Control Systems Engineering ( Chapter 8. Root Locus Techniques ) Prof. KwangChun Ho Tel: Fax:


 Rudolf Holland
 1 years ago
 Views:
Transcription
1 Control Systems Engineering ( Chapter 8. Root Locus Techniques ) Prof. KwangChun Ho Tel: Fax:
2 Introduction In this lesson, you will learn the following : The definition of a root locus How to sketch root locus How to use the root locus to find the poles of a closed loop system How to use root locus to describe the changes in transient response How to use root locus to design a parameter value to meet a desired transient response 2
3 Introduction What is Root Locus? Root Locus is graphical presentation of the closedloop poles as the parameter is varied A powerful method of analysis and design for stability and transient response Ability to provide solution for system of order higher than two A graphic representation of a system s stability Why do we need to use Root Locus? We use Root Locus to analyze the transient qualitatively 3
4 Introduction E.g. we can use Root Locus to analyze qualitatively the effect of varying gain upon percent overshoot, settling time and peak time We can also use Root Locus to check the stability of feedback control system qualitatively The control system problem: The poles of the closedloop transfer function are more difficult to find The closedloop poles change with changes in system gain The poles of T(s) are not immediately known without factoring the denominator, and they are a function of K 4
5 Introduction Since the system transient response and stability are dependent upon the poles of T(s), we have no knowledge of the system performance unless we factor the denominator for specific values of K The Root Locus will be used to give us a vivid picture of the poles of T(s) as K varies 5
6 Introduction Complex Numbers and Their Representation as Vectors: Any complex number, j,described in Cartesian coordinates can be graphically represented by a vector, as shown in the following figure (a) The complex numbers also can be described in the polar form with magnitude M, and angle, as M If the complex number is substituting into a complex function, F(s), another complex number will result For example, if F(s)=(s+a), then substituting the complex number s j yields F() s ( a) j which is another complex number, as shown in figure (b) 6
7 Introduction F(s)=(s+a) Since F(s) has a zero at a, we ( s 7) have alternate representation s 5 j 2 7
8 Introduction Example: Evaluation of a complex function via vectors Magnitude and angle of F(s) at any point ( s 3 j4) Solution: s 3 j4 ( 3 j41) F( 3 j4) ( 3 j4)( 3 j4 2) ( )( ) 20 M ( )
9 Defining the Root Locus Consider security camera that can automatically follow a subject (auto tracking) Can get by 2 s 10s K 9
10 Defining the Root Locus (Pole plot from Table 8.1) (Representation of the paths of the closedloop poles as the gain is varied ) Call Root Locus! 10
11 Properties of the Root Locus The root locus technique can be used to analyze and design the effect of a loop gain upon the system s transient response and stability The closed loop transfer function for a system with a gain, K, is (For control system of slide 5) From this equation, a pole, s, exists when the characteristic polynomial in the denominator becomes zero, or 11
12 Properties of the Root Locus Here, 1 represented in polar form as Alternatively, a value of s is a closed loop pole if and Let s demonstrate the relationship: For control system of slide 9, KG() s H () s s(s10) From Table 8.1, the closedloop poles exist at and when the gain is 5 Substituting the pole at for s and 5 for K yields If these calculation is repeated for the point 5 j5, the angle equals an odd multiple of 180 K K K K KG() s H () s 180 s(s10) ( 5 j5)(5 j5) ( )( 5045 ) 50 KG() s H () s 1 (Thus, it is a point on the root locus!) 1 (2k 1)180 K 12
13 Properties of the Root Locus That is, 5 j5 is a point on the root locus for some value of gain Now we proceed to evaluate the value of gain K KG() s H () s 1 K For a point 2 j2, the angle does not equal () () K K K K KG s H s s(s10) ( 2 j2)(8 j2) ( 8135 )( 6814 ) Example: (Can confirm from Table 8.1!) Thus, it is not a point on the root locus, or is not a closedloop pole for any gain A part of Root Locus for Gs () T() s 1 Gs ( ) 13
14 Properties of the Root Locus For this system, the openloop transfer function is KG() s H () s K(s3)(s4) (s1)(s2) Consider the point 2 j3 Then, the angle does not equal as follows: Thus, this point is not a point on the root locus!! (That is, this point is not a closedloop pole for any gain) 14
15 Sketching the Root Locus The following five basic rules allow us to sketch the root locus using minimal calculations Rule 1: Number of the branches Each closed loop pole moves as the gain is varied If we define a branch as the path that one pole traverses, then there will be one branch for each closed loop pole Our first rule defines the number of branches of the root locus: The number of the branches of the root locus are equal to the number of closed loop poles Rule 2: Symmetry Physically realizable system can not have the complex coefficients K in their transfer functions Thus, we conclude: The root locus is symmetrical about the real axis 15
16 Sketching the Root Locus Rule 3: Realaxis segment Using some mathematical properties of root locus plot, we get On the real axis, for K>0 the root locus exist to the left of an odd number of real axis, finite open loop poles and/or finite open loop zeros symmetrical 16
17 Sketching the Root Locus Rule 4: Starting and ending points Consider the closed loop transfer function As K approaches to zero, the closed loop poles approaches the poles of G(s)H(s) (Refer to Slide 10!) Similarly, as K approaches to infinity, the closed loop poles approaches the zeros of G(s)H(s) (Refer to Slide 10!) We conclude that The root locus begins at the finite and infinite poles of G(s)H(s) and ends at the finite and infinite zeros of G(s)H(s) 17
18 Sketching the Root Locus In this system, the root locus begins at the poles at 1 and 2 and ends at zeros at 3 and 4 Rule 5: Behavior at infinity A function can have infinite poles and zeros Every function of s has an equal number of poles and zeros For example, consider the function This function has two poles at 2 and 3 and one zeros at 1 Because of the number of poles and zeros must be equal, this function has a zero at infinity! (Ref. G(s)=1/s: G( ) 0+ a pole at s=0) a zero at s This rule tell us the following conclusion: (Complete Root Locus) 18
