# 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 |
+ |
− |
- |
+ |
− |

- The system has three roots in the right-half of s-plane
- The system has three roots in the left -half of s-plane
- The system is stable
- The system is unstable

### 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.