Discussion Caused by Several Exercises in Modern Control Theory

The properties and calculation methods of state transition matrix eAt 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 Φ(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.