Scilab Code for Control Systems Theory and Applications by Smarajit Ghosh · 2015-09-15 · Scilab...

Scilab Code forControl Systems

Theory and Applicationsby Smarajit Ghosh1

Created byAnuradha Singhania

4th Year StudentB. Tech. (Elec. & Commun. )

Sri Mata Vaishno Devi University, Jammu

College TeacherProf. Amit Kumar Garg

Sri Mata Vaishno Devi University, JammuReviewer

Prof. Madhu Belur, IIT Bombay

29 June 2010

1Funded by a grant from the National Mission on Education through ICT,http://spoken-tutorial.org/NMEICT-Intro. This Textbook companion and scilabcodes written in it can be downloaded from www.scilab.in

Book Details

Authors: Smarajit Ghosh

Title: Control Systems

Publisher: Pearson Education

Edition: 2nd

Year: 2009

Place: New Delhi

ISBN: 978-81-317-0828-6

1

Contents

List of Scilab Code 4

2 Laplace Transform and Matrix Algebra 112.1 Scilab Code . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3 Transfer Function 143.1 Scilab Code . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4 Control system Components 174.1 Scilab Code . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

6 Control system Components 216.1 Scilab Code . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

8 Time Domain Analysis of Control Systems 298.1 Scilab Code . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

9 Feedback Characteristics of control Systems 449.1 Scilab Code . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

10 Stability 5410.1 Scilab Code . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

11 Root Locus Method 6411.1 Scilab Code . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

12 Frequency Domain Analysis 9012.1 Scilab Code . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

2

13 Bode Plot 9513.1 Scilab Code . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

15 Nyquist Plot 10315.1 Scilab Code . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

17 State Variable Approach 11217.1 Scilab Code . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

18 Digital Control Systems 12218.1 Scilab Code . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

3

List of Scilab Code

2.01-01 2.01.01.sci . . . . . . . . . . . . . . . . . . . . . . . . . . 112.01-02 2.01.02sci . . . . . . . . . . . . . . . . . . . . . . . . . . 112.01-03 2.01.03.sci . . . . . . . . . . . . . . . . . . . . . . . . . . 112.03 2.03.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.04 2.04.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.05-01 2.05.01.sci . . . . . . . . . . . . . . . . . . . . . . . . . . 122.05-02 2.05.02.sci . . . . . . . . . . . . . . . . . . . . . . . . . . 122.06 2.06.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.11 2.11.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.13 2.13.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.02 3.02.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.03 3.03.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.04 3.04.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.15 3.15.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.16 3.16.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.17 3.17.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.18 3.18.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.19 3.19.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.23 3.23.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.01 4.01.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.02 4.02.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.03 4.03.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.04 4.04.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.05 4.05.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19F1 parallel.sce . . . . . . . . . . . . . . . . . . . . . . . . . 21F2 serial.sce . . . . . . . . . . . . . . . . . . . . . . . . . . . 216.01 6.01.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216.02 6.02.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4

6.03 6.03.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226.04 6.04.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236.05 6.05.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236.06 6.06.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236.07 6.07.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236.08 6.08.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246.09 6.09.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246.10 6.10.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256.11 6.11.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256.12 6.12.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266.13 6.13.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266.14 6.14.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276.15 6.15.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278.02 8.02.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298.03-01 8.03.01.sci . . . . . . . . . . . . . . . . . . . . . . . . . . 298.03-02 8.03.02.sci . . . . . . . . . . . . . . . . . . . . . . . . . . 308.03-03 8.03.03.sci . . . . . . . . . . . . . . . . . . . . . . . . . . 308.04 8.04.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318.05 8.05.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318.06 8.06.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328.07 8.07.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328.08 8.08.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338.09 8.09.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348.10 8.10.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348.11 8.11.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358.12 8.12.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358.13 8.13.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368.14 8.14.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368.15 8.15.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378.16 8.16.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378.17 8.17.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388.19 8.19.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398.20 8.20.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398.23-01 8.23.01.sci . . . . . . . . . . . . . . . . . . . . . . . . . . 408.23-02 8.23.02.sci . . . . . . . . . . . . . . . . . . . . . . . . . . 408.23-03 8.23.03.sci . . . . . . . . . . . . . . . . . . . . . . . . . . 418.23-04 8.23.04.sci . . . . . . . . . . . . . . . . . . . . . . . . . . 418.24 8.24.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

5

8.32 8.32.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429.01 9.01.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449.02 9.02.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459.03 9.03.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459.04 9.04.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469.05 9.05.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479.06 9.06.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479.07 9.07.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489.08 9.08.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499.09 9.09.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509.10 9.10.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519.11 9.11.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5210.02-01 10.02.01.sci . . . . . . . . . . . . . . . . . . . . . . . . . 5410.02-02 10.02.02.sci . . . . . . . . . . . . . . . . . . . . . . . . . 5410.02-03 10.02.03.sci . . . . . . . . . . . . . . . . . . . . . . . . . 5510.03 10.03.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . 5510.04 10.04.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . 5610.05-01 10.05.01.sci . . . . . . . . . . . . . . . . . . . . . . . . . 5610.05-02 10.05.02.sci . . . . . . . . . . . . . . . . . . . . . . . . . 5710.06 10.06.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . 5710.07 10.07.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . 5810.08 10.08.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . 5910.09 10.09.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . 5910.10 10.10.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . 6010.11 10.11.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . 6110.12 10.12.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . 6110.13 10.13.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . 6210.14 10.14.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . 6210.15 10.15.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . 6310.16 10.16.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . 6311.01 11.01.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . 6411.02 11.02.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . 6411.03 11.03.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . 6411.04 11.04.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . 6511.05 11.05.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . 6711.06 11.06.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . 6911.08 11.08.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . 7011.09 11.09.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

6

11.10 11.10.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . 7111.11 11.11.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . 7311.12 11.12.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . 7311.13 11.13.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . 7411.14 11.14.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . 7411.15 11.15.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . 7411.16 11.16.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . 7511.17 11.17.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . 7611.19 11.19.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . 7811.20 11.20.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . 7911.21 11.21.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . 7911.22 11.22.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . 8011.23 11.23.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . 8111.25 11.25.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . 8211.26 11.26.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . 8311.27 11.27.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . 8411.28 11.28.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . 8712.01 12.01.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . 9012.02 12.02.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . 9112.03 12.03.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . 9112.04 12.04.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . 9212.05 12.05.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . 9213.01 13.01.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . 9513.02 13.02.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . 9513.03 13.03.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . 9713.04 13.04.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . 9813.05 13.05.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . 9813.06 13.06.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . 10015.01 15.01.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . 10315.02 15.02.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . 10315.03 15.03.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . 10415.04 15.04.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . 10515.05 15.05.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . 10615.06 15.06.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . 10715.07 15.07.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . 10815.08 15.08.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . 10917.03 17.03.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . 11217.04 17.04.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

7

17.06 17.06.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . 11217.07 17.07.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . 11317.08 17.08.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . 11317.09 17.09.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . 11317.10 17.10.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . 11317.11 17.11.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . 11417.12 17.12.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . 11417.13 17.13.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . 11517.14 17.14.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . 11617.16 17.16.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . 11617.17 17.17.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . 11717.18 17.18.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . 11717.19 17.19.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . 11717.20 17.20.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . 11817.21 17.21.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . 11917.22 17.22.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . 11917.23 17.23.sci . . . . . . . . . . . . . . . . . . . . . . . . . . . 120F3 ztransfer.sce . . . . . . . . . . . . . . . . . . . . . . . . . 12218.01 18.01.01.sci . . . . . . . . . . . . . . . . . . . . . . . . . 12218.08 18.08.01.sci . . . . . . . . . . . . . . . . . . . . . . . . . 122

8

List of Figures

11.1 Output Graph of S 11.01 . . . . . . . . . . . . . . . . . . . . 6511.2 Output Graph of S 11.02 . . . . . . . . . . . . . . . . . . . . 6611.3 Output Graph of S 11.03 . . . . . . . . . . . . . . . . . . . . 6711.4 Output of S 11.04 . . . . . . . . . . . . . . . . . . . . . . . . 6811.5 Output of S 11.05 . . . . . . . . . . . . . . . . . . . . . . . . 6911.6 Output of S 11.06 . . . . . . . . . . . . . . . . . . . . . . . . 7011.7 Output of S 11.08 . . . . . . . . . . . . . . . . . . . . . . . . 7111.8 Output of S 11.09 . . . . . . . . . . . . . . . . . . . . . . . . 7211.9 Output of S 11.13 . . . . . . . . . . . . . . . . . . . . . . . . 7511.10Output of S 11.14 . . . . . . . . . . . . . . . . . . . . . . . . 7611.11Output of S 11.15 . . . . . . . . . . . . . . . . . . . . . . . . 7711.12Output of S 11.16 . . . . . . . . . . . . . . . . . . . . . . . . 7811.13Output of S 11.17 . . . . . . . . . . . . . . . . . . . . . . . . 7911.14Output of S 11.19 . . . . . . . . . . . . . . . . . . . . . . . . 8011.15Output of S 11.20 . . . . . . . . . . . . . . . . . . . . . . . . 8111.16Output of S 11.21 . . . . . . . . . . . . . . . . . . . . . . . . 8211.17Output of S 11.22 . . . . . . . . . . . . . . . . . . . . . . . . 8311.18Output of S 11.23 . . . . . . . . . . . . . . . . . . . . . . . . 8411.19Output of S 11.25 . . . . . . . . . . . . . . . . . . . . . . . . 8511.20Output of S 11.26 . . . . . . . . . . . . . . . . . . . . . . . . 8611.21Output of S 11.27 . . . . . . . . . . . . . . . . . . . . . . . . 8811.22Output of S 11.28 . . . . . . . . . . . . . . . . . . . . . . . . 89

13.1 Output of S 13.01 . . . . . . . . . . . . . . . . . . . . . . . . 9613.2 Output of S 13.02 . . . . . . . . . . . . . . . . . . . . . . . . 9713.3 Output of S 13.03 . . . . . . . . . . . . . . . . . . . . . . . . 9913.4 Output of S 13.04 . . . . . . . . . . . . . . . . . . . . . . . . 10013.5 Output of S 13.05 . . . . . . . . . . . . . . . . . . . . . . . . 10113.6 Output of S 13.06 . . . . . . . . . . . . . . . . . . . . . . . . 102

9

15.1 Output of S 15.01 . . . . . . . . . . . . . . . . . . . . . . . . 10415.2 Output of S 15.02 . . . . . . . . . . . . . . . . . . . . . . . . 10515.3 Output of S 15.03 . . . . . . . . . . . . . . . . . . . . . . . . 10615.4 Output of S 15.04 . . . . . . . . . . . . . . . . . . . . . . . . 10715.5 Output of S 15.05 . . . . . . . . . . . . . . . . . . . . . . . . 10815.6 Output of S 15.06 . . . . . . . . . . . . . . . . . . . . . . . . 10915.7 Output of S 15.07 . . . . . . . . . . . . . . . . . . . . . . . . 11015.8 Output of S 15.08 . . . . . . . . . . . . . . . . . . . . . . . . 111

10

Chapter 2

Laplace Transform and MatrixAlgebra

Install Symbolic Toolbox.Refer the spoken tutorial on the link (www.spoken-tutorial.org) for the installation of Symbolic Toolbox.

2.1 Scilab Code

Example 2.01-01 2.01.01.sci

1 syms t s ;2 y=l ap l a c e ( ’ 13 ’ , t , s ) ;3 disp (y , ”ans=” )

Example 2.01-02 2.01.02sci

1 syms t s ;2 y=l ap l a c e ( ’ 4+5∗%eˆ(−3∗ t ) ’ , t , s ) ;3 disp (y , ”ans=” )

Example 2.01-03 2.01.03.sci

1 syms t s ;2 y=l ap l a c e ( ’ 2∗ t−3∗%eˆ(− t ) ’ , t , s ) ;3 disp (y , ”ans=” )

11

Example 2.03 2.03.sci

1 p=poly ( [ 9 1 ] , ’ s ’ , ’ c o e f f ’ )2 q=poly ( [ 3 7 1 ] , ’ s ’ , ’ c o e f f ’ )3 f=p/q ;4 disp ( f , ”F( s )=” )5 x=s∗ f ;6 y=l im i t (x , s , 0 ) ; / / f i n a l v a l u e t h e o r e m

7 disp (y , ” f ( i n f )=” )8 z=l im i t (x , s , %inf ) ; / / i n i t i a l v a l u e t h e o r e m

9 disp ( z , ” f (0 )=” )


1 s=%s2 syms t ;3 disp ( ( s+3) /( ( s+1)∗( s+2)∗( s+4) ) , ”F( s )=” )4 [A]=pfss ( ( s+3) /( ( s+1)∗( s+2)∗( s+4) ) ) / / p a r t i a l f r a c t i o n

o f F ( s )

5 F1= i l a p l a c e (A(1) , s , t )6 F2= i l a p l a c e (A(2) , s , t )7 F3= i l a p l a c e (A(3) , s , t )8 F=F1+F2+F3 ;9 disp (F , ” f ( t )=” )

Example 2.05-01 2.05.01.sci

1 syms t s ;2 y= l ap l a c e ( ’%eˆ(− t )+5∗t+6∗%eˆ(−3∗ t ) ’ , t , s ) ;3 disp (y , ”ans=” )

Example 2.05-02 2.05.02.sci

1 syms t s ;2 y=l ap l a c e ( ’ 5+6∗ t ˆ2+3∗%eˆ(−2∗ t ) ’ , t , s ) ;3 disp (y , ”ans=” )

12


1 syms t s ;2 y=l ap l a c e ( ’3− %eˆ(−3∗ t ) ’ , t , s ) ;3 disp (y , ”ans=” )


1 p=poly ( [ 0 . 3 8 ] , ’ s ’ , ’ c o e f f ’ ) ;2 q=poly ( [ 0 0 .543 2 .48 1 ] , ’ s ’ , ’ c o e f f ’ ) ;3 F=p/q ;4

5 syms s ;6 x=s∗F;7 y=l im i t (x , s , 0 ) ; / / f i n a l v a l u e t h e o r e m

8 y=dbl ( y ) ;9 disp (y , ” f ( i n f )=” )

10 z=l im i t (x , s , %inf ) / / / / i n i t i a l v a l u e t h e o r e m

11 z=dbl ( z ) ;12 disp ( z , ” f (0 )=” )


1 s=%s ;2 p=poly ( [ 3 1 ] , ’ s ’ , ’ c o e f f ’ ) ;3 q=poly ( [ 0 −1 −1 −2] , ’ s ’ , ’ r o o t s ’ ) ;4 f=p/q ;5 syms t s ;6 y=i l a p l a c e ( f , s , t ) ;7 disp (y , ” f ( t )=” )

13

Chapter 3

Transfer Function


3.1 Scilab Code


1 syms t s ;2 f=%eˆ(−3∗ t ) ;3 y=l ap l a c e ( ’%eˆ(−3∗ t ) ’ , t , s ) ;4 disp (y , ”G( s )=” )


1 syms t s ;2 disp (%eˆ(−3∗ t ) , ”g ( t )=” ) ;3 y1=l ap l a c e ( ’%eˆ(−3∗ t ) ’ , t , s ) ;4 disp ( y1 , ”G( s )=” )5 disp (%eˆ(−4∗ t ) , ” r ( t )=” ) ;6 y2=l ap l a c e ( ’%eˆ(−4∗ t ) ’ , t , s ) ;7 disp ( y2 , ”R( s )=” )8 disp ( y1∗y2 , ”G( s )R( s )=” )


14

1 s=%s ;2 H=sysl in ( ’ c ’ , ( 4∗ ( s+2) ∗ ( ( s +2.5) ˆ3) ) / ( ( s+6) ∗ ( ( s+4)ˆ2) ) ) ;3 plzr (H)


1 syms t s2 y=l ap l a c e (%eˆ(−3∗ t )∗ sin (2∗ t ) , t , s ) ;3 disp (y , ”ans=” )


1 syms t s ;2 x=6−4∗%eˆ(−5∗ t )/5+%eˆ(−3∗ t ) / / G i v e n s t e p R e s p o n s e o f

t h e s y s t e m

3 printf ( ” Der iva t i ve o f s tep response g i v e s impulsere sponse \n” )

4 y=d i f f (x , t ) ; / / D e r i v a t i v e o f s t e p r e s p o n s e

5 printf ( ”Laplace Transform o f Impulse Response g i v e s theTrans fe r Function \n ” )

6 p=l ap l a c e (y , t , s ) ;7 disp (p , ” Trans fe r Function=” )


1 printf ( ”Given : Poles are s=−1,(−2+ i ) ,(−2− i ) ; z e r o s s=−3+i ,−3− i , Gain f a c t o r ( k )=5 \n” )

2 num=poly([−3+%i,−3−%i ] , ’ s ’ , ’ r o o t s ’ ) ;3 den=poly([−1,−2+%i,−2−%i ] , ’ s ’ , ’ r o o t s ’ ) ;4 G=(5∗num)/den ;5 disp (G, ”G( s )=” )


