<i>H</i>-Controllability and Observability of Linear Periodic Systems

Abstract

With reference to linear periodic systems, the classical notion of controllability introduced by Kalman (K-controllability) is shown to be equivalent to the characterization of controllability (H-controllability) first proposed by Hewer as a natural extension to the periodic case of an eigenvalue-eigenvector controllability condition independently introduced by several authors for time-invariant systems only. The proof of such an equivalence also leads to the conclusion that if a T-periodic system S is K-controllable and k is the degree of the minimal polynomial of the monodromy matrix associated with S, then, at any time t, S is controllable over $(t,t + kT)$. Since k is lower than or equal to the system order, a well-known result due to Brunovsky is slightly strengthened. By duality, the corresponding observability results follow.