A GEOMETRIC APPROACH TO OPTIMAL STATE-SPACE SOLUTIONS WITH MINIMAL REALIZATION FOR STANDARD DISCRETE-TIME H2 CONTROL PROBLEM

This paper shows that the controllable and unobservable subspaces of the discrete-time H-2 optimal controller can be characterized by the image and kernel spaces of two matrices Z(2) and W-2, where Z(2) and W-2 are positive semi-definite solutions of two pertinent Lyapunov equations whose coefficients involve the stabilizing solutions of two celebrated discrete-time algebraic Riccati equations (DAREs) used in solving the H-2 optimal control problem. By suitably choosing the bases adapted to Z(2) and W-2, a minimal order state-space realization of an H-2 optimal controller is then given via an elegant geometric approach. In terms of geometric language, all the results and proofs given are clear and simple.