On Dual of LMIs for Absolute Stability Analysis of Nonlinear Feedback Systems with Static O'Shea-Zames-Falb Multipliers

TL;DR

This paper develops a dual LMI-based approach to identify non-absolute stability in nonlinear feedback systems with static O'Shea-Zames-Falb multipliers, complementing existing sufficient conditions.

Contribution

It introduces a dual LMI condition for detecting non-absolute stability, providing a new tool for stability analysis when traditional LMIs are infeasible.

Findings

Identifies destabilizing nonlinearities via dual LMIs.

Provides a method to detect non-zero equilibrium points.

Demonstrates effectiveness through numerical examples.

Abstract

This study investigates the absolute stability criteria based on the framework of integral quadratic constraint (IQC) for feedback systems with slope-restricted nonlinearities. In existing works, well-known absolute stability certificates expressed in the IQC-based linear matrix inequalities (LMIs) were derived, in which the input-to-output characteristics of the slope-restricted nonlinearities were captured through static O'Shea-Zames-Falb multipliers. However, since these certificates are only sufficient conditions, they provide no clue about the absolute stability in the case where the LMIs are infeasible. In this paper, by taking advantage of the duality theory of LMIs, we derive a condition for systems to be not absolutely stable when the above-mentioned LMIs are infeasible. In particular, we can identify a destabilizing nonlinearity within the assumed class of slope-restricted nonlinearities as well as a non-zero equilibrium point of the resulting closed-loop system, by which the system is proved to be not absolutely stable. We demonstrate the soundness of our results by numerical examples.