Exact Instability Margin Analysis and Minimum-Norm Strong Stabilization -- phase change rate maximization --

TL;DR

This paper introduces a new optimization problem called phase change rate maximization, linking robust instability analysis and minimum-norm strong stabilization for LTI systems, with exact RIR characterization.

Contribution

It transforms the RIR computation into a phase change rate maximization problem and derives conditions for exact RIR characterization using all-pass functions.

Findings

Maximum phase change rate is attained by simple all-pass functions.

Exact RIR can be characterized under specific phase conditions.

Applications demonstrate practical utility of the theoretical results.

Abstract

This paper is concerned with a new optimization problem named "phase change rate maximization" for single-input-single-output linear time-invariant systems. The problem relates to two control problems, namely robust instability analysis against stable perturbations and minimum-norm strong stabilization. We define an index of the instability margin called "robust instability radius (RIR)" as the smallest $H_\infty$-norm of a stable perturbation that stabilizes a given unstable system. This paper has two main contributions. It is first shown that the problem of finding the exact RIR via the small-gain condition can be transformed into the problem of maximizing the phase change rate at the peak frequency with a phase constraint. Then, we show that the maximum is attained by a constant or a first-order all-pass function and derive conditions, under which the RIR can be exactly characterized, in terms of the phase change rate. Two practical applications are provided to illustrate the utility of our results.