An overview of generalized KYP lemma based methods for finite frequency analysis and design

Frequency-domain methods are a fundamental research approach in control theory and engineering. Many control problems can be viewed as analysis and design issues of finite frequency specifications. The generalized Kalman-Yakubovich-Popov (KYP) lemma, which bridges frequency-domain methods (transfer functions) and time-domain methods (state-space models), has been one of the hotspots in systems and control theory in recent years. In this paper, the background and significance of finite frequency analysis and design are first introduced from signal and system perspectives, respectively. Three main research methods (classical control theory, frequency-weighting strategy and generalized system specification based methodology) are discussed with respect to their individual advantages and disadvantages. The body of the generalized KYP lemma is then introduced briefly, which is followed by a detailed summary of main directions and recent progresses in finite frequency analysis and design based on the generalized KYP lemma. Finally, a few notes are presented, which are important but commonly overlooked in applying the generalized KYP lemma, and a few critical problems in the field are also pointed out, which are worth future investigation.