Design and Build a DC Motor Control System Using Simulink Blocks $-$ Exam Question: A DC motor drives a factory conveyor belt system, and its speed needs to be controlled to ensure consistent product delivery. The motor is part of a feedback control system where the speed is controlled using a PID controller. However, the system currently only uses proportional control and experiences slow response and steady$-$state error when load conditions change. You are tasked with analyzing the current system, improving its performance, and building the circuit to control the conveyor belt$$s speed using appropriate Simulink blocks. Tasks: $1.$ System Analysis : $-$ You are provided with the following parameters for the DC motor system: $-$ Motor Inertia J $=0\mathrm{.}02$ kg$$m$^2-$ Motor Resistance R $=1-$ Motor Inductance L $=0\mathrm{.}5$ H $-$ Back EMF constant K$\_$b $=0\mathrm{.}01$ V$$s$/$rad $-$ Damping coefficient B $=0\mathrm{.}05$ N$$m$$s $-$ Desired speed $\_$desired $=100$ rad$/$s $-$ Predict how the current proportional$-$only control system will behave when there is a change in load. $-$ Discuss the role of inertia and damping in the motor$$s performance and how they impact the system$$s response time and stability. $2.$ Build the Control Circuit: You are now tasked with building a control circuit to improve the conveyor belt system$$s performance. The system should regulate the speed of the DC motor using a PID controller with feedback from the motor$$s velocity sensor$.-$ Build the DC motor control system using appropriate Simulink blocks to: $-$ Use a PID controller to control the motor$$s speed. $-$ Provide an error signal based on the difference between the desired and actual speed of the conveyor. $-$ Visualize the system$$s performance using appropriate blocks to monitor the motor$$s speed over time. $3.$ Tuning the Controller: After building the control system, tune the PID controller to improve system performance. Specifically: a$.$ Explain how increasing the proportional gain K$\_$p affects the system$$s response. b$.$ What role does the integral gain K$\_$i play in reducing steady$-$state error? c$.$ How would increasing the derivative gain K$\_$d affect the system$$s stability and overshoot? $4.$ System Behavior Prediction: Once you$$ve tuned the PID controller, predict how the system will behave if the load on the conveyor suddenly increases: $-$ How will the control system react to maintain the desired speed? $-$ Will the system return to the desired speed, and how long will it recover? Governing Equations: $1.$ DC Motor Electrical Equation: V $=$ L di$/$dt $+$ R i $+$ K$\_$b Where V $=$ input voltage $($V$),$ L $=$ motor inductance $($H$),$ R $=$ motor resistance $(),$ i $=$ motor current $($A$),$ Kb $=$ back EMF constant $($V$$s$/$rad$),$ $=$ angular velocity $($rad$/$s$).2.$ Torque Equation for Motor: T$\_$m $=$ J d$/$dt $+$ B $+$ T$\_$l Where Tm $=$ motor torque $($N$$m$),$ J $=$ rotational inertia $($kg$$m$^2),$ B $=$ damping coefficient $($N$$m$$s$),$ Tl $=$ load torque $($N$$m$).3.$ Rotational Dynamics: $=$ dt Where, $=$ angular position $($rad$),$ $=$ angular velocity $($rad$/$s$).4.$ Control System Equation: V$\_$control $=$ K$\_$p e$($t$)+$ K$\_$i $$e$($t$)$ dt $+$ K$\_$d de$($t$)/$dt Where: e$($t$)=$ error signal $($difference between desired and actual position$),$ K$\_$p$,$ K$\_$i$,$ K$\_$d $=$ PID controller gains.