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

It is known that positive operators Ω on a Hilbert space admit a factorization of the form Ω = 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 Ω 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. More generally, for strictly stable systems it is shown that the Riccati recursion, when started from zero initial conditions, will strongly converge to the exact smallest positive solution, so that the outer factor can be computed in arbitrary precision for any finite interval in time. The results can also be formulated in terms of a time-varying positive real lemma. Finally, some connections are provided with Riccati equations that occur in related problems in time-varying systems theory, such as inner-outer factorization, orthogonal embedding and the time-varying bounded-real lemma.