Closed Form Expressions for the Matrix Exponential and Solutions to State Equations based on a Fundamental Solution

Closed form expressions for solutions to MIMO systems are typically derived by means of Laplace transforms. Central to these expressions is the matrix exponential e ^At and they usually include the eigenvalues and the coefficients of the characteristic equation. In this paper, we derive such expressions by focusing, instead, on the fundamental solution of the underlying differential equation. The advantage of this is threefold. First, it simplifies the derivation of such expressions. Second, it provides a useful insight into the nature of these expressions. Third, since we present an effective procedure for the evaluation of the fundamental solution and its derivatives, it can be used as a basis for procedures to evaluate these expressions both numerically and symbolically. We first consider expressions for e ^At , when A is in the controller form, where the coefficients of the characteristic equation appear explicitly. We then make use of these to derive general O(n ⁴ ) matrix polynomial expressions for e ^At that leads to a recursive O(n ³ ) expression for e ^At b. Finally, we use these to make expressions for solutions to state equations for the SISO as well as the MIMO cases.