H∞ Control with Multi-Saddles

Abstract

AbstractThis paper investigates the H∞ control problems with multi-saddles for linear autonomous systems and affine nonlinear autonomous systems. From the viewpoint of (sub)optimal control, one should find the saddles of the Hamilton function of the considered system in order to solve the H∞ control problem. Usually, solving the H∞ control problem is based on an assumption to assure the existence and uniqueness of the saddle, which is not generally true. This paper uses the theory of generalized inverses (esp. the group inverse) of matrices over a general field to represent the generic saddle solutions of the Hamilton functions of the considered systems, shows the saddle sets are linear manifolds, and each Hamilton function is constant with respect to its saddles. Moreover, some solvability conditions and state feedback solutions for the H∞ control problems with multi-saddles are given via the generic saddle solution representations. In addition, the above results obtained are extended to the case without the existence of saddles.