Parameter selection for best $H_{2}$ moment matching-based model approximation through gradient optimization

In this paper we compute a family of reduced order models, parameterized in a matrix of free parameters, that match a prescribed set of $\nu$ moments of a highly dimensional linear time-invariant system. Based on the parametrization and using as objective function the $H_{2}$-norm of the approximation error, we propose a non-convex optimization problem, which, for a fixed set of interpolation points, is seeking the optimal free parameters yielding the minimal $H_{2}$-norm error. Furthermore, we write the necessary first-order optimality conditions explicitly in terms of the controllability and the observability Gramians of a minimal realization of the error system. Based on the optimality conditions we propose a gradient method-based solution, accompanied by mathematical guarantees about its convergence. We illustrate the efficiency of our results on the test example of a CD player.