19 Sketching the Root Locus a The root locus approaches straight lines as asymptotes as the locus approaches infinity Further, the equation of the asymptotes is given by the real axis intercept,, and angle,, as follows: a where k = 0, ±1, ±2, ±3 and the angle is given in radians with respect to the positive extension of the real axis Example: Sketch the root locus for the system shown in figure a finite poles finite zeros (2k 1), a #finite poles #finite zeros #finite poles #finite zeros 19
20 Sketching the Root Locus Solution: Begin by calculating the asymptotes: If the values for k continued to increase, the angels would begin repeat The number of lines obtained equals the difference between the number of finite poles and the number of finite zeros Rule 4 states that the root locus begins at the open loop poles and ends at the open loop zeros 20
21 Sketching the Root Locus For this example, there are more open loop poles than open loop zeros Thus, there must be zeros at infinity The asymptotes tell us how we get to these zeros at infinity! This figure shows the complete root locus as well as the asymptotes that were just calculated Root locus with the 3 asymptotes 21
22 Sketching the Root Locus Notice that we have made use of all the rules learned so far The real axis segment lie to the left of an odd number of poles and/or zeros The locus starts at the openloop poles and ends at the openloop zeros For this example, there is only one open loop finite zero and three infinite zeros Rule 5, then, tell us that the three zeros at infinity are at the ends of the asymptotes Now, consider four additional rules for Refining the Sketch Once we sketch the root locus using the five rules discussed in the previous slides, we may want to accurately locate points on the root locus as well as find their associated gain 22
23 Refining the Sketch Rule 6: Real Axis Breakaway and Breakin Points Numerous root loci appear to break away from the real axis as the system poles move from the real axis to the complex plane At other times, the loci appear to return to the real axis as a pair of complex poles becomes real This is illustrated in the figure The breakaway point is found at the maximum point, and the breakin point is found at the minimum point (Variation of gain along the real axis ) 23
24 Refining the Sketch The figure shows a root locus leaving the real axis between 1 and 2 and returning to the real axis between +3 and +5 The point where the locus leaves the real axis, 1, is called the breakaway point, and the point where the locus returns to the real axis, 2, is called breakin point At the breakaway or breakin point, the branches of the root locus form an angle of 180/n with the real axis, where n is the number of closed loop poles arriving or departing from the single breakaway or breakin point on the real axis Thus for the poles shown in the figure the branches at the breakaway point form 90 angles with the real axis We now show how to find the breakaway and the breakin points There is a simple method for finding the points at which the root locus breaks away from and breaks into real axis The first method is to maximize and minimize the gain, K, using differential calculus 24
25 Refining the Sketch For all points on the root locus, because of the KG(s)H(s)= 1 as we pointed out before, K = 1 / G(s)H(s) For points along the real axis segment of the root locus where breakaway and breakin points could exist, s Hence along the real axis, K 1 1 GsHs () () G( ) H( ) s Hence if we differentiate this equation with respect to and set the derivative equal to zero, we can find the points of maximum and minimum gain and hence breakaway and breakin points dk 0,? d 25
26 Refining the Sketch Example: Find the breakaway and breakin points for the root locus of the figure using differential calculus Solution: Using the open loop poles and zeros, we present the open loop system whose root locus is shown in figure as follows But for all points along the root locus, KG(s)H(s)=1, andalong the real axis,. Hence s Solving for K, we find 26
27 Refining the Sketch Differentiating K with respect to and setting the derivative equal to zero yields Solving for, we find =1.45 (breakaway point) and 3.82 (breakin point), which are the breakaway and breakin points (Data Sheet for breakaway and breakin points) 27
28 Refining the Sketch Rule 7: j Axis Crossings j The axis crossing is a point on the root locus that separates the stable operation of the system from the unstable operation The value of at the axis crossing yields the frequency of oscillation The gain at the jaxis crossing yields the maximum positive gain for system stability To find j axis crossing, we can use the RouthHurwitz criterion as follows: Forcing a row of zeros in the Routh table will yield the gain; going back one row to the even polynomial equation and solving for the roots yields the frequency at the imaginary axis crossing 28
29 Refining the Sketch Example: For the system of the following figure, find the frequency and gain, K, for which the root locus crosses the imaginary axis For what range of K is the system stable? Solution: The closed loop transfer function for the system is Simplifying some of the entries by multiplying any row by a constant, we obtain the Routh array shown in table 29
30 Refining the Sketch A complete row of zeros yields the possibility for imaginary axis roots For positive value of gain only, the s 1 row can yield a row of zeros for K>0 Thus,, and K = 9.65 Forming the even polynomial by using the s 2 row with K =9.65 We obtain, and s is found to be ±j1.59 Thus, the root locus crosses the j axis at ±j1.59 at a gain of 9.65 We conclude that the system is stable for 0 K
31 Refining the Sketch Rule 8: Angles of Departure and Arrival In order to sketch the root locus more accurately, we want to calculate the root locus departure angle from the complex poles and the arrival angle to the complex zeros If we assume a point on the root locus close to the point, we assume that all angles drawn from all other poles and zeros are drawn directly to the pole that is near the point Thus, the only unknown angle in the sum is the angle drawn from the pole that is close We can solve for this unknown angle, which is also the angle of departure from this complex pole Hence from the figure(a) (Angles of Departure) 31
32 Refining the Sketch (Angles of Departure) If we assume a point on the root locus close to a complex zero, the sum of angles drawn from all finite poles and zeros to this point is an odd multiple of 180 ( ) Except for the zero that is close to the point, we can assume all angles drawn from all other poles and zeros are drawn directly to the zero that is near the point 32
33 Refining the Sketch Thus, the only unknown angle in the sum is the angle drawn from the zeros that is close We can solve for this unknown angle, which is also the angle of arrival to this complex zero Hence from the figure(b) (Angles of Arrival) 33
