GATE ECE 1997 | Question 6

Question

 

The figure is shows the block diagram representation of control system. The system in block A has an impulse response $h_{A}(t)=e^{-t} u (t)$. The system in block $B$ has an impulse response $h_{\mathrm{B}}(t)=e^{{-2 f}} u(t)$. The block ' $\mathrm{K}$ ' amplifies its inputs by a factor $k$. For the overall system with input $x(t)$ and output $y(t)$
(a) Find the transfer function $\frac{\mathrm{Y}(s)}{\mathrm{X}(s)}$ when $k=1$
(b) Find the impulse response when $k=0$
(c) Find the values of $k$ for which the system becomes unstable
Note:
\[
\begin{aligned}
u (t) &=01 \leq 9 \\
&=1 t>0
\end{aligned}
\]