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ESAIM: COCV
DOI: 10.1051/cocv/2009021
Switching and stability properties of conewise linear systems
Jinglai Shen1, Lanshan Han2 and Jong-Shi Pang21 Department of Mathematics and Statistics, University of Maryland Baltimore County, Baltimore, Maryland 21250, USA. shenj@umbc.edu
2 Department of Industrial and Enterprise Systems Engineering, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA. hanlsh@illinois.edu; jspang@illinois.edu
Received September 19, 2008. Revised February 15, 2009. Published online July 2nd, 2009.
Abstract
Being a unique phenomenon in hybrid systems, mode switch
is of fundamental importance in dynamic and control analysis. In
this paper, we focus on global long-time switching and stability
properties of conewise linear systems (CLSs), which are a class of
linear hybrid systems subject to state-triggered switchings
recently introduced for modeling piecewise linear systems. By
exploiting the conic subdivision structure, the “simple switching
behavior” of the CLSs is proved. The infinite-time mode switching
behavior of the CLSs is shown to be critically dependent on two
attracting cones associated with each mode; fundamental properties
of such cones are investigated. Verifiable necessary and
sufficient conditions are derived for the CLSs with infinite mode
switches. Switch-free CLSs are also characterized by exploring
the polyhedral structure and the global dynamical properties. The
equivalence of asymptotic and exponential stability of the CLSs is
established via the uniform asymptotic stability of the CLSs that
in turn is proved by the continuous solution dependence on initial
conditions. Finally, necessary and sufficient stability conditions
are obtained for switch-free CLSs.
Mathematics Subject Classification. 93B12, 93D05, 93D20
Key words: Variable structure systems, Lyapunov and other classical stabilities, asymptotic stability
© EDP Sciences, SMAI 2009
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