Question

The the feedback system given below, the transfer function $G\left(s\right)=\frac{1}{(s+1{)}^2}$. The system CANNOT be stabilized with

• A. $C\left(s\right)=1+\frac{3}{s}$
• B. $C\left(s\right)=3+\frac{7}{s}$
• C. $C\left(s\right)=3+\frac{9}{s}$
• D. $C\left(s\right)=\frac{1}{s}$

Answer 1

The correct option is C $C\left(s\right)=3+\frac{9}{s}$

$ForC\left(s\right)=3+\frac{9}{s}$

characteristic equation

$1+C\left(s\right)G\left(s\right)=0$

$1+(3+\frac{9}{s})\left(\frac{1}{(s+1{)}^2}\right)=0$

$1+\left(\frac{3s+9}{s}\right)\left(\frac{1}{s^2+2s+1}\right)=0$

$s^3+2s^2+4s+9=0$

Using RH criteria
$\because 1^{st}$ column of RH table does not contains positive sign only

$\therefore$ System becoms unstable.