# Course Outline. Closed Loop Stability. Stability. Amme 3500 : System Dynamics & Control. Nyquist Stability. Dr. Dunant Halim

Save this PDF as:
Size: px
Start display at page:

Download "Course Outline. Closed Loop Stability. Stability. Amme 3500 : System Dynamics & Control. Nyquist Stability. Dr. Dunant Halim"

## Transcription

1 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 4 26 Mar Block Diagrams 5 2 Apr Feedback System Characteristics Assign 1 BREAK 6 16 Apr Root Locus 7 23 Apr Root Locus Apr Bode Plots Assign May Bode Plots May Nyquist Assign May State Space Design Techniques May Advanced Control Topics 13 4 June Review Assign 4 14 Spare Dr. Dunant Halim Amme 3 : Nyquist Slide 2 Stability In this lecture we will examine the stability of closed loop systems in more detail We will examine the relationship between the open-loop frequency response and closed-loop stability Methods for estimating closed loop stability will be presented in the form of the Nyquist diagram Dr. Dunant Halim Amme 3 : Nyquist Slide 3 Closed Loop Stability In general, a closed-loop transfer function will take the form Gs ( T( s 1 + GsHs ( ( Note: negative feedback is considered here. Be careful with the feedback sign! Dr. Dunant Halim Amme 3 : Nyquist Slide 4

2 Closed Loop Stability We will often have analytical expressions for the components of the system G(s and H(s The poles of the closed loop system, or roots of 1+G(sH(s, are generally more difficult to find Closed Loop Stability Consider a system with The open-loop transfer function is and NG ( s NH ( s G( s and H ( s D ( s D ( s NG( snh( s GsHs ( ( D ( s D ( s G G DG( s DH( s + NG( s NH( s 1 + GsHs ( ( D ( s D ( s H G H H Dr. Dunant Halim Amme 3 : Nyquist Slide 5 Dr. Dunant Halim Amme 3 : Nyquist Slide 6 Closed Loop Stability The closed-loop transfer function will be Gs ( NG( s DH( s T( s 1 + GsHs ( ( D( sd ( s + N ( sn ( s G H G H This leads us to conclude that The poles of 1+G(sH(s are the same as the poles of G(sH(s The zeros of 1+G(sH(s are the same as the poles of T(s Closed Loop Stability We would like to find a method for evaluating the closed-loop stability based on the location of open-loop poles and zeros The root locus provides us with one approach for estimating the stability of the system The Bode plots examined in the last lecture provide another means for evaluating stability For some system types, however, the frequency response may have multiple cross-over frequencies. This may make the rules for evaluating stability ambiguous Dr. Dunant Halim Amme 3 : Nyquist Slide 7 Dr. Dunant Halim Amme 3 : Nyquist Slide 8

3 Contour Mapping We will introduce a concept called contour mapping Given a series points on a contour A in the s- plane, the function F(s will map these points to another contour B Contour Mapping For example, if we have a function F(s of the form 2 Fs ( s + 2s+ 1 Then a point s 4+j3 will yield a mapping to the complex variable Fs j j 16 + j3 2 ( ( ( F(s s α.5 F(s Dr. Dunant Halim Amme 3 : Nyquist Slide 9 Dr. Dunant Halim Amme 3 : Nyquist Slide 1 For poles or zeros outside of the contour, the angle will not undergo a net change of 36 o This will result in a contour that does not encircle the origin Contour Mapping Dr. Dunant Halim Amme 3 : Nyquist Slide 11 Assuming clockwise contour A, for contours containing poles and/or zeros, the resulting contour will encircle the origin NP-Z times in the counter clockwise direction (when N> Contour Mapping P,Z1,N-1 P1,Z,N1 P1,Z1,N Dr. Dunant Halim Amme 3 : Nyquist Slide 12

4 Contour Mapping We can apply this result to find the number of poles of the closed loop system in the RHP This will allow us to determine the stability of the closed-loop system P1,Z2, so N1-2-1 We extend the contour to enclose the entire RHP Applying the contour mapping techniques we can determine if any closed loop poles exist in the RHP by examining the encirclements of the origin for 1+KG(sH(s Im(s Contour at infinity Re(s Dr. Dunant Halim Amme 3 : Nyquist Slide 13 Dr. Dunant Halim Amme 3 : Nyquist Slide 14 P, Z so N No encirclement around -1 P, Z2 so N clockwise encirclements around -1 Dr. Dunant Halim Amme 3 : Nyquist Slide 15 This is equivalent to finding the zeros of 1+KG(sH(s (Remember: they are also closed loop poles! To simplify this procedure, we can map through KG(sH(s since the poles and zeros of this quantity are often known The resulting contour is the same as mapping through 1+KG(sH(s but translated one unit to the left Any clockwise encirclement of the point -1 indicates a closed loop pole in RHP, i.e. unstable closed loop! Dr. Dunant Halim Amme 3 : Nyquist Slide 16

