分段线性动态系统周期轨道的时域法求解及其稳定性分析

A New Method of Time Domain Solution and Stability Analysis for Periodic Orbits of Piecewise Linear Dynamic Systems

该文提出了分段线性动态系统周期轨道的时域法求解及稳定性判断的新方法。分段线性动态系统的状态空间被切换面分割成若干个线性子区间。借助MATLAB,联合求解周期轨道在各子区间的状态转移方程,可得该周期轨道在各切换面的切换点坐标及在各子区间的运行时间,从而得到该周期轨道的分段时间表达式。由该表达式,可导出该周期轨道在某一切换面的庞加莱映射方程及其雅可比矩阵,根据其特征值可判断周期轨道的稳定性。以三阶、四阶蔡氏电路为例,用该方法求出了它们的多个周期轨道,进行了稳定性判断,数字仿真表明该文所提出的新方法是可行的和正确的。

This paper proposes a new method to get time solutions of periodic orbits and to determine their stability for piecewise linear dynamic systems. The state space of piecewise linear dynamic system is cut into some linear subspaces by several switching surfaces. By solving together all the equations of periodic orbit in these subspaces with MATLAB, the coordinates of periodic orbit on each switching surface and the running time on each subs,pace are obtained, from which the time expressions in sections of periodic orbit can be derived. Based on these expressions, the Poincare mapping equation and the Jacobian matrix of periodic orbits can be deduced. According to the eigenvalues of the Jacobian matrix, the stability of the periodic orbit can be determined. Using 3rd-order and 4th-order Chau＇s circuits as examples, the time expressions of many periodic orbits are obtained and their stability is determined respectively by the new method. The results are exact the same as that of digital simulations, which shows the new method is correct and practical.