Generalized Kalman–Yakubovich–Popov Lemma for 2-D FM LSS Model

This paper is concerned with the generalized Kalman-Yakubovich-Popov (KYP) lemma for 2-D Fornasini- Marchesini local state-space (FM LSS) systems. By carefully analyzing the feature of the states in 2-D FM LSS models, a linear matrix inequality (LMI) characterization for a rectangular finite frequency region is constructed and then by combining this characterization with
S
-procedure, a generalized KYP lemma is proposed for 2-D FM LSS models. As applications of the developed KYP lemma, new conditions are further derived for designing controllers guaranteeing finite frequency positive realness of the closed-loop systems where both state-feedback and dynamic output-feedback control laws are considered. Finally, two illustrative examples clearly show the effectiveness of the proposed results. The main contribution of the paper is trifold: 1) a novel characterization for a finite frequency region has been developed; 2) a generalized KYP lemma has been proposed, including the existing bounded realness and positive realness results as special cases; and 3) systematic methods for finite frequency positive real control of 2-D systems have been presented.