5.1 Routh Stability Analysis

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

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

Limitations of Routh Test:

  • Can’t be used for dead-time systems.

  • Only indicates stability; doesn’t tell about location of roots.