2. Routh-Hurwitz Criterion for Stability

The criterion of stability for the closed-loop systems does not require calculations of the actual values of the roots of the characteristic polynomial. It only requires that we know if any root is to the right of the imaginary axis.

For the characteristic equation (with \(a_0\) as positive) \[a_0s^n + a_1s^{n-1}+\cdots+a_{n-1}s + a_n = 0\]

First test: If any of the coefficients of the characteristic equation \(a_1,a_2,\ldots,a_{n-1},a_n\) is negative, then there is at least one root of the characteristic equation which has a positive real part, and the corresponding system is unstable. No further analysis is needed.

Second test: If all coefficients \(a_1,a_2,\ldots,a_{n-1},a_n\) are positive, then the following analysis by using Routh array is to be made.

The elements of Routh array are written as

Row 1 \(\quad a_0\) \(a_2\) \(a_4\) \(a_6\) \(\qquad \cdots\)
2 \(\quad a_1\) \(a_3\) \(a_5\) \(a_7\) \(\qquad \cdots\)
3 \(\quad A_1\) \(A_2\) \(A_3\) \(\cdot\) \(\qquad \cdots\)
4 \(\quad B_1\) \(B_2\) \(B_3\) \(\cdot\) \(\qquad \cdots\)
5 \(\quad C_1\) \(C_2\) \(C_3\) \(\cdot\) \(\qquad \cdots\)
\(\cdot\) \(\quad \cdot\) \(\cdot\) \(\cdot\) \(\cdot\) \(\qquad \cdots\)
\(\cdot\) \(\quad \cdot\) \(\cdot\) \(\cdot\) \(\cdot\) \(\qquad \cdots\)
\(n+1\) \(\quad V_1\) \(V_2\) \(\cdot\) \(\cdot\) \(\qquad \cdots\)

where

\( A_1=\frac{a_1a_2-a_0a_3}{a_1}\) \( \quad A_2=\frac{a_1a_4-a_0a_5}{a_1}\) \( \quad A_3=\frac{a_1a_6-a_0a_7}{a_1} \quad \cdots\)
 \( B_1=\frac{A_1a_3-a_1A_2}{A_1}\) \( \quad B_2=\frac{A_1a_5-a_1A_3}{A_1}\) \( \cdots\)
 \( C_1=\frac{B_1A_2-A_1B_2}{B_1}\) \( \quad C_2=\frac{B_1A_3-A_1B_3}{B_1}\) \( \cdots\)
etc.

Examine the elements of the first column of the Routh array: \[a_0 \quad a_1 \quad A_1 \quad B_1 \quad C_1 \quad \ldots \quad V_1\]

  • If any of these elements is negative, we have atleast one root to the right of the imaginary axis and the system is unstable.

  • The number of sign changes in the elements of the first column is equal to the number of roots to the the right of the imaginary axis.

Therefore, a system is stable if all the elements in the first column of the Routh array are positive.