Absolute Stability via Lifting and Interpolation

TL;DR

This paper introduces a novel lifting and interpolation method to certify the absolute stability of LTI systems with nonlinear feedback, avoiding traditional frequency-domain techniques and using LMIs for Lyapunov function construction.

Contribution

It presents an alternative approach to stability analysis that constructs Lyapunov functions directly through lifting and interpolation, improving upon classical multiplier-based methods.

Findings

Recovers state-of-the-art results on benchmark problems

Uses LMIs to efficiently search for Lyapunov functions

Provides a direct, frequency-free stability certification method

Abstract

We revisit the classical problem of absolute stability; assessing the robust stability of a given linear time-invariant (LTI) plant in feedback with a nonlinearity belonging to some given function class. Standard results typically take the form of sufficient conditions on the LTI plant, the least conservative of which are based on O'Shea--Zames--Falb multipliers. We present an alternative analysis based on lifting and interpolation that directly constructs a Lyapunov function that certifies absolute stability without resorting to frequency-domain inequalities or integral quadratic constraints. In particular, we use linear matrix inequalities to search over Lyapunov functions that are quadratic in the iterates and linear in the corresponding function values of the system in a lifted space. We show that our approach recovers state-of-the-art results on a set of benchmark problems.