Unit 7: Part 1: Sketching the Root Locus


 Augustus Willis
 3 years ago
 Views:
Transcription
1 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 Locus Techniques
2 Root Locus 1 Root Locus Vector Representation of Complex Numbers Defining the Root Locus Properties of the Root Locus Sketching the Root Locus ENGI 5821 Unit 7: Root Locus Techniques
3 Root Locus The Root Locus is a graphical method for depicting location of the closedloop poles as a system parameter is varied. It is applicable for first and secondorder systems, but also to higher order systems. Changes in a parameter, such as the gain K, affects the location of the equivalent closedloop system poles. Root locus allows us to determine the movement of these poles as K is varied.
4 Vector Representation of Complex Numbers We will need to represent complex numbers and complex functions F (s) as vectors. Typically the complex functions we are concerned with have the following form: F (s) = (s + z 1)(s + z 2 ) (s + p 1 )(s + p 2 ) Consider the complex number s + q, where q will stand in for either a zero or a pole. We can represent s + q as a vector in the complex plane. The magnitude and angle of this vector are given by the complex exponential representation, s + q = re jθ Where r is the vector length and θ is the angle from the realaxis.
5 Normally we would draw s + q as a vector out of the origin. Normal: s + q Complex number drawn from its own zero: s q s + q s q However, we can recognize s = q as a zero of s + q and draw the same vector with its tail at q (see above right).
6 F (s) = (s + z 1)(s + z 2 ) (s + p 1 )(s + p 2 ) We can replace each term s + z i or s + p i with their corresponding complex exponential forms: F (s) = r z 1 e θz 1 r z2 e θz 2 r p1 e θp 1 r p2 e θp 2 = r z 1 r z2 r p1 r p2 eθz 1 +θz 2 + θp 1 θp 2 The magnitude and phase of F (s) are as follows: F (s) = r z 1 r z2 r p1 r p2 = s + z 1 s + z 2 s + p 1 s + p 2 F (s) = θ z1 + θ z2 + θ p1 θ p2 = (s + z 1 ) + (s + z 2 ) + (s + p 1 ) (s + p 2 )
7 e.g. Evaluate the following complex function when s = 3 + j4, F (s) = (s + 1) s(s + 2) The magnitude and phase of the zero is: o. The pole at the origin evalutes to o. The pole at 2 evaluates to o. 20 F (s) F (s) = 5 17 [116.9o o o ]
8 Defining the Root Locus Consider the following system, designed to track a visual target. For a unity feedback system such as this we will refer to K 1K 2 s(s+10) as the openloop transfer function (if the unity feedback signal were cut the system would be openloop). Thus, the openloop poles are at s = 0 and s = 10. However, depending upon K we will obtain different closedloop poles, which are the roots of s s + K. We can utilize the quadratic formula to obtain these roots for values of K 0. We can then plot the positions of these poles...
9
10 The path of the closedloop poles as K varies is the root locus. Observations: for K < 25 the system is overdamped for K = 25 the system is critically damped for K > 25 the system is underdamped
11 More Observations: In the underdamped portion the realpart of the pole, σ d is constant. Therefore so is T s = 4/σ d. The root locus never crosses the jω axis. Therefore the system is stable.
12 Properties of the Root Locus The transfer function for a general closedloop system is, T (s) = KG(s) 1 + KG(s)H(s) We are concerned with the poles of T (s). We will have a pole whenever KG(s)H(s) = 1. The root locus is the locus of points in the splane for which this is true. We can express this equality as follows, KG(s)H(s) KG(s)H(s) = 1 (2k+1)180 o k = 0, ±1, ±2, ±3,... A particular s is on the root locus if its magnitude is unity and its angle is an odd multiple of 180. Satisfying both of these requirements means that s is a closedloop pole of the system.
13 e.g. Consider the following system, We will test a couple of points to see if they are closedloop poles. We evaluate KG(s)H(s) graphically for s = 2 + j3, θ 1 + θ 2 θ 3 θ 4 = = Therefore s = s + j3 is not on the root locus.
14 If we test a point and find it is on the RL (e.g. s = 2 + j 2/2) we may then want to determine the corresponding value of K. Since KG(s)H(s) = 1 on the RL, K = 1 G(s)H(s) = pole lengths zero lengths
15 Sketching the Root Locus One possibility is to sweep through a sampling of points in the splane and test each one for inclusion in the RL. It is much more preferable to utilize some insights about the RL to identify its major characteristics and therefore obtain a rough sketch. The following rules will help achieve this Number of branches: Consider a branch to be the path that one pole traverses. There will be one branch for every closedloop pole. e.g. Two branches for the previous quadratic example. 2. Symmetry: The poles for real physical systems either lie on the realaxis or come in conjugate pairs. Hence... The root locus is symmetrical about the real axis.
