In this unity feedback system, plant transfer function is given as $P=\frac{10}{(s+1)(s+10)}$ Design a controller $C(s)$ satisf$-$

ying these conditions for the closed loop system:

$1$ i$)$ closed loop system should be stable

ii$)$ steady$-$state value of error $)=(r(t)-y(t)$ for a unit step function $)=(u(t)$ must be zero, e S S $=0$

$1$ iii$)$ maximum overshoot of the step response should be $\%16,0S=0\mathrm{.}16$

iv$)$ peak time $(t_p)$ of the step response should be less than $2$ seconds. $,tp<mn>2</mn>$

Try each of these controllers for this system in the given order.

Proportional controller $($P$),C_1=K$

$c_1=k$

$c_2=\frac{k}{s}$

$c_3=k+\frac{kz}{s}{}^{z}0$

Integral controller $($I$),C_2=\frac{K}{s}$

Proportional and integral controller $($PI$),)(0$

Follow these steps for your design project.

$c_3=k+\frac{kz}{s}{r}^{z}0$

A$-$Theoretic Calculations

Sketch the desired region of closed loop poles on the complex plane in order to satisfy design conditions $($i$),($iii$)$

and $($iv$).,$ wd bulunacak. Isteren alani, kompleks d$$clende tarayocolesin.

For the controller $C_1$; $)=(2$

Show that if it is possible to satisfy condition $($ii$).$ ess $=0=>$ type $0,$ type $1,$ typ e$2$ find volve theopen

$(4)$ Find the range of $K$ values satisfying condition $($iv$),dvec(K)a_{\text{r}\text{e}\text{l}\text{i}}$vec$(g{)}^n$ bul.

Find the range of $K$ values satisfying conditions $($iii$)$ and $($iv$).$

aratigini

For the controller $C_2$;

$KRH=0C_{3}KzK$

Find the range $ofK$ values satisfying condition $(i).$ Stoble $RH$

Show that $ifitis$ possible $to$ satisfy condition $(ii).$ ess $=0$ type$0,tpe1,$ type $2$

Sketch the root locus $by$ hand.

For the controller $C_3$;

Find the range $ofK$ values satisfying condition $(i)in$ terms $ofz.$

$K$ aralig i bul.