On Kalman canonical decomposition of linear periodic continuous-time systems with real coefficients

In this note, structural decomposition of linear periodic continuous-time systems is discussed. A fundamental problem to decompose a state of a periodic system into controllable and uncontrollable parts is conjectured to be achieved by a continuously differentiable and periodic coordinate transformation with the same period of the system, however there is a counterexample to this conjecture. Hence we derive a condition for the existence of such a coordinate transformation. We also prove that, by relaxing a class of coordinate transformation, it is always possible to construct a periodic coordinate transformation with the double period of the periodic system.