1 s=%s ;2 G=sysl in ( ’ c ’ , ( 5∗ ( s+2) ) / ( ( s+3)∗( s+4) ) ) ;3 disp (G, ”G( s )=” )

15

4 x=denom(G) ;5 disp (x , ” Cha r a c t e r i s t i c s Polynomial=” )6 y=roots ( x ) ;7 disp (y , ” Poles o f a system=” )


1 printf ( ”Given : Poles are s=−3, Zeros are s=−2, GainFactor ( k )=5 \n ” )

2 num=poly ( [ −2 ] , ’ s ’ , ’ r o o t s ’ ) ;3 den=poly ( [ −3 ] , ’ s ’ , ’ r o o t s ’ ) ;4 G=5∗num/den ;5 disp (G, ”G( s )=” )6 disp ( ” Input i s Step Function ” )7 syms t s ;8 R=lap l a c e (1 , t , s ) ;9 disp (R, ”R( s )=” )

10 printf ( ”C( s )=R( s )G( s ) \n” )11 C=R∗G;12 disp (C, ”C( s )=” )13 c=i l a p l a c e (C, s , t ) ;14 disp ( c , ”c ( t )=” )


1 / / p o l e z e r o p l o t f o r g ( s ) = ( s ˆ 2 + 3 s + 2 ) / ( s ˆ 2 + 7 s + 1 2 )

2 s=%s ;3 p=poly ( [ 2 3 1 ] , ’ s ’ , ” c o e f f ” )4 q=poly ( [ 1 2 7 1 ] , ’ s ’ , ” c o e f f ” )5 V=sysl in ( ’ c ’ ,p , q )6 plzr (V)7 syms s t ;8 v =i l a p l a c e ( ’ (2+(3∗ s )+s ˆ2) /( s ˆ2+(7∗ s )+12) ’ , s , t )9 disp (v , ”V( t ) = ’)

16

Chapter 4

Control system Components

4.1 Scilab Code


1 printf ( ”Given a ) Exc i ta t i on vo l tage ( Ein )=2V \n b)Se t t i ng Ratio ( a )= 0 .4 \n” )

2 Ein=2;3 disp ( Ein , ”Ein=” )4 a=0.4 ;5 disp ( a , ”a=” )6 Rt=10ˆ3;7 disp (Rt , ”Rt=” )8 Rl=5∗10ˆ3;9 disp (Rl , ”Rl=” )

10 printf ( ”Eo = ( a∗Ein ) /(1+(a∗(1−a )∗Rt) /Rl ) \n” )11 Eo = (a∗Ein ) /(1+(a∗(1−a )∗Rt) /Rl ) ;12 disp (Eo , ” output vo l tage (E0)=” )13 printf ( ”e=((a ˆ2)∗(1−a ) ) / ( ( a∗(1−a ) )+(Rl/Rt) ) \n” )14 e=((a ˆ2)∗(1−a ) ) / ( ( a∗(1−a ) )+(Rl/Rt) ) ;15 disp ( e , ” l oad ing e r r o r=” )16 printf ( ”E=Ein∗e \n” )17 E=Ein∗e ; / / V o l t a g e e r r o r = E x c i t a t i o n v o l t a g e ( E i n ) ∗

L o a d i n g e r r o r ( e )

18 disp (E, ”Voltage e r r o r=” )

17


1 printf ( ”n=5 , He l i c a l turn \n” )2 n=5; / / H e l i c a l t u r n

3 disp (n , ”n=” )4 printf ( ”N=9000 ,Winding Turn \n” )5 N=9000; / / W i n d i n g T u r n

6 disp (N, ”N=” )7 printf ( ”R=10000 , Potent iometer Res i s tance \n” )8 R=10000; / / P o t e n t i o m e t e r R e s i s t a n c e

9 disp (R, ”R=” )10 printf ( ”Ein=90 , Input vo l tage \n” )11 Ein=90; / / I n p u t v o l t a g e

12 disp ( Ein , ”Ein=” )13 printf ( ” r=5050 , Res i s tance at mid po int \n” )14 r =5050; / / R e s i s t a n c e a t m i d p o i n t

15 disp ( r , ” r=” )16 printf ( ”D=r−5000 , Deviat ion from nominal at mid−point \

n” )17 D=r−5000; / / D e v i a t i o n f r o m n o m i n a l a t m i d − p o i n t

18 disp (D, ”D=” )19 printf ( ”L=D/R∗100 , L in ea r i t y \n” )20 L=D/R∗100 ; / / L i n e a r i t y

21 disp (L , ”L=” )22 printf ( ”R=Ein/N , Reso lut ion \n” )23 R=Ein/N; / / R e s o l u t i o n

24 disp (R, ”R=” )25 printf ( ”Kp=Ein /(2∗ pi ∗n) , Potent iometer Constant \n” )26 Kp=Ein /(2∗%pi∗n) ; / / P o t e n t i o m e t e r C o n s t a n t

27 disp (Kp, ”Kp=” )


1 printf ( ” s i n c e S2 i s the r e f e r an c e s t a t o r winding , Es2=KVcos0 \n where Es2 & Er are rms vo l t ag e s \n ’ )

2 k=13 Theta=60;4 di sp (Theta , ”Theta=” )

18

5 V=28;6 di sp (V, ”V( app l i ed )=” )7 p r i n t f ( ”Es2=V∗cos ( Theta ) \n” )8 Es2=k∗V∗ cos ( Theta ∗(%pi/180) ) ;9 di sp (Es2 , ”Es2=” )

10 p r i n t f ( ”Es1=k∗V∗cos (Theta−120)\n” )11 Es1=k∗V∗ cos ( ( Theta−120)∗(%pi/180) ) ; // Given Theta=60

in degree s12 di sp (Es1 , ”Es1=’ )13 p r i n t f (”Es3=k∗V∗ cos ( Theta+120) \n”)14 Es3=k∗V∗ cos ( ( Theta+120) ∗(%pi/180) ) ;15 di sp (Es3 , ” Es3=’ )16 printf ( ”Es31=sq r t (3 ) ∗k∗Er∗ s i n ( Theta ) ” )17 Es31=sqrt (3 ) ∗k∗V∗ sin ( Theta ∗(%pi/180) ) ;18 disp (Es31 , ”Es31= ’)19 p r i n t f ( ”Es12=sqrt (3 ) ∗k∗Er∗ sin ( ( Theta−120)” )20 Es12=sq r t (3 ) ∗k∗V∗ s i n ( ( Theta−120)∗(%pi/180) ) ;21 di sp (Es12 , ”Es12=’ )22 p r i n t f (” Es23=sq r t (3 ) ∗k∗Er∗ s i n ( ( Theta+120) ”)23 Es23=sq r t (3 ) ∗k∗V∗ s i n ( ( Theta+120) ∗(%pi/180) ) ;24 di sp (Es23 , ” Es23=’ )


1 printf ( ” S e n s i t i v i t y=5v/1000rpm \n” )2 Vg=5;3 disp (Vg , ”Vg=” )4 printf ( ”w( in rad ians / sec ) =(1000/60) ∗2∗ pi \n” )5 w=(1000/60) ∗2∗%pi ;6 disp (w, ”w=” )7 printf ( ”Kt=Vg/w \n ’ )8 Kt=Vg/w;9 di sp (Kt , ”Gain constant (Kt)=” )


1 printf ( ”Torque=KmVm=2 \n” )2 t=2;

19

3 disp ( t , ”Torque ( t )=” )4 Fm=0.2;5 disp (Fm, ” Co e f f i c i e n t o f Viscous f r i c t i o n (Fm)=” )6 N=47 I =0.28 F1=0.059 printf ( ”Wnl=t /Fm” )

10 Wnl=t /Fm;11 disp (Wnl , ”No Load Speed (Wnl)=” )12 printf ( ”Fwt=I+(Nˆ2∗F1) \n” )13 Fwt=I+(Nˆ2∗F1) ;14 disp (Fwt , ”Total Viscous F r i c t i on (Fwt)=” )15 printf ( ”Te=t−(Fwt∗w) \n” )16 Te=0.8 / / l o a d

17 w=(t−Te) /Fwt ;18 disp (w, ”Speed o f Motor (w)=” )

20

Chapter 6

Control system Components


To Run the following codes use the same destination for the function aswell as the main code.

6.1 Scilab Code

Example F1 parallel.sce

1 function [ y]= p a r a l l e l ( sys1 , sys2 )2 y=sys1+sys23 endfunction

Example F2 serial.sce

1 function [ y]= s e r i e s ( sys1 , sys2 )2 y=sys1 ∗ sys23

4 endfunction


1 exec s e r i e s . s c e ;2 s=%s ;

21

3 sys1=sysl in ( ’ c ’ , ( s+3)/( s+1) )4 sys2=sysl in ( ’ c ’ , 0 . 2 / ( s+2) )5 sys3=sysl in ( ’ c ’ ,50/( s+4) )6 sys4=sysl in ( ’ c ’ ,10/( s ) )7 a=s e r i e s ( sys1 , sys2 ) ;8 b=s e r i e s ( a , sys3 ) ;9 y=s e r i e s (b , sys4 ) ;

10 y=simp( y ) ;11 disp (y , ”C( s ) /R( s )=” )

This code requires S F1.


1 exec p a r a l l e l . s c e ;2 s=%s ;3 sys1=sysl in ( ’ c ’ ,1/ s )4 sys2=sysl in ( ’ c ’ , 2/( s+1) )5 sys3=sysl in ( ’ c ’ , 3/( s+3) )6 a=p a r a l l e l ( sys1 , sys2 ) ;7 y=p a r a l l e l ( a , sys3 ) ;8 y=simp( y ) ;9 disp (y , ”C( s ) /R( s )=” )

This code requires S F2.


1 exec s e r i e s . s c e ;2 s=%s ;3 sys1=sysl in ( ’ c ’ , 3/( s ∗( s+1) ) )4 sys2=sysl in ( ’ c ’ , s ˆ2/(3∗( s+1) ) )5 sys3=sysl in ( ’ c ’ , 6/( s ) )6 a=(−1)∗ sys3 ;7 b=s e r i e s ( sys1 , sys2 ) ;8 y=b/ . a / / f e e d b a c k o p e r a t i o n

9 y=simp( y )10 disp (y , ”C( s ) /R( s )=” )

22


1 exec p a r a l l e l . s c e ;2 syms G1 G2 G3 H;3 a=s e r i e s (G1,G2) ;4 b=p a r a l l e l ( a ,G3) ;5 y=b/ .H / / n e g a t i v e f e e d b a c k o p e r a t i o n

6 y=simple ( y )7 disp (y , ”C( s ) /R( s )=” )


1 exec s e r i e s . s c e ;2 syms G1 G2 H1 H2 s ;3 a=G1/ .H1 ; / / n e g a t i v e f e e d b a c k o p e r a t i o n

4 b=a / .H2 ; / / n e g a t i v e f e e d b a c k o p e r a t i o n

5 y=s e r i e s (b ,G2) ;6 y=simple ( y ) ;7 disp (y , ”C( s ) /R( s )=” )


1 exec p a r a l l e l . s c e ;2 exec s e r i e s . s c e ;3 syms G1 G2 G3 G4 G5 G6 H1 H2 ;4 a=p a r a l l e l (G3,G5) ;5 b=p a r a l l e l ( a,−G4) ;6 c=s e r i e s (G1,G2) ;7 d=c / .H1 ;8 e=s e r i e s (b , d) ;9 f=e / .H2 ;

10 y=s e r i e s ( f ,G6) ;11 y=simple ( y ) ;12 disp (y , ”C( s ) /R( s )=” )


23

1 exec s e r i e s . s c e ;2 syms G1 G2 G3 H1 H2 R X;3 / / p u t t i n g x = 0 , t h e n s o l v i n g t h e b l o c k

4 a=G3/ .H1 ;5 b=s e r i e s (G1,G2) ;6 c=s e r i e s ( a , b ) ;7 x1=c / .H2 ;8 C1=R∗x1 ;9 disp ( x1 , ”C1( s ) /R( s )=” )

10 / / p u t t i n g r =0 , t h e n s o l v i n g t h e b l o c k

11 d=s e r i e s (G1,G2) ;12 e=s e r i e s (d ,H2) ;13 f=G3/ .H1 ;14 x2=f / . e ;15 C2=X∗x2 ;16 disp ( x2 , ”C2( s ) /X( s )=” )17 / / r e s u l t a n t o u t p u t C= C1 + C2

18 C=C1+C2 ;19 C=simple (C) ;20 disp (C, ”Resultant Output=” )


1 exec p a r a l l e l . s c e ;2 exec s e r i e s . s c e ;3 syms G1 G2 G3 H1 H2 ;4 / / s h i f t t h e t a k e − o f f p o i n t a f t e r t h e b l o c k G2

5 a=G3/G2 ;6 b=p a r a l l e l ( a , 1 ) ;7 c=s e r i e s (G1,G2) ;8 d=c / .H1 / / n e g a t i v e f e e d b a c k o p e r a t i o n

9 e=s e r i e s (d , b) ;10 y=e / .H2 ;11 y=simple ( y ) ;12 disp (y , ”C( s ) /R( s )=” )


24

1 exec s e r i e s . s c e2 syms G1 G2 G3 H1 H2 H3 ;3 / / R e m o v e t h e f e e d b a c k l o o p h a v i n g f e e d b a c k p a t h

t r a n s f e r f u n c t i o n H2

4 a=G3/ .H2 ;5 / / I n t e r c h a n g e t h e s u m m e r . a s w e l l a s r e p l a c e t h e

c a s c a d e b l o c k b y i t s e q u i v a l e n t b l o c k

6 b=s e r i e s (G1,G2) ;7 c=b/ .H1 ; / / N e g a t i v e F e e d b a c k O p e r a t i o n

8 d=s e r i e s ( c , a ) ;9 y=d/ .H3 ; / / N e g a t i v e F e e d b a c k O p e r a t i o n

10 y=simple ( y ) ;11 disp (y , ”C( s ) /R( s )=” )


1 exec p a r a l l e l . s c e ;2 exec s e r i e s . s c e ;3 syms G1 G2 G3 G4 G5 H1 H2 ;4 a=G2/ .H1 ; / / n e g a t i v e f e e d b a c k o p e r a t i o n

5 b=s e r i e s (G1, a ) ;6 c=s e r i e s (b ,G3) ;7 d=p a r a l l e l ( c ,G4) ;8 e=s e r i e s (d ,G5) ;9 y=e / .H2 ; / / n e g a t i v e f e e d b a c k o p e r a t i o n

10 y=simple ( y ) ;11 disp (y , ”C( s ) /R( s )=” )


1 exec p a r a l l e l . s c e ;2 exec s e r i e s . s c e ;3 syms G1 G2 G3 G4 G5 G6 G7 H1 H2 H3 ;4 a=p a r a l l e l (G1,G2) ;5 b=p a r a l l e l ( a ,G3) ;6 / / s h i f t t h e t a k e o f f p o i n t t o t h e r i g h t o f t h e b l o c k G4

7 c=G4/ .H1 ; / / n e g a t i v e f e e d b a c k o p e r a t i o n

8 d=G5/G4 ; / / n e g a t i v e f e e d b a c k o p e r a t i o n

25

9 e=p a r a l l e l (d , 1 ) ;10 f=G6/ .H2 ; / / n e g a t i v e f e e d b a c k o p e r a t i o n

11 g=s e r i e s (b , c ) ;12 h=s e r i e s ( g , e ) ;13 i=s e r i e s (h , f ) ;14 j=s e r i e s ( i ,G7) ;15 y=j / .H3 ;16 y=simple ( y ) ;17 disp (y , ”C( s ) /R( s )=” )


1 exec s e r i e s . s c e ;2 exec p a r a l l e l . s c e ;3 syms G1 G2 G3 G4 H1 H2 H3 ;4 / / s h i f t t h e t a k e − o f f p o i n t a f t e r t h e b l o c k G1

5 a=G3/G1 ;6 b=p a r a l l e l ( a ,G2) ;7 c=G1/ .H1 ; / / N e g a t i v e F e e d b a c k O p e r a t i o n

8 d=1/b ; / / N e g a t i v e F e e d b a c k O p e r a t i o n

9 e=p a r a l l e l (d ,H3) ;10 f=s e r i e s ( e ,H2) ;11 g=s e r i e s ( c , b ) ;12 h=g / . f ; / / N e g a t i v e F e e d b a c k O p e r a t i o n

13 y=s e r i e s (h ,G4) ;14 y=simple ( y ) ;15 disp (y , ”C( s ) /R( s )=” )


1 exec s e r i e s . s c e ;2 exec p a r a l l e l . s c e ;3 syms G1 G2 G3 G4 H1 H2 ;4 / / r e d u c e t h e i n t e r n a l f e e d b a c k l o o p

5 a=G2/ .H2 ;6 / / p l a c e t h e s u m m e r l e f t t o t h e b l o c k G1

7 b=G3/G1 ;8 / / e x c h a n g e t h e s u m m e r

26

9 c=p a r a l l e l (b , 1 ) ;10 d=s e r i e s ( a ,G1) ;11 e=s e r i e s (d ,G4) ;12 f=e / .H1 ;13 y=s e r i e s ( c , f ) ;14 y=simple ( y ) ;15 disp (y , ”C( s ) /R( s )=” )