34 Refining the Sketch Example: Given the unity feedback system of following figure, find the angle of departure from the complex poles and sketch the root locus Solution: Using the poles and zeros of G(s)= (s+2)/[(s+3)(s 2 +2s+2)] as plotted in the figure, we calculate the sum of angles drawn to a point close to the complex pole, 1+j1, in the second quadrant Thus, 34
35 Refining the Sketch 35
36 Refining the Sketch Rule 9: Plotting and calibration the Root locus We might want to know the exact coordinates of the root locus as it crosses the radial line representing an overshoot value Overshoot value can be represented by a radial line on the s plane We learnt in Chapter 4, the value of zeta on the splane is cos Given percent overshoot value, we can calculate the zeta value ln(%os /100) 2 2 ln (%OS /100) Radial line representing overshoot value on the splane 36
37 Refining the Sketch Example: Draw a radial line on an splane that represents 20% overshoot Solution: Overshoot is represented by zeta (damping ratio) on the Splane So, the first step is to find the value of zeta OS OS ln % /100 ln(20 /100) ln 2 % /100 2 ln 2 20 /100 Next step is to find the angle of the radial line cos cos 1 cos j 37
38 Refining the Sketch The point where our root locus intersect with the n percent overshoot radial line is the point when the gain value produces a transient response with n percent overshoot Our root locus intersect with the radial line. Meaning the gain at the intersection produces transient response with zeta = 0.45 A point on the radial line is on the root locus if the angular sum (zero angle pole angles) in reference to the point on the radial line add up to an odd multiple of 180, Odd multiple of 180 : (2k+1)180, k = 1, 2, 3, 4,. 180, 540, 900, 1260, 38
39 Refining the Sketch Then, we must then calculate the value of gain Refer to the previous root locus that we have calculated, we will find the exact coordinate as it crosses the radial line representing 20% overshoot
40 Refining the Sketch We can find the point on the radial line that crosses the root locus by selecting a point with radius value then add the angles of the zeros and subtract the angles of the poles Theory : Odd multiple of , 540, 900, 1260 point on the radial line is on the locus At a point s, 40
41 Refining the Sketch Example: Sketch the root locus for the system shown in figure and find the breakaway point on the real axis and the range of K within which the system is stable Solution: The root locus of the system is shown in following figure The real axis segment is found to be between 2 and 4, using Rule 3 The root locus starts at the open loop poles and ends at the open loop zeros, using Rule 4 The root locus crosses the axis at ±j3.9, using Rule 7 j 41
42 Refining the Sketch The exact point and gain where the locus crosses the 0.45 damping ratio line: K=0.47 at , using Rule 9 To find breakaway point use the root locus algorithm to search the real axis between 2 and 4 for the point that yields maximum gain, using Rule 6 Naturally all points will have the sum of their angles equal to an odd multiple of 180 A maximum gain of is found to at the point Thus, the breakaway point is between the open loop poles on the real axis at Finally, the system is stable for K between 0 and 15 42
43 Root Locus Plot using matlab Example: Plot the rootlocus of the following system Gs () 2 2 s s s s 2 ( s 2)( s 4) s 6s % matlab script 5 4 Root Locus clear all num = [14 20]; den = [1 6 8]; G = tf(num,den); rlocus(g); axis('equal'); Real Axis (seconds 1 ) 43
44 Homework Assignment #8 44
45 Homework Assignment #8 45
46 Homework Assignment #8 46
47 Homework Assignment #8 47
Software Engineering 3DX3. Slides 8: Root Locus Techniques
Software Engineering 3DX3 Slides 8: Root Locus Techniques Dr. Ryan Leduc Department of Computing and Software McMaster University Material based on Control Systems Engineering by N. Nise. c 2006, 2007
More informationRoot Locus Techniques
Root Locus Techniques 8 Chapter Learning Outcomes After completing this chapter the student will be able to: Define a root locus (Sections 8.1 8.2) State the properties of a root locus (Section 8.3) Sketch
More informationLecture 1 Root Locus
Root Locus ELEC304Alper Erdogan 1 1 Lecture 1 Root Locus What is RootLocus? : A graphical representation of closed loop poles as a system parameter varied. Based on RootLocus graph we can choose the
More informationI What is root locus. I System analysis via root locus. I How to plot root locus. Root locus (RL) I Uses the poles and zeros of the OL TF
EE C28 / ME C34 Feedback Control Systems Lecture Chapter 8 Root Locus Techniques Lecture abstract Alexandre Bayen Department of Electrical Engineering & Computer Science University of California Berkeley
More informationECE 345 / ME 380 Introduction to Control Systems Lecture Notes 8
Learning Objectives ECE 345 / ME 380 Introduction to Control Systems Lecture Notes 8 Dr. Oishi oishi@unm.edu November 2, 203 State the phase and gain properties of a root locus Sketch a root locus, by
More information7.4 STEP BY STEP PROCEDURE TO DRAW THE ROOT LOCUS DIAGRAM
ROOT LOCUS TECHNIQUE. Values of on the root loci The value of at any point s on the root loci is determined from the following equation G( s) H( s) Product of lengths of vectors from poles of G( s)h( s)
More informationa. Closedloop system; b. equivalent transfer function Then the CLTF () T is s the poles of () T are s from a contribution of a
Root Locus Simple definition Locus of points on the s plane that represents the poles of a system as one or more parameter vary. RL and its relation to poles of a closed loop system RL and its relation
More informationSECTION 5: ROOT LOCUS ANALYSIS
SECTION 5: ROOT LOCUS ANALYSIS MAE 4421 Control of Aerospace & Mechanical Systems 2 Introduction Introduction 3 Consider a general feedback system: Closed loop transfer function is 1 is the forward path
More informationControl Systems Engineering ( Chapter 6. Stability ) Prof. KwangChun Ho Tel: Fax:
Control Systems Engineering ( Chapter 6. Stability ) Prof. KwangChun Ho kwangho@hansung.ac.kr Tel: 027604253 Fax:027604435 Introduction In this lesson, you will learn the following : How to determine
More informationUnit 7: Part 1: Sketching the Root Locus
Root Locus Unit 7: Part 1: Sketching the Root Locus Engineering 5821: Control Systems I Faculty of Engineering & Applied Science Memorial University of Newfoundland March 14, 2010 ENGI 5821 Unit 7: Root
More informationCHAPTER # 9 ROOT LOCUS ANALYSES
F K א CHAPTER # 9 ROOT LOCUS ANALYSES 1. Introduction The basic characteristic of the transient response of a closedloop system is closely related to the location of the closedloop poles. If the system
More informationIntroduction to Root Locus. What is root locus?
