Lab #4 Objective: To evaluate the effect of pole and zero location upon the time response of first- and second-order systems. Minimum Required Software Packages: MATLAB, Simulink, and the Control System Toolbox Part 1 We work out example in the posted file named, “Linear System Analyzer) We consider the following two transfer functions 𝑇1 (𝑠) = 𝑇2 (𝑠) = 𝑠2 24.542 + 4𝑠 + 24.542 245.42 (𝑠 + 10)(𝑠 2 + 4𝑠 + 24.542) To analyze these two systems in terms of rise time, overshoot, and settling time, we can write a MATLAB script as follows: 'Example 1' T1 = tf(24.542, [1 4 24.542]) T2 = tf(245.42, conv([1 10],[1 4 24.542])) linearSystemAnalyzer(T1, T2) 'Example 1' Part 2: Prelab Step 1 25 a. Given the transfer function 𝐺(𝑠) = 𝑠2 +4𝑠+25, evaluate the percent overshoot, settling time, peak time, and rise time. Also, plot the poles. b. We add a pole at -200 to the system of Prelab 1a. Estimate whether the transient response in Prelab 1a will be appreciably affected. c. We repeat Prelab 1b with the pole successively placed at -20; -10, and -2. Step 2: a. A zero is added to the system of Prelab 1a at -200 and then moved to -50; -20; -10; -5, and -2. List the values of zero location in the order of the greatest to the least effect upon the pure second-order transient response. Step 3: (25𝑏⁄𝑎)(𝑠+𝑎) Given the transfer function 𝐺(𝑠) = (𝑠+𝑏)(𝑠2 +4𝑠+25). Let 𝑎 = 3 and 𝑏 = 3.01, 3.1, 3.3, 3.5 and 4.0. Step 4: Given the transfer function 𝐺(𝑠) = (2500𝑏⁄𝑎)(𝑠+𝑎) . let 𝑎 = 30 and 𝑏 = 30.01, 30.1, (𝑠+𝑏)(𝑠2 +4𝑠+25) 30.3, 30.5, 31, 35 and 40. Which value of 𝑏 will have a minimal effect upon the pure secondorder transient response? T1 = 24.54 ----------------s^2 + 4 s + 24.54 Continuous-time transfer function. T2 = 245.4 -----------------------------s^3 + 14 s^2 + 64.54 s + 245.4 Continuous-time transfer function. data1 = struct with fields: RiseTime: 0.2970 SettlingTime: 1.6974 SettlingMin: 0.9375 SettlingMax: 1.2500 Overshoot: 24.9980 Undershoot: 0 Peak: 1.2500 PeakTime: 0.6908 data2 = struct with fields: RiseTime: 0.3396 SettlingTime: 1.7940 SettlingMin: 0.9099 SettlingMax: 1.2165 Overshoot: 21.6457 Undershoot: 0 Peak: 1.2165 PeakTime: 0.8105 s >> Figure 1step1b Figure 2step1c Figure 3step2A Figure 4step2A Figure 5step2A Figure 6step2A Figure 7step2A Figure 8step3 Let 𝑎 = 3 and 𝑏 = 3.01, 3.1, 3.3, 3.5 and 4.0. We conclude that b=4.0 has a minimal effect upon the pure second-order transient response. Lab 1. Using Simulink, we add a pole to the second-order system of Prelab 1a and plot the step responses of the system when the higher-order pole is nonexistent, at -200; -20; -10, and -2. 2. Using Simulink, we add a zero to the second-order system of Prelab 1a and plot the step responses of the system when the zero is nonexistent, at -200; -50;-20; -10; -5, and -2. 3. Using Simulink and the transfer function of Prelab 3 with 𝑎 = 3, plot the step responses of the system when the value of 𝑏 is 3, 3.01, 3.1, 3.3, 3.5, and 4.0. 4. Using Simulink and the transfer function of Prelab 4 with 𝑎 = 30, plot the step responses of the system when the value of 𝑏 is 30, 30.01, 30.1, 30.5, 31, 35, and 40. Department of Electrical and Computer Engineering Intro to Control System and Applications Fall 2018 Lab #4 Objective: To evaluate the effect of pole and zero location upon the time response of first- and second-order systems. Minimum Required Software Packages: MATLAB, Simulink, and the Control System Toolbox Part 1 Work out example E-1 from appendix E in your text book (also available as the LTI viewer in the course Blackboard page) Deliverables: Show evidence of your work and resulted