Invariance Principles and Observability in Switched Systems with an Application in Consensus

TL;DR

This paper extends invariance principles to switched nonlinear time-varying systems using virtual outputs and weak observability, providing new tools for analyzing stability and consensus in multi-robot systems.

Contribution

It introduces a novel extension of LaSalle's invariance principle and an integral invariance principle for switched NLTV systems, enabling analysis without dwell-time assumptions.

Findings

Extended invariance principles for switched NLTV systems.

New control strategy for leaderless consensus in mobile robots.

Proved convergence without dwell-time or uniform Lyapunov stability.

Abstract

Using any nonnegative function with a nonpositive derivative along trajectories to define a virtual output, the classic LaSalle invariance principle can be extended to switched nonlinear time-varying (NLTV) systems, by considering the weak observability (WO) associated with this output. WO is what the output informs about the limiting behavior of state trajectories (hidden in the zero locus of the output). In the context of switched NLTV systems, WO can be explored using the recently established framework of limiting zeroing-output solutions. Adding to this, an extension of the integral invariance principle for switched NLTV systems with a new method to guarantee uniform global attractivity of a closed set (without assuming uniform Lyapunov stability or dwell-time conditions) is proposed. By way of illustrating the proposed method, a leaderless consensus problem for nonholonomic mobile robots with a switching communication topology is addressed, yielding a new control strategy and a new convergence result.