16 3. Realaxis segments: Consider evaluating the anglular contribution of the openloop poles and zeros at points P 1, P 2, P 3, and P 4 below: The angular contribution of a pair of complex openloop poles (or zeros) is zero The contribution of poles or zeros to the left of the point is zero Using only the realaxis poles or zeros to the right of the point, we find that the angle sum alternates between 0 o and 180 o. On the real axis, for K > 0 the root locus exists to the left of an odd number of realaxis, finite openloop poles and/or finite openloop zeros.
17 e.g. G(s) = K(s+3)(s+4) (s+1)(s+2), H(s) = 1
18 4. Starting and ending points: Where does the RL begin and end? It begins at K = 0 and ends at K =. Consider the closedloop transfer function: T (s) = = = KG(s)H(s) 1 + KG(s)H(s) K N G (s) N H (s) D G (s) D H (s) N H (s) D G (s) D H (s) 1 + K N G (s) KN G (s)n H (s) D G (s)d H (s) + KN G (s)n H (s) If we let K 0 the poles of T (s) approach the combined poles of G(s) and H(s). If K the poles of T (s) approach the combined zeros of G(s) and H(s). Thus, the RL begins at the openloop poles of G(s)H(s) and ends at the zeros of G(s)H(s).
19 e.g. G(s) = K(s+3)(s+4) (s+1)(s+2), H(s) = 1 Note that we don t know yet what the exact trajectory of the root locus will be.
20 What if the number of openloop poles and zeros is mismatched? e.g. K F (s) = s(s + 1)(s + 2) A function can have both poles and zeros at infinity. For example, as s. F (s) K s s s We therefore consider F (s) to have three zeros at infinity. If we include both finite and infinite poles and zeros every function has an equal number of poles and zeros. The root locus for F (s) (i.e. KG(s)H(s) = F (s)) would start at the three finite poles and go towards the zeros at infinity. Yet how do we get to these zeros at infinity?
21 5. Behaviour at infinity: The root locus approaches straight lines as asymptotes for zeros at infinity. These asymptotes are defined as lines with realaxis intercept σ a and angle θ a. σ a = θ a = finite poles finite zeros #finite poles #finite zeros (2k + 1)π #finite poles #finite zeros where k is an integer and θ a is the angle (in radians) to the positive realaxis. We get as many asymptotes as there are branches corresponding to zeros at infinity. The derivation for these formulae can be found at under Appendix L.1.
22 e.g. Sketch the RL for the following system: We first apply rules 14:
23 We now compute the realaxis intercept and the angles of all asymptotes: σ a = finite poles finite zeros #finite poles #finite zeros = ( ) ( 3) = 4/3 4 1 θ a = (2k + 1)π #finite poles #finite zeros = π/3 for k = 0 = π for k = 1 = 5π/3 for k = 2 Notice that we have one realaxis intercept but multiple angles. There are three asymptotes one for each infinite zero. We obtain three unique values for θ a before the angles start to repeat.
24 The following is our complete RL sketch.
Unit 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 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 informationSoftware 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 informationControl Systems Engineering ( Chapter 8. Root Locus Techniques ) Prof. KwangChun Ho Tel: Fax:
Control Systems Engineering ( Chapter 8. Root Locus Techniques ) Prof. KwangChun Ho kwangho@hansung.ac.kr Tel: 027604253 Fax:027604435 Introduction In this lesson, you will learn the following : 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 informationProblem Set 2: Solution Due on Wed. 25th Sept. Fall 2013
EE 561: Digital Control Systems Problem Set 2: Solution Due on Wed 25th Sept Fall 2013 Problem 1 Check the following for (internal) stability [Hint: Analyze the characteristic equation] (a) u k = 05u k
More informationSystems Analysis and Control
Systems Analysis and Control Matthew M. Peet Arizona State University Lecture 13: Root Locus Continued Overview In this Lecture, you will learn: Review Definition of Root Locus Points on the Real Axis
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 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 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 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 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 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 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 informationTime Response Analysis (Part II)
Time Response Analysis (Part II). A critically damped, continuoustime, second order system, when sampled, will have (in Z domain) (a) A simple pole (b) Double pole on real axis (c) Double pole on imaginary
