Question

③ The transfer function T(s): represents a GPI> System with a unit slep response of 25%overshoot a setling iime T isec. Th ess -o. Determine the constants K ,k2-_-)k3= nd e steady sae error

Answer 1

Ans)

Given $T(s)=rac{K_{1}}{s^{2}+K_{2}s+K_{3}}$

The steady state error is given as

For step input R(s)=1/s

K1 ess (1 0.1

$rac{K_{1}}{K_{3}}=1-0.1=0.9$

$rac{K_{1}}{K_{3}}=0.9$

From the denominator of transfer function i.e $s^{2}+K_{2}s+K_{3}$ comparing with standard second order equation i.e

Given settling time

$rac{K_{2}}{2sqrt{K_{3}}}*sqrt{K_{3}}=4$

$rac{K_{2}}{2}=4$

$mathbf{K_{2}=8}$

Given overshoot

$zeta=0.404=rac{K_{2}}{2sqrt{K_{3}}}$

using $K_{2}=8$

$zeta=0.404=rac{8}{2sqrt{K_{3}}}$

$K_{3}=(rac{8}{2*0.404})^{2}=98.17$

$mathbf{K_{3}=98.17}$

Now using $rac{K_{1}}{K_{3}}=0.9$

$K_{1}=0.9K_{3}=0.9*98.17=88.35$

$mathbf{K_{1}=88.35}$$mathbf{K_{2}=8}$$mathbf{K_{3}=98.17}$