Incremental stability of Lur'e systems through piecewise-affine approximations
For control theorists, this provides a practical method to analyze incremental stability of Lur'e systems with reduced conservatism compared to the incremental circle criterion.
The paper proposes less conservative conditions for assessing incremental asymptotic stability of Lur'e systems by approximating the nonlinearity with a piecewise-affine function, leading to LMI-based sufficient conditions. Numerical examples demonstrate the effectiveness.
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.