SALMAH THUKIMAN ROSSIELYANA ABDUL RAHIM POLITEKNIK IBRAHIM SULTAN CONTROL SYSTEM THE ROOT LOCUS
CONTROL SYSTEM THE ROOT LOCUS
Chief Editor Mazlan bin Karim @ Hussein Editor Salmah binti Thukiman Rossielyana binti Abdul Rahim Proof Reader Siti Noraini binti Hamzah POLITEKNIK IBRAHIM SULTAN KM. 10, JALAN KONG-KONG 81700 PASIR GUDANG, JOHOR DARUL TAKZIM COPYRIGHTS 2022, POLITEKNIK IBRAHIM SULTAN Published by: POLITEKNIK IBRAHIM SULTAN KM. 10, JALAN KONG KONG, 81700 PASIR GUDANG, JOHOR MALAYSIA Contact No.: 07-261 2488 / 07-261 2404 Fax No.: 07-261 2404 www.pis.edu.my i
Acknowledgment ii I would like to express my special thanks and gratitude to my leader Encik Mazlan bin Karim @ Hussein, who gave the golden opportunity to us to do this wonderful e-book 'THE ROOT LOCUS', which also helped us in doing a lot of research and a guidance in writing this e-book. I am really thankful to them. Secondly, I would also like to thank my family, friends and also my students who inspired me a lot in finishing this ebook. This e-book is not only for students general use but also to strengthen our knowledge. Thanks again to all who helped us.
preface This e-book contains 1 topic, which is Analysis and Root Locus Design Method. In this topic, the explanation of the Root Locus Analysis with knowledge and rules of Root Locus in graphical such as Number of branches, Symmetry, Angle of Asymptotes, Centroid of the Asymptotes (Centroid point), Root Locus on real axis, Angle of Departure and Angle of arrival of the Root Locus, Intersection of the root locus with the Imaginary Axis and Breakaway points. Lastly, the root locus analysis plotted in graph paper. All the rules of the root locus analysis contains the example and exercises to help the students. This e-book was developed by the teaching staff of the Jabatan Kejuruteraan Elektrik, Politeknik Ibrahim Sultan. We are indebted to our colleagues who have contributed greatly to production of this e-book. iii
L I S T O F C O N T E N T iv CONTENT PAGES eISBN i Acknowledgement ii Preface iii Introduction 1 Advantages of Root Locus Technique 2 Analysis and Root Locus Design Method 3 Definition of Root Locus 4 Root Locus Method Foundations 6 Two conditions are required to sketch Root locus 7 Before sketch the Root Locus 8 Transfer function of open loop 9 Basic Rules of Root Locus 10 - 17 Exercises and Solution 18 - 21 Structure Question 22 Essay Question 23 - 24 Answer 25 References 26 Meet My Team 27
In control and stability theory, Root Locus analysis is a graphical method for examining how the roots of a system change with variation of a certain system parameter, commonly a gain within a feedback system. This is a technique used as a stability criterion in the field of classical control theory developed by Walter R. Evans which can determine stability of the system. The Root Locus plots the poles of the closed loop transfer function in the complex s-plane as a function of a gain parameter (see pole–zero plot). INTRODUCTION 1
Root locus technique in control system is easy to implement as compared to other methods. With the help of root locus we can easily predict the performance of the whole system. Root locus provides better ways to indicate the parameters. Advantages of Root Locus Technique 2
OBJECTIVE : Identify the usage of Root Locus method in analyzing and designing a system 3 Analysis and Root Locus Design Method
The Root Locus is the locus of the closed-loop poles when a specific parameter (usually gain, K is varied from 0 to infinity. DDEEFFIINNAATTIIOONN OOFF RROOOOTT L O C U S 4
The closed-loop poles of the negative feedback control: are the roots of the characteristic equation: 1 + KG(s)H(s) = 0 The root locus is the locus of the closed-loop poles when a specific parameter (usually gain, K) is varied from 0 to infinity 5
Root Locus Method Foundations The value of s in the s-plane that make the loop gain KG(s)H(s) equal to -1 are the closed-loop poles KG(s)H(s) = -1 can be split into two equations by equating the magnitudes and angles of both sides of the equation. (i.e. 1 + KG(s)H(s) = 0 => KG(s)H(s) = -1 ) 6
