Retrospect and prospect on integration of gain and phase for feedback stability analysis

In this paper,we review existing approaches to integrating small gain and small phase analysis for feedback stability of dynamical systems,and give a brief outlook for possible future directions in exploring this topic.Small gain analysis has been very successful and popular in control theory since 1960s,while the small phase analysis for multiple-input-multiple-output systems has not been well understood until recently and is now gradually taking shape.Nevertheless,there have been attempts to analyzing feedback stability via the integration of gain and phase information over decades,including the combination of small gain with positive realness as well as that with negative imaginariness.Such combinations can be subsumed into a recently proposed framework for gain-phase integration,which brings in new geometrical methods and also sheds new lights on several future directions.