MAE 106 Laboratory Exercise #5MAE 106 Laboratory Exercise #5PD Control of Motor Position University of California, IrvineDepartment of Mechanical and Aerospace EngineeringREQUIRED PARTS:Qty Parts Equipment2 1k resistor, ¼ W (brown/black/red) Breadboard2 10k resistor, ¼ W (brown/black/orange) Oscilloscope2 100kresistor, ¼ W (brown/black/yellow) Function Generator1 LM 324 quad op amp chip Motor-Amp-Tach Console4 1F capacitors Position-sensing “pot”1 BNC cable IC puller1 breakout (BNC to alligator clips) wrist grounding strap2 banana-to-banana cable (1 black, 1 red) multimeter2 banana-to-alligator clip cable (1 black, 1 red) scope probevar wire, 22AWG1 1/4” shaft coupling (c.1/2”lg)1 IntroductionIn this lab you will build a control system to make a motor shaft move to a position thatyou command. Controlling motor position is a common goal in automation (e.g. multi-joint robot arms, radars, numerically controlled milling machines, manufacturingsystems). In addition, you will need a position controller for your final project. The controller that you will build is called a “Proportional Plus Derivative (PD) PositionFeedback System,” and is the most common controller found in industry. The PD controllaw is:ddpKK  )((1)Where  = actual motor angular positiond = desired motor angular position= actual motor angular velocityKp = position error gainKd = derivative gain= desired motor torqueNote that the controller has two terms – one proportional to the position error (the “P”part), and one proportional to the derivative of position (i.e. velocity, the “D” part). Thus,it is called a “PD” controller.1Kp 1/Js2 Kds d  + + - - -(-d) -Kp(-d) Kd . -Kp(-d) - Kd . C=1F + - 100k + - 100k Function Generator Amp motor pot + - R1= 1k R1= 1k R2= 10k d -d -Kp(-d) - Kd .  Op amp 1 Op amp 2 Op amp 3 Figure 1 – PD Motor Position Control System (Block Diagram and Circuit)Figure 1 shows the block diagram and op-amp circuit that you will implement to makethe PD control law for the motor. J is the inertia of the motor shaft. The lab has the following four parts. You can do Parts 1 and 2 before coming to lab.Part 1: What is the theoretical behavior of the controlled system?The key point to understand here is that the controlled system obeys the samedifferential equation as a mass-spring-damper system. Thus, the controlled system actsdynamically like a mass-spring-damper system. The control gains Kp and Kd determinethe equivalent stiffness and damping of the system. The desired angular position of themotor (d) is equivalent to the rest length of the spring.Part 2: How can a circuit implement the control law?Op-amp circuits (adder, gain, inverter derivative circuits) can be used to implement thecontrol law. The resistors and capacitors set the control gains Kp and Kd.Part 3: What is the step response of the actual system? One way to characterize the system behavior is to measure how it performs when it iscommanded to move rapidly from one position to another (i.e. to follow a step functioninput). You will find that the motor will overshoot and oscillate if the damping is too small.Part 4: What is the frequency response of the actual system? Another common way to characterize the system behavior is to measure how it performswhen it is commanded to follow a sinusoidal position. You will find that the controlleracts like a low pass filter. It tracks low input frequencies well, and high frequenciespoorly. Also, if the controller has low enough damping, it will resonate just like thespring-mass-damper system you experimented with in Lab 4 (i.e. the vibrating beam).22 What is the theoretical behavior of the controlledsystem? In this section, you will derive the theoretical behavior the PD position controlled motor.In the time domain, the theoretical behavior is described by a differential equation. Inthe frequency domain, the theoretical behavior is described by the frequency response.Q1 Derive the dynamical equation that describes how  evolves with time when thecontroller is attached to the motor. Assume d is the input. Q2 Derive the differential equation for a spring-mass-damper system (assume force isthe input and position is the output). The differential equation for Q1 should besimilar to the equation for Q2. This means that the PD position control system hasthe same dynamics as a spring-mass-damper system; i.e. it follows the sameequations of motion. Thus, you can use your intuition about how the spring-mass-damper system works to design the PD controller. Explain what the mass (m),spring (k), damper (c) in the mechanical system correspond to in the PD system. Q3 Derive the closed-loop transfer function, G(s), for the controlled system (the input isd, the output is ). Use either block diagram algebra (applied to the block diagramfrom Figure 1) or take the Laplace Transform of the differential equation that youderived in Q1.Q4 Express the damping ratio and natural frequency of the system in terms of thecontrol gains and motor inertia. The damping ratio is important because itdetermines whether the system oscillates. The natural frequency determines thefrequency at which it oscillates.P1 Plot the predicted response of the system to a step change in d from 0 to 1radians, for damping ratios of 0.1, 1.0, and 2.0. P2 Plot the predicted frequency response (both scaling and phase shift) for dampingratios of 0.1, 1.0, and 2.0. Do this on a Bode plot by plotting {20log(outputamplitude/input amplitude)} vs. {input frequency on a log scale}, and {phase shift}vs. {input frequency on a log scale}. 3 How can a circuit implement the control law? To implement the PD control law, you need to build the circuit shown in Figure 1.Q5 By applying the op-amp golden rules, show that the input to the motor amplifier is:CRRRVoutd 212)( The derivation will be easier if you substitute the impedance 1/sC for thecapacitor then treat it as a resistor in the frequency domain, then transform backto the time domain.3Q6 Compare this equation with the control law of equation (1). What are KP and Kd interms of the electronic components (resistor and capacitor