The general inner-outer factorization problem for discrete-time systems

In this paper we give a theoretical and a computational solution to the most general inner-outer factorization problem formulated for a discrete-time system G. Our method is based on descriptor state-space computations and relies on an efficient dislocation of the minimal indices and of the “unstable” zeros of G by left multiplication with all-pass factors. The minimal indices are dislocated by solving for the stabilizing solution an algebraic Riccati equation of order n_ℓ (the sum of left minimal indices) while the n_b unstable zeros are dislocated by solving a Lyapunov equation of order n_b. The results reported here are a non-trivial extension of a recently developed approach to the continuous-time inner-outer factorization problem.