5 Effectively we are observing the change in magnitude and phase as a function of increasing frequency by varying s along the jω axis The resulting polar plot represents this change in a slightly different fashion to that of the Bode plot Sketching the Nyquist plot consists of: Plot KG(sH(s for -j s j starting with KG(jωH(jω for ω The magnitude (and hence radius from the origin will be small at high frequency for any physical system Reflect the plot about the real axis as the plot will be symmetric Dr. Dunant Halim Amme 3 : Nyquist Slide 17 Dr. Dunant Halim Amme 3 : Nyquist Slide 18 Once the plot is complete Evaluate the number of counter clockwise encirclements, N, of -1 Determine the number, P, of unstable poles of G(s or G(sH(s, i.e. the open loop system The number of unstable closed-loop roots Z will be given by Z P - N Example 1 Dr. Dunant Halim Amme 3 : Nyquist Slide 19 Dr. Dunant Halim Amme 3 : Nyquist Slide 2

6 Example 1 For this system we find Gs ( ( s+ 1( s+ 3( s+ 1 The closed-loop transfer function is T( s ( s+ 1( s+ 3( s+ 1 + It is not immediately clear if this is stable Dr. Dunant Halim Amme 3 : Nyquist Slide 21 Example 1 We can consider the evaluation of the contour as complex arithmetic using the vectors of G(s drawn to the points along the contour We consider sections AC and CD. Note: Analysis for section AD is not required since the Nyquist plot is symmetrical about the real axis Dr. Dunant Halim Amme 3 : Nyquist Slide 22 Example 1: Mapping of AC For s along the imaginary axis G( jω ( s+ 1( s+ 3( s+ 1 s jω ω + + j ω ω 2 3 ( 14 3 (43 Multiplying by the complex conjugate of the denominator yields 2 3 ( 14ω + 3 j(43 ω ω G( jω ( 14 ω (43 ω ω Example 1: Mapping of AC We can evaluate this as a function of ω 2 3 ( 14ω + 3 j(43 ω ω G( jω ( ( ω + + ω ω For zero frequency G(jω /3 5/3 (Point A As ω increases, the real part remains + ve and the imaginary part remains - ve At ωsqrt(3/14 the real part becomes ve, the imaginary part remains - ve At ωsqrt(43 the plot crosses the ve real axis, the imaginary part becomes + ve. Substitute this ω value to give G(jω-.874 At infinite frequency G(jω is close to j/ω 3 (approximately zero (Point C Dr. Dunant Halim Amme 3 : Nyquist Slide 23 Dr. Dunant Halim Amme 3 : Nyquist Slide 24

7 Example 1: Mapping of CD We can also obtain G(s via the contribution of each complex number in polar form: G( s ( R e 1 jθ 1 jθ 3 ( R e ( R ( θ + θ + θ e jθ 1 ( θ 1 + θ 3 + θ 1 Fortunately, Matlab allows us to generate these plots systf(,[1,1]*tf(1,[1,3]* tf(1,[1,1] nyquist(sys Example 1 jθ i where R i e ( s + i consisting of the magnitude and angle of the complex number At point C, all angles are 9 o o, so point C is 27 At point D, all angles are -9 o o, so point D is 27 Dr. Dunant Halim Amme 3 : Nyquist Slide 25 Dr. Dunant Halim Amme 3 : Nyquist Slide 26 Example 1 Since all open loop poles of -1, -3, -1 are stable (i.e. not in RHP, we can conclude that this system is stable as there is no encirclement of -1 That is: P, ZP-N-. No unstable closed loop poles. Example 2: What if there are open loop poles along the imaginary axis? To avoid undetermined solution in the mapping, we have to make an infinitesimally small detour around the poles Dr. Dunant Halim Amme 3 : Nyquist Slide 27 Dr. Dunant Halim Amme 3 : Nyquist Slide 28

8 * N.S. Nise (24 Control Systems Engineering Wiley & Sons Example 2: System: ( s + 2 G( s 2 s We consider sections AB, BCD and EFA, being mapped into A B, B C D and E F A An infinitesimally small detour is made around 2 poles at the origin Dr. Dunant Halim Amme 3 : Nyquist Slide 29 Ex 2: Mapping of AB & BCD Section AB is along the + ve imaginary axis jω with ω from to infinity. Thus, substitute s jω into G(s: ( jω G( jω 2 ω 9 For section BCD: G( s Very low frequency, point A Very high frequency, point B 9 9 ( 9 ( 9 R 2 θ 2 ( R θ ( R θ ( ( 9 9 ( 9 ( 9 Dr. Dunant Halim Amme 3 : Nyquist Slide 3 Point B Point C Point D Ex 2: Mapping of EFA Mapping of section DE is trivial because it is a mirror image of the section AB. For section EFA (Note: ε is an infinitesimal positive number: 2 18 Point E ( ε 9 ( ε 9 R 2 θ 2 2 G( s Point F ( R θ ( R θ ( ε ( ε 2 18 Point A ( ε 9 ( ε 9 Note that point A has been calculated before using a different approach. Point E does not need to be calculated again since it is a mirror image of point A with respect to real axis. Range of gain for Stability What if we had a control parameter for the previous system? Can the Nyquist plot tell us anything about the range of gains for which the system is stable? K Dr. Dunant Halim Amme 3 : Nyquist Slide 31 Dr. Dunant Halim Amme 3 : Nyquist Slide 32