Introduction to Root Locus What is root locus? A graphical representation of the closed loop poles as a system parameter (Gain K) is varied Method of analysis and design for stability and transient response
More informationRoot locus Analysis. P.S. Gandhi Mechanical Engineering IIT Bombay. Acknowledgements: Mr Chaitanya, SYSCON 07
Root locus Analysis P.S. Gandhi Mechanical Engineering IIT Bombay Acknowledgements: Mr Chaitanya, SYSCON 07 Recap R(t) + _ k p + k s d 1 s( s+ a) C(t) For the above system the closed loop transfer function
More informationAutomatic Control Systems, 9th Edition
Chapter 7: Root Locus Analysis Appendix E: Properties and Construction of the Root Loci Automatic Control Systems, 9th Edition Farid Golnaraghi, Simon Fraser University Benjamin C. Kuo, University of Illinois
More informationChapter 7 : Root Locus Technique
Chapter 7 : Root Locus Technique By Electrical Engineering Department College of Engineering King Saud University 1431143 7.1. Introduction 7.. Basics on the Root Loci 7.3. Characteristics of the Loci
More informationUnit 7: Part 1: Sketching the Root Locus. Root Locus. Vector Representation of Complex Numbers
Root Locus Root Locus Unit 7: Part 1: Sketching the Root Locus Engineering 5821: Control Systems I Faculty of Engineering & Applied Science Memorial University of Newfoundland 1 Root Locus Vector Representation
More informationExample on Root Locus Sketching and Control Design
Example on Root Locus Sketching and Control Design MCE44  Spring 5 Dr. Richter April 25, 25 The following figure represents the system used for controlling the robotic manipulator of a Mars Rover. We
More informationDr Ian R. Manchester Dr Ian R. Manchester AMME 3500 : Root Locus
Week Content Notes 1 Introduction 2 Frequency Domain Modelling 3 Transient Performance and the splane 4 Block Diagrams 5 Feedback System Characteristics Assign 1 Due 6 Root Locus 7 Root Locus 2 Assign
More informationModule 07 Control Systems Design & Analysis via RootLocus Method
Module 07 Control Systems Design & Analysis via RootLocus Method Ahmad F. Taha EE 3413: Analysis and Desgin of Control Systems Email: ahmad.taha@utsa.edu Webpage: http://engineering.utsa.edu/ taha March
More informationRoot Locus Methods. The root locus procedure
Root Locus Methods Design of a position control system using the root locus method Design of a phase lag compensator using the root locus method The root locus procedure To determine the value of the gain
More informationRoot Locus Techniques
4th Edition E I G H T Root Locus Techniques SOLUTIONS TO CASE STUDIES CHALLENGES Antenna Control: Transient Design via Gain a. From the Chapter 5 Case Study Challenge: 76.39K G(s) = s(s+50)(s+.32) Since
More informationRoot Locus. Signals and Systems: 3C1 Control Systems Handout 3 Dr. David Corrigan Electronic and Electrical Engineering
Root Locus Signals and Systems: 3C1 Control Systems Handout 3 Dr. David Corrigan Electronic and Electrical Engineering corrigad@tcd.ie Recall, the example of the PI controller car cruise control system.
More informationRoot Locus U R K. Root Locus: Find the roots of the closedloop system for 0 < k < infinity
Background: Root Locus Routh Criteria tells you the range of gains that result in a stable system. It doesn't tell you how the system will behave, however. That's a problem. For example, for the following
More informationChemical Process Dynamics and Control. Aisha Osman Mohamed Ahmed Department of Chemical Engineering Faculty of Engineering, Red Sea University
Chemical Process Dynamics and Control Aisha Osman Mohamed Ahmed Department of Chemical Engineering Faculty of Engineering, Red Sea University 1 Chapter 4 System Stability 2 Chapter Objectives End of this
More information(b) A unity feedback system is characterized by the transfer function. Design a suitable compensator to meet the following specifications:
1. (a) The open loop transfer function of a unity feedback control system is given by G(S) = K/S(1+0.1S)(1+S) (i) Determine the value of K so that the resonance peak M r of the system is equal to 1.4.
