ee 350 continuous-time linear systems recitation...
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![Page 1: EE 350 Continuous-Time Linear Systems Recitation 15courses.ee.psu.edu/schiano/EE350/Recitations/Rec_15_EE350_f16.pdf• Derive the Final Value Theorem 3 0 lim ( ) lim ( ) Provided](https://reader035.vdocuments.net/reader035/viewer/2022071412/6109a4ffdd2db1620e1e42be/html5/thumbnails/1.jpg)
Rec 15.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
EE 350
Continuous-Time Linear Systems
Recitation 15
1
![Page 2: EE 350 Continuous-Time Linear Systems Recitation 15courses.ee.psu.edu/schiano/EE350/Recitations/Rec_15_EE350_f16.pdf• Derive the Final Value Theorem 3 0 lim ( ) lim ( ) Provided](https://reader035.vdocuments.net/reader035/viewer/2022071412/6109a4ffdd2db1620e1e42be/html5/thumbnails/2.jpg)
Rec 15.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Recitation 15 Topics
2
• Solved Problems – Analysis of Feedback Control Systems– Bode Plots
• MATLAB Exercises– SIMULINK graphical programming language
![Page 3: EE 350 Continuous-Time Linear Systems Recitation 15courses.ee.psu.edu/schiano/EE350/Recitations/Rec_15_EE350_f16.pdf• Derive the Final Value Theorem 3 0 lim ( ) lim ( ) Provided](https://reader035.vdocuments.net/reader035/viewer/2022071412/6109a4ffdd2db1620e1e42be/html5/thumbnails/3.jpg)
Rec 15.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 1
• Derive the Final Value Theorem
3
0lim ( ) lim ( )
Provided that ( ) has no poles on theimaginary axis or in th right-half lane
t sf t sF s
sF s
![Page 4: EE 350 Continuous-Time Linear Systems Recitation 15courses.ee.psu.edu/schiano/EE350/Recitations/Rec_15_EE350_f16.pdf• Derive the Final Value Theorem 3 0 lim ( ) lim ( ) Provided](https://reader035.vdocuments.net/reader035/viewer/2022071412/6109a4ffdd2db1620e1e42be/html5/thumbnails/4.jpg)
Rec 15.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 1 Solution
4
![Page 5: EE 350 Continuous-Time Linear Systems Recitation 15courses.ee.psu.edu/schiano/EE350/Recitations/Rec_15_EE350_f16.pdf• Derive the Final Value Theorem 3 0 lim ( ) lim ( ) Provided](https://reader035.vdocuments.net/reader035/viewer/2022071412/6109a4ffdd2db1620e1e42be/html5/thumbnails/5.jpg)
Rec 15.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 2
• Determine the final value of f(t) by– Solving for f(t) and taking the limit as t approaches infinity– Applying the final value theorem
5
12F(s)=
2 ( 3)s
s s s
![Page 6: EE 350 Continuous-Time Linear Systems Recitation 15courses.ee.psu.edu/schiano/EE350/Recitations/Rec_15_EE350_f16.pdf• Derive the Final Value Theorem 3 0 lim ( ) lim ( ) Provided](https://reader035.vdocuments.net/reader035/viewer/2022071412/6109a4ffdd2db1620e1e42be/html5/thumbnails/6.jpg)
Rec 15.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 2 Solution
6
![Page 7: EE 350 Continuous-Time Linear Systems Recitation 15courses.ee.psu.edu/schiano/EE350/Recitations/Rec_15_EE350_f16.pdf• Derive the Final Value Theorem 3 0 lim ( ) lim ( ) Provided](https://reader035.vdocuments.net/reader035/viewer/2022071412/6109a4ffdd2db1620e1e42be/html5/thumbnails/7.jpg)
Rec 15.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 2 Solution
7
![Page 8: EE 350 Continuous-Time Linear Systems Recitation 15courses.ee.psu.edu/schiano/EE350/Recitations/Rec_15_EE350_f16.pdf• Derive the Final Value Theorem 3 0 lim ( ) lim ( ) Provided](https://reader035.vdocuments.net/reader035/viewer/2022071412/6109a4ffdd2db1620e1e42be/html5/thumbnails/8.jpg)
Rec 15.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 3• A certain plant has the transfer function representation
• Is the zero-state unit-step response of the plant overdamped, critically damped, or underdamped?
