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