1 exec s e r i e s . s c e ;2 exec p a r a l l e l . s c e ;3 syms G1 G2 G3 G4 H1 H2 ;4 / / s h i f t t h e t a k e − o f f p o i n t t o t h e r i g h t o f t h e b l o c k G3

5 a=H1/G3 ;6 b=s e r i e s (G2,G3) ;7 c=p a r a l l e l (H2 , a ) ;8 d=b/ . c ;9 e=s e r i e s (d ,G1) ;

10 f=e / . a ;11 y=s e r i e s ( f ,G4) ;12 y=simple ( y ) ;13 disp (y , ”C( s ) /R( s )=” )


1 exec s e r i e s . s c e ;2 exec p a r a l l e l . s c e ;3 syms G1 G2 G3 G4 H1 H2 H3 ;4 / / s h i f t t h e t a k e − o f f p o i n t t o t h e r i g h t o f t h e b l o c k H1

5 / / s h i f t t h e o t h e r t a k e − o f f p o i n t t o t h e r i g h t o f t h e

b l o c k H1 &H2

6 a=s e r i e s (H1 ,H2) ;7 b=1/a ;8 c=1/H1 ;9 d=G3/ . a ;

10 / / m o v e t h e s u m m e r t o t h e l e f t o f t h e b l o c k G2

11 e=G4/G2 ;

27

12 f=s e r i e s (d ,G2) ;13 / / e x c h a n g e t h e s u m m e r

14 g=f / .H1 ;15 h=p a r a l l e l (G1, e ) ;16 i=s e r i e s (h , g ) ;17 j=s e r i e s ( a ,H3) ;18 y=i / . j ;19 y=simple ( y ) ;20 disp (y , ”C( s ) /R( s )=” )

28

Chapter 8

Time Domain Analysis ofControl Systems

Install Symbolic Toolbox.Refer the spoken tutorial on the link (www.spoken-tutorial.org) for the installation of Symbolic Toolbox

8.1 Scilab Code


1 s=%s ;2 p=poly ( [ 3 1 ] , ’ s ’ , ’ c o e f f ’ ) ;3 q=poly ( [ 0 1 0 .95 0 . 2 1 ] , ’ s ’ , ’ c o e f f ’ ) ;4 G=p/q ;5 disp (G, ”G( s )=” )6 H=1;7 y=G∗H; / / T y p e 1 S y s t e m

8 disp (y , ”G( s )H( s )=” )9 / / R e f e r i n g t h e t a b l e 8 . 2 g i v e n i n t h e b o o k , F o r t y p e 1

s y s t e m Kp= % i n f & Ka =0

10 printf ( ”For type1 Kp=i n f & Ka=0 \n” )11 syms s ;12 Kv=l im i t ( s∗y , s , 0 ) ; / / Kv= v e l o c i t y e r r o r c o e f f i c i e n t

13 disp (Kv, ” Ve loc i ty Error C o e f f i c i e n t (Kv)=” )

29

Example 8.03-01 8.03.01.sci

1 s=%s ;2 p=poly ( [ 1 0 ] , ’ s ’ , ’ c o e f f ’ ) ;3 q=poly ( [ 0 0 1 ] , ’ s ’ , ’ c o e f f ’ ) ;4 G=p/q ;5 H=0.7;6 y=G∗H; / / t y p e 2

7 disp (y , ”G( s )H( s )=” )8 / / r e f e r i n g t h e t a b l e 8 . 2 g i v e n i n t h e b o o k , f o r t y p e 1

Kp= % i n f & Kv= % i n f

9 printf ( ”For type1 Kp=i n f & Kv=i n f \n” )10 syms s ;11 Ka=l im i t ( s ˆ2∗y , s , 0 ) ; / / Ka= a c c e l a r a t i o n e r r o r

c o e f f i c i e n t

12 disp (Ka, ”Ka=” )

Example 8.03-02 8.03.02.sci

1 p=poly ( [ 5 ] , ’ s ’ , ’ c o e f f ’ ) ;2 q=poly ( [ 5 3 1 ] , ’ s ’ , ’ c o e f f ’ ) ;3 G=p/q4 H=0.65 y=G∗H / / t y p e 0

6 / / r e f e r i n g t h e t a b l e 8 . 2 g i v e n i n t h e b o o k , f o r t y p e 1

Ka =0 & Kv =0

7 syms s8 Kp=l im i t ( s∗y/s , s , 0 ) / / Kp= p o s i t i o n a l e r r o r c o e f f i c i e n t

Example 8.03-03 8.03.03.sci

1

2 p=poly ( [ 1 0 .13 0 . 4 ] , ’ s ’ , ’ c o e f f ’ ) ;3 q=poly ( [ 0 0 5 3 1 ] , ’ s ’ , ’ c o e f f ’ ) ;4 G=10∗p/q / / g a i n FACTOR = 1 0

5 H=0.86 y=G∗H / / t y p e 2

30



8 syms s9 Ka=l im i t ( s ˆ2∗y , s , 0 ) / / Ka= a c c e l a r a t i o n e r r o r c o e f f i c i e n t


1

2 p=poly ( [ 4 1 ] , ’ s ’ , ’ c o e f f ’ ) ;3 q=poly ( [ 0 0 6 5 1 ] , ’ s ’ , ’ c o e f f ’ ) ;4 syms K real ;5 y=K∗p/q / / g a i n FACTOR=K

6 disp (y , ”G( s )H( s )=” )7 / / G ( s ) H ( s ) = y , a n d i t i s o f t y p e 2



9 printf ( ”For type1 Kp=i n f & Kv=i n f \n” )10 syms A s t ;11 Ka=l im i t ( s ˆ2∗y , s , 0 ) ; / / Ka= a c c e l a r a t i o n e r r o r


12 disp (Ka, ”Ka=” )13 / / g i v e n i n p u t i s r ( t ) =A ( t ˆ 2 ) / 2 & R ( s ) = l a p l a c e ( r ( t ) )

14 printf ( ”Given r ( t )=A( t ˆ2) /2 \n” )15 R=lap l a c e ( ’A∗ t ˆ2/2 ’ , t , s ) ;16 disp (R, ”R( s )=” )17 / / s t e a d y s t a t e e r r o r = L t s −>0 s R ( S ) / 1 + G ( s ) H ( S )

18 e=l im i t ( s∗R/(1+y) , s , 0 )19 disp ( e , ”Ess=” )


1 p=poly ( [ 6 0 ] , ’ s ’ , ’ c o e f f ’ ) ;2 q=poly ( [ 1 2 7 1 ] , ’ s ’ , ’ c o e f f ’ ) ;3 G=p/q ;4 disp (G, ”G( s )=” )5 H=1;6 y=G∗H7 F=1/(1+y) ;

31

8 disp (F , ”1/(1+G( s )H( s ) )=” )9 syms t s ;

10 Ko=l im i t ( s∗F/s , s , 0 ) / / Ko= L t s −>0 ( 1 / ( 1 + G ( s ) H ( S ) )

11 d=d i f f ( s∗F/s , s ) ;12 K1=l im i t ( d i f f ( s∗F/s , s ) , s , 0 ) / / K1= L t s −>0 ( dF ( s ) / d s )

13 K2=l im i t ( d i f f (d , s ) , s , 0 ) / / K2= L t s −>0 ( d 2 F ( s ) / d s )

14 / / g i v e n i n p u t i s r ( t ) = 4 + 3 ∗ t + 8 ( t ˆ 2 ) / 2 & R ( s ) = l a p l a c e ( r ( t

) )

15 a=(4+3∗ t+8∗( t ˆ2) /2)16 b=d i f f (4+3∗ t+8∗( t ˆ2) /2 , t )17 c=d i f f (b , t )18 e=Ko∗a+K1∗b+K2∗c / / e r r o r b y d y n a m i c c o e f f i c i e n t m e t h o d

19 disp ( e , ” e r r o r ” )


1 s=poly (0 , ’ s ’ ) ; / / D e f i n e s s a s p o l y n o m i a l v a r i a b l e

2 F=sysl in ( ’ c ’ , [ 2 5 / ( ( s+1)∗ s ) ] ) ; / / C r e a t e s t r a n s f e r

f u n c t i o n i n f o r w a r d p a t h

3 B=sysl in ( ’ c ’ ,(1+0∗ s ) /(1+0∗ s ) ) ; / / C r e a t e s t r a n s f e r

f u n c t i o n i n b a c k w a r d p a t h

4 CL=F/ .B / / C a l c u l a t e s c l o s e d − l o o p t r a n s f e r f u n c t i o n

5 / / c o m p a r e CL w i t h Wn ˆ 2 / ( s ˆ 2 + 2 ∗ z e t a ∗Wn+Wn ˆ 2 )

6 y=denom(CL) / / e x t r a c t i n g t h e d e n o m i n a t o r o f CL

7 z=coeff ( y ) / / e x t r a c t i n g t h e c o e f f i c i e n t s o f t h e

d e n o m i n a t o r p o l y n o m i a l

8 / / Wn ˆ 2 = z ( 1 , 1 ) , c o m p a r i n g t h e c o e f f i c i e n t s

9 Wn=sqrt ( z (1 , 1 ) ) / / Wn= n a t u r a l f r e q u e n c y

10 / / 2 ∗ z e t a ∗Wn= z ( 1 , 2 )

11 zeta=z (1 , 2 ) /(2∗Wn) / / z e t a = d a m p i n g f a c t o r

12 Wd=Wn∗sqrt(1− zeta ˆ2)13 Tp=%pi/Wd14 Mp=100∗exp((−%pi∗ zeta ) /sqrt(1− zeta ˆ2) )



32

2 F=sysl in ( ’ c ’ , [ 2 0 / ( s ˆ2+5∗ s+5) ] ) ; / / C r e a t e s t r a n s f e r











10 / / 2 ∗ z e t a ∗Wn= z ( 1 , 2 )


12 Wd=Wn∗sqrt(1− zeta ˆ2)13 Tp=%pi/Wd / / Tp= p e a k t i m e

14 Mp=100∗exp((−%pi∗ zeta ) /sqrt(1− zeta ˆ2) ) / / p e a k

o v e r s h o o t

15 Td=(1+0.7∗ zeta ) /Wn / / Td= d e l a y t i m e

16 a=atan ( sqrt(1− zeta ˆ2) / zeta )17 Tr=(%pi−a ) /Wd / / T r = r i s e t i m e

18 Ts=4/( zeta ∗Wn) / / T s = s e t t l i n g t i m e


1 p=poly ( [ 1 4 0 , 3 5 ] , ’ s ’ , ’ c o e f f ’ ) ;2 q=poly ( [ 0 ,10 ,7 , 1 ] , ’ s ’ , ’ c o e f f ’ ) ;3 G=p/q4 H=15 y=G∗H / / t y p e 1


Kp= % i n f & Ka =0

7 syms s8 Kv=l im i t ( s∗y , s , 0 ) / / Kv= v e l o c i t y e r r o r c o e f f i c i e n t

9 / / g i v e n i n p u t i s r ( t ) = 5 ∗ t & R ( s ) = l a p l a c e ( r ( t ) )

10 R=lap l a c e ( ’ 5∗ t ’ , t , s )11 / / s t e a d y s t a t e e r r o r = L t s −>0 s R ( S ) / 1 + G ( s ) H ( S )

12 e=l im i t ( s∗R/(1+y) , s , 0 ) / / e = e r r o r f o r r a m p i n p u t

13 disp ( e , ” steady s t a t e e r r o r ” )

33


1 p=poly ( [ 4 0 , 2 0 ] , ’ s ’ , ’ c o e f f ’ ) ;2 q=poly ( [ 0 , 0 , 5 , 6 , 1 ] , ’ s ’ , ’ c o e f f ’ ) ;3 G=p/q ;4 H=1;5 y=G∗H; / / t y p e 2

6 disp (y , ”G( s )H( s=” )7 / / r e f e r i n g t h e t a b l e 8 . 2 g i v e n i n t h e b o o k , f o r t y p e 2


8 syms s t ;9 Ka=l im i t ( s ˆ2∗y , s , 0 ) / / Ka= a c c e l a r a t i o n e r r o r


10 / / g i v e n i n p u t i s r ( t ) =1+3 t + t ˆ 2 / 2 & R ( s ) = l a p l a c e ( r ( t ) )

11 R=lap l a c e ( ’ (1+3∗ t+(t ˆ2/2) ) ’ , t , s )12 / / s t e a d y s t a t e e r r o r = L t s −>0 s R ( S ) / 1 + G ( s ) H ( S )

13 e=l im i t ( s∗R/(1+y) , s , 0 ) / / e = e r r o r f o r r a m p i n p u t

14 disp ( e , ” steady s t a t e e r r o r ” )


1 s = poly ( 0 , ’ s ’ )2 sys = sysl in ( ’ c ’ , ( 20 ) /( s ˆ2+7∗ s+10) )3 a=sys / . ( 2∗ ( s+1) ) / / s i m p l i f y i n g t h e i n t e r n a l f e e d b a c k

l o o p

4 b=20∗2∗a ;5 disp (b , ”G( s ) ” )6 c=1;7 disp ( c , ”H( s ) ” )8 OL=b∗c ;9 disp (OL, ”G( s )H( s ) ” )

10 , ”G( s )∗H( s ) ” )11 syms t s ;12 Kp=l im i t ( s∗OL/s , s , 0 ) / / Kp= p o s i t i o n e r r o r c o e f f i c i e n t

13 Kv=l im i t ( s∗OL, s , 0 ) / / Kv= v e l o c i t y e r r o r c o e f f i c i e n t

14 Ka=l im i t ( s ˆ2∗OL, s , 0 ) / / Ka= a c c e l a r a t i o n e r r o r


34

15 / / g i v e n i n p u t r ( t ) =6

16 R=lap l a c e ( ’ 6 ’ , t , s )17 / / s t e a d y s t a t e e r r o r = L t s −>0 s R ( S ) / 1 + G ( s ) H ( S )

18 e1=l im i t ( s∗R/(1+OL) , s , 0 ) ; / / e = e r r o r f o r g i v e n i n p u t

19 disp ( e1 , ” e r r o r ” )20 / / g i v e n i n p u t r ( t ) =8 t

21 M=lap l a c e ( ’ 8∗ t ’ , t , s )22 / / s t e a d y s t a t e e r r o r = L t s −>0 s R ( S ) / 1 + G ( s ) H ( S )

23 e2=l im i t ( s∗M/(1+OL) , s , 0 ) ; / / e = e r r o r f o r g i v e n i n p u t

24 disp ( e2 , ” e r r o r ” )25 / / g i v e n i n p u t r ( t ) = 1 0 + 4 t +3 t ˆ 2 / 2

26 N=lap l a c e ( ’ 10+4∗ t+(3∗ t ˆ2) /2 ’ , t , s )27 / / s t e a d y s t a t e e r r o r = L t s −>0 s R ( S ) / 1 + G ( s ) H ( S )

28 e3=l im i t ( s∗N/(1+OL) , s , 0 ) ; / / e = e r r o r f o r g i v e n i n p u t

29 disp ( e3 , ” e r r o r ” )


1 s = poly ( 0 , ’ s ’ )2 sys1 = sysl in ( ’ c ’ , ( s ) /( s+6) ) ;3 sys2 = sysl in ( ’ c ’ , ( s+2)/( s+3) ) ;4 sys3 = sysl in ( ’ c ’ , ( 5 ) / ( ( s+3)∗ s ˆ3) ) ;5 a=sys2+sys3 ;6 disp ( a , ”H( s ) ” )7 b=sys1 ;8 disp (b , ”G(S) ” )9 y=a∗b ;

10 disp (y , ”G(S)H(S) ” )11 syms s12 Kp=l im i t ( s∗y/s , s , 0 ) / / Kp= p o s i t i o n e r r o r c o e f f i c i e n t

13 Kv=l im i t ( s∗y , s , 0 ) / / Kv= v e l o c i t y e r r o r c o e f f i c i e n t

14 Ka=l im i t ( s ˆ2∗y , s , 0 ) / / Ka= a c c e l a r a t i o n e r r o r



1 s=%s ;2 syms k t ;

35

3 y=k /( ( s+1)∗ s ˆ2∗( s+4) ) ;4 disp (y , ”G( s )H( s )=” )5 r=1+(8∗ t )+(18∗ t ˆ2/2) ;6 disp ( r , ” r ( t )=” )7 R=lap l a c e ( r , t , s ) ;8 disp (R, ”R( s )=” )9 e=l im i t ( ( s∗R)/(1+y) , s , 0 )

10 disp ( e , ”Ess=” )11 printf ( ’ Given Ess = 0 .8 \n”)12 e =0.8 ;13 k=72/e ;14 di sp (k , ” k=”)


1 syms s t k ;2 s = poly ( 0 , ’ s ’ ) ;3 y=k/( s ∗( s+2) ) ; / / G ( s ) H ( s )