More informationProblems XO («) splane. splane *~8 X 5. id) X splane. splane. * Xtg) FIGURE P8.1. jplane. JO) k JO)
Problems 1. For each of the root loci shown in Figure P8.1, tell whether or not the sketch can be a root locus. If the sketch cannot be a root locus, explain why. Give all reasons. [Section: 8.4] *~8 XO
More informationIf you need more room, use the backs of the pages and indicate that you have done so.
EE 343 Exam II Ahmad F. Taha Spring 206 Your Name: Your Signature: Exam duration: hour and 30 minutes. This exam is closed book, closed notes, closed laptops, closed phones, closed tablets, closed pretty
More information"APPENDIX. Properties and Construction of the Root Loci " E1 K ¼ 0ANDK ¼1POINTS
AppendixE_1 5/14/29 1 "APPENDIX E Properties and Construction of the Root Loci The following properties of the root loci are useful for constructing the root loci manually and for understanding the root
More informationSECTION 8: ROOTLOCUS ANALYSIS. ESE 499 Feedback Control Systems
SECTION 8: ROOTLOCUS ANALYSIS ESE 499 Feedback Control Systems 2 Introduction Introduction 3 Consider a general feedback system: Closedloop transfer function is KKKK ss TT ss = 1 + KKKK ss HH ss GG ss
More informationAlireza Mousavi Brunel University
Alireza Mousavi Brunel University 1 » Control Process» Control Systems Design & Analysis 2 OpenLoop Control: Is normally a simple switch on and switch off process, for example a light in a room is switched
More informationCHAPTER 7 : BODE PLOTS AND GAIN ADJUSTMENTS COMPENSATION
CHAPTER 7 : BODE PLOTS AND GAIN ADJUSTMENTS COMPENSATION Objectives Students should be able to: Draw the bode plots for first order and second order system. Determine the stability through the bode plots.
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Mechanical Engineering Dynamics and Control II Fall K(s +1)(s +2) G(s) =.
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Mechanical Engineering. Dynamics and Control II Fall 7 Problem Set #7 Solution Posted: Friday, Nov., 7. Nise problem 5 from chapter 8, page 76. Answer:
More informationEE302  Feedback Systems Spring Lecture KG(s)H(s) = KG(s)
EE3  Feedback Systems Spring 19 Lecturer: Asst. Prof. M. Mert Ankarali Lecture 1.. 1.1 Root Locus In control theory, root locus analysis is a graphical analysis method for investigating the change of
More informationControls Problems for Qualifying Exam  Spring 2014
Controls Problems for Qualifying Exam  Spring 2014 Problem 1 Consider the system block diagram given in Figure 1. Find the overall transfer function T(s) = C(s)/R(s). Note that this transfer function
More informationLecture 3: The Root Locus Method
Lecture 3: The Root Locus Method Venkata Sonti Department of Mechanical Engineering Indian Institute of Science Bangalore, India, 56001 This draft: March 1, 008 1 The Root Locus method The Root Locus method,
More informationRoot locus 5. tw4 = 450. Root Locus S51 S O L U T I O N S
Root Locus S51 S O L U T I O N S Root locus 5 Note: All references to Figures and Equations whose numbers are not preceded by an "S" refer to the textbook. (a) Rule 2 is all that is required to find the
More informationControl Systems. University Questions
University Questions UNIT1 1. Distinguish between open loop and closed loop control system. Describe two examples for each. (10 Marks), Jan 2009, June 12, Dec 11,July 08, July 2009, Dec 2010 2. Write
More informationStep input, ramp input, parabolic input and impulse input signals. 2. What is the initial slope of a step response of a first order system?
IC6501 CONTROL SYSTEM UNITII TIME RESPONSE PARTA 1. What are the standard test signals employed for time domain studies?(or) List the standard test signals used in analysis of control systems? (April
More informationSchool of Mechanical Engineering Purdue University. DC Motor Position Control The block diagram for position control of the servo table is given by:
Root Locus Motivation Sketching Root Locus Examples ME375 Root Locus  1 Servo Table Example DC Motor Position Control The block diagram for position control of the servo table is given by: θ D 0.09 See
More informationLecture 5 Classical Control Overview III. Dr. Radhakant Padhi Asst. Professor Dept. of Aerospace Engineering Indian Institute of Science  Bangalore
Lecture 5 Classical Control Overview III Dr. Radhakant Padhi Asst. Professor Dept. of Aerospace Engineering Indian Institute of Science  Bangalore A Fundamental Problem in Control Systems Poles of open
More informationKINGS COLLEGE OF ENGINEERING DEPARTMENT OF ELECTRONICS AND COMMUNICATION ENGINEERING
KINGS COLLEGE OF ENGINEERING DEPARTMENT OF ELECTRONICS AND COMMUNICATION ENGINEERING QUESTION BANK SUB.NAME : CONTROL SYSTEMS BRANCH : ECE YEAR : II SEMESTER: IV 1. What is control system? 2. Define open
More informationModule 3F2: Systems and Control EXAMPLES PAPER 2 ROOTLOCUS. Solutions
Cambridge University Engineering Dept. Third Year Module 3F: Systems and Control EXAMPLES PAPER ROOTLOCUS Solutions. (a) For the system L(s) = (s + a)(s + b) (a, b both real) show that the rootlocus
More informationHomework 7  Solutions
Homework 7  Solutions Note: This homework is worth a total of 48 points. 1. Compensators (9 points) For a unity feedback system given below, with G(s) = K s(s + 5)(s + 11) do the following: (c) Find the
More informationCHAPTER 1 Basic Concepts of Control System. CHAPTER 6 Hydraulic Control System
CHAPTER 1 Basic Concepts of Control System 1. What is open loop control systems and closed loop control systems? Compare open loop control system with closed loop control system. Write down major advantages
