A second order open$-$loop system is given by

X$($s$)=$ G$($s$)$U $($s$),$ G$($s$)=1$

s$2$ s $+1.(1)$

$1.$ According to Routh$$s necessary conditions for stability, is the open$-$loop system stable?

$2.$ Consider the PID control

U $($s$)=$ C$($s$)($R$($s$)$ X$($s$)),$ C$($s$)=$ kp $+$ kds $+$ ki

s $.(2)$

Obtain the closed$-$loop transfer function X$($s$)$

R$($s$)=$ GCL$($s$)=$ BCL$($s$)$

ACL$($s$).$ You must write down the

polynomials BCL$($s$)$ and ACL$($s$)$ explicitly.

$3.$ Apply Routh$$s necessary conditions for stability to the closed$-$loop characteristic equation

ACL$($s$)=0.(3)$

$4.$ Indicate which term of kp$,$ kd and ki must be included in the PID control to stabilize the

open$-$loop system.

$5.$ Consider only PD control, i$.$e$.$ C$($s$)=$ kp $+$ kds$.$ Find the relationship between kp and kd

such that the closed$-$loop system is critically damped