Given a mass$-$spring$-$damper system with $m=1kg,c=8\mathrm{.}8Kg\frac{s}{m},$ and $k=40\frac{N}{m}.$ Obtain

The transfer function $G(s).$

The open loop transfer function G $1($s$)$ considering a proportional controller

The closed$-$loop transfer function if $Kp=10$

The steady$-$state for $3).$

The steady$-$state error for $3).$

The pole$-$zero plane and depict the poles of the system

The closed$-$loop transfer function if $Kp=50$

The steady$-$state for $3).$

The steady$-$state error for $3).$ Is the error bigger or $($arger than $5)$

Based on the pole$-$zero plane for each case, which value of $Kp$ will result in oscillations, $50$ or $10?$ Explain.

The open loop transfer function for a Proportional$-$Integral controller.

The closed$-$loop transfer function for a Proportional$-$Integral controller.

Determine the steady state error of the closed$-$loop transfer function.