Result: Stability Criteria for Switched and Hybrid Systems

Title:
Stability Criteria for Switched and Hybrid Systems
Authors:
SHORTEN, Robert, WIRTH, Fabian, MASON, Oliver, WULFF, Kai, KING, Christopher
Source:
SIAM review (Print). 49(4):545-592
Publisher Information:
Philadelphia, PA: Society for Industrial and Applied Mathematics, 2007.
Publication Year:
2007
Physical Description:
print, 191 ref
Original Material:
INIST-CNRS
Subject Terms:
Control theory, operational research, Automatique, recherche opérationnelle, Mathematics, Mathématiques, Sciences exactes et technologie, Exact sciences and technology, Sciences et techniques communes, Sciences and techniques of general use, Mathematiques, Mathematics, Algèbre, Algebra, Géométrie algébrique, Algebraic geometry, Analyse mathématique, Mathematical analysis, Equations différentielles, Ordinary differential equations, Analyse numérique. Calcul scientifique, Numerical analysis. Scientific computation, Analyse numérique, Numerical analysis, Algèbre linéaire numérique, Numerical linear algebra, Méthodes de calcul scientifique (y compris calcul symbolique, calcul algébrique), Methods of scientific computing (including symbolic computation, algebraic computation), Article synthèse, Review, Artículo síntesis, Calcul scientifique, Scientific computation, Computación científica, Commutation, Switching, Conmutación, Concept, Concepto, Contrainte, Constraint, Coacción, Fonction Lyapunov, Lyapunov function, Función Lyapunov, Fonction quadratique, Quadratic function, Función cuadrática, Inclusion différentielle, Differential inclusion, Inclusión diferencial, Indécidabilité, Undecidability, Indecidibilidad, Mathématiques appliquées, Applied mathematics, Matemáticas aplicadas, Recherche, Research, Investigación, Résolution (math), Solving, Resolución (matemática), Stabilité numérique, Numerical stability, Estabilidad numérica, Superficie, Area, Système hybride, Hybrid system, Sistema híbrido, Système linéaire, Linear system, Sistema lineal, Taux croissance, Growth rate, Tasa crecimiento, Théorème existence, Existence theorem, Teorema existencia, 14C20, 34A60, 34Dxx, 37B25, 37L45, 65F05, 34D08, 34D10, 37C75, 93D09, 93D30, Lur'e problem, common quadratic Lyapunov functions, converse Lyapunov theorem, dwell-time, growth rates, hybrid systems, stability radii, stability, switched systems
Document Type:
Academic journal Article
File Description:
text
Language:
English
Author Affiliations:
NUI Maynooth, The Hamilton Institute, Kilcock Rd, Maynooth, Ireland
Institut für Mathematik, Universität Würzburg, Am Hubland, 97074 Würzburg, Germany
Fachgebiet Regelungssysteme, Technische Universität Berlin, EN 11, Einsteinufer 17, 10587 Berlin, Germany
Department of Mathematics, Northeastern University, 360 Huntington Ave, Boston, MA, United States
ISSN:
0036-1445
Access URL:
http://pascal-francis.inist.fr/vibad/index.php?action=search&terms=19896747
Rights:
Copyright 2008 INIST-CNRS
CC BY 4.0
Sauf mention contraire ci-dessus, le contenu de cette notice bibliographique peut être utilisé dans le cadre d’une licence CC BY 4.0 Inist-CNRS / Unless otherwise stated above, the content of this bibliographic record may be used under a CC BY 4.0 licence by Inist-CNRS / A menos que se haya señalado antes, el contenido de este registro bibliográfico puede ser utilizado al amparo de una licencia CC BY 4.0 Inist-CNRS
Notes:
Mathematics
Accession Number:
edscal.19896747
Database:
PASCAL Archive

Further Information

The study of the stability properties of switched and hybrid systems gives rise to a number of interesting and challenging mathematical problems. The objective of this paper is to outline some of these problems, to review progress made in solving them in a number of diverse communities, and to review some problems that remain open. An important contribution of our work is to bring together material from several areas of research and to present results in a unified manner. We begin our review by relating the stability problem for switched linear systems and a class of linear differential inclusions. Closely related to the concept of stability are the notions of exponential growth rates and converse Lyapunov theorems, both of which are discussed in detail. In particular, results on common quadratic Lyapunov functions and piecewise linear Lyapunov functions are presented, as they represent constructive methods for proving stability and also represent problems in which significant progress has been made. We also comment on the inherent difficulty in determining stability of switched systems in general, which is exemplified by NP-hardness and undecidability results. We then proceed by considering the stability of switched systems in which there are constraints on the switching rules, through both dwell-time requirements and state-dependent switching laws. Also in this case the theory of Lyapunov functions and the existence of converse theorems are reviewed. We briefly comment on the classical Lur'e problem and on the theory of stability radii, both of which contain many of the features of switched systems and are rich sources of practical results on the topic. Finally we present a list of questions and open problems which provide motivation for continued research in this area.