Linear time-varying systems—state transition matrix

The state transition matrix of a linear time-varying system cannot, in general, be expressed in a closed form and has, therefore, to be evaluated numerically. For the commutative class of linear time-varying systems, the state-transition matrix is the exponential matrix. A numerical procedure is developed for the evaluation of this matrix to any desired degree of accuracy by the method of series expansion. For linear time-varying systems, which are not restricted to belonging to the commutative class, an efficient computational algorithm is developed for the evaluation of the state-transition matrix. This is based on the minimum m.s.e. approximation of a time function in terms of a set of block-pulse functions, which are orthogonal in the speficied interval. The algorithms developed in the paper are illustrated by appropriate examples.