Low‐rank approximations of frequency‐limited discrete‐time Gramians

Abstract We consider the approximate computation of Gramians of large‐scale linear discrete‐time state‐space systems. Of particular interest are the Gramians which arise in frequency‐limited balanced truncation model order reduction. While there are many results for the continuous‐time situation, the discrete‐time scenario is usually considered to a much lesser extent. The Gramians are solutions of discrete‐time Lyapunov matrix equations (also called Stein matrix equations) which, if the reduction is restricted to smaller frequency intervals, also incorporate matrix functions in the form of matrix logarithms. We discuss the efficient numerical handling of these matrix function as well as the computation of low‐rank Gramian approximation by rational Krylov subspace and related iterative methods.