4 disp (y , ”G( s )H( s ) ” )5 / / R= l a p l a c e ( ’ 0 . 2 ∗ t ’ , t , s )

6 R=lap l a c e ( ’ 0 .2∗ t ’ , t , s )7 e=l im i t ( s∗R/(1+y) , s , 0 )8 / / g i v e n e < = 0 . 0 2

9 a = [ 0 . 0 2 ] ;10 b=[−0.4 ] ;11 m=l insolve ( a , b ) ; / / S o l v e s T h e L i n e a r E q u a t i o n

12 disp (m, ”k” )


1 syms s , t , k ;2 s=%s ;3 y=k/( s ∗( s+2)∗(1+0.5∗ s ) ) / / G ( s ) H ( s )

4 disp (y , ”G( s )H( s ) ” )5 / / R= l a p l a c e ( ’ 3 ∗ t ’ , t , s )

6 R=lap l a c e ( ’ 3∗ t ’ , t , s )7 e=l im i t ( s∗R/(1+y) , s , 0 ) ;8 disp ( e , ” steady s t a t e e r r o r ” )

36

9 k=4; / / g i v e n

10 y=e ;11 disp (y , ” s t a t e s t a t e e r r o r when k=4” )


1 s = poly ( 0 , ’ s ’ ) ;2 sys = sysl in ( ’ c ’ ,180/( s ∗( s+6) ) ) / / G ( s ) H ( s )

3 disp ( sys , ”G( s )H( s ) ” )4 syms t s ;5 / / R= l a p l a c e ( ’ 4 ∗ t ’ , t , s )

6 R=lap l a c e ( ’ 4∗ t ’ , t , s ) ;7 e=l im i t ( s∗R/(1+ sys ) , s , 0 ) ;8 y=dbl ( e ) ;9 disp (y , ” steady s t a t e e r r o r ” )

10 syms k real ;11 / / v a l u e o f k i f e r r o r r e d u c e d b y 6% ;

12 e1=l im i t ( s∗R/(1+k/( s ∗( s+6) ) ) , s , 0 ) / / −−−−−−−1

13 e1=0.94∗ e / / −−−−−−−−2

14 / / n o w s o l v i n g t h e s e t w o e q u a t i o n s

15 a = [47 ] ;16 b=[−9000];17 m=l insolve ( a , b ) ;18 disp (m, ”k” )


1 s=%s ;2 F=sysl in ( ’ c ’ , ( 81 ) /( s ˆ2+6∗ s ) ) ; / / C r e a t e s t r a n s f e r










37


10 / / 2 ∗ z e t a ∗Wn= z ( 1 , 2 )




o v e r s h o o t


1 s=%s ;2 F=sysl in ( ’ c ’ , ( 25 ) /( s ˆ2+7∗ s ) ) ; / / C r e a t e s t r a n s f e r




4 k=20/25; / / k = g a i n f a c t o r

5 CL=k∗(F/ .B) / / C a l c u l a t e s c l o s e d − l o o p t r a n s f e r f u n c t i o n







11 / / 2 ∗ z e t a ∗Wn= z ( 1 , 2 )




o v e r s h o o t




20 / / y ( t ) = e x p r e s s i o n f o r o u t p u t

21 y=(20/25)∗(1−(exp(−1∗ zeta ∗Wn∗ t ) /sqrt(1− zeta ˆ2) )∗ sin (Wd∗t+atan ( ze ta /sqrt(1− zeta ˆ2) ) ) ) ;

22 disp (y , ”Y( t ) ” )

38


1 s=%s ;2 F=sysl in ( ’ c ’ , ( 144) /( s ˆ2+12∗ s ) ) ; / / C r e a t e s t r a n s f e r




4 k=20/25; / / k = g a i n f a c t o r

5 CL=k∗(F/ .B) / / C a l c u l a t e s c l o s e d − l o o p t r a n s f e r f u n c t i o n







11 / / 2 ∗ z e t a ∗Wn= z ( 1 , 2 )




o v e r s h o o t





1 ieee (2 ) ;2 syms k T;3 num=k ;4 den=s∗(1+s∗T) ;5 G=num/den ;6 disp (G, ”G( s )=” )7 H=1;8 CL=G/ .H;9 CL=simple (CL) ;

39

10 disp (CL, ”C( s ) /R( s )=” ) / / C a l c u l a t e s c l o s e d − l o o p

t r a n s f e r f u n c t i o n


12 [ num, den]=numden(CL) / / e x t r a c t s num & d e n o f s y m b o l i c

f u n c t i o n ( CL )

13 den=den/T;14 c o f a 0 = c o e f f s ( den , ’ s ’ , 0 ) / / c o e f f o f d e n o f s y m b o l i c


15 c o f a 1 = c o e f f s ( den , ’ s ’ , 1 )16 / / Wn ˆ 2 = c o f a 0 , c o m p a r i n g t h e c o e f f i c i e n t s

17 Wn=sqrt ( c o f a 0 )18 disp (Wn, ” natura l f requency Wn” ) / / Wn= n a t u r a l

f r e q u e n c y

19 / / c o f a 1 = 2 ∗ z e t a ∗Wn

20 zeta=co f a 1 /(2∗Wn)

Example 8.23-01 8.23.01.sci

1 s= poly ( 0 , ’ s ’ ) ;2 sys = sysl in ( ’ c ’ ,10/( s+2) ) ; / / G ( s ) H ( s )

3 disp ( sys , ”G( s )H( s ) ” )4 F=1/(1+sys )5 syms t s ;6 Co=l im i t ( s∗F/s , s , 0 ) / / Ko= L t s −>0 ( 1 / ( 1 + G ( s ) H ( S ) )

7 a=(3) ;8 e=Co∗a ;9 disp ( e , ” steady s t a t e e r r o r ” )

Example 8.23-02 8.23.02.sci



7 d=d i f f ( s∗F/s , s )8 C1=l im i t ( d i f f ( s∗F/s , s ) , s , 0 ) / / K1= L t s −>0 ( dF ( s ) / d s )

40

9 a=(2∗ t ) ;10 b=d i f f ( (2∗ t ) , t ) ;11 e=Co∗a+C1∗b ;12 disp ( e , ” s t eadt s t a t e e r r o r ” )

Example 8.23-03 8.23.03.sci




9 C2=l im i t ( d i f f (d , s ) , s , 0 ) / / K2= L t s −>0 ( d 2 F ( s ) / d s )

10 a=(( t ˆ2) /2) ;11 b=d i f f ( ( t ˆ2) /2 , t ) ;12 c=d i f f (b , t ) ;13 e=Co∗a+C1∗b+C2∗c ;14 disp ( e , ” steady s t a t e e r r o r ” )

Example 8.23-04 8.23.04.sci





10 / / g i v e n i n p u t i s r ( t ) = 1 + 2 ∗ t + ( t ˆ 2 ) / 2 & R ( s ) = l a p l a c e ( r ( t )

)

11 a=(1+2∗ t+(t ˆ2) /2) ;12 b=d i f f ( a , t ) ;13 c=d i f f (b , t ) ;

41

14 e=Co∗a+C1∗b+C2∗c / / e r r o r b y d y n a m i c c o e f f i c i e n t m e t h o d


1 s= poly ( 0 , ’ s ’ ) ;2 sys1 = sysl in ( ’ c ’ , ( s+3)/( s+5) ) ;3 sys2= sysl in ( ’ c ’ , ( 100) /( s+2) ) ;4 sys3= sysl in ( ’ c ’ , ( 0 . 1 5 ) /( s+3) ) ;5 G=sys1 ∗ sys2 ∗ sys3 ∗2∗56 H=1;7 y=G∗H; / / G ( s ) H ( s )

8 disp (y , ”G( s )H( s ) ” )9 F=1/(1+y)

10 syms t s ;11 Co=l im i t ( s∗F/s , s , 0 ) / / Ko= L t s −>0 ( 1 / ( 1 + G ( s ) H ( S ) )



15 a=(1+(2∗ t ) +(5∗( t ˆ2/2) ) ) ;16 b=d i f f ( a , t ) ;17 c=d i f f (b , t ) ;18 e=Co∗a+C1∗b+C2∗c ;19 disp ( e , ” s t eadt s t a t e e r r o r ” )


1 s=%s ;2 sys=sysl in ( ’ c ’ ,(9∗(1+2∗ s ) ) /( s ˆ2+0.6∗ s+9) ) ;3 disp ( sys , ”C( s ) /R( s )=” )4 / / g i v e n r ( t ) =u ( t )

5 syms t s ;6 R=lap l a c e ( ’ 1 ’ , t , s ) ;7 disp (R, ”R( s )=” )8 C=R∗ sys ;9 disp (C, ”C( s )=” )

10 c=i l a p l a c e (C, s , t )11 disp ( c , ”c ( t )=” )12 G=9/( s ˆ3+0.6∗ s ˆ2) ;

42

13 disp (G, ”G( s )=” )14 H=1;15 y=1+G∗H;16 syms t s ;17 Kp=l im i t ( s∗G/s , s , 0 )18 Kv=l im i t ( s∗G, s , 0 )19 Ka=l im i t ( s ˆ2∗G, s , 0 )20 R=lap l a c e ( ’ (1+t+(t ˆ2/2) ) ’ , t , s )21 / / s t e a d y s t a t e e r r o r = L t s −>0 s R ( S ) / 1 + G ( s ) H ( S )

22 e=l im i t ( s∗R/(1+y) , s , 0 ) ; / / e = e r r o r f o r r a m p i n p u t

23 disp ( e , ” steady s t a t e e r r o r ( Ess ) ” )

43

Chapter 9

Feedback Characteristics ofcontrol Systems


9.1 Scilab Code


1 s=%s ;2 G=sysl in ( ’ c ’ ,20/( s ∗( s+4) ) )3 H=0.35;4 y=G∗H;5

6 S=1/(1+y) ;7 disp (S , ”1/(1+G( s )∗H( s ) ) ” )8

9 / / g i v e n w = 1 . 2

10 w=1.211 s=%i∗w12 S=horner (S , s ) / / c a l c u l a t e s v a l u e o f S a t s

13 a=abs (S)14 disp ( a , ” s e n s i t i v i t y o f open loop ” )15

44

16 F=−y/(1+y)17 disp (F , ”(−G( s )∗H( s ) ) /(1+G( s )∗H( s ) ) ” )18 S=horner (F , s ) / / c a l c u l a t e s v a l u e o f F a t s

19 b=abs (S)20 disp (b , ” s e n s i t i v i t y o f c l o s ed loop ” )


1 s=%s ;2 sys1=sysl in ( ’ c ’ , 9/( s ∗( s +1.8) ) ) ;3 syms Td ;4 sys2=1+(s∗Td) ;5 sys3=sys1 ∗ sys2 ;6 H=1;7 CL=sys3 / .H; / / C a l c u l a t e s c l o s e d − l o o p t r a n s f e r f u n c t i o n

8 disp (CL, ”C( s ) /R( s ) ” )9 / / c o m p a r e CL w i t h Wn ˆ 2 / ( s ˆ 2 + 2 ∗ z e t a ∗Wn+Wn ˆ 2 )

10 [ num, den]=numden(CL) / / e x t r a c t s num & d e n o f s y m b o l i c

f u n c t i o n CL

11 den=den /5 ;12 c o f a 0 = c o e f f s ( den , ’ s ’ , 0 ) / / c o e f f o f d e n o f s y m b o l i c

f u n c t i o n CL


15 Wn=sqrt ( c o f a 0 )16 disp (Wn, ” natura l f requency Wn” ) / / Wn= n a t u r a l

f r e q u e n c y

17 / / c o f a 1 = 2 ∗ z e t a ∗Wn

18 zeta=co f a 1 /(2∗Wn)19 zeta =1;disp ( zeta , ” f o r c r i t i c a l y damped func t i on zeta ” )20 Td=((2∗Wn) −1.8) /921 Ts=4/( zeta ∗Wn) ;22 Ts=dbl (Ts ) ;23 disp (Ts , ” s e t t l i n g time Ts” )


1 s=%s ;

45

2 G=sysl in ( ’ c ’ ,40/( s ∗( s+4) ) )3 H=0.50;4 y=G∗H;5 S=1/(1+y) ;6 disp (S , ”1/(1+G( s )∗H( s ) ) ” )7 / / g i v e n w = 1 . 3

8 w=1.39 s=%i∗w

10 S=horner (S , s )11 a=abs (S)12 disp ( a , ” s e n s i t i v i t y o f open loop ” )13 F=−y/(1+y)14 disp (F , ”(−G( s )∗H( s ) ) /(1+G( s )∗H( s ) ) ” )15 S=horner (F , s )16 b=abs (S)17 disp (b , ” s e n s i t i v i t y o f c l o s ed loop ” )


1 s=%s ;2 syms s ;3 syms Wn zeta A H real ;4 T=6.28;5 Wn=(8∗%pi ) /T;6 zeta =0.37 n=Wnˆ2 ;8 d=sˆ2+2∗ zeta ∗Wn∗ s+Wnˆ2 ;9 G1=n/d ;

10 disp (G1, ”G1( s ) ” )11 G=A∗G1;12 disp (G, ”G( s ) ” )13 S1=(d i f f (G,A) ) ∗(A/G) ;14 a=simple ( S1 ) ;15 disp ( a , ”open loop s e n s i t i v i t y f o r changes in A” )16 M=G/ .H;17 p=simple (M)18 S2=(d i f f (p ,A) ) ∗(A/p) ;19 b=simple ( S2 ) ;

46

20 disp (b , ” c l o s ed loop s e n s i t i v i t y f o r changes in A” )21 S3=(d i f f (p ,H) ) ∗(H/p) ;22 c=s imple ( S3 ) ;23 disp ( c , ” c l o s ed loop s e n s i t i v i t y f o r changes in H” )


1 s=%s ;2 sys1=sysl in ( ’ c ’ , ( s+3)/ s ) ;3 syms u rp k H RL;4 num2=u∗RL∗ s ∗( s+2) ;5 den2=(rp+RL) ∗( s+3) ;6 sys2=num2/den2 ;7 num3=k ;8 den3=s+2;9 sys3=num3/den3 ;

10 sys=sys1 ∗ sys2 ∗ sys3 ;11 disp ( sys , ”G( s )=” ) ;12 RL=10∗10ˆ3;13 rp =4∗10ˆ3;14 sys=eval ( sys )15 sys=f l o a t ( sys )16 disp ( sys , ” sys=” ) ;17 disp (H, ”H( s ) ” ) ;18 M=sys / .H / / G ( s ) / 1 + G ( s ) H ( S )

19 M=simple (M)20 S=(d i f f (M, u) ) ∗(u/M) ;21 S=simple (S) ;22 disp (S , ” system s e n s i t i v i t y due to va r i a t i o n o f u=” ) ;23 H=0.3;24 u=12;25 S=eval (S) / / −−−−−−−−− e q 1

26 S=0.04;27 k=((7/S)−7)/18 ; / / f r o m e q 1

28 disp (k , ”K=” )


47

1 G=210;2 H=0.12;3 syms Eo Er4 printf ( ” f o r c l o s ed loop system” )5 sys=G/ .H; / / Eo / E r =G / ( 1 + GH )

6 disp ( sys , ”Eo/Er=” )7 Eo=240 / / g i v e n

8 Er=Eo/8 . 0152 ;9 disp (Er , ”Er=” )

10 printf ( ” f o r open loop system” )11 disp (G, ”Eo/Er=” )12 Er=Eo/G;13 G=210;14 disp (Er , ”Er=” )15 / / p r i n t f ( ” s i n c e G i s r e d u c e d b y 1 2 % , t h e n e w v a l u e o f

g a i n i s 7 8 4 . 8 V ” ) ;

16 S=1/(1+G∗H)17 disp (S , ” (%change in M) /(%change in G)=” )18 disp (12 , ”%CHANGE IN G=” )19 y=12∗0.03869;20 disp (y , ”%CHANGE IN M=” )21 printf ( ” f o r open loop system” )22 disp (28 .8∗100/240 , ”%change in Eo” )


1 s=%s ;2 sys1=sysl in ( ’ c ’ ,20/( s ∗( s+2) ) ) ;3 syms Kt ;4 sys2=Kt∗ s ;5 sys3=sys1 / . sys2 ;6 p=simple ( sys3 ) ;7 disp (p , ”G( s )=” )8 H=1;9 sys=sys3 / .H;

10 sys=simple ( sys ) ;11 disp ( sys , ”C( s ) /R( s )=” )12 [ num, den]=numden( sys )

48

13 c o f a 0 = c o e f f s ( den , ’ s ’ , 0 ) / / c o e f f o f d e n o f s y m b o l i c

f u n c t i o n CL


16 Wn=sqrt ( c o f a 0 )17 Wn=dbl (Wn) ;18 disp (Wn, ” natura l f requency Wn=” ) / / Wn=

n a t u r a l f r e q u e n c y

19 / / c o f a 1 = 2 ∗ z e t a ∗Wn

20 zeta=co f a 1 /(2∗Wn)21 zeta =0.6 ;22 Kt=((2∗ zeta ∗Wn)−2) /20 ;23 disp (Kt , ”Kt=” )24 Wd=Wn∗sqrt(1− zeta ˆ2) ;25 disp (Wd, ”Wd=” )26 Tp=%pi/Wd;27 disp (Tp, ”Tp=” )28 Mp=100∗exp((−%pi∗ zeta ) /sqrt(1− zeta ˆ2) ) ;29 disp (Mp, ”Mp=” )30 Ts=4/( zeta ∗Wn) ;31 disp (Ts , ”Ts=” )


