Incremental analysis of nonlinear systems with efficient methods for piecewise-affine systems

TL;DR

This paper introduces efficient methods for analyzing incremental stability of nonlinear systems, especially piecewise-affine systems, using dissipativity and LMI-based techniques that improve upon previous conservative approaches.

Contribution

It proposes less conservative LMI-based conditions for incremental stability and L2-gain analysis of PWA systems, linking incremental L2-gain to asymptotic stability under certain conditions.

Findings

New LMI conditions for incremental L2-gain bounds

Less conservative than previous methods

Established connection between L2-gain and asymptotic stability

Abstract

This paper is concerned with incremental stability properties of nonlinear systems. We propose conditions to compute an upper bound on the incremental L2-gain and to assess incremental asymptotic stability of piecewise-affine (PWA) systems. The conditions are derived from dissipativity analysis, and are based on the construction of piecewise-quadratic functions via linear matrix inequalities (LMI) that can be efficiently solved numerically. The developments are shown to be less conservative than previous results, and are illustrated with numerical examples. In the last part of this paper, we study the connection between incremental L2-gain stability and incremental asymptotic stability. It is shown that, with appropriate observability and reachability assumptions on the input-output operator, incremental L2-gain implies incremental asymptotic stability. Finally, it is shown that the converse implication follows provided some regularity conditions on the state space representation are met.