graphs. Comment on the use of the Simulink and the LTI viewer in solving control problems. Part 2: Prelab Step 1 25 a. Given the transfer function 𝐺(𝑠) = 𝑠2 +4𝑠+25, evaluate the percent overshoot, settling time, peak time, and rise time. Also, plot the poles. b. Add a pole at -200 to the system of Prelab 1a. Estimate whether the transient response in Prelab 1a will be appreciably affected. c. Repeat Prelab 1b with the pole successively placed at -20; -10, and -2. Step 2: a. A zero is added to the system of Prelab 1a at -200 and then moved to -50; -20; -10; -5, and -2. List the values of zero location in the order of the greatest to the least effect upon the pure second-order transient response. Step 3: (25𝑏⁄𝑎)(𝑠+𝑎) Given the transfer function 𝐺(𝑠) = (𝑠+𝑏)(𝑠2 +4𝑠+25). Let 𝑎 = 3 and 𝑏 = 3.01, 3.1, 3.3, 3.5 and 4.0. which value of 𝑏 will have a minimal effect upon the pure second-order transient response? Step 4: (2500𝑏⁄𝑎)(𝑠+𝑎) Given the transfer function 𝐺(𝑠) = (𝑠+𝑏)(𝑠2 +4𝑠+25). let 𝑎 = 30 and 𝑏 = 30.01, 30.1, 30.3, 30.5, 31, 35 and 40. Which value of 𝑏 will have a minimal effect upon the pure second-order transient response? Lab 1. Using Simulink, add a pole to the second-order system of Prelab 1a and plot the step responses of the system when the higher-order pole is nonexistent, at -200; -20; -10, and -2. Make your plots on a single graph, using the Simulink LTI Viewer. Normalize all plots to a steady-state value of unity. Record percent overshoot, settling time, peak time, and rise time for each response. 2. Using Simulink, add a zero to the second-order system of Prelab 1a and plot the step responses of the system when the zero is nonexistent, at -200; -50;-20; -10; -5, and -2. Make your plots on a single graph, using the Simulink LTI Viewer. Normalize all plots to a steadystate value of unity. Record percent overshoot, settling time, peak time, and rise time for each response. 3. Using Simulink and the transfer function of Prelab 3 with 𝑎 = 3, plot the step responses of the system when the value of 𝑏 is 3, 3.01, 3.1, 3.3, 3.5, and 4.0. Make your plots on a single graph using the Simulink LTI Viewer. Record percent overshoot, settling time, peak time, and rise time for each response. 4. Using Simulink and the transfer function of Prelab 4 with 𝑎 = 30, plot the step responses of the system when the value of 𝑏 is 30, 30.01, 30.1, 30.5, 31, 35, and 40. Make your plots on a single graph, using the Simulink LTI Viewer. Record percent overshoot, settling time, peak time, and rise time for each response. Postlab 1. Discuss the effect upon the transient response of the proximity of a higher-order pole to the dominant second-order pole pair. 2. Discuss the effect upon the transient response of the proximity of a zero to the dominant second-order pole pair. Explore the relationship between the length of the vector from the zero to the dominant pole and the zero’s effect upon the pure second-order step response. 3. Discuss the effect of pole-zero cancellation upon the transient response of a dominant second-order pole pair. Allude to how close the canceling pole and zero should be and the relationships of (1) the distance between them and (2) the distance between the zero and the dominant second-order poles. Deliverables Your report should include objectives of the lab, problem statements, all your hand analysis and calculations of the prelab section, all your Simulink models and rested graphs of the lab section, answers to the Postlab section, and your comments, lessons learned and conclsions
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