A Rational Krylov Iteration for Optimal H 2 Model Reduction

In the sequel, we will construct the reduced order models Gr(s) through Krylov projection methods. Toward this end, we construct matrices V ∈ R and Z ∈ R that span certain Krylov subspaces with the property that Z V = Ir. The reduced order model Gr(s) will then be obtained as Ar = Z T AV, Br = Z T B, and Cr = CV. (2) The corresponding oblique projection is given by V Z . Many researchers have worked on the problem (1); see [18], [16], [7], [5], [3], [17], [4] and references therein. Since obtaining a global minimum is a hard task, the goal is to generate a reduced-order model that satisfies the firstorder conditions for (1). However, these methods require solving a (sequence) of large-scale Lyapunov equations and hence dense matrix operations including inversion, which rapidly become intractable as the dimension increases in large-scale settings. Indeed some of these methods are unsuitable even for medium scale problems. Here, we propose an iterative rational Krylov algorithm which efficiently seeks a minimizer to the problem (1). The method is based on the computationally proven approaches utilizing Krylov subspaces. The proposed method is suitable for large-scale settings where the order of the system, n, can grow to the order of many thousands of state variables. Our starting point is the interpolation based first-order conditions for the optimal H2 approximation obtained by Meier and Luenberger [5]: Theorem 1: Let Gr(s) solve the optimal H2 problem and let λ̂i denote the eigenvalues of Ar, i.e. λ̂i are the Ritz values. For simplicity, assume that λ̂i has multiplicity