On Spectral Factorization and Riccati Equations for Time-Varying Systems in Discrete Time

It is known that positive operators W on a Hilbert space admit a factorization of the form W = W * W, where W is an outer operator whose matrix representation is upper. As upper Hilbert space operators have an interpretation of transfer operators of linear time-varying systems in discrete time, this proves the existence of a spectral factorization for time-varying systems. In this paper, the above result is translated from operator theory into control theory language, by deriving how such a factorization can be actually computed if a state realization of the upper part of W is known. The crucial step in this algorithm is the solution of a Riccati recursion with time-varying coefficients. It is shown that, under conditions, positive solutions exists, and that the smallest positive solution lead to a factor which is outer (`minimum-phase'). The outer factor can be computed numerically in a number of cases, e.g., if the system is initially time-invariant, periodically time-varying. Mor..