A complete solution to the general H∞ control problem: LMI existence conditions and state space formulas

This paper presents all controllers for the general H_∞ control problem (with no assumptions on the plant matrices). Necessary and sufficient conditions for the existence of an H_∞ suboptimal controller of any order are given in terms of three Linear Matrix Inequalities (LMIs). Furthermore, we provide the set of all H_∞ suboptimal controllers explicitly parametrized in the state space using the positive definite solutions to the LMIs. The inequality formulation converts the existence conditions to a convex feasibility problem, and also a free matrix parameter in the controller formula defines a finite dimensional design space, as opposed to the infinite dimensional space associated with the Q-parametrization.