Question

A system with impulse response $h\left(t\right)=e^{t}u\left(t\right)$ is connected as a feedback system shown in the figure.

The above system is BIBO stable for a > ______

• A. 1

Answer 1

The correct option is A 1

$y\left(t\right)=h\left(t\right)\ast \left[x\right(t)-ay(t\left)\right]$

Taking Laplace transform :

$Y\left(s\right)=H\left(s\right)\left[X\right(s)-aY(s\left)\right]$

$Y\left(s\right)[1+aH(s\left)\right]=H\left(s\right)X\left(s\right)$

$\frac{Y\left(s\right)}{X\left(s\right)}=\frac{H\left(s\right)}{1+aH\left(s\right)}$

$H\left(s\right)=L\left[h\right(t\left)\right]=\frac{1}{s-1}$

So, $\frac{Y\left(s\right)}{X\left(s\right)}=\frac{\frac{1}{s-1}}{1+\frac{a}{s-1}}=\frac{1}{s-1+a}=\frac{1}{s+a-1}$

For stability $a-1>0\left(or\right)a>1$