Optimal H∞ model reduction via linear matrix inequalities: continuous and discrete-time cases

Necessary and sufficient conditions are derived for the existence of a solution to the continuous-time and discrete-time H_∞ model reduction problems. These conditions are expressed in terms of linear matrix inequalities (LMIs) and a coupling non-convex rank constraint set. In addition, an explicit parametrization of all reduced order models that correspond to a feasible solution are provided in terms of a contractive matrix. These results follow from a previous solution of the H_∞ control design problem using LMIs. Particularly simple conditions and a simple parametrization of all solutions are obtained for the zeroth-order H_∞ approximation problem, and the convexity of this problem is demonstrated. Computational issues are discussed and an iterative procedure is proposed to solve the H_∞ model reduction problem using alternating projections, although global convergence of the algorithm is not guaranteed