More informationEXAMPLE PROBLEMS AND SOLUTIONS
Similarly, the program for the fourthorder transfer function approximation with T = 0.1 sec is [num,denl = pade(0.1, 4); printsys(num, den, 'st) numlden = sa42o0sa3 + 1 80O0sA2840000~ + 16800000 sa4
More informationProportional plus Integral (PI) Controller
Proportional plus Integral (PI) Controller 1. A pole is placed at the origin 2. This causes the system type to increase by 1 and as a result the error is reduced to zero. 3. Originally a point A is on
More information5 Root Locus Analysis
5 Root Locus Analysis 5.1 Introduction A control system is designed in tenns of the perfonnance measures discussed in chapter 3. Therefore, transient response of a system plays an important role in the
More informationMAK 391 System Dynamics & Control. Presentation Topic. The Root Locus Method. Student Number: Group: IB. Name & Surname: Göksel CANSEVEN
MAK 391 System Dynamics & Control Presentation Topic The Root Locus Method Student Number: 9901.06047 Group: IB Name & Surname: Göksel CANSEVEN Date: December 2001 The RootLocus Method Göksel CANSEVEN
More informationCourse roadmap. ME451: Control Systems. What is Root Locus? (Review) Characteristic equation & root locus. Lecture 18 Root locus: Sketch of proofs
ME451: Control Systems Modeling Course roadmap Analysis Design Lecture 18 Root locus: Sketch of proofs Dr. Jongeun Choi Department of Mechanical Engineering Michigan State University Laplace transform
More informationEE C128 / ME C134 Fall 2014 HW 6.2 Solutions. HW 6.2 Solutions
EE C28 / ME C34 Fall 24 HW 6.2 Solutions. PI Controller For the system G = K (s+)(s+3)(s+8) HW 6.2 Solutions in negative feedback operating at a damping ratio of., we are going to design a PI controller
More informationECEN 605 LINEAR SYSTEMS. Lecture 20 Characteristics of Feedback Control Systems II Feedback and Stability 1/27
1/27 ECEN 605 LINEAR SYSTEMS Lecture 20 Characteristics of Feedback Control Systems II Feedback and Stability Feedback System Consider the feedback system u + G ol (s) y Figure 1: A unity feedback system
More informationTest 2 SOLUTIONS. ENGI 5821: Control Systems I. March 15, 2010
Test 2 SOLUTIONS ENGI 5821: Control Systems I March 15, 2010 Total marks: 20 Name: Student #: Answer each question in the space provided or on the back of a page with an indication of where to find the
More informationCompensator Design to Improve Transient Performance Using Root Locus
1 Compensator Design to Improve Transient Performance Using Root Locus Prof. Guy Beale Electrical and Computer Engineering Department George Mason University Fairfax, Virginia Correspondence concerning
More informationSome special cases
Lecture Notes on Control Systems/D. Ghose/2012 87 11.3.1 Some special cases Routh table is easy to form in most cases, but there could be some cases when we need to do some extra work. Case 1: The first
More informationAutomatic Control (TSRT15): Lecture 4
Automatic Control (TSRT15): Lecture 4 Tianshi Chen Division of Automatic Control Dept. of Electrical Engineering Email: tschen@isy.liu.se Phone: 13282226 Office: Bhouse extrance 2527 Review of the last
More informationROOT LOCUS. Consider the system. Root locus presents the poles of the closedloop system when the gain K changes from 0 to. H(s) H ( s) = ( s)
C1 ROOT LOCUS Consider the system R(s) E(s) C(s) + K G(s)  H(s) C(s) R(s) = K G(s) 1 + K G(s) H(s) Root locus presents the poles of the closedloop system when the gain K changes from 0 to 1+ K G ( s)
More informationClass 12 Root Locus part II
Class 12 Root Locus part II Revising (from part I): Closed loop system K The Root Locus the locus of the poles of the closed loop system, when we vary the value of K Comple plane jω ais 0 real ais Thus,
More informationEE 380 EXAM II 3 November 2011 Last Name (Print): First Name (Print): ID number (Last 4 digits): Section: DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO
EE 380 EXAM II 3 November 2011 Last Name (Print): First Name (Print): ID number (Last 4 digits): Section: DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SO Problem Weight Score 1 25 2 25 3 25 4 25 Total
More informationRemember that : Definition :
Stability This lecture we will concentrate on How to determine the stability of a system represented as a transfer function How to determine the stability of a system represented in statespace How to
More informationSoftware Engineering/Mechatronics 3DX4. Slides 6: Stability
Software Engineering/Mechatronics 3DX4 Slides 6: Stability Dr. Ryan Leduc Department of Computing and Software McMaster University Material based on lecture notes by P. Taylor and M. Lawford, and Control
More informationVALLIAMMAI ENGINEERING COLLEGE SRM Nagar, Kattankulathur
VALLIAMMAI ENGINEERING COLLEGE SRM Nagar, Kattankulathur 603 203. DEPARTMENT OF ELECTRONICS & COMMUNICATION ENGINEERING SUBJECT QUESTION BANK : EC6405 CONTROL SYSTEM ENGINEERING SEM / YEAR: IV / II year
More informationCourse Outline. Closed Loop Stability. Stability. Amme 3500 : System Dynamics & Control. Nyquist Stability. Dr. Dunant Halim
Amme 3 : System Dynamics & Control Nyquist Stability Dr. Dunant Halim Course Outline Week Date Content Assignment Notes 1 5 Mar Introduction 2 12 Mar Frequency Domain Modelling 3 19 Mar System Response
More informationEE402  Discrete Time Systems Spring Lecture 10
EE402  Discrete Time Systems Spring 208 Lecturer: Asst. Prof. M. Mert Ankarali Lecture 0.. Root Locus For continuous time systems the root locus diagram illustrates the location of roots/poles of a closed
