A Cyclic Small Phase Theorem

TL;DR

This paper introduces a new segmental phase concept for MIMO systems, establishing a cyclic small phase theorem that aids stability analysis of multi-component feedback systems.

Contribution

It proposes a novel matrix segmental phase definition with a key eigen-phase bound, enabling a cyclic small phase theorem for stability analysis of complex feedback systems.

Findings

Introduces the segmental phase for MIMO systems.

Establishes a cyclic small phase theorem with phase bounds.

Provides a generalized theorem with reduced conservatism.

Abstract

This paper introduces a brand-new phase definition called the segmental phase for multi-input multi-output linear time-invariant systems. The underpinning of the definition lies in the matrix segmental phase which, as its name implies, is graphically based on the smallest circular segment covering the matrix normalized numerical range in the unit disk. The matrix segmental phase has the crucial product eigen-phase bound, which makes itself stand out from several existing phase notions in the literature. The proposed bound paves the way for stability analysis of a single-loop cyclic feedback system consisting of multiple subsystems. A cyclic small phase theorem is then established as our main result, which requires the loop system phase to lie between $-\pi$ and $\pi$. The proposed theorem complements a cyclic version of the celebrated small gain theorem. In addition, a generalization of the proposed theorem is made via the use of angular scaling techniques for reducing conservatism.