On the geometric and dynamic structures of the H2 optimal and H∞ central controllers

In this paper, the geometric structure of observer-based controllers is investigated in order to characterize the controllable and unobservable subspaces of the H2 optimal and the H∞ central controllers. It is shown that the controllable and unobservable subspaces of the H2 optimal and the H∞ central controllers can be characterized by the kernel and image subspaces of the solutions of two Lyapunov equations. Under this characterization, the connection between the geometric subspaces and the dynamic behavior of the plant and those of the H2 optimal and H∞ controllers is derived. It is also shown that the H2 optimal and the H∞ central controllers inherit a certain part of the given plant dynamics in the geometric sense. A numerical example is also given for illustration.