Generalized Kalman-Yakubovich-Popov Lemma for 2-D FM LSS Model and Its Application to Finite Frequency Positive Real Control

This paper is concerned with the development of the generalized Kalman-Yakubovich-Popov (KYP) lemma for two-dimensional (2-D) Fornasini-Marchesini local state-space (FM LSS) systems and its application to state-feedback positive realness control with finite frequency specifications. An linear matrix inequality (LMI) characterization for a rectangular finite frequency region is firstly technically constructed and then a generalized KYP lemma is proposed for 2-D FM LSS models. This lemma provides sufficient conditions in terms of LMI for general quadratic properties of the transfer function over a rectangular finite frequency region, including the extensively investigated bounded realness and positive realness as special cases. Based on this result, a new condition is further derived for designing controllers guaranteeing the finite frequency positive realness of the closed-loop systems. The presented numerical example shows the advantage of the proposed design method.