Department of Electrical Engineering. Indian Institute of Technology Dharwad EE 303: Control Systems Practical Assignment - 6

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1 Department of Electrical Engineering Indian Institute of Technology Dharwad EE 303: Control Systems Practical Assignment - 6 Adapted from Take Home Labs, Oklahoma State University Root Locus Design 1 OBJECTIVE This assignment is about designing a feedback controller for a motor positioning system. Using the model of the motor developed in Assignment 4A and root locus plot, you will identify best closed loop pole locations and design both P and PD controllers to obtain the appropriate closed loop poles. The design should be verified with a simulation and through experiment. 2 SETUP 2.1 REQUIRED MATERIALS HARDWARE All the required materials is issued to you SOFTWARE Matlab/Simulink 2017a or later Matlab/Simulink files provided along with this document PREREQUISITE EXPERIMENTS You must have completed all the assignments give till now. 2.2 SOFTWARE SETUP All the software setup is completed in the previous assignments. You may modify and use the slx file you have generated in the previous assignment. 2.3 HARDWARE SETUP Connect the hardware component as per Assignment 5. 1

2 3 EXPERIMENTAL PROCEDURES 3.1 EXERCISE 1: Control Design (Proportional) In this exercise, you will design a proportional controller for position control of a DC motor i.e., the controller signal! u(t) (motor voltage) will be proportional to the error signal (difference between the reference signal! r (t) and the motor position! θ(t) (or! y(t) )). 1. Using block diagram manipulation, obtain the transfer functions G(s)! and H(s)! for the equivalent block diagram shown in the figure below. Plug in the values of K! m and τ! m obtained in Assignment 4A. Y(s) 2.Find the closed loop transfer function!. Find the closed loop poles as a function of K!. R(s) For values of K! = 0.002, 0.01, 0.1, 10, compute location of the closed loop poles! % OS, T s (5%), T p for the step response. Draw the pole-zero plot showing the locations of poles of G(s)H(s)! and the closed loop poles for the given values of K!. Indicate the value of! K near corresponding poles. 3.Plot the root locus as K! varies from 0! to!. Explain briefly and specifically, how the step response of the system will change as K! varies from a very small value to a very large value. Draw sketches of expected step-response for a very small value of K! and a very large value of! K. 2

3 4.You need to select K! so that the step response of the system has the smallest settling time (! 5 % ) with! % OS < 5 %. Where should the closed loop poles lie? Explain. 3.2 EXERCISE 2: Verification by Simulation In Simulink, create a simulation model similar to that shown in the figure below: Using the parameter values as: sampling time Ts = 0.01, rp = 2*pi/(64*gear constant), ref = pi/ 2, pulse period of pulse generator T = 20, pulse width of pulse generator D = 50, gains K = as computed in Exercise 1, K2 = 0.0 and Km and tau computed in Assignment 4A, run the simulation to collect the data. Plot Angular position vs time for both the values of K2 in separate plots. Compute the settling time, percent overshoot and frequency of oscillations from the simulation plots. Compare with values obtained by calculations in Section EXERCISE 3: Experiment with motor Modify the simulink file of Assignment 5 so that it looks as shown in the figure below: 3

4 Use the parameter values given in Section 3.2. By setting appropriate ports, build the model on Arduino. Collect the data through Plot Data single by setting appropriate port and 8000 samples to plot. Connect the power and collect the data. Using procedure used in previous assignments, plot the data of angular displacement vs time. Compare the plots with those obtained by simulation (Section 3.2) and hand-drawn plots (Section 3.1). Explain the similarities and differences with reasons. Compute the settling time, percent overshoot and frequency of oscillations from the simulation plots. Compare with values obtained by calculations in Section 3.1 and the values observed by simulations in Section 3.2. Comment on the Steady-State Error observed in hand-calculating, simulations and experiments. Can you adjust K! and improve the experimental response? Optional (10% additional marks of this assignment provided the answer is not copied): Find the root mean square error (RMSE) between the reference and motor angle. Plot the RMSE. How small can you make it by varying K!. Can you improve RMSE by compromising on any other parameter? 3.4 EXERCISE 4: Control Design (Proportional and Derivative) In this exercise, you will design a PD controller for position control of a DC motor i.e., the controller signal u(t)! (motor voltage) will be K! times r! (k 2 ω + k 1 θ ). Set k! 1 = 1. Thus K! multiplies to r! θ k 2 θ = e k2 θ. The term Ke! is proportional feedback and the term! Kk 2 θ is the derivative feedback, and has a damping like effect. The block diagram is shown in the figure below: 4

5 1. Using block diagram manipulation, obtain the transfer functions G(s)! and H(s)! for the equivalent block diagram shown earlier. Plug in the values of K! m and τ! m obtained in Assignment 4A.! H(s) should be of the form! k 2 (s + b). Y(s) 2.Set k! 2 = 0.1 and find the closed loop transfer function!. Find the closed loop poles as a R(s) function of K!. For values of K! = 0.002, 0.01, 0.1, 10, compute location of the closed loop poles! % OS, T s (5%), T p for the step response. Draw the pole-zero plot showing the locations of poles of G(s)H(s)! and the closed loop poles for the given values of K!. Indicate the value of! K near corresponding poles. 3.Using k! 2 = 0.1, plot the root locus as K! varies from 0! to!. Explain briefly and specifically, how the step response of the system will change as K! varies from a very small value to a very large value. Draw sketches of expected step-response for a very small value of K! and a very large value of! K. 4.You need to select K! so that the step response of the system has the smallest settling time (! 5 % ) with! % OS < 5 %. Where should the closed loop poles lie? Explain. 5.If k! 2 is changed, how is the root locus affected? Sketch the root locus for various values of k! 2 and illustrate. By adjusting both K! and k! 2, how much flexibility do you have in placing the closed loop poles? Are there in theoretical limits on the closed loop pole locations? What about the practical limits. Discuss in detail, but be precise. 3.4 EXERCISE 5: Verification by simulation and experiment (Proportional and Derivative) Repeat Exercises 2 and 3 with the values of K! and k! 2 chosen in Exercise 4. Can you reduce the steady-state error (compared to only proportional controller) while maintaining the! % OS and the settling time specifications by adjusting the values of K! and k! 2? Justify your design with an explanation about your tuning process. Optional (10% additional marks of this assignment provided the answer is not copied): Find the controller that produces the minimum RMSE. How much can RMSE be reduces compared to the proportional controller? Can you now have better specifications in terms of! % OS, settling time AND RMSE compared to the proportional controller? 4. SUBMISSION INSTRUCTIONS Create a document in the format explained in previous assignments. Answer all the questions asked in this document. Save this document as a pdf file named as <DepartmentCode><Roll- No>-A5.pdf Compress this document along with all the m-files, simulink files and generated 5

6 plots and figures as a single zip file. Rename the zip file as <Department Code><Roll Number>-A5.zip and upload on the moodle portal created for this. 6