When Small Gain Meets Small Phase

TL;DR

This paper develops a new stability criterion for MIMO LTI systems that combines gain and phase information, introducing a small vase theorem and a geometric approach for better analysis and controller design.

Contribution

It introduces a novel small vase theorem and a geometric method for combined gain and phase stability analysis, along with a state-space characterization via a bounded & sectored real lemma.

Findings

Established a stability condition for systems with small phase and gain.

Proposed a mixed small gain and phase condition with necessity.

Presented a computationally efficient state-space characterization.

Abstract

In this paper, we investigate the feedback stability of multiple-input multiple-output linear time-invariant systems with combined gain and phase information. To begin with, we explore the stability condition for a class of so-called easily controllable systems, which have small phase at low frequency ranges and low gain at high frequency. Next, we extend the stability condition via frequency-wise gain and phase combination, based on which a mixed small gain and phase condition with necessity, called a small vase theorem, is then obtained. Furthermore, the fusion of gain and phase information is investigated by a geometric approach based on the Davis-Wielandt shell. Finally, for the purpose of efficient computation and controller synthesis, we present a bounded & sectored real lemma, which gives state-space characterization of combined gain and phase properties based on a triple of linear matrix inequalities.