Necessary and Sufficient Conditions for Stability (RH-Criterion)

 

Routh–Hurwitz Stability Criterion

The Routh–Hurwitz criterion is a mathematical test that determines whether all roots of a characteristic polynomial lie in the left half of the s-plane (i.e., whether the system is stable).

Routh–Hurwitz Polynomials

For a characteristic polynomial:

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

• The system is stable if all roots have negative real parts.

• The Routh–Hurwitz method constructs a tabular array (Routh array) to check stability without solving for roots explicitly.

Necessary and Sufficient Conditions for Stability

1. Necessary Condition:

   • All coefficients of the characteristic polynomial must be positive and non-zero.

   • If any coefficient is negative or zero, the system is unstable.

2. Sufficient Condition (Hurwitz Determinants):

   • All Hurwitz determinants (principal minors of the Hurwitz matrix) must be positive.

   • This ensures that all poles lie in the left half-plane.

Routh Array Construction

• First row: coefficients of even powers of $s$.

• Second row: coefficients of odd powers of $s$.

• Remaining rows: calculated using the formula:

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

• Stability check: No sign changes in the first column → system is stable.

Example Problem

Characteristic equation:

$$s^4+3s^3+3s^2+2s+1=0$$

Routh Array:

Row	Elements
$s^4$	1 , 3 , 1
$s^3$	3 , 2 , 0
$s^2$	$\frac{(3\cdot 3-1\cdot 2)}{3}=\frac{7}{3}$ , 1
$s^1$	$\frac{(\frac{7}{3}\cdot 2-3\cdot 1)}{\frac{7}{3}}=\frac{5}{7}$ , 0
$s^0$	1

First column: 1, 3, 7/3, 5/7, 1 → all positive → System is stable.

Key Points

• Necessary condition: All coefficients positive.

• Sufficient condition: All Hurwitz determinants positive.

• Routh array provides a practical way to check stability and count unstable poles.