Topic: Control Systems
The unit impulse response of a system is $h(t)=e^{-t}, t\ge 0$. For this system, the steady-state value of the output for unit step input is equal to
- A. -1
- B. 0
- C. 1
- D. $\infty$
Answer: C. 1
Explanation: Input $x(t)=u(t) \Rightarrow X(s)=\frac{1}{s}$.
$h(t)=e^{-t} \Rightarrow H(s)=\frac{1}{s+1}$.
Output $Y(s) = X(s)H(s) = \frac{1}{s(s+1)} = \frac{1}{s} - \frac{1}{s+1}$.
$y(t) = 1 - e^{-t}$.
For steady state, $y(\infty) = \lim_{t\to\infty} (1 - e^{-t}) = 1$.