Two conditions are required to sketch root locus : KG(s)H(s) = -1 1) MAGNITUDE CONDITION K l G(s) H(s) l = 1 2) ANGLE CONDITION G(s) H (S) = ± 180 ° (2K + 1) where ( k = 0,1,2,……) 7
Before sketching the root locus, remember !!!! To identify the position of poles and zeros of G(s) H(s) {open loop } The angles from poles and zeros of G(s) H(s) {open loop } to s test point are measured anticlockwise 8
TRANSFER FUNCTION OF OPEN LOOP G(s)H(s) = Where Z = m, zeros of G(s)H(s) P = n, poles of G(s)H(s) K = system gain The n-m branches terminate at infinity = n - m 9
BASIC RULES OF ROOT LOCUS METHOD
RULE #1 : Number of branches The n branches of the root locus start at the n (poles). The n branches end on the m (zeros) The n-m (The number of branches) terminate at infinity . Assuming n poles and m zeros for G(s)H(s): Example 1 Example 1 Example 1 10
RULE#2 : Symmetry of the Root Locus The root locus is symmetrical to the real axis. This is a result of the fact that complex poles will always occur in conjugate pairs. RULE#3 : Angle of Asymptotes Is a straight line indicating the direction of the pole to infinity The number of asymptote is given by the difference by poles and zeros α = 11
CONT … Asymptote that drives locus to infinity Example 2 Example 2 Example 2 12
RULE #4 : Centroid of Asymptotes The Centroid point of Asymptotes are on real axis only and given by Example 3 Example 3 Example 3 13
RULE #5 : Breakaway points Breakaway point exists when two or more locus meet and then split It must be exists between the two poles 14
Example 4 Example 4 Example 4 15
RULE #6 : Angles of Departure and Arrival Departure angle from a complex pole = 180ᵒ – (sum of the angles of vectors to a complex pole in question from other poles) + (sum of the angles of vectors to a complex pole in question from zeros) Arrival angle at a complex zero = 180ᵒ – (sum of the angles of vectors to a complex zero in question from other zeros) + (sum of the angles of vectors to a complex zero in question from poles) 16
The objective is to find the intersection point at which the locus crosses the imaginary axis. Points on root locus satisfy: RouthHurwitz Criterion :- Find the range of K for stable system Take the range of Kmax and insert into additional equation ( at S² row : refer table R-H) to get the value of S1, S2 on imaginary axis RULE #7 : Intersection with Imaginary Axis jω? jω? 17
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SOLUTION 19
SOLUTION 20
SOLUTION 21
STRUCTURE QUESTION 22
essay question question 23
24 essay question question
25 answers
1. Bakshi, V.U., S.C., Bakshi, U.A (2010). Control Systems (1st ed). Technical Publications Pune. 2. Golnaraghi, F., Kuo, B.C. Automatic Control System (10th ed). McGraw-Hill Education. 3. Katsuhiko Ogata (5th ed). Modern Control Engineering. Press Hall India. 4. Lopez, C. (2014). MATLAB Control Systems Engineering. Berkley, United States: aPress. 5. Nise, N. S. (2015). Control Systems Engineering. New York, United States: John Wiley & Sons Inc. 6. Robiah Ahmad (2011). Introduction to Control Engineering. UTM Press. 7. Syed Najib Syed Salim, Maslan bin Zainon (2010). Control System Engineering (1st Published). Universiti Teknikal Melaka. 8. Palamides, A., Veloni, A. (2012). Control Systems Problems: Formulas, Solutions & Simulation Tools. Taylor & Francis Group, LL 9. en.wikipeia.org/wiki/Root_locus 10. https://www.electrical4u.com/root-locus-technique-in-control-systemroot-locus-plot 26 R E F E R E N C E S
MEET THE TEAM Salmah binti Thukiman is a lecturer at Jabatan Kejuruteraan Elektrik, Politeknik Ibrahim Sultan. She has an experience of more than 5 years in teaching Control System. Rossielyana binti Abdul Rahim is a lecturer at Jabatan Kejuruteraan Elektrik, Politeknik Ibrahim Sultan. She has an experience at least 1 year in teaching Control System. 26
This book is suitable for students who are majoring in engineering field. It introduces students to the concept of control system and analysis method by using Root Locus. Upon this book, students should be able to apply the concept and principles of control system fundamental in various type of control system engineering applications. CONTROL SYSTEM The Root Locus Penerbit Politeknik Ibrahim Sultan www.pis.edu.my