Closed form expressions for linear MIMO system responses and solutions of the Lyapunov equation

In this paper, closed form expressions of linear MIMO system responses are presented. These expressions make use of matrix polynomial formulations of e^tA. They depend on the eigenvalues of A, the characteristic polynomial of A, as well as the partial fraction expansion coefficients of the corresponding unity numerator transfer function. The general partial fraction coefficients may be calculated using a computationally efficient recursive formula that has been derived in the context of SISO system responses. The final expressions are presented in a form that enhances the understanding of linear systems as such and emphasizes efficient computational implementation and the resulting time complexity. Finally, a closed form solution of the Lyapunov equation, the Gramian, is presented.