ALL CONTROLLERS FOR THE GENERAL H-INFINITY CONTROL PROBLEM - LMI EXISTENCE CONDITIONS AND STATE-SPACE FORMULAS

This paper presents all controllers for the general H(infinity) control problem (with no assumptions on the plant matrices). Necessary and sufficient conditions for the existence of an H(infinity) controller of any order are given in terms of three Linear Matrix Inequalities (LMIs). Our existence conditions are equivalent to Scherer's results, but with a more elementary derivation. Furthermore, we provide the set of all H(infinity) controllers explicity parametrized in the state space using the positive definite solutions to the LMIs. Even under standard assumptions (full rank, etc.), our controller parametrization has an advantage over the Q-parametrization. The freedom Q (a real-rational stable transfer matrix with the H(infinity) norm bounded above by a specified number) is replaced by a constant matrix L of fixed dimension with a norm bound, and the solutions (X, Y) to the LMIs. The inequality formulation converts the existence conditions to a convex feasibility problem, and also the free matrix L and the pair (X, Y) define a finite dimensional design space, as opposed to the infinite dimensional space associated with the Q-parametrization.