# A linear time-invariant system, initially at rest, when subjected to a unit step input at t = 0

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

### Right Answer is: S/(S + 1)2

#### SOLUTION

To get the transfer function from step response we need to get the impulse response of the system.

That is, the impulse response h(t) of a system is equal to the derivative of its step response s(t). Therefore, the impulse response of a system can be determined from its step response simply through differentiation.

$\begin{array}{l}h(t) = \dfrac{d}{{dt}}(yt)\\\\= \dfrac{d}{{dt}}(t{e^{ – t}})\end{array}$

h(t) = e−t − te−t

Laplace transform of e−t = 1/(S + 1)

Laplace transform of te−t = 1/(S + 1)2

$\begin{array}{l}h(t) = \dfrac{1}{{S + 1}} – \dfrac{1}{{{{(S + 1)}^2}}}\\\\h(t) = \dfrac{S}{{{{(S + 1)}^2}}}\end{array}$

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