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Switching and stability propertiesof conewise linear systems

Published online by Cambridge University Press:  02 July 2009

Jinglai Shen ,
Lanshan Han  and
Jong-Shi Pang
Jinglai Shen
Affiliation:
Department of Mathematics and Statistics, University of Maryland Baltimore County, Baltimore, Maryland 21250, USA. shenj@umbc.edu
Lanshan Han
Affiliation:
Department of Industrial and Enterprise Systems Engineering, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA. hanlsh@illinois.edu; jspang@illinois.edu
Jong-Shi Pang
Affiliation:
Department of Industrial and Enterprise Systems Engineering, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA. hanlsh@illinois.edu; jspang@illinois.edu
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Abstract

Being a unique phenomenon in hybrid systems, mode switchis of fundamental importance in dynamic and control analysis. Inthis paper, we focus on global long-time switching and stabilityproperties of conewise linear systems (CLSs), which are a class oflinear hybrid systems subject to state-triggered switchingsrecently introduced for modeling piecewise linear systems. Byexploiting the conic subdivision structure, the “simple switchingbehavior” of the CLSs is proved. The infinite-time mode switchingbehavior of the CLSs is shown to be critically dependent on twoattracting cones associated with each mode; fundamental propertiesof such cones are investigated. Verifiable necessary andsufficient conditions are derived for the CLSs with infinite modeswitches. Switch-free CLSs are also characterized by exploringthe polyhedral structure and the global dynamical properties. Theequivalence of asymptotic and exponential stability of the CLSs isestablished via the uniform asymptotic stability of the CLSs thatin turn is proved by the continuous solution dependence on initialconditions. Finally, necessary and sufficient stability conditionsare obtained for switch-free CLSs.

Keywords

Variable structure systemsLyapunov and other classical stabilitiesasymptotic stability
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 16 , Issue 3 , July 2010 , pp. 764 - 793
Copyright
© EDP Sciences, SMAI, 2009

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