Mean uniformly stable function and its application to almost sure stability analysis of randomly switched time-varying systems

TL;DR

This paper introduces a novel probabilistic stability analysis method for randomly switched time-varying systems using mode-dependent Lyapunov functions and mean uniformly stable functions, accommodating unstable subsystems.

Contribution

It proposes the mean uniformly stable function (MUSF) concept and integrates it with indefinite Lyapunov functions to analyze almost sure stability of complex systems.

Findings

MUSF condition ensures mean-boundedness of subsystems during sojourns

The approach handles unstable or non-exponentially decaying subsystems

Numerical examples validate the effectiveness of the method

Abstract

This paper investigates uniform almost sure stability of randomly switched time-varying systems. Mode-dependent indefinite multiple Lyapunov functions (iMLFs) are introduced to assess stability properties of diverse time-varying subsystems. To realize the stability conditions establishment based on iMLFs, we present a novel condition so-called mean uniformly stable function for time-varying parameters of iMLFs' derivatives. Our approach provides a probabilistic perspective, making iMLFs well-suited for randomly switched time-varying systems. Moreover, the MUSF condition reveals an essential insight: ensuring that each time-varying subsystem remains mean-bounded during its corresponding sojourn time interval is a prerequisite for the almost sure stability of the entire system. Additionally, the combination of iMLFs and MUSFs is able to accommodate stability analysis scenarios where some subsystems are unstable or exhibit non-exponential decay. Numerical examples are provided to demonstrate the effectiveness and advantages of our approach.