1 s=%s ;2 printf ( ” 1) ze ta & Wn without Kd” )3 G=60∗ sysl in ( ’ c ’ , 1/( s ∗( s+4) ) ) ;4 disp (G, ”G(S)=” )5 CL=G/ .H;6 disp (CL, ”C( s ) /R( s )=” )7 y=denom(CL) / / e x t r a c t i n g t h e d e n o m i n a t o r o f CL





11 / / 2 ∗ z e t a ∗Wn= z ( 1 , 2 )

12 zeta=z (1 , 2 ) /(2∗Wn)13 sys1=sysl in ( ’ c ’ , 1/( s ∗( s+4) ) ) ;

49

14 syms Kd;15 printf ( ” 2)Kd f o r ze ta =0.60 with c o n t r o l l e r ” )16 sys2=s∗Kd;17 sys3=sys1 / . sys2 ;18 G=sys3 ∗60 ;19 disp (G, ”G( s )=” )20 H=1;21 sys=G/ .H;22 disp ( sys , ”C( s ) /R( s )=” )23 [ num, den]=numden( sys )24 c o f a 0 = c o e f f s ( den , ’ s ’ , 0 )25 c o f a 1 = c o e f f s ( den , ’ s ’ , 1 )26 / / Wn ˆ 2 = c o f a 0 , c o m p a r i n g t h e c o e f f i c i e n t s

27 Wn=sqrt ( c o f a 0 )28 Wn=dbl (Wn) ;29 disp (Wn, ” natura l f requency Wn=” )30 / / c o f a 1 = 2 ∗ z e t a ∗Wn

31 zeta =0.6032 Kd=(2∗ zeta ∗Wn)−4


1 s=%s ;2 printf ( ” 1) without c o n t r o l l e r ” )3 G=sysl in ( ’ c ’ ,120/( s ∗( s +12.63) ) ) ;4 disp (G, ”G( s )=” )5 H=1;6 CL=G/ .H;7 disp (CL, ”C( s ) /R( s )=” )8 y=denom(CL) / / e x t r a c t i n g t h e d e n o m i n a t o r o f CL




11 Wn=sqrt ( z (1 , 1 ) ) ;12 disp (Wn, ”Wn=” ) / / Wn= n a t u r a l f r e q u e n c y

13 / / 2 ∗ z e t a ∗Wn= z ( 1 , 2 )

14 zeta=z (1 , 2 ) /(2∗Wn) ;15 disp ( zeta , ” zeta=” )

50

16 printf ( ” 2) with c on t r o l l e r ’ )17 G=s y s l i n ( ’ c ’ , (120∗(30+ s ) ) /( s ∗( s +12.63) ∗30) ) ;18 di sp (G, ”G( s )=” )19 CL=G/ .H;20 di sp (CL, ”C( s ) /R( s )=” )21 den=denom(CL)22 den=den/30 // ex t r a c t i n g the denominator o f CL23 z=c o e f f ( den ) // ex t r a c t i n g the c o e f f i c i e n t s o f the

denominator polynomial24 //Wn̂ 2=z (1 , 1 ) , comparing the c o e f f i c i e n t s25 Wn=sqr t ( z (1 , 1 ) ) ;26 di sp (Wn, ”Wn=” ) // Wn=natura l f requency27 //2∗ zeta ∗Wn=z (1 , 2 )28 zeta=z (1 , 2 ) /(2∗Wn) ;29 Mp=100∗exp((−%pi∗ zeta ) / sq r t (1− zeta ˆ2) ) ;30 di sp (Mp, ”Mp=” )31 Ts=4/( zeta ∗Wn) ;32 di sp (Ts , ”Ts=” )


1 s=%s ;2 printf ( ” 1) without c o n t r o l l e r ” )3 G=64∗ sysl in ( ’ c ’ , 1/( s ∗( s+4) ) ) ;4 disp (G, ”G( s )=” )5 H=1;6 CL=G/ .H;7 disp (CL, ”C( s ) /R( s )=” )8 / / E x t r a c t i n g t h e d e n o m i n a t o r o f CL

9 y=denom(CL)10 / / E x t r a c t i n g t h e c o e f f i c i e n t s o f t h e d e n o m i n a t o r

p o l y n o m i a l

11 z=coeff ( y )12 / / Wn ˆ 2 = z ( 1 , 1 ) , c o m p a r i n g t h e c o e f f i c i e n t s

13 Wn=sqrt ( z (1 , 1 ) ) ;14 / / Wn= n a t u r a l f r e q u e n c y

15 disp (Wn, ”Wn=” )16 printf ( ” 2) with c on t r o l l e r ’ )

51

17 syms K;18 sys1=s y s l i n ( ’ c ’ , 1 / ( s ∗( s+4) ) ) ;19 sys2=sys1 / . ( s∗K) ;20 G=64∗ sys221 di sp (G, ”G( s )=” )22

23 sys=G/ .H;24 sys=simple ( sys ) ;25 di sp ( sys , ”C( s ) /R( s )=” )26 [ num, den]=numden( sys )27 // Coef f o f den o f symbol ic func t i on CL28 c o f a 0 = c o e f f s ( den , ’ s ’ , 0 )29 c o f a 1 = c o e f f s ( den , ’ s ’ , 1 )30 //Wn̂ 2= co f a 0 , comparing the c o e f f i c i e n t s31 Wn=sqr t ( c o f a 0 )32 Wn=dbl (Wn) ;33 //Wn=natura l f requency34 di sp (Wn, ” natura l f requency Wn=” )35 // c o f a 1=2∗zeta ∗Wn36 zeta=co f a 1 /(2∗Wn)37 zeta =0.6 ;38 k=(16∗ zeta )−439 di sp (k , ”K=” )


1 printf ( ” 2) with c on t r o l l e r ’ )2 syms K;3 sys1=s y s l i n ( ’ c ’ , 1 / ( s ∗( s +1.2) ) ) ;4 sys2=sys1 / . ( s∗K) ;5 G=16∗ sys2 ;6 G=simple (G)7 di sp (G, ”G( s )=” )8 sys=G/ .H;9 sys=simple ( sys ) ;

10 di sp ( sys , ”C( s ) /R( s )=” )11 [ num, den]=numden( sys )12 den=den /5 ; // so that c o e f f o f sˆ2=1

52

13 c o f a 0 = c o e f f s ( den , ’ s ’ , 0 ) // c o e f f o f den o f symbol icfunc t i on CL

14 c o f a 1 = c o e f f s ( den , ’ s ’ , 1 )15 //Wn̂ 2= co f a 0 , comparing the c o e f f i c i e n t s16 Wn=sqr t ( c o f a 0 )17 Wn=dbl (Wn) ;18 di sp (Wn, ” natura l f requency Wn=” ) // Wn=

natura l f requency19 // c o f a 1=2∗zeta ∗Wn20 // zeta=co f a 1 /(2∗Wn)21 zeta =0.56;22 k=(8∗ zeta )−1.223 di sp (k , ”K=” )24 Wd=Wn∗ s q r t (1− zeta ˆ2) ;25 di sp (Wd, ”Wd=” )26 Tp=%pi/Wd;27 di sp (Tp, ”Tp=” )28 Mp=100∗exp((−%pi∗ zeta ) / sq r t (1− zeta ˆ2) ) ;29 di sp (Mp, ”Mp=” )30 Ts=4/( zeta ∗Wn) ;31 di sp (Ts , ”Ts=” )

53

Chapter 10

Stability


You can also refer the inbuilt function(routh-t) for generating Routh Ta-ble.

10.1 Scilab Code

Example 10.02-01 10.02.01.sci

1 s = poly (0 , ” s ” ) ;2 p=poly ( [ 1 2 ] , ’ s ’ , ’ c o e f f ’ ) ;3 q=poly ( [ 0 2 4 1 ] , ’ s ’ , ’ c o e f f ’ ) ;4 G=p/q ;5 H=poly ( [ 0 . 5 ] , ’ s ’ , ’ c o e f f ’ ) ;6 / / c h a r a c t e r i s t i c e q u a t i o n i s 1+G ( s ) H ( s ) =0

7 y=1+G∗H8 r=numer( y )9 disp ( ’=0 ’ , r , ” c h a r a c t e r i s t i c s equat ion i s ” )

Example 10.02-02 10.02.02.sci

1 s = poly (0 , ” s ” ) ;2 p=poly ( [ 7 ] , ’ s ’ , ’ c o e f f ’ ) ;3 q=poly ( [ 2 3 1 ] , ’ s ’ , ’ c o e f f ’ ) ;

54

4 G=p/q ;5 H=poly ( [ 0 1 ] , ’ s ’ , ’ c o e f f ’ ) ;6 / / c h a r a c t e r i s t i c e q u a t i o n i s 1+G ( s ) H ( s ) =0


Example 10.02-03 10.02.03.sci

1 s = poly (0 , ” s ” ) ;2 G=sysl in ( ’ c ’ , 2/( s ˆ2+2∗ s ) )3 H=sysl in ( ’ c ’ ,1/ s ) ;4 / / c h a r a c t e r i s t i c e q u a t i o n i s 1+G ( s ) H ( s ) =0



1 s=%s ;2 m=sˆ3+5∗ s ˆ2+10∗ s+3;3 r=coeff (m)4 n=length ( r ) ;5 routh=[ r ( [ 4 , 2 ] ) ; r ( [ 3 , 1 ] ) ] ;6 routh=[ routh ;−det ( routh ) / routh (2 , 1 ) , 0 ] ;7 t=routh ( 2 : 3 , 1 : 2 ) ; / / e x t r a c t i n g t h e s q u a r e s u b b l o c k o f

r o u t h m a t r i x

8 routh=[ routh ;−det ( t ) / t ( 2 , 1 ) , 0 ]9 c=0;

10 for i =1:n11 i f ( routh ( i , 1 )< 0)12 c=c+1;13 end14 end15 i f ( c>=1)16 printf ( ” system i s unstab le ” )17 else ( ” system i s s t ab l e ” )18 end

55


1 s=%s ;2 m=sˆ3+2∗ s ˆ2+3∗ s +10;3 r=coeff (m)4 n=length ( r ) ;5 routh=[ r ( [ 4 , 2 ] ) ; r ( [ 3 , 1 ] ) ] ;6 routh=[ routh ;−det ( routh ) / routh (2 , 1 ) , 0 ] ;7 t=routh ( 2 : 3 , 1 : 2 ) ; / / e x t r a c t i n g t h e s q u a r e s u b b l o c k o f


8 routh=[ routh ;−det ( t ) / t ( 2 , 1 ) , 0 ]9 c=0;

10 for i =1:n11 i f ( routh ( i , 1 ) <0)12 c=c+1;13 end14 end15 i f ( c>=1)16 printf ( ” system i s unstab le ” )17 else ( ” system i s s t ab l e ” )18 end

Example 10.05-01 10.05.01.sci

1 ieee (2 )2 s=%s ;3 m=sˆ4+6∗ s ˆ3+21∗ s ˆ2+36∗ s+204 r=coeff (m)5 n=length ( r ) ;6 routh=[ r ( [ 5 , 3 , 1 ] ) ; r ( [ 4 , 2 ] ) , 0 ]7 routh=[ routh ;−det ( routh ( 1 : 2 , 1 : 2 ) ) / routh (2 , 1 ) ,−det ( routh

( 1 : 2 , 2 : 3 ) ) / routh (2 , 2 ) , 0 ] ;8 routh=[ routh ;−det ( routh ( 2 : 3 , 1 : 2 ) ) / routh (3 , 1 ) ,−det ( routh

( 2 : 3 , 2 : 3 ) ) / routh (3 , 2 ) , 0 ] ;9 routh=[ routh ;−det ( routh ( 3 : 4 , 1 : 2 ) ) / routh (4 , 1 ) , 0 , 0 ] ;

10 disp ( routh , ” routh=” )

56

11 c=0;12 for i =1:n13 i f ( routh ( i , 1 ) <0)14 c=c+1;15 end16 end17 i f ( c>=1)18 printf ( ” system i s unstab le ” )19 else ( ” system i s s t ab l e ” )20 end

Example 10.05-02 10.05.02.sci

1 s=%s ;2 m=sˆ5+6∗ s ˆ4+3∗ s ˆ3+2∗ sˆ2+s+13 r=coeff (m)4 n=length ( r )5 routh=[ r ( [ 6 , 4 , 2 ] ) ; r ( [ 5 , 3 , 1 ] ) ]6 routh=[ routh ;−det ( routh ( 1 : 2 , 1 : 2 ) ) / routh (2 , 1 ) ,−det ( routh

( 1 : 2 , 2 : 3 ) ) / routh (2 , 2 ) , 0 ]7 routh=[ routh ;−det ( routh ( 2 : 3 , 1 : 2 ) ) / routh (3 , 1 ) ,−det ( routh

( 2 : 3 , 2 : 3 ) ) / routh (3 , 2 ) , 0 ]8 routh=[ routh ;−det ( routh ( 3 : 4 , 1 : 2 ) ) / routh (4 , 1 ) ,−det ( routh

( 3 : 4 , 2 : 3 ) ) / routh (4 , 2 ) , 0 ]9 routh=[ routh ;−det ( routh ( 4 : 5 , 1 : 2 ) ) / routh (5 , 1 ) , 0 , 0 ]

10 c=0;11 for i =1:n12 i f ( routh ( i , 1 ) <0)13 c=c+1;14 end15 end16 i f ( c>=1)17 printf ( ” system i s unstab le ” )18 else ( ” system i s s t ab l e ” )19 end


57

1 ieee (2 )2 s=%s ;3 m=sˆ5+2∗ s ˆ4+4∗ s ˆ3+8∗ s ˆ2+3∗ s+14 r=coeff (m) ; / / E x t r a c t s t h e c o e f f i c i e n t o f t h e

p o l y n o m i a l

5 n=length ( r ) ;6 routh=[ r ( [ 6 , 4 , 2 ] ) ; r ( [ 5 , 3 , 1 ] ) ]7 syms eps ;8 routh=[ routh ; eps ,−det ( routh ( 1 : 2 , 2 : 3 ) ) / routh (2 , 2 ) , 0 ] ;9 routh=[ routh ;−det ( routh ( 2 : 3 , 1 : 2 ) ) / routh (3 , 1 ) ,−det ( routh


( 3 : 4 , 2 : 3 ) ) / routh (4 , 2 ) , 0 ] ;11 routh=[ routh ;−det ( routh ( 4 : 5 , 1 : 2 ) ) / routh (5 , 1 ) , 0 , 0 ] ;12 disp ( routh , ” routh=” )13 / / To c h e c k t h e s t a b i l i t y

14 routh (4 , 1 )=8− l im i t (5/ eps , eps , 0 ) ; / / P u t t i n g t h e v a l u e o f

e p s =0 i n r o u t h ( 4 , 1 )

15 disp ( routh (4 , 1 ) , ” routh (4 , 1 )=” )16 routh (5 , 1 )= l im i t ( routh (5 , 1 ) , eps , 0 ) ; / / P u t t i n g t h e

v a l u e o f e p s =0 i n r o u t h ( 5 , 1 )

17 disp ( routh (5 , 1 ) , ” routh (5 , 1 ) = ’)18 routh19 p r i n t f ( ”There are two sign changes o f f i r s t column ,

hence the system i s uns tab l e \n” )


1 s=%s ;2 m=sˆ5+2∗ s ˆ4+2∗ s ˆ3+4∗ s ˆ2+4∗ s+83 routh=routh t (m) / / T h i s F u n c t i o n g e n e r a t e s t h e R o u t h

t a b l e

4 c=0;5 for i =1:n6 i f ( routh ( i , 1 ) <0)7 c=c+1;8 end9 end

58

10 i f ( c>=1)11 printf ( ” system i s unstab le ” )12 else ( ” system i s s t ab l e ” )13 end


1 s=%s ;2 m=sˆ4+5∗ s ˆ3+2∗ s ˆ2+3∗ s+23 r=coeff (m)4 n=length ( r ) ;5 routh=[ r ( [ 5 , 3 , 1 ] ) ; r ( [ 4 , 2 ] ) , 0 ]6 routh=[ routh ;−det ( routh ( 1 : 2 , 1 : 2 ) ) / routh (2 , 1 ) ,−det ( routh


( 2 : 3 , 2 : 3 ) ) / routh (3 , 2 ) , 0 ] ;8 routh=[ routh ;−det ( routh ( 3 : 4 , 1 : 2 ) ) / routh (4 , 1 ) , 0 , 0 ]9 c=0;

10 for i =1:n11 i f ( routh ( i , 1 ) <0)12 c=c+1;13 end14 end15 i f ( c>=1)16 printf ( ” system i s unstab le ” )17 else ( ” system i s s t ab l e ” )18 end


