What you should know for this experiment
You should write and submit a report for this preparation. This report will be marked making a contribution to your final mark. During the experiment you may be asked to answer questions or demonstrate your skills for the experiment for the marking.
This report must be submitted in the laboratory to the laboratory demonstrator at the beginning of your week 8 laboratory if you are in an even week laboratory (at the beginning of your week 9 laboratory if you are in an odd week laboratory). You are not allowed to do the experiment before submitting the report.
The report should answer each questions as concise as possible. Tables and graphs should be used as much as possible. The report should be written in such a way that another person in the same laboratory can understand and reproduce your results in minimum time. Make sure you include all the programs.
Before submitting your report you should attach a copy of your report to your lab logbook for your experiment.
You can ask a demonstrator to give your full mark (or a mark proportional to the work completed), in the same laboratory session that your submitted your report, by demonstrating that you have successfully completed this experiment by following your report.
Gain control is one of the most simple and widely used control methods for industrial applications. The objective of this control system is to drive the motor to a desired position. The ideal position is specified using the signal generator and the motor position is measured using the encoder. The difference between the reference signal from the signal generator and the feedback from the encoder is the error signal. The controller issues control signal in order to reduce the error signal to zero. The ideal position will be reached by the motor when the error signal is zero.
Model reduction and P controller design
(1) Visit the Control Tutorials at http://ctms.engin.umich.edu/ and study the DC Motor Position: CONTROL Root locus section for proportional controller design by clicking on MOTOR POSITION at the top and root locus on the left under CONTROL.
(2) Consider the DC motor transfer function for motor position in radians obtained in Experiment 3. If the transfer function is of third order or higher, reduce it to a second order transfer function by following the example given in the Model Reduction section of the Tutorial in order to eliminate the least dominant pole.
(3) Determine the reduced transfer function for the velocity output.
(4) Determine the rise time, settling time and steady state gain for the velocity to see if there is any difference between the original estimated model and the reduced model.
(5) Compare the bode plot of the two models for the velocity and comment on their similarity and difference.
(6) Plot root loci for the original 3rd order system for position output and the reduced transfer function for position output and comment on wether the order can be reduced for the design of a proportional controller.
(7) Draw an equal damping ratio line for an overshoot of 20%. Find and record the gain for the intersection of the equal damping ratio line and the root locus. This gain is the P controller.
Design a proportional controller for 20% overshoot using Matlab
1) Type sisotool in Matlab Command Window to activate sisotool for controller design. Import the transfer function for motor position obtained in Experiment 3 and zoom the root loci for details of the dominant poles. Click on Analysis-->Response to Step Command on the toolbar of sisotool in order open LTI viewer. Move the poles for 20% overshoot by clicking on and dragging one pole in default mode. Check the designed proportional controller by click on Design-->Edit Compensator...-->Compensator Editor in order to have the gain of the controller. Compare this controller with the manually designed controller and make comments.
2) Change the gain of the controller in sisotool in order to study the effects of the P controller on characteristics of the control system. Comment on the effects of the controller gain on rise time, settling time, overshoot, phase margin and gain margin. Summarize your data using a table or graph in order to support your comments.
3) Open Simulink
and construct a Simulink program as shown below, where the PID controller
is replaced using the P controller to be designed and the Motor Position block
is replaced using the estimated DC motor transfer function for position output
in radians obtained in Experiment 3.
4) Measure the rise time (10%-90%), settling time (5%) and overshoot of the closed loop system using a square wave input and the display of the scope block. Compare them with the corresponding results obtained using sisotool. If the overshoot is not 20% fine tune the controller so that the overshoot is 20%
5) Passing a step input through an integrator in Simulink to produce unit ramp. Passing a the unit ramp through an integrators in Simulink to produce a parabolic parabolic signal. Apply them to the closed loop system and comment on how the steady state error of the closed loop system are affected by the gain of the controller based on the simulation results. Compare the simulation results with theoretical predictions.
(6) Comment on how the static, velocity and acceleration error constant can be determined experimentally for the DC motor control system using only the measurements of input and output from the scope. You can change the gain of the controller and compare the corresponding behaviour in the error signal in order to figure out the relationship between the error signal and the gain of the controller.
(7) You should not use the transfer function to determine the error constants directly in this question. However, you can add a scope to measure the error signal. Assuming the model is unknown determine the error constants experimentally from the display of a scope.
(8) Compare estimated error constants obtained in question (7) with theoretical predictions obtained from the transfer function and make comments.
9) Assuming the model of the system is unknown and measure the phase margin and gain margin of the control system in the Simulink program by applying a sine wave to the closed loop system and measuring the open loop frequency response, i.e. the error signal (output of the comparator or input to the controller) is the input and the feedback to the comparator is the output. You can display the input of the controller using a scope block. Compare your results with theoretical predictions obtained using the transfer function.
Closed loop frequency response
Revise typical closed loop frequency responses of a DC motor under unit feedback control. You will notice at low frequencies, the gain of the system is unit (as expected in a closed-loop system the output will track the reference signal). As the frequency of the reference signal increases, however, the gain may rise to a certain peak value at a frequency, before falling dramatically. This frequency is known as the resonant frequency and the gain at this value is known as the resonant peak. A typical plot of a closed loop frequency response is shown below:
(1) Draw a bode plot of the closed loop system designed using sisotool in order to see if there is a clear resonant peak or not. If not, increase the gain of the P controller until there is a clear resonant peak.
(2) Apply a sine wave to the Simulink program of the closed loop system with
the P controller adjusted for the clear resonant peak. Adjust
the frequency of the sine wave input for the closed loop system until the output peak-to-peak value is at a maximum.
(3) Measure the frequency response of the closed-loop control system for the closed loop bode plot as follows. Set the amplitude of the sine wave from the signal generator block to 3 and adjust the frequency to 2 rad/sec. For a number of frequencies ranging from 2 rad/sec up to the frequency at least 10 times higher than the estimated resonance frequency make a table through experiment in terms of the followings: frequency (rad/sec), frequency (Hz), time difference between adjacent peaks (s), phase shift (degree), input (peak to peak), output (peak to peak) and the gain. Taking extra measurements close to the resonant frequency and draw a bode plot for the closed loop system.
(4) Draw a bode plot using the table. Compare your results with theoretical predictions.