Algebraic theory of linear periodic discrete-time systems in their polynomial matrix and module descriptions

An algebraic theory of linear periodic discrete-time (LPDT) systems is developed in terms of a periodic polynomial matrix description (PMD) and in a dynamics (module) structure defined over a non-commutative, non-integral and non-principal ideal ring ℛ (the ring of periodic polynomials). Various properties of matrices and modules over ℛ are explored, and links between algebraic properties of LPDT systems such as reachability, controllability are established directly in the periodic PMD and ℛ-module descriptions; i.e., without using the technique of lifting a LPDT system to its associated linear time-invariant (LTI) ones. The advantage of such characterizations is to show that LPDT systems are not index-invariant only for a class-that we characterize-of non-reversible time systems and that, except for this class of systems, the parametrization of all stabilizing controllers can be constructed similarly to LTI systems. Furthermore, one shows that for the class of non-reversible time systems above-mentioned, it is not possible to extract a canonical part like for LTI systems