Question

The closed-loop transfer function of a system is \(\frac{{C\left( s \right)}}{{R\left( s \right)}} = \frac{{s - 2}}{{{s^3} + 8{s^2} + 19s + 12}}.\)

The system is
1. Stable
2. Unstable
3. Conditionally stable
4. Critically stable

Answer 1

Correct Answer - Option 1 : Stable

Concept:

• Routh Hurwitz criterion is a good tool to measure the stability of the system.
• It makes use of the Routh table in which the coefficients of the characterstic equation of the system are placed
• If there is no sign change in the first column of routh table (all positive entries) the given system is stable
• The number of sign changes in the first column of routh table indicates the number of poles that exist in the right half of the s-plane.

Calculation:

Transfer function: \(\frac{{C\left( s \right)}}{{R\left( s \right)}} = \frac{{s - 2}}{{{s^3} + 8{s^2} + 19s + 12}}.\)

Characteristic equation of the given transfer function is

s³ + 8s² + 19s + 12 = 0

Routh table:

S3	1	19
S2	8	12
S1	\(\frac{{\left( {8 \times 19} \right) - 12}}{8}\)
S0	12

 

All entries of first column of Routh table are positive. Hence the given system is stable.