In this paper, the problem of obtaining a periodic description in state-space form of a linear process which can be modeled by linear difference equations with periodic coefficients is considered. On the basis of a polynomial time-invariant description of a linear periodic process, system equivalence between two such processes is introduced and studied. For a given periodic causal process, under an additional assumption, a periodic state-space description is found which is system equivalent to it. It is shown that the order, the characteristic multipliers, and the stacked transfer matrix at any initial time of the periodic system thus obtained coincide with those of the original periodic process, and that the asymptotic stability, the reachability, the observability, the controllability, the reconstructibility, the stabilizability, the detectability, and even the Jordan form of the monodromy matrix of such a system are determined by the original periodic model, as well as the existence of a solution of the robust tracking and regulation problem.
Grasselli, O.m., Longhi, S., & Tornambe, A. (1995). System equivalence for periodic models and systems. SIAM JOURNAL ON CONTROL AND OPTIMIZATION, 33(2), 455-468 [10.1137/S0363012992234578].