THE ROUTH TEST AND COVARIANCE CONTROL

It is well known that positive definiteness of a suitable Liapunov fun ction is equivalent to the Hurwitz-Routh test. This paper uses the sta te covariance matrix in a quadratic Liapunov function to prove the Hur witz-Routh test and to show a conservation principle relating stabilit y (the characteristic coefficients) to L(2) performance. Our proof req uires the positive definite test of two matrices approximately half th e size of the state, a savings over the usual positive definite test o f an n x n Liapunov matrix.