Gain and phase type multipliers for feedback robustness

TL;DR

This paper links feedback robustness to the existence of specific gain- or phase-type multipliers, enhancing the understanding of stability conditions in linear systems and unifying known theorems.

Contribution

It establishes the existence of gain- or phase-type multipliers for robust stability, connecting matrix phases with integral quadratic constraints.

Findings

Existence of phase-type multipliers for gain-robust stability

Existence of gain-type multipliers for phase-robust stability

Unification of small-gain and small-phase theorems

Abstract

It is known that the stability of a feedback interconnection of two linear time-invariant systems implies that the graphs of the open-loop systems are quadratically separated. This separation is defined by an object known as the multiplier. The theory of integral quadratic constraints shows that the converse also holds under certain conditions. This paper establishes that if the feedback is robustly stable against certain structured uncertainty, then there always exists a multiplier that takes a corresponding form. In particular, if the feedback is robustly stable to certain gain-type uncertainty, then there exists a corresponding multiplier that is of phase-type, i.e., its diagonal blocks are zeros. These results build on the notion of phases of matrices and systems, which was recently introduced in the field of control. Similarly, if the feedback is robustly stable to certain phase-type uncertainty, then there exists a gain-type multiplier, i.e., its off-diagonal blocks are zeros. The results are meaningfully instructive in the search for a valid multiplier for establishing robust closed-loop stability, and cover the well-known small-gain and the recent small-phase theorems.