Routh’s Stability Criterion

 

Routh’s Stability Criterion

Routh’s criterion is a mathematical test used to determine the stability of a linear time-invariant (LTI) system without explicitly calculating the roots of the characteristic equation.

Steps and Procedure

1. Write the characteristic equation of the system:

$$a_n{s}^n+a_{n-1}{s}^{n-1}+\cdots +a_{1}s+a_0=0$$

2. Construct the Routh array:

   • First row: coefficients of even powers of $s$ (starting from $a_n$).

   • Second row: coefficients of odd powers of $s$ (starting from $a_{n-1}$).

   • Remaining rows: computed using the formula:

$$b=\frac{(a\cdot c)-(d\cdot e)}{a}$$

where terms are taken from the previous two rows.

• a = the element at the top of the first column of the previous row.

• c = the element immediately to the right of a (same row).

• d = the element just below a (first column of the next row).

• e = the element immediately to the right of d (same row as d).

So, each new entry in the Routh array is computed using elements from the two rows above.

3. Check the first column of the Routh array:

   • If all elements are positive → system is stable.

   • If any element is negative → system is unstable.

   • If sign changes occur → number of sign changes = number of roots in RHP (unstable poles).

4. Special cases:

   • Zero in the first column: Replace with a small $ϵ$ and continue.

   • Entire row of zeros: Use auxiliary polynomial derived from the row above.

Numerical Problems

Example 1

Characteristic equation:

$$s^3+2s^2+3s+5=0\mathrm{<br>}\mathrm{<br>}\mathrm{<br>}\mathrm{<br>}\mathrm{<br>}$$

First column: 1, 2, 0.5, 5 → all positive → Stable system.

Example 2

Characteristic equation:

$$s^4+2s^3+3s^2+4s+5=0$$

First column: 1, 2, 1, -6, 5 → sign changes occur → Unstable system with 2 poles in RHP.

Key Takeaways

• Routh’s criterion avoids solving for roots directly.

• Stability depends on the signs of the first column.

• Number of sign changes = number of unstable poles.