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 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 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 information27. The pole diagram and the Laplace transform
124 27. The pole diagram and the Laplace transform When working with the Laplace transform, it is best to think of the variable s in F (s) as ranging over the complex numbers. In the first section below
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 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 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 informationControl Systems I. Lecture 9: The Nyquist condition
Control Systems I Lecture 9: The Nyquist condition Readings: Åstrom and Murray, Chapter 9.1 4 www.cds.caltech.edu/~murray/amwiki/index.php/first_edition Jacopo Tani Institute for Dynamic Systems and Control
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 informationDue Wednesday, February 6th EE/MFS 599 HW #5
Due Wednesday, February 6th EE/MFS 599 HW #5 You may use Matlab/Simulink wherever applicable. Consider the standard, unityfeedback closed loop control system shown below where G(s) = /[s q (s+)(s+9)]
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 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 informationK(s +2) s +20 K (s + 10)(s +1) 2. (c) KG(s) = K(s + 10)(s +1) (s + 100)(s +5) 3. Solution : (a) KG(s) = s +20 = K s s
321 16. Determine the range of K for which each of the following systems is stable by making a Bode plot for K = 1 and imagining the magnitude plot sliding up or down until instability results. Verify
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 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 information6.302 Feedback Systems Recitation 7: Root Locus Prof. Joel L. Dawson
To start with, let s mae sure we re clear on exactly what we mean by the words root locus plot. Webster can help us with this: ROOT: A number that reduces and equation to an identity when it is substituted
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 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 informationUnit 8: Part 2: PD, PID, and Feedback Compensation
Ideal Derivative Compensation (PD) Lead Compensation PID Controller Design Feedback Compensation Physical Realization of Compensation Unit 8: Part 2: PD, PID, and Feedback Compensation Engineering 5821:
More informationfor nonhomogeneous linear differential equations L y = f y H
Tues March 13: 5.45.5 Finish Monday's notes on 5.4, Then begin 5.5: Finding y P for nonhomogeneous linear differential equations (so that you can use the general solution y = y P y = y x in this section...
More informationSignals and Systems. Lecture 11 Wednesday 22 nd November 2017 DR TANIA STATHAKI
Signals and Systems Lecture 11 Wednesday 22 nd November 2017 DR TANIA STATHAKI READER (ASSOCIATE PROFFESOR) IN SIGNAL PROCESSING IMPERIAL COLLEGE LONDON Effect on poles and zeros on frequency response
More informationThe degree of a function is the highest exponent in the expression
L1 1.1 Power Functions Lesson MHF4U Jensen Things to Remember About Functions A relation is a function if for every xvalue there is only 1 corresponding yvalue. The graph of a relation represents a function
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 informationFrequency Response Analysis
Frequency Response Analysis Consider let the input be in the form Assume that the system is stable and the steady state response of the system to a sinusoidal inputdoes not depend on the initial conditions
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 informationTransient Response of a SecondOrder System
Transient Response of a SecondOrder System ECEN 830 Spring 01 1. Introduction In connection with this experiment, you are selecting the gains in your feedback loop to obtain a wellbehaved closedloop
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 informationControl of Manufacturing Processes
Control of Manufacturing Processes Subject 2.830 Spring 2004 Lecture #19 Position Control and Root Locus Analysis" April 22, 2004 The Position Servo Problem, reference position NC Control Robots Injection
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 informationRadar Dish. Armature controlled dc motor. Inside. θ r input. Outside. θ D output. θ m. Gearbox. Control Transmitter. Control. θ D.
Radar Dish ME 304 CONTROL SYSTEMS Mechanical Engineering Department, Middle East Technical University Armature controlled dc motor Outside θ D output Inside θ r input r θ m Gearbox Control Transmitter
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 informationControl Systems. Frequency Method Nyquist Analysis.
