Switching and stability properties of conewise linear systems

Jinglai Shen¹, Lanshan Han² and Jong-Shi Pang²

¹ 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: 19 September 2008
Revised: 15 February 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.