Real Floquet Factors of Linear Time-Periodic Systems

Authors: 

Pierre Montagnier
Christopher C. Paige
Raymond J. Spiteri

Author Addresses: 

Pierre Montagnier (pierrem@cim.mcgill.ca)
Centre for Intelligent Machines,
McGill University,
Montreal, Quebec,
H3A 2K6 Canada.

Christopher C. Paige (paige@cs.mcgill.ca)
School of Computer Science,
McGill University,
Montreal, Quebec,
H3A 2A7 Canada.

Raymond J. Spiteri (spiteri@cs.dal.ca)
Faculty of Computer Science
Dalhousie University
6050 University Ave.
PO Box 15000
Halifax, Nova Scotia, Canada
B3H 4R2

Abstract: 

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.

Tech Report Number: 
CS-2002-01
Report Date: 
February 16, 2002
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