Closed form solutions of the Sylvester and the Lyapunov equations - closed form Gramians

In this paper, closed form expressions of linear MIMO system responses are used to express the solution of the Sylvester equation in closed form. The solution makes use of matrix polynomial formulations of e^tA. The solutions to the input and the output Lyapunov equations, i.e. the input and the output Gramians, are then presented as a special case. From these the Hankel singular values can be computed. The final expressions are presented in a form that emphasizes efficient computational implementation and the resulting time complexity.