• Choose the proportional gain KP so that the zero-state unit-step response of the closed-loop system is critically damped, and determine the resulting steady-state error
8
2
10G (s) = 7 10p s s
R
YPK pG sE
![Page 9: EE 350 Continuous-Time Linear Systems Recitation 15courses.ee.psu.edu/schiano/EE350/Recitations/Rec_15_EE350_f16.pdf• Derive the Final Value Theorem 3 0 lim ( ) lim ( ) Provided](https://reader035.vdocuments.net/reader035/viewer/2022071412/6109a4ffdd2db1620e1e42be/html5/thumbnails/9.jpg)
Rec 15.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 3 Solution
9
![Page 10: EE 350 Continuous-Time Linear Systems Recitation 15courses.ee.psu.edu/schiano/EE350/Recitations/Rec_15_EE350_f16.pdf• Derive the Final Value Theorem 3 0 lim ( ) lim ( ) Provided](https://reader035.vdocuments.net/reader035/viewer/2022071412/6109a4ffdd2db1620e1e42be/html5/thumbnails/10.jpg)
Rec 15.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 3 Solution
10
![Page 11: EE 350 Continuous-Time Linear Systems Recitation 15courses.ee.psu.edu/schiano/EE350/Recitations/Rec_15_EE350_f16.pdf• Derive the Final Value Theorem 3 0 lim ( ) lim ( ) Provided](https://reader035.vdocuments.net/reader035/viewer/2022071412/6109a4ffdd2db1620e1e42be/html5/thumbnails/11.jpg)
Rec 15.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 3 Solution
11
![Page 12: EE 350 Continuous-Time Linear Systems Recitation 15courses.ee.psu.edu/schiano/EE350/Recitations/Rec_15_EE350_f16.pdf• Derive the Final Value Theorem 3 0 lim ( ) lim ( ) Provided](https://reader035.vdocuments.net/reader035/viewer/2022071412/6109a4ffdd2db1620e1e42be/html5/thumbnails/12.jpg)
Rec 15.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 4• Determine the closed-loop transfer function Y(s)/R(s) for
the following feedback system
• Is the closed-loop system BIBO stable? If so, is the zero-state unit-step response underdamped, overdamped, or critically damped?
12
R
Y6
11s
E
14s
![Page 13: EE 350 Continuous-Time Linear Systems Recitation 15courses.ee.psu.edu/schiano/EE350/Recitations/Rec_15_EE350_f16.pdf• Derive the Final Value Theorem 3 0 lim ( ) lim ( ) Provided](https://reader035.vdocuments.net/reader035/viewer/2022071412/6109a4ffdd2db1620e1e42be/html5/thumbnails/13.jpg)
Rec 15.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 4 Solution
13
![Page 14: EE 350 Continuous-Time Linear Systems Recitation 15courses.ee.psu.edu/schiano/EE350/Recitations/Rec_15_EE350_f16.pdf• Derive the Final Value Theorem 3 0 lim ( ) lim ( ) Provided](https://reader035.vdocuments.net/reader035/viewer/2022071412/6109a4ffdd2db1620e1e42be/html5/thumbnails/14.jpg)
Rec 15.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 4 Solution
14
![Page 15: EE 350 Continuous-Time Linear Systems Recitation 15courses.ee.psu.edu/schiano/EE350/Recitations/Rec_15_EE350_f16.pdf• Derive the Final Value Theorem 3 0 lim ( ) lim ( ) Provided](https://reader035.vdocuments.net/reader035/viewer/2022071412/6109a4ffdd2db1620e1e42be/html5/thumbnails/15.jpg)
Rec 15.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 4 Solution
15
![Page 16: EE 350 Continuous-Time Linear Systems Recitation 15courses.ee.psu.edu/schiano/EE350/Recitations/Rec_15_EE350_f16.pdf• Derive the Final Value Theorem 3 0 lim ( ) lim ( ) Provided](https://reader035.vdocuments.net/reader035/viewer/2022071412/6109a4ffdd2db1620e1e42be/html5/thumbnails/16.jpg)
Rec 15.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 5• Sketch the Bode magnitude and phase plot of the
