Incremental stability of Lur'e systems through piecewise-affine approximations

TL;DR

This paper introduces less conservative conditions for incremental stability of Lur'e systems by approximating nonlinearities with piecewise-affine functions and employing LMIs for stability analysis.

Contribution

It proposes a novel approach using piecewise-affine approximations and LMIs to assess incremental stability more effectively than existing criteria.

Findings

Conditions for incremental stability are less conservative.

Method is computationally efficient via LMIs.

Numerical examples validate the approach.

Abstract

Lur'e-type nonlinear systems are virtually ubiquitous in applied control theory, which explains the great interest they have attracted throughout the years. The purpose of this paper is to propose conditions to assess incremental asymptotic stability of Lur'e systems that are less conservative than those obtained with the incremental circle criterion. The method is based on the approximation of the nonlinearity by a piecewise-affine function. The Lur'e system can then be rewritten as a so-called piecewise-affine Lur'e system, for which sufficient conditions for asymptotic incremental stability are provided. These conditions are expressed as linear matrix inequalities (LMIs) allowing the construction of a continuous piecewise-quadratic incremental Lyapunov function, which can be efficiently solved numerically. The results are illustrated with numerical examples.