Computation of general inner-outer and spectral factorizations

In this paper we solve two problems in linear systems theory: the computation of the inner–outer and spectral factorizations of a continuous–time system considered in the most general setting. We show that these factorization problems rely essentially on solving for the stabilizing solution a standard algebraic Riccati equation of order usually much smaller than the McMillan degree of the transfer function matrix of the system. The proposed procedures are completely general being applicable for a polynomial/proper/improper system whose transfer function matrix could be rank deficient and could have poles/zeros on the imaginary axis or at infinity. As an application we discuss the extension to rational matrices of the complete orthogonal decomposition of a constant matrix. Numerical refinements are discussed in detail. To illustrate the proposed approach several numerical examples are also given. Keywords— Inner–outer factorization; spectral factorization; rational matrices; descriptor systems; system zeros; singular matrix pencils; numerical algorithms