following frequency response functions
Verify your answers using the MATLAB command bode
16
2
1000H( ) = 10
100H( ) = 10
jj
jjj
![Page 17: EE 350 Continuous-Time Linear Systems Recitation 15courses.ee.psu.edu/schiano/EE350/Recitations/Rec_15_EE350_f16.pdf• Derive the Final Value Theorem 3 0 lim ( ) lim ( ) Provided](https://reader035.vdocuments.net/reader035/viewer/2022071412/6109a4ffdd2db1620e1e42be/html5/thumbnails/17.jpg)
Rec 15.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 5 Solution
17
![Page 18: EE 350 Continuous-Time Linear Systems Recitation 15courses.ee.psu.edu/schiano/EE350/Recitations/Rec_15_EE350_f16.pdf• Derive the Final Value Theorem 3 0 lim ( ) lim ( ) Provided](https://reader035.vdocuments.net/reader035/viewer/2022071412/6109a4ffdd2db1620e1e42be/html5/thumbnails/18.jpg)
Rec 15.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 5 Solution
18
![Page 19: EE 350 Continuous-Time Linear Systems Recitation 15courses.ee.psu.edu/schiano/EE350/Recitations/Rec_15_EE350_f16.pdf• Derive the Final Value Theorem 3 0 lim ( ) lim ( ) Provided](https://reader035.vdocuments.net/reader035/viewer/2022071412/6109a4ffdd2db1620e1e42be/html5/thumbnails/19.jpg)
Rec 15.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 6
• Simulate the closed-loop system in Problem 3 using SIMULINK
• Write a MATLAB M-file that– Represents the system using a transfer function– Plots the zero-state unit-step response
19
![Page 20: EE 350 Continuous-Time Linear Systems Recitation 15courses.ee.psu.edu/schiano/EE350/Recitations/Rec_15_EE350_f16.pdf• Derive the Final Value Theorem 3 0 lim ( ) lim ( ) Provided](https://reader035.vdocuments.net/reader035/viewer/2022071412/6109a4ffdd2db1620e1e42be/html5/thumbnails/20.jpg)
Rec 15.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
EE 350
Continuous-Time Linear Systems
Recitation 15
1
![Page 21: EE 350 Continuous-Time Linear Systems Recitation 15courses.ee.psu.edu/schiano/EE350/Recitations/Rec_15_EE350_f16.pdf• Derive the Final Value Theorem 3 0 lim ( ) lim ( ) Provided](https://reader035.vdocuments.net/reader035/viewer/2022071412/6109a4ffdd2db1620e1e42be/html5/thumbnails/21.jpg)
Rec 15.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Recitation 15 Topics
2
• Solved Problems – Analysis of Feedback Control Systems– Bode Plots
• MATLAB Exercises– SIMULINK graphical programming language
![Page 22: EE 350 Continuous-Time Linear Systems Recitation 15courses.ee.psu.edu/schiano/EE350/Recitations/Rec_15_EE350_f16.pdf• Derive the Final Value Theorem 3 0 lim ( ) lim ( ) Provided](https://reader035.vdocuments.net/reader035/viewer/2022071412/6109a4ffdd2db1620e1e42be/html5/thumbnails/22.jpg)
Rec 15.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 1
• Derive the Final Value Theorem
3
![Page 23: EE 350 Continuous-Time Linear Systems Recitation 15courses.ee.psu.edu/schiano/EE350/Recitations/Rec_15_EE350_f16.pdf• Derive the Final Value Theorem 3 0 lim ( ) lim ( ) Provided](https://reader035.vdocuments.net/reader035/viewer/2022071412/6109a4ffdd2db1620e1e42be/html5/thumbnails/23.jpg)
Rec 15.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 1 Solution
4
![Page 24: EE 350 Continuous-Time Linear Systems Recitation 15courses.ee.psu.edu/schiano/EE350/Recitations/Rec_15_EE350_f16.pdf• Derive the Final Value Theorem 3 0 lim ( ) lim ( ) Provided](https://reader035.vdocuments.net/reader035/viewer/2022071412/6109a4ffdd2db1620e1e42be/html5/thumbnails/24.jpg)
