Finite-dimensional H∞ controllers for a class of infinite-dimensional discrete-time systems

In this paper, we consider H_∞ control problem with measurement feedback for a class of infinite-dimensional discrete-time systems. The aim is to construct a finite-dimensional stabilizing controller for the infinite-dimensional discrete-time system that makes the H_∞ norm of the closed-loop transfer function less than a given positive number δ. For that purpose, we first derive a finite-dimensional reduced-order system for the infinite-dimensional discrete-time system. Then, a stabilizing controller that makes the H_∞ norm of the closed-loop transfer function less than another positive number is constructed for the reduced-order model. The finite-dimensional controller together with a residual mode filter plays a role of a finite-dimensional stabilizing controller that makes the H_∞ norm of the closed-loop transfer function less than δ for the original infinite-dimensional discrete-time system, if the order of residual mode filter is chosen sufficiently large.