Dynamics and Stability of Continuous-Time Switched Linear Systems

Linear switched systems are a fascinating field of research, with many theoretical questions arising from applications which require sophisticated mathematical tools for their resolution. This monograph presents a unified theoretical approach for the analysis of stability of continuous-time linear switched systems, organizing and optimizing results scattered throughout literature. Emphasis is placed on the development of a rigorous and complete mathematical theory.

In addition to fundamental tools such as common Lyapunov functions, converse Lyapunov theorems, and maximal Lyapunov exponents, the concept of Barabanov norm is also discussed. While this is now well understood from a theoretical point of view, it has not received much attention in more application-focused settings, likely because this fundamental object was developed in the context of arbitrary switches but has no immediate equivalent for classes of switching signals subject to various constraints (dwell time, persistent excitation, etc.). One of the aims of this text is to bridge this gap as far as possible by explaining how the main features of Barabanov norms can be generalized for classes of constrained switchings. Throughout the text, the authors maintain a general point of view, rather than treating classes of switching signals separately, by developing an axiomatic approach and identifying structural properties of these classes that allow crucial aspects of the Barabanov norm to be extended.

This monograph will be a valuable resource for mathematicians and control engineers interested in continuous-time switched linear systems, as well as a definitive reference for more experienced researchers.

Presents a unified theoretical approach for the analysis of stability of continuous-time linear switched systems Introduces a new method to study classes of arbitrary and non-concatenable switching signals Includes a systematic treatment of Barabanov norms to bridge the gap between theory and applications

Autorentext

Yacine Chitour was born in Algiers, Algeria, in 1968. He received his PhD from Rutgers, New Jersey, in 1996. He was with the Mathematics Department of Université Paris-Sud, Orsay, France, from 1997 to 2004. Since then, he is Professor of control theory at Université Paris-Saclay, France, and a member of the Laboratoire des Signaux et Systèmes, Université Paris-Saclay, CentraleSupélec and CNRS. He is interested in geometric and optimal control, delay and switched systems, sliding mode and control of partial differential equations.

Paolo Mason was born in Dolo, Italy, in 1978. He received the Laurea degree in mathematics from the University of Padova, Italy, in 2002 and his PhD from SISSA, Trieste, Italy, in 2006. Since 2009, he works as a "chargé de recherche'' (researcher) for CNRS at the Laboratoire des Signaux et Systèmes, Université Paris-Saclay, CentraleSupélec and CNRS. His research interests include geometric control theory, quantum control and hybrid systems.

Mario Sigalotti was born in Udine, Italy, in 1975. He received the Laurea degree in mathematics from the University of Trieste, Trieste, Italy, in 1999, and his PhD from SISSA, Trieste, Italy, in 2003. In 2005, he became an Associate Scientist (chargé de recherche) at Inria, Nancy, France. After joining the Inria center of Saclay from 2011 to 2017, he moved to the Inria center of Paris, where he is Research Director (directeur de recherche) and heads the team CAGE. He is member of the Laboratoire Jacques-Louis Lions, Sorbonne Université, Paris. His research interests include hybrid systems, geometric control theory, and control of quantum systems.

Inhalt

Preface.- Introduction.- Switched linear systems: Definitions and basic facts.- Stability analysis for two-dimensional switched systems.- Direct and converse Lyapunov functions.- Barabanov norms.- Switched systems in dimension three.- Restrained switching signals.- Switched controlled dynamics.- Appendix: Complementary definitions and results.