On Incremental Stability of Interconnected Switched Systems

TL;DR

This paper investigates the incremental stability of interconnected switched nonlinear systems with state-dependent switching, providing sufficient conditions using contraction theory, and extends results to input-to-state stability and feedback interconnections.

Contribution

It introduces new contraction-theoretic conditions for incremental stability of interconnected switched systems, including input-to-state stability and feedback configurations.

Findings

Derived computationally tractable stability conditions.

Extended stability analysis to input-to-state stability with external inputs.

Validated results through numerical examples and simulations.

Abstract

In this paper, the incremental stability of interconnected switched nonlinear systems is discussed. The nature of switching considered is state-dependent. The incremental stability of the switched interconnected system is a stronger property compared to the conventional notion of stability. Even if individual systems in the interconnected setting are stable, guaranteeing stability for the overall system is challenging. However, one of the important features of incremental stability is that the notion is preserved over interconnection. Here, leveraging the contraction-theoretic tools, we derive a set of sufficient conditions for the overall interconnection consisting of bimodal switched systems. To showcase the wider usability of our proposed results, we have also included the effect of external input, which leads to the study of incremental input-to-state stability ($\delta$-ISS). For the case of feedback interconnection, the small gain characterisation is presented for the overall system's $\delta$-ISS. Further, for a special case of feedback, i.e., cascade interconnection, the results are derived. The derived conditions are based on the matrix measure, making it computationally tractable and general. To make the results more general, a generalised interconnection of bimodal switched systems is studied and corresponding sufficient condition for $\delta$-ISS are presented. Two numerical examples are demonstrated and supported with simulation results to verify the theoretical claims.