Uniform asymptotic stability of switched nonlinear time-varying systems and detectability of reduced limiting control systems

TL;DR

This paper develops new criteria for the uniform asymptotic stability of switched nonlinear time-varying systems using detectability of reduced limiting control systems, extending classical stability theorems without dwell-time constraints.

Contribution

It introduces the concept of reduced limiting control systems and weakly zero-state detectability, providing necessary and sufficient conditions for stability without dwell-time assumptions.

Findings

Criteria for stability based on detectability are established.

The results extend Krasovskii-LaSalle theorem to switched NLTV systems.

Applications include stability analysis of a semi-quasi-Z-source inverter.

Abstract

This paper is concerned with the study of both, local and global, uniform asymptotic stability for switched nonlinear time-varying (NLTV) systems through the detectability of output-maps. With this aim the notion of reduced limiting control systems for switched NLTV systems whose switchings verify time/state dependent constraints, and the concept of weakly zero-state detectability for those reduced limiting systems are introduced. Necessary and sufficient conditions for the (global)uniform asymptotic stability of families of trajectories of the switched system are obtained in terms of this detectability property. These sufficient conditions in conjunction with the existence of multiple weak Lyapunov functions, yield a criterion for the (global) uniform asymptotic stability of families of trajectories of the switched system. This criterion can be seen as an extension of the classical Krasovskii-LaSalle theorem. An interesting feature of the results is that no dwell-time assumptions are made. Moreover, they can be used for establishing the global uniform asymptotic stability of switched NLTV system under arbitrary switchings. The effectiveness of the proposed results is illustrated by means of various interesting examples, including the stability analysis of a semi-quasi-Z-source inverter.