Correct Answer - Option 3 : 1
Concept:
Laplace's initial value theorem:
\(\mathop {\lim }\limits_{t \to 0} f\left( t \right)\;\; = \mathop {\lim }\limits_{s \to ∞ } sL\left[ {f\left( t \right)} \right]\;\;\)
or
\(\mathop {\lim }\limits_{t \to 0} f\left( t \right)\; = \mathop {\lim }\limits_{s \to ∞ \;} sF\left( s \right)\;\)
Laplace's final value theorem:
\(\mathop {\lim }\limits_{t \to ∞ } f\left( t \right)\; = \mathop {\lim }\limits_{s \to 0} sF\left( s \right)\;\)
or
\(\mathop {\lim }\limits_{t \to ∞ } f\left( t \right)\; = \mathop {\lim }\limits_{s \to 0} sF\left( s \right)\;\)
Explanation:
The unit impulse response is given as:
h(t) = e-t ; t > 0
The closed-loop transfer function of a control system
\(\frac{{{\rm{C}}\left( {\rm{s}} \right)}}{{{\rm{R}}\left( {\rm{s}} \right)}}{\rm{\;}} = \frac{{\rm{1}}}{{1{\rm{\;}}+{\rm{\;s}}}}\)
r(t) = Unit step input
Laplace of u(t)
\(R\left( s \right) = \frac{1}{s}\)
\({\rm{C}}\left( {\rm{s}} \right){\rm{\;}} = \frac{{\rm{1}}}{{1{\rm{\;}}+{\rm{\;s}}}}R\left( s \right)\)
Laplace final value theorem:
\(\mathop {\lim }\limits_{t \to ∞ } f\left( t \right)\; = \mathop {\lim }\limits_{s \to 0} sF\left( s \right)\;\)
\(\mathop {\lim }\limits_{t \to ∞ } c\left( t \right) = c\left( ∞ \right) = \;\mathop {\lim }\limits_{s \to 0} sC\left( s \right)\;\)
\(c\left( ∞ \right) = \;\mathop {\lim }\limits_{s \to 0} \frac{{s}}{{s\left( {1 + s} \right)}}\;\)
c(∞) = 1
