On the Geometric Structure of the H∞ Central Controller

This paper shows that the controllable and unobservable subspaces of the H∞ central controller for a linear continuous-time system can be characterized by the image and kernel spaces of two matrices ZL and WL, where ZL and WL are positive semidefinite solutions of two pertinent Lyapunov equations whose coefficients involve X∞ and Z∞, the stabilizing solutions of two celebrated algebraic Riccati equations used in solving the H∞ control problem. Furthermore, under this characterization, it is shown that the unobservable subspace of the central controller contains the intersection of KerX∞ and the unobservable subspace of the plant. In addition, it is also shown that the central controller's controllable subspace is a subspace of the sum of ImZ∞ and the plant's controllable subspace. A numerical example is also given for illustration. In terms of geometric language, all the results and proofs given are clear and simple.