On the existence of minimal and canonical realizations of linear periodic discrete time systems

This note deals with the problem of the existence of a canonical (i.e. completely reachable and observable) realization (CR) for a linear periodic discrete-time (LPDT) system given by its periodic polynomial matrix description (PPMD). Contrary to linear time-invariant (LTI) systems, it is shown here that a CR exists only for a class of LPDT systems and that consequently minimal and canonical ones are not equivalent: namely minimal (i.e. with the least possible dimension) realization (MR) of a LPDT system is linked with controllability and reconstructability, CR is linked with reachability and observability. Minimal order, necessary and sufficient conditions to ensure the existence of a MR and a CR of LPDT systems are characterized