(Generalized) KYP lemma and applications

Abstract

Various properties of dynamical systems can be characterized in terms of inequality conditions on their frequency responses. The Kalman-Yakubovich-Popov (KYP) lemma shows equivalence of such frequency domain inequality (FDI) and a linear matrix inequality (LMI). The fundamental result has been a basis for robust and optimal control theories in the past several decades. The KYP lemma has recently been generalized to the case where an FDI on a possibly improper transfer function is required to hold in a (semi)finite frequency range. The generalized KYP lemma allows us to directly deal with practical situations where design parameters are sought to satisfy FDIs in multiple (semi)finite frequency ranges. Various design problems, including FIR filter and PID controller, reduce to LMI problems which can be solved via semidefinite programming.