5.5 Routh Stability Criterion

The stability of the system is determined from the values of the roots (poles) of its characteristic equation. Finding roots is easy for first- and second-order equations (and not too hard for third) since there is an analytical solution for the roots of polynomials through third order. If the polynomial is fourth order or higher, the roots must be determined numerically. There is a method for determining whether any of the roots are positive (unstable) without actually calculating the roots (Routh, 1905). This method involves an analysis of the coefficients of the characteristic polynomial by setting up the Routh array. The test of the coefficients in the Routh array is called the Routh stability criterion.

The Routh (pronounced like truth) stability criterion is based on a polynomial equation that has the following form

The necessary condition for all the roots to be negative is that all a_i > 0 (you can multiply all coefficients by –1 if needed). The sufficiency test requires the Routh array

where the elements of rows 3 and higher have the form

If all the coefficients in the first column of the Routh array are positive, then the sufficient condition for stability is satisfied. If all the coefficients are not positive, then we can determine the number of unstable (positive) roots by the number of changes in the sign of the coefficients as we move down the first column. An example of the use of the Routh array to determine limits of a tuning parameter for closed-loop stability is shown next.

Example 5.3: Third-Order Process with a P-Only Controller

Find the upper bound on proportional gain (for P-only control) for closed-loop stability of the following process

The closed-loop transfer function is

We must use the Routh stability criterion on the characteristic polynomial. The coefficients are

The Routh array is

Inserting our values into the Routh array,

we see from the necessary condition that 1 + k_c > 0, or k_c > 0. The sufficient condition that must be checked is b₁ > 0

which leads to k_c < 10 for stability. The range of tuning parameters for stability is then

As a practical matter, however, the controller gain should be the same sign as the process gain (recall that in Example 5.1 a controller gain with the wrong sign led to a process output response in the opposite direction of the setpoint change). Simulation results for a unit step setpoint change are shown in Figure 5-15. As k_c is increased, the closed-loop response becomes more oscillatory. The closed-loop system will lose stability at k_c = 10.

Indeed, in Table 5-1 we show the relationship between k_c and the closed-loop poles. Notice that one pole stays real and becomes more negative (faster) as k_c increases. The other two poles become complex and the magnitude of the imaginary portion becomes larger as k_c increases, indicating that the magnitude of oscillation is increasing. At k_c = 10, the complex poles cross from the left-half plane to the RHP, indicating that the closed-loop system is unstable for k_c > 10.

Figure 5-16 is a plot of the roots of the characteristic equation, as a function of k_c. It shows vividly the results presented in Table 5-1. The closed-loop poles start as three real, distinct poles at k_c = 0. Two become complex, and move toward the RHP as k_c is increased. When k_c > 10, the two complex poles move into the RHP; at k_c = 10, a continuous oscillation is formed.