IV.3.3 Routh Hurwitz Criterion for Stability

The Routh Hurwitz criterion does not calculate the actual values of closed loop poles, rather it calculates whether any of the poles is on the left hand side of the imaginary axis of complex plane. In other words, the Routh Hurwitz criterion find whether a closed loop transfer function is stable. In this process, it also finds a limiting condition for controller parameters which would ensure stability for closed loop system. The denominator of a closed loop process transfer function is termed as characteristic equation . The Routh Hurwitz criterion works on the characteristic equation as follows:

Step 1.  Expand the characteristic equation into a polynomial

(IV.44)

Step 2.  If is negative, multiply the whole polynomial with

Step 3.  If any of the coefficients is negative, then there exists at least one unstable pole in the closed loop system. No further analysis is required.

Step 4.  If all the coefficients are positive, then form the following table, called Routh Array .