Switching and stability properties of conewise linear systems

• Published in: ESAIM. Control, optimisation and calculus of variations Vol. 16; no. 3; pp. 764 - 793
• Main Authors: Shen, Jinglai, Han, Lanshan, Pang, Jong-Shi
• Format: Journal Article
• Language: English
• Published: Les Ulis EDP Sciences 01.07.2010
• Subjects: 93B12; 93D05; 93D20; Asymptotic properties; asymptotic stability; Cones; Dynamical systems; Hybrid systems; Linear systems; Lyapunov and other classical stabilities; Stability; Switches; Switching; Variable structure systems
• ISSN: 1292-8119; 1262-3377
• DOI: 10.1051/cocv/2009021

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.