Closed-form solution of the continuous-time Lyapunov matrix equation

A simple algorithm is proposed for a closed-form solution to the continuous-time Lyapunov matrix equation using the Routh array, which is useful for the computation of controllability, observability and cross-Gramians for both SISO and MIMO systems. The proposed algorithm provides the Gramians directly from the Routh array and thus is more efficient than existing algorithms. The closed-form solutions can be obtained even for unstable nonminimal systems, provided that all the leading coefficients of the Routh array are nonzero. Numerical examples are included to demonstrate the computational procedure