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

Abstract

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 function. In the former case we obtain a stronger notion of ISS, for which a necessary and sufficient Lyapunov characterization is available. The use of the tools developed herein is illustrated through examples from a Micro-Electro-Mechanical System (MEMS) oscillator and a problem of remote estimation over a communication network.

Impulsively-controlled systems and reverse dwell time: A linear programming approach

“…In this context, it is assumed that the continuous time system has the tendency to become unstable, while the applied impulses have a stabilizing effect upon the system. Under the aforestated assumption, previous results in [15] establish that the system will be ISS as long as the exerted impulses are separated by a time interval bounded by above. This last condition is known as reverse dwell time condition.…”

“…The present paper improves the results in [3] in at least three directions: i) a simplified reformulation of the problem based on previous results on reverse dwell time impulse control [15], ii) a completely new rearrangement of the receding horizon framework and iii) additional numerical illustrations of a queueing system [11,19].…”

“…Here a controller has to stabilize asymptotically (the control horizon is infinite) an "unstable" dynamics through "infinitely often" and not "too rare" controlled impulses [15].…”

“…We cast the problem within the framework of Input to State Stabilizability (ISS) of impulsively controlled systems with reverse dwell time [15]. Actually, we start from the observation that the problem of driving and keeping the system in a safe operating interval can be reviewed as an ISS problem over an infinite horizon.…”

The stability of impulsive stochastic Cohen–Grossberg neural networks with mixed delays and reaction–diffusion terms

“…Moreover, those perturbations often may make stable systems unstable or unstable systems stable. Therefore, impulsive effects should also be taken into account Fu and Li 2011;Wang and Xu 2009;Hespanha et al 2008;Wan and Zhou 2008;). Fu and Li (2011) investigated the asymptotic stability of impulsive stochastic CGNNs with mixed time delays by using Lyapunov-Krasovskii functional and LMI technology.…”