论文标题
线性时变和周期系统的分析
Analysis of Linear Time-Varying & Periodic Systems
论文作者
论文摘要
本论文将浮点理论应用于分析线性周期性时变（LPTV）系统，该系统由依赖时间变量t的普通微分方程（ODE）系统表示，并且具有周期t> 0的系数矩阵。由平方周期性功能矩阵A（t）= a（t+t）表示的LPTV系统的过渡矩阵可以表示为平方周期函数矩阵p（t）= p（t+t）的乘积和rt的凸起的平方矩阵，其中r是恒定矩阵（独立于t）。尽管Floquet理论的有效性，但是当过渡矩阵φ_A（t，t_0）未知时，很难找到矩阵p（t）和r的分析闭合形式。从本质上讲，很难为LPTV系统找到分析解决方案（即其过渡矩阵的封闭形式）。研究结果表明，对于给定的周期性矩阵a（t）家族，我们可以比较将谐波乘积的函数（即，ω是偶数[奇数]表示中的含量[SINE]因子的一部分，或者在[奇数]表示中的含量[SINE]因子的一部分，或在复杂表示中的指示因子中的频率均与序列相关的频率（t）和R的相关性。时间不变的系统，该系统由零频率（ω= 0）定义。
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 Φ_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 ω that multiply the harmonics (i.e., ω 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 ω and the associated linear time-invariant system, which is defined by having zero frequency (ω=0).
