Solutions to the general inner-outer and spectral factorization problems

In this paper we solve two open problems in linear system theory: the computation of the inner-outer and spectral factorizations of a descriptor continuous-time system considered in the most general setting. Our method is based on descriptor state-space computations and relies on three techniques developed for rational matrices: compression of G to a surjective (full row rank) matrix, and dislocation of the unstable zeros/poles of G to stable locations, all achieved by left multiplication with all-pass factors. The proposed procedures are completely general being applicable for G polynomial/proper/improper, of arbitrary rank, with poles/zeros on the imaginary axis.