Stability of Time-delay Systems By

Stability of Time-Delay Systems by Keqin Gu, Vladimir L. Kharitonov, and Jie Chen, Birkhäuser, 2003, 353 pp., ISBN 0-8176-4212-9, $79.95. Reviewed by Vladimir Rasvan and Dan Popescu. The destabilizing effects of time delays are well known to control engineers. In fact, robustness against time delay has been an important and challenging problem from the earliest days of classical control. The last 10–15 years have witnessed a renewed interest in time delay control systems because of new problems, new models, and new techniques. The advent of linear matrix inequalities and other computationally feasible methods has stimulated research in mature areas such as linear system stability and stabilization. The Routh-Hurwitz problem of locating the roots of a characteristic polynomial in the stability zone, which is either the open left-half plane or the interior of the unit disk, can be tackled by frequency or matrix methods, the latter being related to quadratic Lyapunov functions. The book covers these topics in the context of linear time delay systems, where the characteristic equations are quasi-polynomials and the Lyapunov functions are LyapunovKrasovskii functionals. The authors of this book are well known to those who are active in the field of time delay systems. Chen has developed frequency-sweeping tests for root location of quasi-polynomials; Gu has introduced spatial-discretization techniques for LyapunovKrasovskii functionals; and Kharitonov has investigated quadratic LyapunovKrasovskii functionals for stability domain estimation. This book is a product of these specialists’ research in the field of linear time delay systems and contains the essence of their contributions. The book consists of two main parts: the Frequency Domain Approach, dealing with classical stability, frequency sweeping, and constant matrix tests; and the Time Domain Approach, dealing with Lyapunov-Krasovskii functionals and their discretized versions. The cases of single delay, commensurate delays, and incommensurate delays are discussed separately.