The Small Phase Condition is Necessary for Symmetric Systems

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.