The Small Phase Condition is Necessary for Symmetric Systems

TL;DR

This paper establishes that the small phase condition is both necessary and sufficient for feedback stability in symmetric systems, filling a key gap in phase-based stability criteria and exploring its relevance to asymmetric systems.

Contribution

It proves the necessity of the small phase condition for symmetric systems and extends insights to asymmetric systems, advancing stability analysis methods.

Findings

Small phase condition is necessary and sufficient for symmetric systems.

Transformation matrices in symmetric cases can be real, simplifying analysis.

Insights into the necessity of small phase condition for asymmetric systems.

Abstract

In this paper, we show that the small phase condition is both sufficient and necessary to ensure the feedback stability when the interconnected systems are symmetric. Such symmetric systems arise in diverse applications. The key lies in that, for a complex symmetric and semi-sectorial matrix, the transformation matrix in its generalized sectorial decomposition can be taken to be real. Such a result fills the gap of phase based necessary condition for the feedback stability of symmetric systems, and serves as a counterpart of the necessity result for small gain condition. Moreover, we explore the necessity of small phase condition for general asymmetric systems. Some insightful results are presented, which help to clarify the main challenge in the general case.