Introduction to LPV Time-Delay Systems

Abstract

Time-delay often appears in many real-world engineering systems either in the state, the control input, or the measurements. Delays are strongly involved in challenging areas of communication and information technologies: in stabilization of networked controlled systems and in high-speed communication networks. Time-delay is, in many cases, a source of instability. However, for some systems, the presence of delay can have a stabilizing effect. The stability analysis and robust control of time- delay systems (TDSs) are, therefore, of theoretical and practical importance. As in systems without delay, an efficient method for stability analysis of TDSs is the Lyapunov method. For TDSs, there exist two main Lyapunov methods: the Krasovskii method of Lyapunov functionals (1956) and the Razumikhin method of Lyapunov functions (1956). The Krasovskii method is applicable to a wider range of problems and it leads usually to less conservative results, than the Razumikhin method. The Lyapunov stability criterion for linear systems without delay can be for- mulated in terms of a linear matrix inequality (LMI). The realization that LMI may be treated as a convex optimization problem and the development of the efficient interior point method led to formulation of many control problems and their solutions in the form of LMIs [17]. The LMI approach to analysis and design of TDSs provides constructive finite-dimensional conditions, in spite of significant model uncertainties. Modeling of continuous-time systems with digital control in the form of continuous-time systems with time-varying delay [166] and the extension of Krasovskii method to TDSs without any constraints on the delay derivative [72] and to discontinuous delays [76] have allowed the development of the time-delay approach to sampled-data and to network-based control. The beginning of the twenty-first century can be characterized as the “time-delay boom” leading to numerous important results. The books that have been published so far have been restricted to detailed presentation of certain specific time-delay topics. The purpose of this book is twofold, to familiarize the non-expert reade with TDSs and to provide a systematic treatment of modern ideas and techniques for experts. The book leads the reader from some basic classical results to recent developments on Lyapunov-based analysis and design with applications to the hot topics of sampled-data and network-based control. It should be of interest for researchers working in the field, for graduate students in engineering and applied mathematics, and for practicing engineers. It may also be used as a textbook for a graduate course on TDSs. The book is based on the course “Introduction to time- delay systems” for graduate students in Engineering and Applied Mathematics that I taught in Tel Aviv University in 2011–2012 and 2012–2013 academic years. The sufficient background to follow most of the material are the undergraduate courses in mathematics and an introduction to control. Chapters 1 and 2 are introductory and are aimed at illustrating the new features that are brought by the time-delay. Chapter 1 discusses models with time-delays and gives some mathematical background: solution concept, the step method, and the state of TDS. Chapter 2 presents solution to linear TDSs and treats characteristic equation of LTI systems. Section 2.3 discusses the effects of delay on stability and presents the classical direct frequency domain method for stability analysis. Section 2.4 considers controllability and observability of TDSs. The emphasis in this book is on the Lyapunov-based analysis and design (Chaps. 3–7). Chapters 3 and 4 present stability and performance analysis of continuous-time TDSs, starting from the simple stability conditions and showing the ideas and tools that essentially improve the results. The objective is to provide useful techniques that will allow the reader not only to apply the existing methods but also to develop new ones. Chapter 5 provides solutions to the classical linear quadratic regulator problem for LTI systems in terms of Riccati equations, and to robust control of systems with time-varying delays in terms of LMIs. The LMI-based design conditions are derived via the descriptor method [52], having almost the same form for the continuous and for the discrete-time systems. Therefore, the LMI- based analysis and design for the discrete-time systems in Chap. 6 are presented as a simple extension of the continuous-time results of Chaps. 3–5. Chapter 7 develops a time-delay approach to the hot topic of sampled-data and networked control systems. I would like to strongly encourage readers to send me suggestions, comments, and corrections by e-mail (emilia@eng.tau.ac.il). I wish to acknowledgemy great debt to many colleagues and students who helped me in writing this book. My special thanks to former students Kun Liu, Christophe Fiter, Oren Solomon, and Vladimir Suplin for their great help. Of the friends and colleagues with whom I have had worked on problems directly pertinent to this book, it is a pleasure to acknowledge: Uri Shaked, Jean-Pierre Richard, Michel Dambrine, Silviu Niculescu, Yury Orlov, Valery Glizer, Sabine Mondie, Alexander Seuret, Laurentiu Hetel, and Frederic Gouaisbaut. I am happy to acknowledge Leonid Mirkin for fruitful discussions on time-delay and sampled-data systems. Great thanks to Springer Editor Donna Chernyk who invited me to write a book in the correct time (when I was preparing a course on TDSs in Tel Aviv University).Heartfelt thanks to my family, Eugenii and Boris Shustin, for moral and emotional support, and to my brother, Leonid Fridman, for encouragement and inspiring experience. This book is dedicated to my parents, Izabella Goldreich and Moisei Fridman, and to my teachers, Vadim Vasil’evich Strygin and Vladimir Borisovich Kolmanovskii. I am grateful to Tel Aviv University for an environment that allowed me to write this book, and to the Israel Science Foundation for supportingmy research on TDSs. Tel Aviv, Israel June 2014