, where... uniformly convergent if the sequence of partial product ρn defined by ρn (z) = n ∏ (1 + f m (z)) = (1 + f 1 (z)) (1 + f 2 (z)) m =1 (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 is uniformly convergent ∑ | f n (z)| n =1 Definition 6 (Entire function) A function... real (or complex) numbers In particular, if the λξ (ξ = 0, , m) are commensurable real numbers and 0 = λ0 < 1 , < λm , then the quasipolynomial (12 ) can be written in the form (11 ) studied by Pontryagin Notice that, some trigonometric functions, e.g., sin and cos are quasipolynomials since √ 1 1 1 1 sin(mz) = e jmz − e− jmz and cos(nz) = e jnz + e− jnz , where j = 1, and m, n ∈ IN 2j 2j 2 2 Remark... interlacing in high frequencies of the class of time delay systems considered 5 The dynamic behavior of many industrial plants may be mathematically modeled by a linear time invariant system with time delay The problem of stability of linear time invariant systems with time delay make necessary a method for localization of the roots of analytic functions These systems are described by the linear differential... be 1 1− 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) zm Remark 3 The decomposition (8) is not unique Remark 4 In the Theorem (3), if the sequence { pn }n∈IN of nonnegative integers is constant, i.e., pn = ρ for all n ∈ IN, then the following infinite product: e g(z) ∞ z ∏ Eρ ( α n ) (9) n =1 ∞ R 1 ∑ ( |αn | )ρ +1 converges for ρ + 1 n =1. .. application to a problem of stabilization in area of Control Theory 8 Time- Delay Systems We can denominate the equation (18 ) as an equation with delayed argument, if the coefficient an0 = 0 and the remaining coefficients anν = 0(ν = 1, , m), that is, an0 u ( n) (t) + n 1 m ∑ ∑ aμν u(μ) (t − τν ) = h(t); analogously, the equation (18 ) is denominated μ =0 ν =0 an equation with advanced argument, if the... factorization of the following manner: f (z) = zm e g( z) ∏ E p ( ), where g(z) is a polynomial αn n =1 function of degree q, and max{ p, q } ≤ ϑ The first example of infinite product representation was given by Euler in 17 48, viz., ∞ z2 sin(πz) = πz ∏ (1 − 2 ), where m = 1, p = 1, q = 0 [ g(z) ≡ 0], and ϑ = 1 n n =1 3 Zeros of quasipolynomials due to Pontryagin’s theorem We know that, under the analytic standpoint... < | z − β| < R} but not in B ( β, R) = {z ∈ C: | z − β| < R} 1 2 See Levin (19 64) 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 See Ahlfors (19 53) and Titchmarsh (19 39) for a detailed exposition 2 Time- Delay Systems Definition 3 (Pole) Let Ω be a region A value β ∈ Ω is . Input Delayed Robotic Manipulators 211 Haiping Du and Weihua Li Chapter 10 Chapter 11 Pref ac e The problem of investigation of time delay systems has been explored over many years. Time delay. for a Class of Multiple Time- Delay Systems 16 1 Tung-Sheng Chiang and Peter Liu Contents Contents VI Recent Progress in Synchronization of Multiple Time Delay Systems 18 1 Thang Manh Hoang T-S. TIME DELAY SYSTEMS Edited by Dragu n Debeljković Time- Delay Systems Edited by Dragutin Debeljković Published by InTech Janeza Trdine 9, 510 00 Rijeka, Croatia Copyright © 2 011 InTech All