Lyapunov conditions for input-to-state stability of impulsive systems

This paper introduces appropriate concepts of input-to-state stability (ISS) and integral-ISS for impulsive systems, i.e., dynamical systems that evolve according to ordinary differential equations most of the time, but occasionally exhibit discontinuities (or impulses). We provide a set of Lyapunov-based sufficient conditions for establishing these ISS properties. When the continuous dynamics are ISS but the discrete dynamics that govern the impulses are not, the impulses should not occur too frequently, which is formalized in terms of an average dwell-time (ADT) condition. Conversely, when the impulse dynamics are ISS but the continuous dynamics are not, there must not be overly long intervals between impulses, which is formalized in terms of a novel reverse ADT condition. We also investigate the cases where (i) both the continuous and discrete dynamics are ISS and (ii) one of these is ISS and the other only marginally stable for the zero input, while sharing a common Lyapunov funct...

• Type: Journal
• Year: 2008
• Where: AUTOMATICA
• Authors: João Pedro Hespanha, Daniel Liberzon, Andrew R. Teel
• Keywords: AUTOMATICA 2008, Continuous Dynamics, Discrete Dynamics, Impulses