Analysis of Linear Time-Varying & Periodic Systems

TL;DR

This thesis applies Floquet theory to analyze linear periodic time-varying systems, providing insights into their transition matrices and relations to LTI systems, despite the challenge of finding closed-form solutions.

Contribution

It introduces a method to compare harmonic frequencies in LPTV systems to determine their Floquet matrices and relates these systems to associated LTI systems.

Findings

Comparison of harmonic powers to determine Floquet matrices

Relations established between LPTV and LTI systems at different frequencies

Insights into the analytical challenges of transition matrix solutions

Abstract

This thesis applies Floquet theory to analyze linear periodic time-varying (LPTV) systems, represented by a system of ordinary differential equations (ODEs) that depend on a time variable t and have a matrix of coefficients with period T>0. The transition matrix of an LPTV system represented by a square periodic-function matrix A(t)=A(t+T) can be expressed as the product of a square periodic function matrix P(t)=P(t+T) and an exponentiated square matrix of the form Rt, where R is a constant matrix (independent of t). Despite the validity of Floquet theory, it is difficult to find an analytical closed form for the matrices P(t) and R when the transition matrix {\Phi}_A (t,t_0 ) is unknown. In essence, it is difficult to find an analytical solution for an LPTV system (i.e., a closed form for its transition matrix). The research results show that for a given family of periodic matrices A(t), we can compare the powers of {\omega} that multiply the harmonics (i.e., {\omega} is part of the coefficients multiplying the cosine [sine] factors in even [odd] representations or of the exponential factors in complex representations) to determine the matrices P(t) and R. In addition, the results lead to relations between LPTV systems at frequency {\omega} and the associated linear time-invariant system, which is defined by having zero frequency ({\omega}=0).