DEPARTMENT OF ELECTRICAL ENGINEERING UNIVERSITY OF MINNESOTA EE 4237 State Space Control Laboratory Experiment 5: PD/PID control of rigid body dynamics Objective: 1. To study the PD control of the system (Model 205a) 2. To study PID control of the system References: 1. ECP systems manual (Model 205a) Apparatus: 1. Torsion control model 205a 2. PC 3. Control Box 4. Disks and brass weights Prelab Report: 1. Read sections 6.2. 2. Make a list of steps giving some brief detail of what you want to achieve in this set of experiments (very brief). 3. Explain differences between Figures 6.2-1a and 6.2-1b. 4. Find the characteristic roots for both systems. Postlab Report: • Answer the questions asked within or at the end of the procedureecpChapter 6. Experiments 78 6.2 Rigid Body PD & PID Control This experiment demonstrates some key concepts associated with proportional plus derivative (PD) control and subsequently the effects of adding integral action (PID). This control scheme, acting on plants modeled as rigid bodies finds broader application in industry than any other. It is employed in such diverse areas as machine tools, automobiles (cruise control), and spacecraft (attitude and gimbal control). The block diagram for forward path PID control of a rigid body is shown in Figure 6.2-1a where friction is neglected.1 Figure 6.2-1b shows the case where the derivative term is in the return path. Both implementations are found commonly in application and –as the student should verify – have identical characteristic roots. They therefore have identical stability properties and vary only in their response to dynamic inputs. The closed loop transfer functions for the respective cases are: c(s) = θ(s)r(s) = khw/J kds2+kps+kis3+ khw/J kds2+kps+ki (6.2-1a) c(s) = θ(s)r(s) = khw/J kps+kis3+ khw/J kds2+kps+ki (6.2-1b) 1The student may want to later verify that for the relatively high amount of control damping in the scheme that follows – induced via the parameter kd – that the plant damping is very small.ecpChapter 6. Experiments 79PID ControllerΣ–r(s)Plantθ(s)HardwareGainkp+kis+kds1Js2khwReference Input(E.g. InputTrajectory)Output(Mass position)PIDControllerΣ–r(s)Plantθ(s)HardwareGainkp+kis1Js2khwReference Input(E.g. InputTrajectory)Output(Mass position)a) PID In Forward Pathb) PI In Forward Path, D In Return PathΣkds– Figure 6.2-1. Rigid Body PID Control – Control Block Diagram For the first portion of this exercise we shall consider PD control only (ki=0). For the case of kd in the return path the transfer function reduces to: c(s) = kpkhw/Js2+ khw/J kds+kp (6.2-2) By defining: ωn =∆ kpkhwJ (6.2-3) ζ =∆ kdkhw2Jωn = kdkhw2 Jkpkhw (6.2-4) we may express: c(s) = ωn2s2+2ζωns +ωn2 (6.2-5) The effect of kp and kd on the roots of the denominator (damped second order oscillator) of Eq (6.2-2) is studied in the work that follows.ecpChapter 6. Experiments 80Procedure : Proportional & Derivative Control Actions 1. Using the results of Section 6.1 construct a model of the plant with two mass pieces at 9.0 cm radial center distance on the bottom disk – both other disks removed. You may neglect friction. 2. Set-up the plant in the configuration described in Step 1. 3. From Eq (6.2-3) determine the value of kp (kd=0) so that the system behaves like a 1 Hz spring-inertia oscillator. 4. Set-up to collect Encoder #1 and Commanded Position information via the Set-up Data Acquisition box in the Data menu. Set up a closed-loop step of 0 (zero) counts, dwell time = 5000 ms, and 1 (one) rep (Trajectory in the Command menu). 5. Enter the Control Algorithm box under Set-up and set Ts=0.00442 s and select Continuous Time Control. Select PI + Velocity Feedback (this is the return path derivative form) and Set-up Algorithm. Enter the kp value determined above for 1 Hz oscillation (kd & ki = 0, do not input values greater than kp = 0.082) and select OK. In this and all future work, be sure to stay clear of the mechanism before doing the next step. Selecting Implement Algorithm immediately implements the specified controller; if there is an instability or large control signal3, the plant may react violently. If the system appears stable after implementing the controller, first displace the disk with a light, non sharp object (e.g. a plastic ruler) to verify stability prior to touching plant Select Implement Algorithm, then OK. 6. Select Execute under Command. Prepare to manually rotate the lower disk roughly 60 deg. Select Run, rotate about 60 deg. and release disk. Do not hold the rotated disk position for longer than 1-2 seconds as this may cause the motor drive thermal protection to open the control loop. 7. Plot encoder #1 output (see Step 6 Section 6.1). Determine the frequency of oscillation. What will happen when proportional gain, kp, is doubled? Repeat Steps 5 & 6 and verify your prediction. (Again, for system stability, do not input values greater than kp = 0.08). 2Here due to friction the system, which is ideally quasi-stable (characteristic roots on the jω axis), remains stable for small kp. For larger values, the time delay associated with sampling may cause instability. 3E.g. a large error at the time of implementation.ecpChapter 6. Experiments 818. Determine the value of the derivative gain, kd, to achieve kdkhw= 0.1N-m/(rad/s).4. Repeat Step 5, except input the above value for kd and set kp & ki = 0. (Do not input values greater than kd = 0.1). 9. After checking the system for stability by displacing it with a ruler, manually move the disk back and forth to feel the effect of viscous damping provided by kd. Do not excessively coerce the disk as this will again cause the motor drive thermal protection to open the control loop. 10. Repeat Steps 8 & 9 for a value of kd five times as large (Again, kd ≤ 0.1). Can you feel the increased damping? PD Control Design 11. From Eq's (6.2-3,-4) design controllers (i.e. find kp & kd) for a system natural frequency ωn = 1 Hz, and three damping cases: 1) ζ = 0.2 (under-damped), 2) ζ = 1.0 (critically damped), 3) ζ = 2.0 (over-damped).5 Step Response 12. Implement the underdamped controller (via PI + Velocity Feedback) and set up a trajectory for a 2500 count closed-loop Step with 2000 ms dwell time and 1 rep. 13. Execute this trajectory and plot the commanded position and encoder position (Plot them both on the