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 [...]... 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... Oliveira, V.A., Teixeira, M.C.M & Cossi, L.V (2003) Stabilizing a class of time delay systems using the Hermite-Biehler theorem Linear Algebra and its Applications, Vol 369, April 2003, pp 203–216, ISSN 0024-3795 Oliveira, V.A.; Cossi, L.V., Silva, A.M.F & Teixeira, M.C.M (2009) Synthesis of PID Controllers for a Class of Time Delay Systems Automatica, Vol 45, Issue 7, July 2009, pp 1778-1782, ISSN 0024-3795... Report No 321 14 Time- Delay Systems Pontryagin, L.S.(1955) On the zeros of some elementary transcendental functions, In: Izv Akad Nauk SSSR Ser Mat 6 (1942), English Transl in American Mathematical Society, Vol 2, pp 95-110 Pontryagin, L.S.(1969) Équations Différentielles Ordinaires, Éditions MIR, Moscou Silva, G J., Datta, A & Bhattacharyya, S.P (2000) Stabilization of Time Delay Systems Proceedings... Descriptor Systems over Infinite and Finite Time Interval Dragutin Lj Debeljković1 and Tamara Nestorović2 1University of Belgrade, Faculty of Mechanical Engineering, 2Ruhr-University of Bochum, 1Serbia 2Germany 1 Introduction 1.1 Classes of systems to be considered It should be noticed that in some systems we must consider their character of dynamic and static state at the same time Singular systems (also... dynamic systems with time delays, that is, for setting a characterization of all stabilizing P, PI or PID controllers for a given plant It should be pointed out that the definition of the formal concept of signature—introduced in the reference Oliveira et al (2003)—allows to extend results of the polynomial case to quasipolynomial case via property of interlacing in high frequencies of the class of time delay. .. results, for so called normal systems, are very well known 16 Time- Delay Systems But in practice one is not only interested in system stability (e.g in sense of Lyapunov), but also in bounds of system trajectories A system could be stable but completely useless because it possesses undesirable transient performances Thus, it may be useful to consider the stability of such systems with respect to certain... differentiable functions of time t which satisfy the considered equations of the model Since for continuous singular systems not all initial values x 0 of x ( t ) will generate smooth solution, those that generate such solutions (continuous to the right) we call consistent Moreover, positive solvability condition guarantees uniqueness and closed form of solutions to (1) 18 Time- Delay Systems Lemma 2.1.1.2... Descriptor Systems over Infinite and Finite Time Interval b 19 there exists a positive-definite, self-adjoin operator P , such that: ( ) xT ( t ) AT PE + ET PA x ( t ) = − xT ( t ) I x ( t ) , (10) for all x ∈W k∗ , where W k∗ denotes the subspace of consistent initial conditions 2.1.2 Continuous singular systems – stability over finite time interval Dynamical behaviour of the system (1) is defined over time. .. regular singular systems are treated with matrices given in (24) (26) 22 Time- Delay Systems ˆ ˆ Theorem 2.1.2.4 For given constant matrix ED A any solution of (1) satisfies the following inequality (Kablar & Debeljkovic 1998) x ( t0 ) e ( ) ˆ ˆ − μ − ED A ( t − t0 ) ≤ x ( t ) ≤ x ( t0 ) e ( ) ˆ ˆ μ ED A ( t − t0 ) , ∀t ∈ ℑ (27) Theorem 2.1.2.5 In order for the system (1) to be finite time stable w.r.t... differential-algebraic systems or semi-state) are those, the dynamics of which are governed by a mixture of algebraic and differential (difference) equations Recently many scholars have paid much attention to singular systems and have obtained many good consequences The complex nature of singular systems causes many difficulties in the analytical and numerical treatment of such systems, particularly . 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,. 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. 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