# 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 $\XX(t,0)$ 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 information on $[0,T]$ alone. Here we give an example illustrating that there are other such factorizations, and delineate all factorizations of this form and 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 information on $[0,T]$ only. This new information can be useful in the analysis and control of linear time-periodic systems.

Attachment | Size |
---|---|

CS-2002-01.pdf | 300.52 KB |