Assuming a PI controller with Kp$,$ i respectively gain and Intergral time for Proportional and integral controller $($PI$).$

Assuming a first order plant model with time constant and K process Gain

$1)$What is the looptransferfunctionofthe systemsince it$$s defined by

where C $($s$)$ and R $($s$)$ are the laplacefunction for output process and

input signal

$2)$ Determine the order of the laplaceTransfer function $($G$($s$))$

$3)$ Assuming i $=0,5$and KpK $=0,5,$ what would be the process response shape $?$

$4)$ By applying a step inputon input signal $($R$)$ with M magnitude value, calculate the related steadystateerroress which means the value of e$($t$)$ while$$ t$$

$5)$ Is the ess value in compliance with your own expectation $?$