Generalized KYP lemma: unified frequency domain inequalities with design applications

The celebrated Kalman-Yakubovic-Popov (KYP) lemma establishes the equivalence between a frequency domain inequality (FDI) and a linear matrix inequality, and has played one of the most fundamental roles in systems and control theory. This paper first develops a necessary and sufficient condition for an S-procedure to be lossless, and uses the result to generalize the KYP lemma in two aspects-the frequency range and the class of systems-and to unify various existing versions by a single theorem. In particular, our result covers FDIs in finite frequency intervals for both continuous/discrete-time settings as opposed to the standard infinite frequency range. The class of systems for which FDIs are considered is no longer constrained to be proper, and nonproper transfer functions including polynomials can also be treated. We study implications of this generalization, and develop a proper interface between the basic result and various engineering applications. Specifically, it is shown that our result allows us to solve a certain class of system design problems with multiple specifications on the gain/phase properties in several frequency ranges. The method is illustrated by numerical design examples of digital filters and proportional-integral-derivative controllers.