Rec 15.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 2
• Determine the final value of f(t) by– Solving for f(t) and taking the limit as t approaches infinity– Applying the final value theorem
5
![Page 25: EE 350 Continuous-Time Linear Systems Recitation 15courses.ee.psu.edu/schiano/EE350/Recitations/Rec_15_EE350_f16.pdf• Derive the Final Value Theorem 3 0 lim ( ) lim ( ) Provided](https://reader035.vdocuments.net/reader035/viewer/2022071412/6109a4ffdd2db1620e1e42be/html5/thumbnails/25.jpg)
Rec 15.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 2 Solution
6
![Page 26: EE 350 Continuous-Time Linear Systems Recitation 15courses.ee.psu.edu/schiano/EE350/Recitations/Rec_15_EE350_f16.pdf• Derive the Final Value Theorem 3 0 lim ( ) lim ( ) Provided](https://reader035.vdocuments.net/reader035/viewer/2022071412/6109a4ffdd2db1620e1e42be/html5/thumbnails/26.jpg)
Rec 15.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 2 Solution
7
![Page 27: EE 350 Continuous-Time Linear Systems Recitation 15courses.ee.psu.edu/schiano/EE350/Recitations/Rec_15_EE350_f16.pdf• Derive the Final Value Theorem 3 0 lim ( ) lim ( ) Provided](https://reader035.vdocuments.net/reader035/viewer/2022071412/6109a4ffdd2db1620e1e42be/html5/thumbnails/27.jpg)
Rec 15.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 3• A certain plant has the transfer function representation
• Is the zero-state unit-step response of the plant overdamped, critically damped, or underdamped?
• Choose the proportional gain KP so that the zero-state unit-step response of the closed-loop system is critically damped, and determine the resulting steady-state error
8
![Page 28: EE 350 Continuous-Time Linear Systems Recitation 15courses.ee.psu.edu/schiano/EE350/Recitations/Rec_15_EE350_f16.pdf• Derive the Final Value Theorem 3 0 lim ( ) lim ( ) Provided](https://reader035.vdocuments.net/reader035/viewer/2022071412/6109a4ffdd2db1620e1e42be/html5/thumbnails/28.jpg)
Rec 15.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 3 Solution
9
![Page 29: EE 350 Continuous-Time Linear Systems Recitation 15courses.ee.psu.edu/schiano/EE350/Recitations/Rec_15_EE350_f16.pdf• Derive the Final Value Theorem 3 0 lim ( ) lim ( ) Provided](https://reader035.vdocuments.net/reader035/viewer/2022071412/6109a4ffdd2db1620e1e42be/html5/thumbnails/29.jpg)
Rec 15.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 3 Solution
10
![Page 30: EE 350 Continuous-Time Linear Systems Recitation 15courses.ee.psu.edu/schiano/EE350/Recitations/Rec_15_EE350_f16.pdf• Derive the Final Value Theorem 3 0 lim ( ) lim ( ) Provided](https://reader035.vdocuments.net/reader035/viewer/2022071412/6109a4ffdd2db1620e1e42be/html5/thumbnails/30.jpg)
Rec 15.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 3 Solution
11
![Page 31: EE 350 Continuous-Time Linear Systems Recitation 15courses.ee.psu.edu/schiano/EE350/Recitations/Rec_15_EE350_f16.pdf• Derive the Final Value Theorem 3 0 lim ( ) lim ( ) Provided](https://reader035.vdocuments.net/reader035/viewer/2022071412/6109a4ffdd2db1620e1e42be/html5/thumbnails/31.jpg)
Rec 15.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 4• Determine the closed-loop transfer function Y(s)/R(s) for
the following feedback system
• Is the closed-loop system BIBO stable? If so, is the zero-state unit-step response underdamped, overdamped, or critically damped?
