A note on the input-output structure of linear periodic continuous-time systems with real-valued coefficients

In this paper, a Kalman canonical decomposition of finite-dimensional linear periodic continuous-time systems is considered. This paper firstly investigates the invariance properties of the controllable subspace and the observable subspace. This paper then illustrates a counterexample to the existence of the periodic Kalman canonical decomposition in a typical setting, where the coefficients are restricted to be real-valued and the period of the transformed system is restricted to be the same as the given system. Motivated by this counterexample, this paper gives a self-contained exposition of the periodic Kalman canonical decomposition in real-valued coefficients.