Real Floquet factors of linear time-periodic systems

Floquet theory plays a ubiquitous role in the analysis and control of time-periodic systems. Its main result is that any fundamental matrix X(t, 0) of a linear system with T -periodic coefficients will have a (generally complex) Floquet factorization with one of the two factors being T -periodic. It is also well known that it is always possible to obtain a real Floquet factorization for the fundamental matrix of a real T -periodic system by treating the system as having 2T -periodic coefficients. The important work of Yakubovich in 1970 and Yakubovich and Starzhinskii in 1975 exhibited a class of real Floquet factorizations that could be found from computations on [0, T ] alone. Here we generalize these results to obtain other such factorizations. We delineate all factorizations of this form and show how they are related. We give a simple extension of the Lyapunov part of the Floquet-Lyapunov theorem in order to provide one way that the full range of real factorizations may be used based on computations on [0, T ] only. This new information can be useful in the analysis and control of linear time-periodic systems.