Algebraic Theory of Linear Time-Varying Systems

An algebraic theory of linear time-varying discrete-time systems is developed in terms of a module structure defined over a noncommutative polynomial ring. The module setup is induced from a semilinear transformation that is derived from the given system. Various structural properties of the module framework are explored including the concepts of cyclicity and $n$-cyclicity. The module theory is then applied to the study of reachability and state feedback. Results on the construction of feedback controllers are obtained that resemble pole or coefficient assignability in the theory of time-invariant systems.