Nonquadratic global asymptotic stability certificates for saturated linear feedbacks

TL;DR

This paper develops new nonquadratic Lyapunov functions with necessary and sufficient conditions for establishing global asymptotic stability of saturated linear feedback systems, including multi-input cases.

Contribution

It introduces nonquadratic Lyapunov functions based on sign-indefinite quadratic forms and provides convex stability tests via linear matrix inequalities for complex feedback systems.

Findings

Conditions are necessary and sufficient for stability in key examples.

The approach applies to multi-input systems with convex stability tests.

Results are non-conservative for certain non-globally-stabilizable plants.

Abstract

We establish sufficient conditions for positive (semi-)definiteness, with or without radial unboundedness, for nonquadratic Lyapunov function constructed as sign-indefinite quadratic forms involving the state and the deadzone of a suitable input. We then use these conditions to build weak nonquadratic Lyapunov functions establishing global asymptotic stability of linear systems in feedback through a saturation, leveraging invariance principles. Our results are shown to be non-conservative (necessary and sufficient) for a family of well known prototypical examples of linear SISO feedbacks that are not globally exponentially stabilizable (the so-called ANCBI plants). Our multi-input extension leads to convex stability analysis tests, formulated as linear matrix inequalities that are applicable to ANCBI non-globally-exponentially-stabilizable plants.