The Routh test and covariance control

It is well known that positive definiteness of a suitable Lyapunov function is equivalent to the Hurwitz-Routh test. The author uses the state covariance matrix in a quadratic Lyapunov function to prove the Hurwitz-Routh test and to show a conservation principle relating stability (the characteristic coefficients) to performance (achievable root-mean-square). The proof requires the positive definite test of two matrices approximately half the size of the state, a savings over the usual positive definite test of an n*n Lyapunov matrix. For third-order all-poles systems, a closed-form expression for all stabilizing output-feedback controllers is given in terms of three arbitrary positive numbers.<>