The geometric approach for linear periodic discrete-time systems

Recently, the disturbance localization problem and the problems of designing disturbance decoupled observers and of giving a geometric characterization of invariant zeros were solved for linear periodic discrete-time systems through an extension of the geometric approach, based on the notions of controlled invariant and conditionally invariant subspaces, to periodic ones. Here the existing periodic geometric theory is supplemented with the notions of outer reachable subspace and controllability subspace, and with some further results on the previously introduced notions of inner reachable (controllable) subspace and outer controllable subspace. In addition, it is shown that any such periodic geometric theory can be restated in an equivalent time-invariant form, which is just the same theory written for a suitable time-invariant system having an extended state space.