Circuit Design Using Simulink and Simscape Blocks $-$ Exam

Question:

You are tasked with designing a control system for an automated crane used in a warehouse

to lift and move objects. The crane operates with two main movements:

$1.$ Vertical movement $($lifting$/$lowering the load$)$ controlled by a DC motor driving a pulley.

$2.$ Horizontal movement $($moving the crane arm$)$ controlled by a hydraulic actuator.

The system should use feedback from position and velocity sensors for both vertical and

horizontal movements. A PID controller should be used to control both movements,

ensuring precise positioning and minimal overshoot.

Governing Equations:

$1.$ DC Motor Electrical Equation $($Vertical Movement$)$:

V $=$ L di$/$dt $+$ R i $+$ K$\_$b

Where: V $=$ input voltage $($V$),$ L $=$ inductance $($H$),$ R $=$ resistance $(),$ i $=$ current $($A$),$ K$\_$b $=$

back EMF constant $($V$$s$/$rad$),$ $=$ angular velocity $($rad$/$s$).$

$2.$ Torque Equation for Motor:

T$\_$m $=$ J d$/$dt $+$ B $+$ T$\_$l

Where: T$\_$m $=$ motor torque $($N$$m$),$ J $=$ inertia $($kg$$m$^2),$ B $=$ damping coefficient $($N$$m$$s$),$ T$\_$l

$=$ load torque $($N$$m$).$

$3.$ Pulley Dynamics $($Vertical Movement$)$:

v$\_$lift $=$ r

Where: v$\_$lift $=$ vertical velocity $($m$/$s$),$ r $=$ pulley radius $($m$),$ $=$ angular velocity $($rad$/$s$).$

$4.$ Hydraulic Actuator Force Equation $($Horizontal Movement$)$:

F $=$ P A

Where: F $=$ force generated $($N$),$ P $=$ pressure difference $($Pa$),$ A $=$ piston area $($m$^2).$

$5.$ Horizontal Velocity Equation:

v$\_$horizontal $=$ F$/$m

Where: v$\_$horizontal $=$ horizontal velocity $($m$/$s$),$ F $=$ force $($N$),$ m $=$ mass $($kg$).$

$6.$ PID Controller 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.

Tasks:

$1.$ Block Selection and Circuit Creation:

Design the system for the following :

$-$ DC Motor block for the vertical movement.

$-$ Hydraulic Actuator block for horizontal movement.

$-$ Maintain the desired position for both axes.

$2.$ Challenge Section $$ Dynamics:

a$.$ For the DC motor: The load mass is $200$ kg$,$ and the pulley radius is $0\mathrm{.}2$ meters. Calculate

the motor torque required to lift the load at a speed of $0\mathrm{.}5$ m$/$s$.$

b$.$ For the hydraulic actuator: The piston area is $0\mathrm{.}01$ m$^2,$ and the pressure difference is

$150$ bar.

$3.$ System Feedback and Stability:

$-$ Explain how the PID controller stabilises the system