$-$imeun auromaton chan operation, complex interaction with the environment and typically a lack of prior information. A sample open$-$loop control transfer function that characterizes steering angle input to path error output can be written as function $G(s)$

$G(s)=\frac{(34\mathrm{.}41s^2-10\mathrm{.}52s+2419)}{s^2(s^2+2\mathrm{.}809s+5\mathrm{.}295)}$

Because $s=0$ is the double pole of the system, the system will be unstable. Therefore, a secondary system function is introduced

$H(s)=10s+1$

which yields a combined system transfer function $H(s)G(s).$ In Part I, you would discover that the output of the system deviate from the initial input signal, which means there exists errors in lateral trajectory of the autonomous vehicle. In Part II$,$ we will introduce a PID controller with transfer function

$G_c(s)=1+\frac{0\mathrm{.}5}{s}+10s$

Problem #$1$: Given the following Closed Loop Control System Model for lateral path error with PID controller $G_c(s)$

Derive the closed$-$loop system transfer function. Detailed mathematical derivation is required

Find the output response $y(t)$ of the system to a sinusoidal position input $r(t)=sin(t),$ where $=\frac{40}{13}ra\frac{d}{second}.$ You should:

$(1)$ Plot the initial input function $sin(t)$ and the output response $y(t)$ on the same figure in Matlab for comparison. You should observe that the output $y(t)$ follows the input $sin(t),$ This is an improvement compared with the output in Part il of the lab $($tab ${?}^{2}2)$