Question

A unit feedback system has the open loop transfer function \(G(s) = \frac {10^4}{s(s+10)^2}\). The closed loop system is
1. stable
2. marginally stable
3. unstable
4. stable at an angular frequency of 10 rad / s

Answer 1

Correct Answer - Option 3 : unstable

Routh-Hurwitz stability criterion: 

• Used to find whether the system is stable or unstable.
• Used to find the how many roots of the given system lies in left and right half of the s-plane. 

 

Calculation:

Given:

Open-loop transfer function ⇒ \(G\left( s \right) = \frac{{{{10}^4}}}{{s{{\left( {s + 10} \right)}^2}}}\)

For unity feedback ⇒ H(s) = 1

Characteristic equation ⇒ 1 + G(s)H(s) = 0

⇒ \(1 + \frac{{{{10}^4}}}{{s{{\left( {s + 10} \right)}^2}}} = 0\)

⇒ s(s+10)² + 10⁴ = 0

⇒ s³ + 20s² +100s + 10000 = 0

Routh Hurwitz table:

s3	1	100
s2	20	10000
s1	\(\frac{{20 \times 100 - 1 \times 10000}}{{20}} = - 400\)
s0	\(\frac{{ - 400 \times 1000 - 0}}{{ - 400}} = 10000\)

 

There are 2 sign change

First from 20 to -400

Second from -400 to 10000

So, two roots will lie on the right half of the s-plane.

∴ System is unstable.