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

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

s$2+0$:$2$s $+1$: $(1)$

Consider the PID control

U$($s$)=$ C$($s$)($R$($s$)$ X$($s$))$;

C$($s$)=$ kP $+$ kDs $+$ kI

s

: $(2)$

$1.$ What is the damping ratio of the open$-$loop system?

$2.$ 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.

$3.$ Find all the necessary and sufficient conditions for stability of the closed$-$loop system. You

must show the Routh array to derive the sufficient conditions.

$4.$ Hard Problem $[10$pts$]$

If you only use PD control, i$.$e$.$ C$($s$)=$ kP $+$ kDs$,$ find the relationship between kP and kD

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