MIT OpenCourseWare http://ocw.mit.edu 2.004 Dynamics and Control II Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.Massachusetts Institute of Technology Department of Mechanical Engineering 2.004 Dynamics and Control II Laboratory Session 6: Closed-Loop Position Control, and the Effect of Derivative Control Action1 Laboratory Objectives: (i) To investigate closed-loop control of the angular position (θ) of the rotational plant. (ii) To investigate the manipulation of closed-loop dynamic response through the use of derivative (D), and proportional + derivative (PD) control. (iii) To compare your experimental results with a Simulink digital simulation. Introduction: In all of our laboratory sessions to date we have investigated closed-loop control of the angular velocity of the rotational plant. In this session we switch to position control, and our goal is to command the flywheel to move to a given angular displacement. We will see that, for this particular plant, proportional control does not generate satisfactory transient behavior, and that the use of PD (proportional + derivative) control allows us to manipulate the closed-loop poles to achieve much improved response characteristics. The Experimental Setup: The set-up is very similar to that of Labs. 4 and 5, except for the substitution of a US Digital EDAC2 position sensor for the ETACH tachometer module: s e t - p P o w e r A m p m o t o r o i n t C o m p u t e r A C H 0 A C H 1 D A 0 O U T p o s i t i o n s e n s o r E D A C 2 r o t a r y e n c o d e r r e s e t The EDAC2 produces an output voltage that is proportional to the angular position of the flywheel. Optical encoders, such as those used in this system, are incremental – that is, they do not have an inherent absolute zero position. You will notice a RESET button on the side of the module; this is used to set the current position of the flywheel as the “home” or zero position. Experiment #1: Calibrate the Position Sensor: Use the DVM to calibrate the EDAC2 position sensor. Obtain the gain factor Kpos (with appropriate units), and investigate the 1March 28, 2008 1r ( t ) y ( t ) d t range of angular displacement over which there is a linear relationship between angle and output voltage. What happens when you exceed this limit? Experiment #2: Proportional Control of Position: Using the information in Ap-pendix A, derive the closed-loop transfer function for the system with proportional control. Set up your plant with proportional control (Ki =0, Kd = 0), with a single damping magnet, using a 1 v DC signal from the function generator, and investigate the step response for Kp =1, 2, 3, 4, 5. Remember to reset the EDAC2 between tests. Make a single plot of all the responses. Would you classify the response as “satisfactory”? Examine your closed-loop characteristic equation. Is it possible to move the closed-loop poles deep into the left half-plane so as to control the settling time Ts? Introduction to PD Control: Proportional + derivative (PD) control is used to manip-ulate the closed-loop poles to improve the transient response of closed-loop systems. A PD controller has a transfer function Gc(s)= Kp + Kds with a block-diagram v c ( t ) = K p e ( t ) + K d d e K - + P D C o n t r o l l e r s e ( t ) K p + + d s e t p o i n t and a time-domain response de(t) vc(t)= Kpe(t)+ Kd dt where vc(t) is the controller output. A description of how the derivative component acts to modify the transient response is given in Appendix B. In practice pure derivative action is rarely used. Appendix B explains why, and briefly describes the use of a band-limited differentiator,or pseudo-differentiator. Please take a few minutes to read through and understand the Appendices. Experiment #3: Verification of the Differentiator Performance The 2.004 PID controller uses a band-limited differentiator to reduce the effects of the high frequency noise at the output of the position and velocity sensors. Verify that this approximate differentia-tion produces satisfactory results using the following steps: (a) Connect the PID computer-based controller, but keep the power amp turned off. (b) Start the controller, and select a sampling rate of 100 samples/sec. (Maintain this value for all parts of the lab.) (c) Disconnect the position sensor from ACH1 and ground the input connection using a 50Ω terminator resistor (NOT the one from your function generator) to prevent noise from contaminating the derivative action. 2(d) Set Kp,Ki = 0 and Kd = 1 on the front panel. For each of the following input waveforms (from the function generator with an offset of 0) start the controller and observe the output trace (grey/blue). Visually confirm that the output trace approximates the derivative of the input. Either save and plot the output, or make a sketch of it. Point out any differences between what you expect to see and the results you obtain. • A 2 v p-p triangular wave at 1 Hz. • A 1 v p-p square wave at 1 Hz. • A 1 v p-p sinusoid at 0.1, 0.5 and 1 Hz. Experiment #4: PD Control: In this experiment, use PD Control, that is with Gc(s)= Kp + Kds (a) Start with Kp =5, Kd =0.5, and a single magnet for damping. Use a 1 v p-p square wave at 0.1 Hz and record and save the step response. What is the peak current driving the motor with the square wave input? Is the controller “saturating”? Use the pan and zoom tools to select a complete positive step section of the response before saving it to MATLAB. Is the response with PD control more satisfactory? (d) Repeat (a) with Kd = 1 and 2. In each case remember to reset the EDAC2 before be-ginning, save the response to MATLAB, and make a plot of the positive step response. Compare your three plots. Briefly describe how the value of Kd has affected 1) any “over-shoot” in the step response, 2) the time to the peak response, and the time to reach the steady-state response. Experiment #5: Compare your results with a Simulink Simulation: Build a Simulink model of the closed-loop system. We suggest that you use the Simulink model given to you for Lab 5 as the starting point, and modify it for this week’s configuration. (a) The Lab. 5 Simulink model is contained in the file PIControl.mdl in the Lab
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