Pole Placement via the Periodic Schur Decomposition

We present a new method for eigenvalue assignment in linear periodic discrete-time systems through the use of linear periodic state feedback. The proposed method uses reliable numerical techniques based on unitary transformations. In essence, it computes the Schur form of the open-loop monodromy matrix via a recent implicit eigen-decomposition algorithm, and shifts its eigenvalues sequentially. Given complete reachability of the open-loop system, we show that we can assign an arbitrary set of eigenvalues to the closed-loop monodromy matrix in this manner. Under the weaker assumption of complete control-lability, this method can be used to place all eigenvalues at the origin, thus solving the so-called deadbeat control problem. The algorithm readily extends to more general situations, such as when the system equation is given in descriptor form.