Question

A linear time-invariant system initially at rest, when subjected to a unit-step input, gives a response y(t) = te^-t, t > 0. The transfer function of the system is:

1. \(\frac{1}{{{{\left( {s + 1} \right)}^2}}}\)
2. \(\frac{1}{{{{s\left( {s + 1} \right)}^2}}}\)
3. \(\frac{s}{{{{\left( {s + 1} \right)}^2}}}\)
4. \(\frac{1}{{s + 1}}\)

Answer 1

Correct Answer - Option 3 : \(\frac{s}{{{{\left( {s + 1} \right)}^2}}}\)

Concept:

A transfer function is defined as the ratio of the Laplace transform of the output to the Laplace transform of the input by assuming initial conditions are zero.

TF = L[output]/L[input]

\(TF = \frac{{C\left( s \right)}}{{R\left( s \right)}}\)

Calculation:

The step response = te-t, t > 0

By applying the Laplace transform, we get

Transfer function \( =\frac{ \frac{1}{({s + 1})^2}}{\frac{1}{s}}\)

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