Frequency Method Nyquist Analysis chibum@seoultech.ac.kr Outline Polar plots Nyquist plots Factors of polar plots PolarNyquist Plots Polar plot: he locus of the magnitude of ω vs. the phase of ω on polar
More informationPositioning Servo Design Example
Positioning Servo Design Example 1 Goal. The goal in this design example is to design a control system that will be used in a pickandplace robot to move the link of a robot between two positions. Usually
More informationDelhi Noida Bhopal Hyderabad Jaipur Lucknow Indore Pune Bhubaneswar Kolkata Patna Web: Ph:
Serial : 0. LS_D_ECIN_Control Systems_30078 Delhi Noida Bhopal Hyderabad Jaipur Lucnow Indore Pune Bhubaneswar Kolata Patna Web: Email: info@madeeasy.in Ph: 04546 CLASS TEST 089 ELECTRONICS ENGINEERING
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 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 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 informationDigital Control: Part 2. ENGI 7825: Control Systems II Andrew Vardy
Digital Control: Part 2 ENGI 7825: Control Systems II Andrew Vardy Mapping the splane onto the zplane We re almost ready to design a controller for a DT system, however we will have to consider where
More informationSolution for Mechanical Measurement & Control
Solution for Mechanical Measurement & Control December2015 Index Q.1) a). 23 b).34 c). 5 d). 6 Q.2) a). 7 b). 7 to 9 c). 1011 Q.3) a). 1112 b). 1213 c). 13 Q.4) a). 1415 b). 15 (N.A.) Q.5) a). 15
More information2.004 Dynamics and Control II Spring 2008
MT OpenCourseWare http://ocw.mit.edu 2.004 Dynamics and Control Spring 2008 or information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. Reading: ise: Chapter 8 Massachusetts
More informationAutomatic Control Systems. Part III: Root Locus Technique
www.pdhcenter.com PDH Coure E40 www.pdhonline.org Automatic Control Sytem Part III: Root Locu Technique By ShihMin Hu, Ph.D., P.E. Page of 30 www.pdhcenter.com PDH Coure E40 www.pdhonline.org VI. Root
More informationTo get horizontal and slant asymptotes algebraically we need to know about end behaviour for rational functions.
Concepts: Horizontal Asymptotes, Vertical Asymptotes, Slant (Oblique) Asymptotes, Transforming Reciprocal Function, Sketching Rational Functions, Solving Inequalities using Sign Charts. Rational Function
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 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 informationAnalysis of SISO Control Loops
Chapter 5 Analysis of SISO Control Loops Topics to be covered For a given controller and plant connected in feedback we ask and answer the following questions: Is the loop stable? What are the sensitivities
More informationChapter 1 Polynomial Functions
Chapter 1 Polynomial Functions Lesson Package MHF4U Chapter 1 Outline Unit Goal: By the end of this unit, you will be able to identify and describe some key features of polynomial functions, and make
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 informationMAE 143B  Homework 9
MAE 43B  Homework 9 7.2 2 2 3.8.6.4.2.2 9 8 2 2 3 a) G(s) = (s+)(s+).4.6.8.2.2.4.6.8. Polar plot; red for negative ; no encirclements of, a.s. under unit feedback... 2 2 3. 4 9 2 2 3 h) G(s) = s+ s(s+)..2.4.6.8.2.4
More informationControl Systems I. Lecture 9: The Nyquist condition
Control Systems I Lecture 9: The Nyquist condition adings: Guzzella, Chapter 9.4 6 Åstrom and Murray, Chapter 9.1 4 www.cds.caltech.edu/~murray/amwiki/index.php/first_edition Emilio Frazzoli Institute
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 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 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 informationRoot Locus (2A) Young Won Lim 10/15/14
Root Locus (2A Copyright (c 2014 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version
More informationMEM 355 Performance Enhancement of Dynamical Systems
MEM 355 Performance Enhancement of Dynamical Systems Frequency Domain Design Harry G. Kwatny Department of Mechanical Engineering & Mechanics Drexel University 5/8/25 Outline Closed Loop Transfer Functions
More informationRoot Locus. 1 Review of related mathematics. Ang Man Shun. October 30, Complex Algebra in Polar Form. 1.2 Roots of a equation
Root Locus Ang Man Shun October 3, 212 1 Review of relate mathematics 1.1 Complex Algebra in Polar Form For a complex number z, it can be expresse in polar form as z = re jθ 1 Im z Where r = z, θ = tan.
More informationMCPS Algebra 2 and Precalculus Standards, Categories, and Indicators*
Content Standard 1.0 (HS) Patterns, Algebra and Functions Students will algebraically represent, model, analyze, and solve mathematical and realworld problems involving functional patterns and relationships.
More informationSystems Analysis and Control
Systems Analysis and Control Matthew M. Peet Illinois Institute of Technology Lecture 12: Overview In this Lecture, you will learn: Review of Feedback Closing the Loop Pole Locations Changing the Gain
More informationIntroduction to Rational Functions
Introduction to Rational Functions The net class of functions that we will investigate is the rational functions. We will eplore the following ideas: Definition of rational function. The basic (untransformed)
More informationReview: transient and steadystate response; DC gain and the FVT Today s topic: systemmodeling diagrams; prototype 2ndorder system
Plan of the Lecture Review: transient and steadystate response; DC gain and the FVT Today s topic: systemmodeling diagrams; prototype 2ndorder system Plan of the Lecture Review: transient and steadystate
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 informationChapter 1 Polynomial Functions
Chapter 1 Polynomial Functions Lesson Package MHF4U Chapter 1 Outline Unit Goal: By the end of this unit, you will be able to identify and describe some key features of polynomial functions, and make
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