1 ieee (2 ) ;2 s=%s ;3 m=sˆ5+sˆ4+3∗ s ˆ3+3∗ s ˆ2+4∗ s+84 r=coeff (m) ; / / E x t r a c t s t h e c o e f f i c i e n t o f t h e

p o l y n o m i a l

5 n=length ( r ) ;6 routh=[ r ( [ 6 , 4 , 2 ] ) ; r ( [ 5 , 3 , 1 ] ) ]7 syms eps ;

59

8 routh=[ routh ; eps ,−det ( routh ( 1 : 2 , 2 : 3 ) ) / routh (2 , 2 ) , 0 ] ;9 routh=[ routh ;−det ( routh ( 2 : 3 , 1 : 2 ) ) / routh (3 , 1 ) ,−det ( routh


( 3 : 4 , 2 : 3 ) ) / routh (4 , 2 ) , 0 ] ;11 routh=[ routh ;−det ( routh ( 4 : 5 , 1 : 2 ) ) / routh (5 , 1 ) , 0 , 0 ] ;12 disp ( routh , ” routh=” )13 / / To c h e c k t h e s t a b i l i t y

14 routh (4 , 1 )=l im i t ( routh (4 , 1 ) , eps , 0 ) ; / / P u t t i n g t h e v a l u e

o f e p s =0 i n r o u t h ( 4 , 1 )

15 disp ( routh (4 , 1 ) , ” routh (4 , 1 )=” )16 routh (5 , 1 )= l im i t ( routh (5 , 1 ) , eps , 0 ) ; / / P u t t i n g t h e

v a l u e o f e p s =0 i n r o u t h ( 5 , 1 )

17 disp ( routh (5 , 1 ) , ” routh (5 , 1 ) = ’)18 routh19 p r i n t f ( ”There are two sign changes o f f i r s t column ,

hence the system i s uns tab l e \n” )


1 ieee (2 ) ;2 syms s k ;3 m=sˆ4+4∗ s ˆ3+7∗ s ˆ2+6∗ s+k ;4 c o f a 0 = c o e f f s (m, ’ s ’ , 0 ) ;5 c o f a 1 = c o e f f s (m, ’ s ’ , 1 ) ;6 c o f a 2 = c o e f f s (m, ’ s ’ , 2 ) ;7 c o f a 3 = c o e f f s (m, ’ s ’ , 3 ) ;8 c o f a 4 = c o e f f s (m, ’ s ’ , 4 ) ;9

10 r=[ c o f a 0 c o f a 1 c o f a 2 c o f a 3 c o f a 4 ]11

12 n=length ( r ) ;13 routh=[ r ( [ 5 , 3 , 1 ] ) ; r ( [ 4 , 2 ] ) , 0 ]14 routh=[ routh ;−det ( routh ( 1 : 2 , 1 : 2 ) ) / routh (2 , 1 ) ,−det ( routh


( 2 : 3 , 2 : 3 ) ) / routh (3 , 2 ) , 0 ] ;16 routh=[ routh ;−det ( routh ( 3 : 4 , 1 : 2 ) ) / routh (4 , 1 ) , 0 , 0 ] ;

60

17 disp ( routh , ” routh=” )


1 ieee (2 ) ;2 syms s k ;3 m=(24/100)∗ sˆ3+sˆ2+s+k ;4 c o f a 0 = c o e f f s (m, ’ s ’ , 0 ) ;5 c o f a 1 = c o e f f s (m, ’ s ’ , 1 ) ;6 c o f a 2 = c o e f f s (m, ’ s ’ , 2 ) ;7 c o f a 3 = c o e f f s (m, ’ s ’ , 3 ) ;8 r=[ c o f a 0 c o f a 1 c o f a 2 c o f a 3 ]9 n=length ( r ) ;

10 routh=[ r ( [ 4 , 2 ] ) ; r ( [ 3 , 1 ] ) ]11 routh=[ routh ;−det ( routh ) / routh (2 , 1 ) , 0 ]12 t=routh ( 2 : 3 , 1 : 2 ) ; / / e x t r a c t i n g t h e s q u a r e s u b b l o c k o f


13 routh=[ routh ;−det ( t ) / routh (3 , 1 ) , 0 ]14 disp ( routh , ” routh=” ) ;15 routh (3 , 1 )=0 / / F o r m a r g i n a l y s t a b l e s y s t e m

16 k=1/0.24;17 disp (k , ”K( marginal )=” )18 disp ( ’=0 ’ , ( s ˆ2)+k , ” a ux i l l a r y equat ion ” )19 s=sqrt(−k ) ;20 disp ( s , ”Frequency o f o s c i l l a t i o n ( in rad/ sec )=” )


1 ieee (2 ) ;2 syms p K s ;3 m=sˆ3+(p∗ s ˆ2)+(K+3)∗ s +(2∗(K+1) )4 c o f a 0 = c o e f f s (m, ’ s ’ , 0 ) ;5 c o f a 1 = c o e f f s (m, ’ s ’ , 1 ) ;6 c o f a 2 = c o e f f s (m, ’ s ’ , 2 ) ;7 c o f a 3 = c o e f f s (m, ’ s ’ , 3 ) ;8 r=[ c o f a 0 c o f a 1 c o f a 2 c o f a 3 ]9 n=length ( r ) ;

10 routh=[ r ( [ 4 , 2 ] ) ; r ( [ 3 , 1 ] ) ] ;

61

11 routh=[ routh ;−det ( routh ) / routh (2 , 1 ) , 0 ] ;12 t=routh ( 2 : 3 , 1 : 2 ) ; / / e x t r a c t i n g t h e s q u a r e s u b b l o c k o f


13 routh=[ routh ;−det ( t ) / routh (3 , 1 ) , 0 ] ;14 disp ( routh , ” routh=” )


1 ieee (2 ) ;2 s=%s ;3 m=2∗s ˆ4+(4∗ s ˆ2)+14 routh=routh t (m) / / G e n e r a t e s t h e R o u t h T a b l e

5 roots (m) / / G i v e s t h e R o o t s o f t h e P o l y n o m i a l

( m )

6 disp (0 , ” the number o f r oo t s l y i n g in the l e f t h a l fp lane o f s−plane=” )

7 disp (0 , ” the number o f r oo t s l y i n g in the r i g h t h a l fp lane o f s−plane=” )

8 disp (4 , ” the number o f r oo t s l y i n g on the imaginary ax i s=” )


1 ieee (2 ) ;2 syms s k ;3 m=sˆ4+6∗ s ˆ3+10∗ s ˆ2+8∗ s+k ;4 c o f a 0 = c o e f f s (m, ’ s ’ , 0 ) ;5 c o f a 1 = c o e f f s (m, ’ s ’ , 1 ) ;6 c o f a 2 = c o e f f s (m, ’ s ’ , 2 ) ;7 c o f a 3 = c o e f f s (m, ’ s ’ , 3 ) ;8 c o f a 4 = c o e f f s (m, ’ s ’ , 4 ) ;9 r=[ c o f a 0 c o f a 1 c o f a 2 c o f a 3 c o f a 4 ]

10 n=length ( r ) ;11 routh=[ r ( [ 5 , 3 , 1 ] ) ; r ( [ 4 , 2 ] ) , 0 ]12 routh=[ routh ;−det ( routh ( 1 : 2 , 1 : 2 ) ) / routh (2 , 1 ) ,−det ( routh


( 2 : 3 , 2 : 3 ) ) / routh (3 , 2 ) , 0 ] ;

62

14 routh=[ routh ;−det ( routh ( 3 : 4 , 1 : 2 ) ) / routh (4 , 1 ) , 0 , 0 ] ;15 disp ( routh , ” routh=” )


1 ieee (2 ) ;2 syms s T;3 m=sˆ2+(2−T)∗ s+14 c o f a 0 = c o e f f s (m, ’ s ’ , 0 ) ;5 c o f a 1 = c o e f f s (m, ’ s ’ , 1 ) ;6 c o f a 2 = c o e f f s (m, ’ s ’ , 2 ) ;7 r=[ c o f a 0 c o f a 1 c o f a 2 ]8 n=length ( r ) ;9 routh=[ r ( [ 3 , 1 ] ) ; r (2 ) , 0 ] ;

10 routh=[ routh ;−det ( routh ) / routh (2 , 1 ) , 0 ] ;11 disp ( routh , ” routh=” )


1 ieee (2 )2 s=%s ;3 m=sˆ6+2∗ s ˆ5+7∗ s ˆ4+10∗ s ˆ3+14∗ s ˆ2+8∗ s+84 routh=routh t (m) ;5 disp ( routh , ” routh=” )6 c=0;7 for i =1:n8 i f ( routh ( i , 1 ) <0)9 c=c+1;

10 end11 end12 i f ( c>=1)13 printf ( ” system i s unstab le ” )14 else ( ” system i s s t ab l e ” )15 end

63

Chapter 11

Root Locus Method


When we will execute the programm we will get the following Graphs

11.1 Scilab Code


1 s=%s ;2 sys1=sysl in ( ’ c ’ , 1/( s+1) ) ;3 evans ( sys1 , 200 )4 printf ( ” I f k i s var i ed from 0 to any value , root l o cu s

v a r i e s from −k to 0 \n ” )


1 s=%s ;2 sys1=sysl in ( ’ c ’ , ( s+1)/( s+4) ) ;3 evans ( sys1 , 100 )4 printf ( ” r o o t l o cu s beg ins at s=−4 & ends at s=−1” )


64

Figure 11.1: Output Graph of S 11.01

1 s=%s ;2 sys1=sysl in ( ’ c ’ , ( s+3−%i) ∗( s+3+%i) / ( ( s+2−%i) ∗( s+2+%i) ) ) ;3 evans ( sys1 , 100 )4 printf ( ”Root locus s t a r t s from s=−2+i & −2− i ends at s

=−3+i &−3− i \n” )


1 s=%s ;2 H=sysl in ( ’ c ’ , ( s+1)/( s+2) ) ;3 evans (H, 100 )

65


4 printf ( ” C l ea r l y from the graph i t observed that g ivenpo int −1+i & −3+i does not l i e on the root l o cu s \n

” )5 / / t h e r e i s a n o t h e r p r o c e s s t o c h e c k w h e t h e r t h e p o i n t s

l i e o n t h e l o c u s o f t h e s y s t e m

6 P=−1+%i ; / / P= s e l e c t e d p o i n t

7 k1=−1/real (horner (H,P) )8 Ns=H( ’num ’ ) ; Ds=H( ’ den ’ ) ;9 roots (Ds+k1∗Ns) / / d o e s n o t c o n t a i n s P a s p a r t i c u l a r

r o o t

10 P=−3+%i ; / / P= s e l e c t e d p o i n t

11 k2=−1/real (horner (H,P) ) ;

66


12 Ns=H( ’num ’ ) ; Ds=H( ’ den ’ )13 roots (Ds+k2∗Ns) / / d o e s n o t c o n t a i n s P a s p a r t i c u l a r

r o o t


1 s=%s ;2 H=sysl in ( ’ c ’ , 1/( s ∗( s+1)∗( s+3) ) ) ;3 evans (H, 100 )4 printf ( ” C l ea r l y from the graph i t observed that g iven

po int −0.85 l i e s on the root l o cu s \n” )

67

Figure 11.4: Output of S 11.04

5 / / t h e r e i s a n o t h e r p r o c e s s t o c h e c k w h e t h e r t h e p o i n t s


6 P=−0.85; / / P= s e l e c t e d p o i n t

7 k=−1/real (horner (H,P) ) ;8 disp (k , ”k= ’)9 Ns=H( ’num’ ) ; Ds=H( ’ den ’ ) ;

10 r oo t s (Ds+k∗Ns) // conta in s P as p a r t i c u l a r root

68



1 s=%s ;2 H=sysl in ( ’ c ’ , 1/ ( ( s+1)∗( s+5) ) ) ;3 evans (H, 100 )4 printf ( ” C l ea r l y from the graph i t observed that g iven

po int −0.85 l i e s on the root l o cu s \n” )5 / / t h e r e i s a n o t h e r p r o c e s s t o c h e c k w h e t h e r t h e p o i n t s


6 P=−3+5∗%i ; / / P= s e l e c t e d p o i n t

7 k=−1/real (horner (H,P) ) ;8 disp (k , ”k= ’)

69


9 Ns=H( ’num’ ) ; Ds=H( ’ den ’ ) ;10 r oo t s (Ds+k∗Ns) // conta in s P as p a r t i c u l a r root


1 s=%s ;2 H=sysl in ( ’ c ’ , ( s+2) / ( ( s+1)∗ s ∗( s+4) ) ) ;3 evans (H, 100 )4 printf ( ”From the graph we observed that , \n a )The no o f

l o c i ending at i n f i s 2 \n b) Three l o c i w i l l s t a r tfrom s= 0,−1 & −4,\n c )One l o c i w i l l end at −2 &

70


remaining two w i l l end at i n f ” )


1 s=%s ;2 H=sysl in ( ’ c ’ , ( s+2) / ( ( s+1)∗ s ∗( s+4) ) ) ;3 evans (H, 100 )


1 n=3;

71


2 disp (n , ”no o f po l e s=” )3 m=1;4 disp (m, ”no o f po l e s=” )5 q=0;6 O=((2∗q )+1)/(n−m) ∗180 ;7 disp (O, ”q=” )8 q=1;9 O=((2∗q )+1)/(n−m) ∗180 ;

10 disp (O, ”q=” )11

12 printf ( ”Centroid =((sum of a l l r e a l part o f po l e s o f G( s)H( s ) )−(sum of a l l r e a l part o f z e r o s o f G( s )H( s ) ) /(

72

n−m) \n” )13 C=((0−1−4)−(−2)) /2 ;14 disp (C, ” c en t r o id=” )


1 n=3;2 disp (n , ”no o f po l e s=” )3 m=0;4 disp (m, ”no o f po l e s=” )5 q=0;6 O=((2∗q )+1)/(n−m) ∗180 ;7 disp (O, ”q=” )8 q=1;9 O=((2∗q )+1)/(n−m) ∗180 ;

10 disp (O, ”q=” )11 q=2;12 O=((2∗q )+1)/(n−m) ∗180 ;13 disp (O, ”q=” )14

15 printf ( ”Centroid =((sum of a l l r e a l part o f po l e s o f G( s)H( s ) )−(sum of a l l r e a l part o f z e r o s o f G( s )H( s ) ) /(n−m) \n” )

16 C=((0−1−1)−(−0)) /3 ;17 disp (C, ” c en t r o id=” )


1 n=4;2 disp (n , ”no o f po l e s=” )3 m=1;4 disp (m, ”no o f po l e s=” )5 q=0;6 O=((2∗q )+1)/(n−m) ∗180 ;7 disp (O, ”q=” )8 q=1;9 O=((2∗q )+1)/(n−m) ∗180 ;

10 disp (O, ”q=” )

73

11 q=2;12 O=((2∗q )+1)/(n−m) ∗180 ;13 disp (O, ”q=” )14

15 printf ( ”Centroid =((sum of a l l r e a l part o f po l e s o f G( s)H( s ) )−(sum of a l l r e a l part o f z e r o s o f G( s )H( s ) ) /(n−m) \n” )

16 C=((0−2−4−5)−(−3)) /3 ;17 disp (C, ” c en t r o id=” )


1 s=%s ;2 H=sysl in ( ’ c ’ , ( s+2) / ( ( s+1)∗ s ∗( s+3) ) ) ;3 plzr (H)4 printf ( ”There are two adjacent p laced po l e s at s=0 &s

=−1 \n” )5 printf ( ”One breakaway po int e x i s t s between s=0 & s=−1 \

n” )


1 s=%s ;2 H=sysl in ( ’ c ’ , ( ( s+2)∗( s+4) ) / ( ( s ˆ2) ∗( s+5) ) ) ;3 plzr (H)4 printf ( ”There are two adjacent p laced z e ro s at s=−2 &s

=−4 \n” )5 printf ( ”One break in po int e x i s t s between s=−2 & s=−4 \n

” )


1 s=%s ;2 H=sysl in ( ’ c ’ , ( s+6) / ( ( s+1)∗( s+3) ) ) ;3 plzr (H)4 printf ( ”There are two adjacent p laced po l e s at s=−3 &s

=−1 \n” )

74


5 printf ( ”One breakaway po int e x i s t s between s=−3 & s=−1\n” )

6 printf ( ”One break in po int e x i s t s to the l e f t o f z e r o sat s=−6 \n” )


1 s=%s ;2 H=sysl in ( ’ c ’ , 1/ ( ( s+1)∗ s ∗( s+3) ) ) ;3 plzr (H)4 printf ( ”There are two adjacent p laced po l e s at s=0 &s

75


=−1 \n” )5 printf ( ”One breakaway po int e x i s t s between s=0 & s=−1 \

n” )


1 s=%s ;2 H=sysl in ( ’ c ’ , 1/ ( ( s+1)∗ s ∗( s+3) ) ) ;3 evans (H, 100 )4 syms k ;5 m=sˆ3+6∗ s ˆ2+8∗ s+k ;

