Question

A system has a complex pole pair of (-1 $\pm$ j2) and a real zero of (-3). The steady state output to a unit step input is 2. The transfer function of the system is.

Answer 1

Transfer function of the system is
$T.F.=\frac{K(s+3)}{(s+1+2j)(s+1-2j)}$
$=\frac{k(s+3)}{\left[\right(s+1{)}^2-\left(2j{)}^2\right]}$
$=\frac{K(s+3)}{(s^2+1+2s+4)}=\frac{K(s+3)}{(s^2+2s+5)}$
at steady state
${lim}_{s\to 0}s.\left(\frac{1}{s}\right)\frac{k(s+3)}{(s^2+2s+5)}=2$
$\Rightarrow \frac{K(0+3)}{(0+0+5)}=2$
$\therefore k=\frac{10}{3}$
$\therefore T.F.=\frac{10(s+3)}{3(s^2+2s+5)}$