Finite frequency positive real control of singularly perturbed systems

This paper discusses the problem of finite frequency positive real control for singularly perturbed systems based on GKYP lemma. The purpose of this study is to design a state feedback controller such that the resulting closed-loop system is stable, and achieves the extended strictly positive realness for singularly perturbed systems at low and high frequencies. By employing GKYP lemma, respectively, on the slow and fast subsystem, the problems of the reduced order subsystems are solved in terms of bilinear matrix inequalities (BMIs). The two frequency-scale solution for the full-order SPS constructed in this paper uses the solutions of two well-defined lower-order problems, and therefore it is numerically better conditioned. An iterative algorithm for the computation of the BMIs is presented, and the efficiency of the method in this paper is illustrated by an example.