76


6 c o f a 0 = c o e f f s (m, ’ s ’ , 0 ) ;7 c o f a 1 = c o e f f s (m, ’ s ’ , 1 ) ;8 c o f a 2 = c o e f f s (m, ’ s ’ , 2 ) ;9 c o f a 3 = c o e f f s (m, ’ s ’ , 3 ) ;

10 r=[ c o f a 0 c o f a 1 c o f a 2 c o f a 3 ]11

12 n=length ( r ) ;13 routh=[ r ( [ 4 , 2 ] ) ; r ( [ 3 , 1 ] ) ] ;14 routh=[ routh ;−det ( routh ) / routh (2 , 1 ) , 0 ] ;15 t=routh ( 2 : 3 , 1 : 2 ) ; / / e x t r a c t i n g t h e s q u a r e s u b b l o c k o f


16 routh=[ routh ;−det ( t ) / t ( 2 , 1 ) , 0 ]

77


17 disp (48 , ”K( marginal )=” )18 disp ( ’=0 ’ , (6∗ s ˆ2)+k , ” a ux i l l a r y equat ion ” )19 k=48;20 s=sqrt(−k/6) ;21 disp ( s , ” s=” )


1 s=%s ;2 H=sysl in ( ’ c ’ , 1/ ( ( s+1+%i )∗ s ∗( s+1−%i) ) ) ;3 evans (H, 100 )

78



1 s=%s ;2 H=sysl in ( ’ c ’ , 1/ ( ( s+3)∗ s ∗( s+5) ) ) ;3 evans (H, 100 )


1 s=%s ;2 H=sysl in ( ’ c ’ , 1/ ( ( s+2+%i ) ∗( s+1)∗( s+2−%i) ) ) ;3 evans (H, 100 )

79



1 s=%s ;2 num=real (poly ( [ 1 ] , ’ s ’ , ” c o e f f ” ) )3 den=real (poly([−1,−2+%i,−2−%i ] , ’ s ’ ) )4 H=num/den5 evans (H, 100 )6 k=1.5 ;7 disp (k , ”K( des ign )=” )8 / / K p u r e c a l c u l a t e s t h e v a l u e o f k a t i m a g i n a r y

c r o s s o v e r

9 [K,Y]=kpure (H)

80


10 GM=K/k ;11 disp (GM, ” value o f k at imaginary c r o s s ov e r /k ( des ign )=” )12 disp (GM, ” gain margin=” )


1 s=%s ;2 H=sysl in ( ’ c ’ , 1/( s ∗ ( ( s+3)ˆ2) ) ) ;3 evans (H, 100 )4 K=25;5 y=K∗H; / / −−−−− e q 1 )

81


6 disp (K∗H, ”G( s )H( s )=” ) ;7 disp ( ’=1 ’ ,K∗H, ”mod(G( s )H( s ) ) ” ) ;8 / / o n s o l v i n g e q 1 f o r s = % i ∗ w t h i s w e g e t a n e q u a t i o n m

9 w=poly (0 , ’w ’ ) ;10 m=wˆ3+9∗w−2511 n=roots (m)12 s=%i∗n (1)13 p=horner (y , s )14 [R, Theta ]=polar (p)15 PM=180+Theta

82



1 s=%s ;2 H=sysl in ( ’ c ’ , 1/ ( ( s+1)∗ s ∗( s+2)∗( s+4) ) ) ;3 evans (H, 100 )


1 s=%s ;2 H=sysl in ( ’ c ’ , ( s+4)∗( s+5) /( ( s+1)∗( s+3) ) ) ;3 evans (H, 100 )

83



1 s=%s ;2 syms k Wn;3 H=sysl in ( ’ c ’ , 1/ ( ( s+3)ˆ2∗ s ) ) ;4 evans (H, 100 ) / / r o o t l o c u s

5 printf ( ”To determine the value o f Wn \n” )6 disp ( k∗H, ”G( s )H( s )=” )7 y=1+(k∗H) ;8 disp ( ’=0 ’ , y , ”1+G( s )H( s ) ” )9 evans (H, 100 )

84


10 [ num, den]=numden(y )11

12 c o f a 0 = c o e f f s (num, ’ s ’ , 0 ) ;13 c o f a 1 = c o e f f s (num, ’ s ’ , 1 ) ;14 c o f a 2 = c o e f f s (num, ’ s ’ , 2 ) ;15 c o f a 3 = c o e f f s (num, ’ s ’ , 3 ) ;16 r=[ c o f a 0 c o f a 1 c o f a 2 c o f a 3 ]17

18 n=length ( r ) ;19 routh=[ r ( [ 4 , 2 ] ) ; r ( [ 3 , 1 ] ) ] ;20 routh=[ routh ;−det ( routh ) / routh (2 , 1 ) , 0 ] ;21 t=routh ( 2 : 3 , 1 : 2 ) ; / / e x t r a c t i n g t h e s q u a r e s u b b l o c k o f

85



22 routh=[ routh ;−det ( t ) / t ( 2 , 1 ) , 0 ]23 / / t o o b t a i n Wn

24 disp ( ’=0 ’ , ( ( 6∗ s ˆ2)+54) , ” a u x i l l a r y eq” )25 p=(6∗( s ˆ2) )+k ;26 s=%i∗Wn27 k=54;28 p=eval (p)29 Wn=sqrt ( k/6)30 printf ( ”With gvn va lue s o f ze ta adding a g r id on root

l o cu s \n” )31

86

32 zeta =0.5 ; / / g i v e n

33 sgrid ( zeta ,Wn, 7 ) / / a d d a g r i d o v e r a n e x i s t i n g

c o n t i n u o u s s − p l a n e r o o t w i t h g i v e n v a l u e s f o r z e t a

a n d wn .

34 printf ( ”NOTE:− c l i c k on the po int where l o cu s i n t e r s e c t sz=0.5 f o r d e s i r ed value o f k \n” )

35 k=−1/real (horner (H, [ 1 , %i ]∗ locate (1 ) ) ) / / To o b t a i n t h e

g a i n a t a g i v e n p o i n t o f t h e l o c u s

36

37

38 p=locate (1 ) / / d e s i r e d p o i n t o n t h e r o o t l o c u s


1 s=%s ;2 H=sysl in ( ’ c ’ , 1/ ( ( s+4)∗ s ∗( s+6) ) ) ;3 evans (H, 100 )

87


88


89

Chapter 12

Frequency Domain Analysis


12.1 Scilab Code



2 F=sysl in ( ’ c ’ , [ 2 2 5 / ( ( s+6)∗ s ) ] ) ; / / C r e a t e s t r a n s f e r











10 / / 2 ∗ z e t a ∗Wn= z ( 1 , 2 )


12 Mr=1/(2∗ zeta ∗sqrt(1− zeta ˆ2) )13 Wr=Wn∗sqrt(1− zeta ˆ2)14 Wc=Wn∗sqrt ((1−2∗ zeta ˆ2)+sqrt (4∗ zeta ˆ4−4∗ zeta ˆ2+2) )

90

15 BW=Wc / / BANDWIDTH



2 F=sysl in ( ’ c ’ , [ 3 6 / ( ( s+8)∗ s ) ] ) ; / / C r e a t e s t r a n s f e r











10 / / 2 ∗ z e t a ∗Wn= z ( 1 , 2 )


12 Mr=1/(2∗ zeta ∗sqrt(1− zeta ˆ2) )13 Wr=Wn∗sqrt(1− zeta ˆ2)14 Wc=Wn∗sqrt ((1−2∗ zeta ˆ2)+sqrt (4∗ zeta ˆ4−4∗ zeta ˆ2+2) )15 BW=Wc / / BANDWIDTH



2 F=sysl in ( ’ c ’ , [ 8 1 / ( s ˆ2+7∗ s ) ] ) ; / / C r e a t e s t r a n s f e r











10 / / 2 ∗ z e t a ∗Wn= z ( 1 , 2 )

91


12 Mr=1/(2∗ zeta ∗sqrt(1− zeta ˆ2) )13 Wr=Wn∗sqrt(1− zeta ˆ2)14 Wc=Wn∗sqrt ((1−2∗ zeta ˆ2)+sqrt (4∗ zeta ˆ4−4∗ zeta ˆ2+2) )15 BW=Wc / / BANDWIDTH


1 / / D e f i n e s s a s p o l y n o m i a l v a r i a b l e

2 s=poly (0 , ’ s ’ ) ;3 / / C r e a t e s t r a n s f e r f u n c t i o n i n f o r w a r d p a t h

4 F=sysl in ( ’ c ’ , [ 8 1 / ( ( s+18)∗ s ) ] ) ;5 / / C r e a t e s t r a n s f e r f u n c t i o n i n b a c k w a r d p a t h

6 B=sysl in ( ’ c ’ ,(1+0∗ s ) /(1+0∗ s ) ) ;7 / / C a l c u l a t e s c l o s e d − l o o p t r a n s f e r f u n c t i o n

8 CL=F/ .B9 / / c o m p a r e CL w i t h Wn ˆ 2 / ( s ˆ 2 + 2 ∗ z e t a ∗Wn+Wn ˆ 2 )


11 / / E x t r a c t i n g t h e c o e f f i c i e n t s o f t h e d e n o m i n a t o r

p o l y n o m i a l

12 z=coeff ( y )13 / / Wn ˆ 2 = z ( 1 , 1 ) , c o m p a r i n g t h e c o e f f i c i e n t s

14 Wn=sqrt ( z (1 , 1 ) )15 / / 2 ∗ z e t a ∗Wn= z ( 1 , 2 )

16 zeta=z (1 , 2 ) /(2∗Wn)17 / / z e t a = d a m p i n g f a c t o r

18 / / NOTE= h e r e s q r t ( 1 − 2 z e t a ˆ 2 ) b e c o m e s c o m p l e x s o t h e

o t h e r s o l u t i o n i s Wr =0 & Mr =1

19 Mr=120 Wr=021 Wc=Wn∗((1−2∗ zeta ˆ2)+sqrt (4∗ zeta ˆ4−4∗ zeta ˆ2+2) )22 BW=Wc / / BANDWIDTH


1 ieee (2 ) ;2 syms k a ;3 num=k ;

92

4 den=s ∗( a+s ) ;5 G=num/den ;6 disp (G, ”G( s )=” )7 H=1;8 CL=G/ .H;9 CL=simple (CL) ;

10 disp (CL, ”C( s ) /R( s )=” ) / / C a l c u l a t e s c l o s e d − l o o p

t r a n s f e r f u n c t i o n


12 [ num, den]=numden(CL) ; / / e x t r a c t s num & d e n o f s y m b o l i c


13 c o f a 0 = c o e f f s ( den , ’ s ’ , 0 ) ; / / c o e f f o f d e n o f

s y m b o l i c f u n c t i o n ( CL )

14 c o f a 1 = c o e f f s ( den , ’ s ’ , 1 ) ;15 / / Wn ˆ 2 = c o f a 0 , c o m p a r i n g t h e c o e f f i c i e n t s

16 Wn=sqrt ( c o f a 0 )17 / / c o f a 1 = 2 ∗ z e t a ∗Wn

18 zeta=co f a 1 /(2∗Wn)19 Mr=1/(2∗ zeta ∗sqrt(1− zeta ˆ2) ) / / −−−−−−−−−−1)

20 printf ( ”Given ,Mr=1.25 \n” ) ;21 / / On s o l v i n g e q ( 1 ) w e g e t k = 1 . 2 5 a ˆ2−−−−−−−2

22 printf ( ”k=1.25∗aˆ2 \n” )23 Wr=Wn∗sqrt (1−2∗ zeta ˆ2) / / −−−−−−−−−−−−−−−3)

24 printf ( ”Given , Wr=12.65 \n” ) ;25 / / o n s o v i n g e q ( 3 ) , w e g e t 2 k− a ˆ2=320−−−−−−−−−−−4)

26 printf ( ”2k−aˆ2=320 \n” )27 / / n o w e q 2 &4 c a n b e s i m u l t a n e o u s l y s o v e d t o t a k e o u t

v a l u e s o f k & a

28 / / L e t k = x & a ˆ 2 = y

29 A=[1 ,−1.25;2 ,−1] ; / / c o e f f i c i e n t m a t r i x

30 b = [ 0 ; 3 2 0 ] ;31 m=A\b ;32 x=m(1 ,1 ) ;33 k=x34 y=m(2 ,1 ) ;35 a=sqrt ( y )36 Wn=dbl ( eval (Wn) ) ;37 disp (Wn, ”Wn=” )

93

38 zeta=dbl ( eval ( ze ta ) ) ;39 disp ( zeta , ” zeta = ’)40 Ts=4/( zeta ∗Wn) ;41 di sp (Ts , ” S e t t l i n g Time (Ts )=” )42 Wc=Wn((1−(2∗ zeta ˆ2) )+sq r t (4∗ zeta ˆ4−4∗ zeta ˆ2+2) )43 di sp (Wc, ”BW=” )

94

Chapter 13

Bode Plot

When we will execute the programm we will get the following Graphs.

13.1 Scilab Code



2 F=sysl in ( ’ c ’ , [ 2 0 / ( s+2) ] ) / / C r e a t e s t r a n s f e r f u n c t i o n i n

f o r w a r d p a t h

3 B=sysl in ( ’ c ’ ,(1+0∗ s ) /(1+0∗ s ) ) / / C r e a t e s t r a n s f e r


4 OL=F∗B / / C a l c u l a t e s o p e n − l o o p t r a n s f e r f u n c t i o n

5 fmin =0.1 ; / / M i n f r e q i n Hz

6 fmax=100; / / Max f r e q i n Hz

7 s c f (1 ) ; c l f ;8 bode(OL, fmin , fmax ) ; / / P l o t s f r e q u e n c y r e s p o n s e o f o p e n −

l o o p s y s t e m i n B o d e d i a g r a m

9 show margins (OL) / / d i s p l a y g a i n a n d p h a s e m a r g i n a n d

a s s o c i a t e d c r o s s o v e r f r e q u e n c i e s



95


2 F=sysl in ( ’ c ’ , [20/((2+ s )∗ s ) ] ) ; / / C r e a t e s t r a n s f e r




4 OL=F∗B; / / C a l c u l a t e s o p e n − l o o p t r a n s f e r f u n c t i o n

5 fmin =0.01; / / M i n f r e q i n Hz


7 s c f (1 ) ; c l f ;

96


8 bode(OL, fmin , fmax ) ; / / P l o t s f r e q u e n c y r e s p o n s e o f o p e n −






2 F=sysl in ( ’ c ’ , [40/((2+ s )∗ s ∗( s+5) ) ] ) / / C r e a t e s t r a n s f e r


97








9 [ GainMargin , freqGM]=g margin (OL) / / C a l c u l a t e s g a i n

m a r g i n [ dB ] a n d c o r r e s p o n d i n g f r e q u e n c y [ H z ]

10 [ Phase , freqPM]=p margin (OL) / / C a l c u l a t e s p h a s e [ d e g ]

a n d c o r r e s p o n d i n g f r e q [ H z ] o f p h a s e m a r g i n

11 PhaseMargin=180+Phase / / C a l c u l a t e s a c t u a l p h a s e m a r g i n

[ d e g ]





2 F=sysl in ( ’ c ’ , [ 4 0 / ( ( s+5)∗(2+s )∗ s ˆ2) ] ) / / C r e a t e s t r a n s f e r













98


2 F=sysl in ( ’ c ’ , [ ( 4 0 0∗ ( s+2) ) / ( ( s+5)∗(10+ s )∗ s ˆ2) ] ) / /

C r e a t e s t r a n s f e r f u n c t i o n i n f o r w a r d p a t h










99




2 F=sysl in ( ’ c ’ , [ ( 2 8 8∗ ( s+4) ) / ( ( s+2)∗(144+4.8∗ s+s ˆ2)∗ s ) ] )/ / C r e a t e s t r a n s f e r f u n c t i o n i n f o r w a r d p a t h






100






101


102

Chapter 15

Nyquist Plot

When we will execute the programm we will get the following Graphs

15.1 Scilab Code


1 s=%s ;2 sys=sysl in ( ’ c ’ , 1/( s+2) )3 nyquist ( sys )4 show margins ( sys , ’ nyqu i s t ’ )5 printf ( ” S ince P=0(no o f po l e s in RHP)=Poles o f G( s )H( s )

\n here the number o f z e r o s o f 1+G( s )H( s ) in theRHP i s zero \n hence the system i s s t ab l e ” )


1 s=%s ;2 sys=sysl in ( ’ c ’ , 1/( s ∗( s+2) ) )3 nyquist ( sys )4 show margins ( sys , ’ nyqu i s t ’ )5 printf ( ” S ince P=0(no o f po l e s in RHP)=Poles o f G( s )H( s )

\n here the number o f z e r o s o f 1+G( s )H( s ) in theRHP i s zero \n hence the system i s s t ab l e ” )

103



1 s=%s ;2 sys=sysl in ( ’ c ’ , 1/( s ˆ2∗( s+2) ) )3 nyquist ( sys )4 show margins ( sys , ’ nyqu i s t ’ )5 printf ( ” S ince P=0(no o f po l e s in RHP)=Poles o f G( s )H( s )

