Question

An open loop TF of a unity feedback system is given by

\(G\left( s \right) = \frac{1}{{{{\left( {s + 2} \right)}^2}}}\)

The closed loop transfer function will have poles at
1.  -2, -2
2. -2, -1
3. -2 + j, - 2 - j
4. – 2, 2

Answer 1

Correct Answer - Option 3 : -2 + j, - 2 - j

Concept:

The closed-loop transfer function for unity feedback is

\(TF = \frac{{G\left( s \right)}}{{1 + G\left( s \right)}}\)

G(s) = open loop gain

for closed loop poles 1 + G(s) = 0

Calculation:

Given that

\(G\left( s \right) = \frac{1}{{{{\left( {s + 2} \right)}^2}}}\)

Now closed-loop transfer function is

\(TF = \frac{{\frac{1}{{{{\left( {s + 2} \right)}^2}}}}}{{1 + \frac{1}{{{{\left( {s + 2} \right)}^2}}}}} = \frac{1}{{{{\left( {s + 2} \right)}^2} + 1}}\)

\(TF = \frac{1}{{{s^2} + 4s + 5}}\)

Now for close loop pole 1 + G(s) = 0

s² + 4s + 5 = 0

s = -2 ± j = -2 + j, -2 - j