12
![Page 32: EE 350 Continuous-Time Linear Systems Recitation 15courses.ee.psu.edu/schiano/EE350/Recitations/Rec_15_EE350_f16.pdf• Derive the Final Value Theorem 3 0 lim ( ) lim ( ) Provided](https://reader035.vdocuments.net/reader035/viewer/2022071412/6109a4ffdd2db1620e1e42be/html5/thumbnails/32.jpg)
Rec 15.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 4 Solution
13
![Page 33: EE 350 Continuous-Time Linear Systems Recitation 15courses.ee.psu.edu/schiano/EE350/Recitations/Rec_15_EE350_f16.pdf• Derive the Final Value Theorem 3 0 lim ( ) lim ( ) Provided](https://reader035.vdocuments.net/reader035/viewer/2022071412/6109a4ffdd2db1620e1e42be/html5/thumbnails/33.jpg)
Rec 15.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 4 Solution
14
![Page 34: EE 350 Continuous-Time Linear Systems Recitation 15courses.ee.psu.edu/schiano/EE350/Recitations/Rec_15_EE350_f16.pdf• Derive the Final Value Theorem 3 0 lim ( ) lim ( ) Provided](https://reader035.vdocuments.net/reader035/viewer/2022071412/6109a4ffdd2db1620e1e42be/html5/thumbnails/34.jpg)
Rec 15.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 4 Solution
15
![Page 35: EE 350 Continuous-Time Linear Systems Recitation 15courses.ee.psu.edu/schiano/EE350/Recitations/Rec_15_EE350_f16.pdf• Derive the Final Value Theorem 3 0 lim ( ) lim ( ) Provided](https://reader035.vdocuments.net/reader035/viewer/2022071412/6109a4ffdd2db1620e1e42be/html5/thumbnails/35.jpg)
Rec 15.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 5• Sketch the Bode magnitude and phase plot of the
following frequency response functions
Verify your answers using the MATLAB command bode
16
![Page 36: EE 350 Continuous-Time Linear Systems Recitation 15courses.ee.psu.edu/schiano/EE350/Recitations/Rec_15_EE350_f16.pdf• Derive the Final Value Theorem 3 0 lim ( ) lim ( ) Provided](https://reader035.vdocuments.net/reader035/viewer/2022071412/6109a4ffdd2db1620e1e42be/html5/thumbnails/36.jpg)
Rec 15.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 5 Solution
17
![Page 37: EE 350 Continuous-Time Linear Systems Recitation 15courses.ee.psu.edu/schiano/EE350/Recitations/Rec_15_EE350_f16.pdf• Derive the Final Value Theorem 3 0 lim ( ) lim ( ) Provided](https://reader035.vdocuments.net/reader035/viewer/2022071412/6109a4ffdd2db1620e1e42be/html5/thumbnails/37.jpg)
Rec 15.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 5 Solution
18
![Page 38: EE 350 Continuous-Time Linear Systems Recitation 15courses.ee.psu.edu/schiano/EE350/Recitations/Rec_15_EE350_f16.pdf• Derive the Final Value Theorem 3 0 lim ( ) lim ( ) Provided](https://reader035.vdocuments.net/reader035/viewer/2022071412/6109a4ffdd2db1620e1e42be/html5/thumbnails/38.jpg)
Rec 15.School of Electrical Engineering and Computer Science
Jeffrey Schiano 2015-2017 All rights reserved.
Problem 6
• Simulate the closed-loop system in Problem 3 using SIMULINK
• Write a MATLAB M-file that– Represents the system using a transfer function– Plots the zero-state unit-step response
19