由《现代控制理论》教材习题引发的讨论

Discussion Caused by Several Exercises in Modern Control Theory

许多文献对状态转移矩阵eAt的性质和计算方法进行了广泛的讨论。但给定一个矩阵函数,如何判断它是否是某一连续时间线性时不变系统的状态转移矩阵,在相关的教材中却没有涉及。本文根据常微分方程解的唯一性得到了判断一个矩阵函数是某一连续时间线性时不变系统的状态转移矩阵的充分必要条件,并求出了其对应的系统矩阵。此外,若状态转移矩阵Φ(t)已知,本文也给出了求解对应的系统矩阵的三种方法。

The properties and calculation methods of state transition matrix e^At have been widely discussed in the previous literatures. However, for a given matrix function, how to judge if it is the state transition matrix of a certain continuous linear time- invariant system has not been involved in the textbooks. In this paper, based on the uniqueness of solutions of the ordinary differential equation, a sufficient and necessary condition for a matrix function to be the state transition matrix is obtained and the corresponding system matrix Ф（t） is solved. Furthermore, if the state transition matrix is known, three solution methods of the corresponding system matrix are given. The above results are illustrated by examples.