More informationFrequency Response Techniques
4th Edition T E N Frequency Response Techniques SOLUTION TO CASE STUDY CHALLENGE Antenna Control: Stability Design and Transient Performance First find the forward transfer function, G(s). Pot: K 1 = 10
More information1 (s + 3)(s + 2)(s + a) G(s) = C(s) = K P + K I
MAE 43B Linear Control Prof. M. Krstic FINAL June 9, Problem. ( points) Consider a plant in feedback with the PI controller G(s) = (s + 3)(s + )(s + a) C(s) = K P + K I s. (a) (4 points) For a given constant
More information2.004 Dynamics and Control II Spring 2008
MT OpenCourseWare http://ocw.mit.edu.004 Dynamics and Control Spring 008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. Massachusetts nstitute of Technology
More informationEEE 184 Project: Option 1
EEE 184 Project: Option 1 Date: November 16th 2012 Due: December 3rd 2012 Work Alone, show your work, and comment your results. Comments, clarity, and organization are important. Same wrong result or same
More informationClass 11 Root Locus part I
Class 11 Root Locus part I Closed loop system G(s) G(s) G(s) Closed loop system K The Root Locus the locus of the poles of the closed loop system, when we vary the value of K We shall assume here K >,
More informationEE 370L Controls Laboratory. Laboratory Exercise #7 Root Locus. Department of Electrical and Computer Engineering University of Nevada, at Las Vegas
EE 370L Controls Laboratory Laboratory Exercise #7 Root Locus Department of Electrical an Computer Engineering University of Nevaa, at Las Vegas 1. Learning Objectives To emonstrate the concept of error
More informationBlock Diagram Reduction
Block Diagram Reduction Figure 1: Single block diagram representation Figure 2: Components of Linear Time Invariant Systems (LTIS) Figure 3: Block diagram components Figure 4: Block diagram of a closedloop
More informationTransient response via gain adjustment. Consider a unity feedback system, where G(s) = 2. The closed loop transfer function is. s 2 + 2ζωs + ω 2 n
Design via frequency response Transient response via gain adjustment Consider a unity feedback system, where G(s) = ωn 2. The closed loop transfer function is s(s+2ζω n ) T(s) = ω 2 n s 2 + 2ζωs + ω 2
More informationINSTITUTE OF AERONAUTICAL ENGINEERING (Autonomous) Dundigal, Hyderabad
INSTITUTE OF AERONAUTICAL ENGINEERING (Autonomous) Dundigal, Hyderabad  500 043 Electrical and Electronics Engineering TUTORIAL QUESTION BAN Course Name : CONTROL SYSTEMS Course Code : A502 Class : III
More informationECE 486 Control Systems
ECE 486 Control Systems Spring 208 Midterm #2 Information Issued: April 5, 208 Updated: April 8, 208 ˆ This document is an info sheet about the second exam of ECE 486, Spring 208. ˆ Please read the following
More information1 (20 pts) Nyquist Exercise
EE C128 / ME134 Problem Set 6 Solution Fall 2011 1 (20 pts) Nyquist Exercise Consider a close loop system with unity feedback. For each G(s), hand sketch the Nyquist diagram, determine Z = P N, algebraically
More informationMODERN CONTROL SYSTEMS
MODERN CONTROL SYSTEMS Lecture 1 Root Locu Emam Fathy Department of Electrical and Control Engineering email: emfmz@aat.edu http://www.aat.edu/cv.php?dip_unit=346&er=68525 1 Introduction What i root locu?
More informationControl Systems I. Lecture 7: Feedback and the Root Locus method. Readings: Jacopo Tani. Institute for Dynamic Systems and Control DMAVT ETH Zürich
Control Systems I Lecture 7: Feedback and the Root Locus method Readings: Jacopo Tani Institute for Dynamic Systems and Control DMAVT ETH Zürich November 2, 2018 J. Tani, E. Frazzoli (ETH) Lecture 7:
More informationMethods for analysis and control of dynamical systems Lecture 4: The root locus design method
Methods for analysis and control of Lecture 4: The root locus design method O. Sename 1 1 Gipsalab, CNRSINPG, FRANCE Olivier.Sename@gipsalab.inpg.fr www.gipsalab.fr/ o.sename 5th February 2015 Outline
More informationLaplace Transform Analysis of Signals and Systems
Laplace Transform Analysis of Signals and Systems Transfer Functions Transfer functions of CT systems can be found from analysis of Differential Equations Block Diagrams Circuit Diagrams 5/10/04 M. J.
More informationDepartment of Mechanical Engineering
Department of Mechanical Engineering 2.14 ANALYSIS AND DESIGN OF FEEDBACK CONTROL SYSTEMS Fall Term 23 Problem Set 5: Solutions Problem 1: Nise, Ch. 6, Problem 2. Notice that there are sign changes in
More informationCourse roadmap. Step response for 2ndorder system. Step response for 2ndorder system
ME45: Control Systems Lecture Time response of ndorder systems Prof. Clar Radcliffe and Prof. Jongeun Choi Department of Mechanical Engineering Michigan State University Modeling Laplace transform Transfer
More informationController Design using Root Locus
Chapter 4 Controller Design using Root Locus 4. PD Control Root locus is a useful tool to design different types of controllers. Below, we will illustrate the design of proportional derivative controllers
More informationINTRODUCTION TO DIGITAL CONTROL
ECE4540/5540: Digital Control Systems INTRODUCTION TO DIGITAL CONTROL.: Introduction In ECE450/ECE550 Feedback Control Systems, welearnedhow to make an analog controller D(s) to control a lineartimeinvariant
More informationSystems Analysis and Control
Systems Analysis and Control Matthew M. Peet Arizona State University Lecture 15: Root Locus Part 4 Overview In this Lecture, you will learn: Which Poles go to Zeroes? Arrival Angles Picking Points? Calculating
More informationCourse Summary. The course cannot be summarized in one lecture.
