Rigid Body PD and PID Control

Objectives Understand the individual effects of proportional, derivative, and integral action. Design PD and PID controllers for the 1DOF rigid-body system. Implement your first controller in LabVIEW!

Pre-Lab Derive the equations of motion for the 1DOF (mass carriage loaded with four 0.5 k g brass weights) rigid-body system that you will control in this lab. The plant configuration is shown below.

Rigid Body Plant Configuration Starting with the EOM from the previous question, rewrite it so that the control effort into the system is u ⁡ t (in units of DAC counts) and the position, velocity, and acceleration are x e , x . e , x .. e where the subscript "e" denotes the encoder. Find the plant's transfer function G ⁡ s . Consider a PID controller G c s = k p + k i s + k d s Find the closed-loop transfer function of the system below where G ⁡ s is the transfer function you found previously.

Transfer Function With k i = 0 = 0 find k p and k d such that the closed-loop system is critically damped and has a natural frequency of ω n = 2 H z . Now with k i = 0 find k p and k d such that the closed-loop system has a natural frequency of ω n = 4 H z and the following damping ratios: ζ = 0.2 ζ = 0.5 ζ = 1.0 ζ = 2.0 Compute k i such that k h w k i = 7500 N / m - sec Using the Control Design Toolkit, write a LabVIEW VI that simulates the step response of the closed-loop system. The controller gains k p , k i , and k d should be controls on the front panel so you can change each one individually and see how it affects the system's response.

Lab Procedure Configure the plant as shown above in Fig. 1 with the first mass carriage free and the other two clamped. Load four 0.5 k g brass weights onto the carriage. No springs should be attached to the carriage. Inset a PID controller structure into the control loop VI. You should implement the control algorithm in two ways: one with the differentiator in the forward loop and the other with the differentiator in an inner feedback loop. A good way to do this is with a case structure. Place the controller gains and case selector as controls on the front panel. Set k i = k d = 0 and enter the value of k p you found in step 5. After carefully checking for stability, acquire and plot the natural response of the system. Find the frequency of the oscillation. Is it what you expected? Now set k p = k i = 0 and enter the value of k d you found in step 5. Manually displace the mass carriage; what do you feel? Do not excessively coerce the carriage as this may cause the motor drive protection to open the control loop. Now enter a value of kd five times as large, and manually displace the carriage again. What do you notice now? Now implement the values of k p and k d you found for the four cases in step 6. Perform a 3000 count step for each case; choose a dwell time long enough for the system to reach steady-state. Does the system's response agree with your simulations? Save these plots and turn them in when you complete the lab. With your critically damped k p and k d parameters from above, add integral action by entering the value of k i you found in step 7. Perform a 3000 count step and save the response plot. Change the control algorithm to the second case (with the differentiator in an inner feedback loop), and plot and save the step response. What differences do you notice between this response and the case with the differentiator in the forward loop? Stop and exit the VI, power off the amplifier and PXI, and return all plant materials to the instructor.

Post-Lab What was the effect of k p in step 3 above? If you doubled your value of k p in this step what would you expect the natural frequency of the system to be? (You should show this mathematically) What was the effect of k d in step 4 above? What differences did you notice between the PD controller and the PID controller? What differences did you notice when the differentiator was in an inner feedback loop as opposed to the forward loop?