\n here the number o f z e r o s o f 1+G( s )H( s ) in the

104


RHP i s not equal to zero \n hence the system i sunstab le ” )



\n here the number o f z e r o s o f 1+G( s )H( s ) in the

105


RHP i s N>0 \n hence the system i s unstab le ” )



\n here the number o f z e r o s o f 1+G( s )H( s ) in theRHP i s N>0 \n hence the system i s unstab le ” )

106



1 s=%s ;2 P1=1;3 P2=2;4 sys=sysl in ( ’ c ’ , 1/ ( ( s+1)∗( s+2) ) )5 nyquist ( sys )6 show margins ( sys , ’ nyqu i s t ’ )7 printf ( ” S ince P=0(no o f po l e s in RHP)=Poles o f G( s )H( s )

\n\n Here the number o f z e r o s o f 1+G( s )H( s ) in theRHP i s zero \n\n Hence the system i s s t ab l e ” )

107



1 s=%s ;2 sys=sysl in ( ’ c ’ ,12/( s ∗( s+1)∗( s+2) ) )3 nyquist ( sys )4 show margins ( sys , ’ nyqu i s t ’ )5 gm=g margin ( sys )6 i f (gm<=0)7 printf ( ” system i s unstab le ” )8 else9 printf ( ” system i s s t ab l e ” ) ; end ;

108



1 s=%s ;2 sys=sysl in ( ’ c ’ , ( 30 ) / ( ( s ˆ2+2∗ s+2)∗( s+3) ) )3 nyquist ( sys )4 gm=g margin ( sys )5 show margins ( sys , ’ nyqu i s t ’ )6 printf ( ” S ince P=0(no o f po l e s in RHP)=Poles o f G( s )H( s )

\n Here the number o f z e r o s o f 1+G( s )H( s ) in theRHP i s zero \n Hence the system i s s t ab l e ” )

7 i f (gm<=0)8 printf ( ” system i s unstab le ” )

109


9 else10 printf ( ” system i s s t ab l e ” )11 end

110


111

Chapter 17

State Variable Approach


17.1 Scilab Code


1 s=%s ;2 / / C r e a t i n g c o n t − t i m e t r a n s f e r f u n c t i o n

3 TFcont=sysl in ( ’ c ’ , 3/( s ˆ4+(2∗ s ˆ3)+(3∗ s )+2) )4 SScont=t f2ss (TFcont )5 / / CCF f o r m

6 [ Ac , Bc ,U, ind ]=canon ( SScont (2 ) , SScon (3 ) )


1 s=%s ;2 TFcont=sysl in ( ’ c ’ , [ ( 7 + 2∗ s + 3∗( s ˆ2) ) /(5 + 12∗ s + 5∗( s

ˆ2) + s ˆ3 ) ] )3 SScont=t f2ss (TFcont )4 [ Ac , Bc ,U, ind ]=canon ( SScont (2 ) , SScont (3 ) )


112


3 TFcont=sysl in ( ’ c ’ , [ ( 5 ∗ ( s+1)∗( s+2) ) / ( ( s+4)∗( s+5) ) ] )4 SScont=t f2ss (TFcont )5 / / CCF f o r m

6 [ Ac , Bc ,U, ind ]=canon ( SScont (2 ) , SScont (3 ) )



3 TFcont=sysl in ( ’ c ’ , ( s+1) / ( ( s+2)∗( s+5)∗( s+3) ) )4 SScont=t f2ss (TFcont )5 / / CCF f o r m




3 TFcont=sysl in ( ’ c ’ , ( 6 ) / ( ( s+2)ˆ2∗( s+1) ) )4 SScont=t f2ss (TFcont )5 / / CCF f o r m



1 A=[0 1;−6 −5]2 [Row Col ]= s ize (A) / / S i z e o f a m a t r i x

3 l=poly (0 , ’ l ’ ) ;4 m=l ∗eye (Row, Col )−A / / l I −A

5 n=det (m) / / To F i n d T h e D e t e r m i n a n t o f l i −A

6 roots (n) / / To F i n d T h e V a l u e O f l


1 A=[−2 1 ;0 −3]

113

2 B=[0 ; 1 ]3 C=[1 1 ]4 s=poly (0 , ’ s ’ ) ;5 [Row Col ]= s ize (A) / / S i z e o f a m a t r i x

6 m=s∗eye (Row, Col )−A / / s I −A

7 n=det (m) / / To F i n d T h e D e t e r m i n a n t o f s i −A

8 p=inv (m) / / To F i n d T h e I n v e r s e O f s I −A

9 y=C∗p∗B ; / / To F i n d C ∗ ( s I −A ) ˆ − 1 ∗ B

10 disp (y , ” Trans fe r Function=” )


1 A=[0 1;−6 −5]2 x = [ 1 ; 0 ] ;3 disp (x , ”x ( t ) = ’)4 s=poly (0 , ’ s ’ ) ;5 [Row Col ]= s i z e (A) // S i z e o f a matrix6 m=s∗ eye (Row, Col )−A // sI−A7 n=det (m) //To Find The Determinant o f s i−A8 p=inv (m) ; // To Find The Inve r s e Of sI−A9 syms t s ;

10 di sp (p , ” phi ( s )=” ) // Resolvent Matrix11 f o r i =1:Row12 f o r j =1:Col13 //Taking Inve r s e Laplace o f each element o f Matrix phi (

s )14 q ( i , j )=i l a p l a c e (p( i , j ) , s , t ) ;15 end ;16 end ;17 di sp (q , ” phi ( t )=” ) // State Trans i t i on Matrix18 r=inv (q ) ;19 r=s imple ( r ) ; //To Find phi(−t )20 di sp ( r , ” phi(−t )=” )21 y=q∗x ; //x ( t )=phi ( t )∗x (0 )22 di sp (y , ” So lu t i on To The given eq .=” )


114

1 A=[0 1;−6 −5]2 B=[0 ; 1 ]3 x = [1 ; 0 ]4 disp (x , ”x ( t ) = ’)5 s=poly (0 , ’ s ’ ) ;6 [Row Col ]= s i z e (A) // S i z e o f a matrix A7 m=s∗ eye (Row, Col )−A // sI−A8 n=det (m) //To Find The Determinant o f s i−A9 p=inv (m) ; // To Find The Inve r s e Of sI−A

10 syms t s m;11 di sp (p , ” phi ( s )=” ) // Resolvent Matrix12 f o r i =1:Row13 f o r j =1:Col14 // Inve r s e Laplace o f each element o f Matrix ( phi ( s ) )15 q ( i , j )=i l a p l a c e (p( i , j ) , s , t ) ;16 end ;17 end ;18 di sp (q , ” phi ( t )=” ) // State Trans i t i on Matrix19 t=(t−m) ;20 q=eva l ( q ) //At t=t−m , eva lua t ing q i . e phi ( t−m)21 // In t e g r a t e q w. r . t m( I n d e f i n i t e I n t e g r a t i on )22 r=in t eg (q∗B,m)23 m=0 //Upper l im i t i s t24 g=eva l ( r ) // Putting the value o f upper l im i t in q25 m=t //Lower Limit i s 026 h=eva l ( r ) // Putt ing the value o f lower l im i t in q27 y=(h−g ) ;28 di sp (y , ”y=” )29 p r i n t f ( ”x ( t )= phi ( t )∗x (0 ) + in t eg ( phi ( t−m)∗B) w. r . t m

from 0 t0 t \n” )30 //x ( t )=phi ( t )∗x (0 )+in t eg ( phi ( t−m)∗B) w. r . t m from 0 t0

t31 y1=(q∗x )+y ;32 di sp ( y1 , ”x ( t )=” )


1 A=[3 0 ;2 4 ]

115

2 B=[0 ; 1 ]3 Cc=cont mat (A,B) ;4 disp (Cc , ” Con t r o l a b i l i t y Matrix=” )5 / / To C h e c k W h e t h e r t h e m a t r i x ( C c ) i s s i n g u l a r i . e

d e t e r m i n t o f C c =0

6 i f determ(Cc)==0;7 printf ( ” S ince the matrix i s S ingular , the system i s

not c o n t r o l l a b l e \n” ) ;8 else ;9 printf ( ”The system i s c o n t r o l l a b l e \n” )

10 end ;


1 A=[−2 1 ;0 −3]2 B=[4 ; 1 ]3 C=[1 0 ]4 [O]=obsv mat (A,C) ;5 disp (O, ” Obse rvab i l i t y Matrix=” )6 / / To C h e c k W h e t h e r t h e m a t r i x ( C c ) i s s i n g u l a r i . e


7 i f determ(O)==0;8 printf ( ” S ince the matrix i s S ingular , the system i s not

Observable \n” ) ;9 else ;

10 printf ( ”The system i s Observable \n” )11 end ;



3 TFcont=sysl in ( ’ c ’ , ( ( 5∗ s ˆ2)+(2∗ s )+6)/( s ˆ3+(7∗ s ˆ2)+(11∗ s )+8) )

4 SScont=t f2ss (TFcont )5 / / CCF f o r m


116



3 TFcont=sysl in ( ’ c ’ , ( 8 ) /( s ∗( s+2)∗( s+3) ) )4 SScont=t f2ss (TFcont )5 / / CCF f o r m



1 A=[0 1;−3 −4]2 x = [ 1 ; 0 ] ;3 disp (x , ”x ( t ) = ’)4 s=poly (0 , ’ s ’ ) ;5 [Row Col ]= s i z e (A) // S i z e o f a matrix6 m=s∗ eye (Row, Col )−A // sI−A7 n=det (m) //To Find The Determinant o f s i−A8 p=inv (m) ;// To Find The Inve r s e Of sI−A9 syms t s ;

10 di sp (p , ” phi ( s )=” ) // Resolvent Matrix11 f o r i =1:Row12 f o r j =1:Col13 //Taking Inve r s e Laplace o f each element o f Matrix phi (

s )14 q ( i , j )=i l a p l a c e (p( i , j ) , s , t ) ;15 end ;16 end ;17 di sp (q , ” phi ( t )=” ) // State Trans i t i on Matrix18 r=inv (q ) ;19 r=s imple ( r ) ; //To Find phi(−t )20 di sp ( r , ” phi(−t )=” )


1 A=[0 1;−3 −4]2 B=[0 ; 1 ]3 C=[1 0 ]

117

4 x = [0 ; 0 ]5 disp (x , ”x ( t ) = ’)6 s=poly (0 , ’ s ’ ) ;7 [Row Col ]= s i z e (A) // S i z e o f a matrix A8 m=s∗ eye (Row, Col )−A // sI−A9 n=det (m) //To Find The Determinant o f s i−A

10 p=inv (m) ; // To Find The Inve r s e Of sI−A11 syms t s m;12 di sp (p , ” phi ( s )=” ) // Resolvent Matrix13 f o r i =1:Row14 f o r j =1:Col15 //Taking Inve r s e Laplace o f each element o f Matrix ( phi (

s ) )16 q ( i , j )=i l a p l a c e (p( i , j ) , s , t ) ;17 end ;18 end ;19 di sp (q , ” phi ( t )=” ) // State Trans i t i on Matrix20 t=(t−m)21 q=eva l ( q ) //At t=t−m , eva lua t ing q i . e phi ( t−m)22 r=in t eg (q∗B,m) // In t e g r a t e q w. r . t m ( I n d e f i n i t e

I n t e g r a t i on )23 m=0 //Upper l im i t i s t24 g=eva l ( r ) // Putting the value o f upper l im i t in q25 m=t //Lower Limit i s 026 h=eva l ( r ) // Putt ing the value o f lower l im i t in q27 y=(h−g ) ;28 p r i n t f ( ”x ( t )= phi ( t )∗x (0 ) + in t eg ( phi ( t−m)∗B) w. r . t m

from 0 t0 t \n” )29 //x ( t )=phi ( t )∗x (0 ) + in t eg ( phi ( t−m)∗B) w. r . t m from 0

t0 t30 y1=(q∗x )+y ;31 di sp ( y1 , ”x ( t )=” )32 y2=C∗y1 ;33 di sp ( y2 , ”Output Response=” )


1 A=[0 1;−2 0 ]

118

2 B=[0 ; 3 ]3 Cc=cont mat (A,B) ;4 disp (Cc , ” Con t r o l a b i l i t y Matrix=” )5 / / To C h e c k W h e t h e r t h e m a t r i x ( C c ) i s s i n g u l a r i . e


6 i f determ(Cc)==0;7 printf ( ” S ince the matrix i s S ingular , the system i s not

c o n t r o l l a b l e \n” ) ;8 else ;9 printf ( ”The system i s c o n t r o l l a b l e \n” )

10 end ;


1 A=[−3 0 ;0 −2]2 B=[4 ; 1 ]3 C=[2 0 ]4 [O]=obsv mat (A,C) ;5 disp (O, ” Obse rvab i l i t y Matrix=” )6 / / To C h e c k W h e t h e r t h e m a t r i x ( C c ) i s s i n g u l a r i . e


7 i f determ(O)==0;8 printf ( ” S ince the matrix i s S ingular , the system i s not

Observable \n” ) ;9 else ;

10 printf ( ”The system i s Observable \n” )11 end ;


1 ieee (2 )2 A=[−3 0 0 ;0 −1 1 ; 0 0 −1]3 B= [ 0 ; 1 ; 0 ]4 s=poly (0 , ’ s ’ ) ;5 [Row Col ]= s ize (A) / / S i z e o f a m a t r i x

6 m=s∗eye (Row, Col )−A / / s I −A

7 n=det (m) / / To F i n d T h e D e t e r m i n a n t o f s i −A

8 p=inv (m) ; / / To F i n d T h e I n v e r s e O f s I −A

119

9 syms t s ;10 disp (p , ” phi ( s )=” ) / / R e s o l v e n t M a t r i x


1 A=[−2 0 ;1 −1]2 B=[0 ; 1 ]3 x = [0 ; 0 ]4 disp (x , ”x ( t ) = ’)5 s=poly (0 , ’ s ’ ) ;6 [Row Col ]= s i z e (A) // S i z e o f a matrix A7 m=s∗ eye (Row, Col )−A // sI−A8 n=det (m) //To Find The Determinant o f s i−A9 p=inv (m) ; // To Find The Inve r s e Of sI−A

10 syms t s m;11 di sp (p , ” phi ( s )=” ) // Resolvent Matrix12 t=(t−m)13 q=eva l ( q ) //At t=t−m , eva lua t ing q i . e phi ( t−m)14 // In t e g r a t e q w. r . t m ( I n d e f i n i t e I n t e g r a t i on )15 r=in t eg (q∗B,m)16 m=0 //Upper l im i t i s t17 g=eva l ( r ) // Putting the value o f upper l im i t in q18 m=t //Lower Limit i s 019 h=eva l ( r ) // Putt ing the value o f lower l im i t in q20 y=(h−g ) ;21 di sp (y , ”y=” )22 p r i n t f ( ”x ( t )= phi ( t )∗x (0 ) + in t eg ( phi ( t−m)∗B) w. r . t m

from 0 t0 t \n” )23 //x ( t )=phi ( t )∗x (0 )+in t eg ( phi ( t−m)∗B)w. r . t m from 0 t0 t24 y1=(q∗x )+y ;25 di sp ( y1 , ”x ( t )=” )26 // CONTROLABILITY OF THE SYSTEM27 Cc=cont mat (A,B) ;28 di sp (Cc , ” Con t r o l a b i l i t y Matrix=” )29 //To Check Whether the matrix (Cc) i s s i n gu l a r i . e

determint o f Cc=030 i f determ (Cc)==0;

120

31 p r i n t f ( ” S ince the matrix i s S ingular , the system i s notc o n t r o l l a b l e \n” ) ;

32 e l s e ;33 p r i n t f ( ”The system i s c o n t r o l l a b l e \n” )34 end ;

121

Chapter 18

Digital Control Systems

For the execution of the programm keep the main programm and the Func-tion in the same folder.

18.1 Scilab Code

Example F3 ztransfer.sce

1 function [ Z t r an s f e r ]= z t r a n s f e r ( sequence )2 z = poly (0 , ’ z ’ , ’ r ’ )3 Zt r an s f e r=sequence ∗(1/ z ) ˆ [ 0 : ( length ( sequence )−1) ] ’4 endfunction

Example 18.01 18.01.01.sci

1 syms n z ;2 x=(−0.5)ˆn3 y=(4∗ ( (0 .2 ) ˆn) )4 f 1=symsum(x∗( zˆ(−n) ) ,n , 0 , %inf )5 f 2=symsum(y∗( zˆ(−n) ) ,n , 0 , %inf )6 y=( f1+f2 ) ;7 disp (y , ”ans=” )

Example 18.08 18.08.01.sci

122

1 exec z t r a n s f e r . s c e ;2 sequence =[0 2 0 0 −3 0 0 8 ]3 y=z t r a n s f e r ( sequence ) ;4 disp (y , ”ans=” )

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Scilab Code for Control Systems Theory and Applications by Smarajit Ghosh · 2015-09-15 · Scilab...

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