Course Summary Unit 1: Introduction Unit 2: Modeling in the Frequency Domain Unit 3: Time Response Unit 4: Block Diagram Reduction Unit 5: Stability Unit 6: SteadyState Error Unit 7: Root Locus Techniques
More informationCONTROL SYSTEMS. Chapter 5 : Root Locus Diagram. GATE Objective & Numerical Type Solutions. The transfer function of a closed loop system is
CONTROL SYSTEMS Chapter 5 : Root Locu Diagram GATE Objective & Numerical Type Solution Quetion 1 [Work Book] [GATE EC 199 IIScBangalore : Mark] The tranfer function of a cloed loop ytem i T () where i
More informationChapter 7. Root Locus Analysis
Chapter 7 Root Locu Analyi jw + KGH ( ) GH ( )  K 0 z O 4 p 2 p 3 p Root Locu Analyi The root of the cloedloop characteritic equation define the ytem characteritic repone. Their location in the complex
More informationECE317 : Feedback and Control
ECE317 : Feedback and Control Lecture : RouthHurwitz stability criterion Examples Dr. Richard Tymerski Dept. of Electrical and Computer Engineering Portland State University 1 Course roadmap Modeling
More informationand a where is a Vc. K val paran This ( suitab value
198 Chapter 5 RootLocus Method One classical technique in determining pole variations with parameters is known as the rootlocus method, invented by W. R. Evens, which will be introduced in this chapter.
More informationCISE302: Linear Control Systems
Term 8 CISE: Linear Control Sytem Dr. Samir AlAmer Chapter 7: Root locu CISE_ch 7 AlAmer8 ١ Learning Objective Undertand the concept of root locu and it role in control ytem deign Be able to ketch root
More informationIC6501 CONTROL SYSTEMS
DHANALAKSHMI COLLEGE OF ENGINEERING CHENNAI DEPARTMENT OF ELECTRICAL AND ELECTRONICS ENGINEERING YEAR/SEMESTER: II/IV IC6501 CONTROL SYSTEMS UNIT I SYSTEMS AND THEIR REPRESENTATION 1. What is the mathematical
More informationChapter 7. Digital Control Systems
Chapter 7 Digital Control Systems 1 1 Introduction In this chapter, we introduce analysis and design of stability, steadystate error, and transient response for computercontrolled systems. Transfer functions,
More informationControl Systems I. Lecture 7: Feedback and the Root Locus method. Readings: Guzzella 9.13, Emilio Frazzoli
Control Systems I Lecture 7: Feedback and the Root Locus method Readings: Guzzella 9.13, 13.3 Emilio Frazzoli Institute for Dynamic Systems and Control DMAVT ETH Zürich November 3, 2017 E. Frazzoli (ETH)
More informationMethods for analysis and control of. Lecture 4: The root locus design method
Methods for analysis and control of Lecture 4: The root locus design method O. Sename 1 1 Gipsalab, CNRSINPG, FRANCE Olivier.Sename@gipsalab.inpg.fr www.lag.ensieg.inpg.fr/sename Lead Lag 17th March
More informationME 375 Final Examination Thursday, May 7, 2015 SOLUTION
ME 375 Final Examination Thursday, May 7, 2015 SOLUTION POBLEM 1 (25%) negligible mass wheels negligible mass wheels v motor no slip ω r r F D O no slip e in Motor% Cart%with%motor%a,ached% The coupled
More informationEC6405  CONTROL SYSTEM ENGINEERING Questions and Answers Unit  I Control System Modeling Two marks 1. What is control system? A system consists of a number of components connected together to perform
More informationLecture Sketching the root locus
Lecture 05.02 Sketching the root locus It is easy to get lost in the detailed rules of manual root locus construction. In the old days accurate root locus construction was critical, but now it is useful
More informationPower System Operations and Control Prof. S.N. Singh Department of Electrical Engineering Indian Institute of Technology, Kanpur. Module 3 Lecture 8
Power System Operations and Control Prof. S.N. Singh Department of Electrical Engineering Indian Institute of Technology, Kanpur Module 3 Lecture 8 Welcome to lecture number 8 of module 3. In the previous
More informationUsing MATLB for stability analysis in Controls engineering Cyrus Hagigat Ph.D., PE College of Engineering University of Toledo, Toledo, Ohio
Using MATLB for stability analysis in Controls engineering Cyrus Hagigat Ph.D., PE College of Engineering University of Toledo, Toledo, Ohio Abstract Analyses of control systems require solution of differential
More informationDIGITAL CONTROL OF POWER CONVERTERS. 3 Digital controller design
DIGITAL CONTROL OF POWER CONVERTERS 3 Digital controller design Frequency response of discrete systems H(z) Properties: z e j T s 1 DC Gain z=1 H(1)=DC 2 Periodic nature j Ts z e jt e s cos( jt ) j sin(
More informationEEE 184: Introduction to feedback systems
EEE 84: Introduction to feedback systems Summary 6 8 8 x 7 7 6 Level() 6 5 4 4 5 5 time(s) 4 6 8 Time (seconds) Fig.. Illustration of BIBO stability: stable system (the input is a unit step) Fig.. step)
More information