Background: Autonomous vehicle is a mobile robot integrating multi$-$sensor navigation and positioning, intelligent decision making and control technology. Path tracking is an important function of an autonomous vehicle. The path$-$tracking control of an autonomous vehicle is one of the most difficult automation challenges because of the constraints on mobility, speed of motion, high$-$speed 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 $H(s)G(s).$

Problem #$1$: Given the following Closed Loop Control System Model for lateral path error $($note: no PID is introduced at this point$),$

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. Ideally, the output $y(t)$ should follow the input $sin(t).$ However, this model does not yield an ideal situation over a long period of time. Try tin$[0,2]$ seconds, tin$[0,5]$ seconds, and tin$[0,20],$ etc.