Multipliers for nonlinearities with monotone bounds

TL;DR

This paper introduces new multipliers for analyzing the stability of Lurye systems with multivalued nonlinearities bounded by monotone functions, enhancing existing methods and applicable to both continuous and discrete systems.

Contribution

It generalizes stability analysis techniques for nonlinear systems using convex search-based multipliers suitable for multivalued nonlinearities and loop transformations.

Findings

Outperforms existing stability tests in continuous-time examples.

Effective for asymmetric saturation nonlinearities.

Applicable to systems with non-zero steady state signals.

Abstract

We consider Lurye (sometimes written Lur'e) systems whose nonlinear operator is characterised by a possibly multivalued nonlinearity that is bounded above and below by monotone functions. Stability can be established using a sub-class of the Zames-Falb multipliers. The result generalises similar approaches in the literature. Appropriate multipliers can be found using convex searches. Because the multipliers can be used for multivalued nonlinearities they can be applied after loop transformation. We illustrate the power of the new mutlipliers with two examples, one in continuous time and one in discrete time: in the first the approach is shown to outperform available stability tests in the literature; in the second we focus on the special case for asymmetric saturation with important consequences for systems with non-zero steady state exogenous signals.