# The first element of each of the following row of a Routh-Hurwitz stability test showed the signs as follows

The first element of each of the following row of a Routh-Hurwitz stability test showed the signs as follows
 Row I II III IV V Sign + − - + −
Consider the following statements:
1. The system has three roots in the right-half of s-plane
2. The system has three roots in the left -half of s-plane
3. The system is stable
4. The system is unstable
Which of the above statements about the system are correct?

### Right Answer is: A and D

#### SOLUTION

Stable System: If all the roots of the characteristic equation lie on the left half of the ‘s’ plane then the system is said to be a stable system.

Unstable System: If some of the roots of the system lie on the right half of the ‘s’ plane then the system is said to be an unstable system.

Routh- Hurwitz Criterion:

Routh Hurwitz criterion states that any system can be stable if and only if all the roots of the first column have the same sign and if it does not has the same sign or there is a sign change then the number of sign changes in the first column is equal to the number of roots of the characteristic equation in the right half of the s-plane i.e. equals to the number of roots with positive real parts.

Application:

Number of sign changes in the given table = 3

Number of poles on right half of s-plane = 3

Number of poles on left half of s-plane = 5 – 3 = 2

As there are three poles on the right half of the s-lane, the system is unstable.

Number of sign change = Number of Poles Lies In RHS of S plane = 2

Hence there are two negative roots, therefore, the system is unstable.

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