9 Stability Range Stability Range Changes in gain will result in a corresponding change in the Nyquist plot K1 K2 ZP-N- for small gain Stable closed loop ZP-N-(-22 for large gain Unstable closed loop K.5 ZP-N2-2 for small gain Unstable closed loop ZP-N2-2 for small gain Stable closed loop Dr. Dunant Halim Amme 3 : Nyquist Slide 33 Dr. Dunant Halim Amme 3 : Nyquist Slide 34 Gain and Phase Margins As we saw, the Nyquist plot will change as a function of the gain parameter In the lecture on Bode plots we also saw that the Bode magnitude will rise or lower based on the system gain K We defined the gain margin as the factor by which the gain can be raised before instability results The phase margin is the change in phase shift at unity gain that will result in instability Gain and Phase Margin The gain and phase margins can also be read from the Nyquist plot * N.S. Nise (24 Control Systems Engineering Wiley & Sons Dr. Dunant Halim Amme 3 : Nyquist Slide 35 Dr. Dunant Halim Amme 3 : Nyquist Slide 36

10 Nyquist and Bode It is apparent from the previous development that there is a strong relationship between the Nyquist and Bode plots The Nyquist plot is effectively a polar plot of the magnitude and phase with frequency encoded along the path ω6.5 ω1.5 Nyquist and Bode ω.6 ω Dr. Dunant Halim Amme 3 : Nyquist Slide 37 Dr. Dunant Halim Amme 3 : Nyquist Slide 38 Nyquist and Bode For simple, LTI systems there is little difference between the Nyquist and Bode plots for estimating the stability of the system For more complex systems, such as multiinput/multi-output systems we can t use the Bode plot but may be able to use Nyquist to determine stability These techniques are important in more advanced control studies Conclusions We have looked at another method of determining the stability of a closed loop system based on its open loop characteristics We have presented rules for sketching the Nyquist plot given the open loop transfer function Dr. Dunant Halim Amme 3 : Nyquist Slide 39 Dr. Dunant Halim Amme 3 : Nyquist Slide 4

11 Nise Sections Franklin & Powell Section Further Reading Dr. Dunant Halim Amme 3 : Nyquist Slide 41

### Systems Analysis and Control

Systems Analysis and Control Matthew M. Peet Illinois Institute of Technology Lecture 23: Drawing The Nyquist Plot Overview In this Lecture, you will learn: Review of Nyquist Drawing the Nyquist Plot Using

### The Nyquist criterion relates the stability of a closed system to the open-loop frequency response and open loop pole location.

Introduction to the Nyquist criterion The Nyquist criterion relates the stability of a closed system to the open-loop frequency response and open loop pole location. Mapping. If we take a complex number

### Systems Analysis and Control

Systems Analysis and Control Matthew M. Peet Arizona State University Lecture 23: Drawing The Nyquist Plot Overview In this Lecture, you will learn: Review of Nyquist Drawing the Nyquist Plot Using the

### Linear Control Systems Lecture #3 - Frequency Domain Analysis. Guillaume Drion Academic year

Linear Control Systems Lecture #3 - Frequency Domain Analysis Guillaume Drion Academic year 2018-2019 1 Goal and Outline Goal: To be able to analyze the stability and robustness of a closed-loop system

### Control 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

### Control 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

### K(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

### Nyquist Stability Criteria

Nyquist Stability Criteria Dr. Bishakh Bhattacharya h Professor, Department of Mechanical Engineering IIT Kanpur Joint Initiative of IITs and IISc - Funded by MHRD This Lecture Contains Introduction to

### Lecture 1 Root Locus

Root Locus ELEC304-Alper Erdogan 1 1 Lecture 1 Root Locus What is Root-Locus? : A graphical representation of closed loop poles as a system parameter varied. Based on Root-Locus graph we can choose the

### 1 (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

### Control 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

### 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

### Introduction 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

### Lecture 15 Nyquist Criterion and Diagram

Lecture Notes of Control Systems I - ME 41/Analysis and Synthesis of Linear Control System - ME86 Lecture 15 Nyquist Criterion and Diagram Department of Mechanical Engineering, University Of Saskatchewan,

### Dr Ian R. Manchester Dr Ian R. Manchester AMME 3500 : Review

Week Date Content Notes 1 6 Mar Introduction 2 13 Mar Frequency Domain Modelling 3 20 Mar Transient Performance and the s-plane 4 27 Mar Block Diagrams Assign 1 Due 5 3 Apr Feedback System Characteristics

### 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

### CONTROL SYSTEM STABILITY. CHARACTERISTIC EQUATION: The overall transfer function for a. where A B X Y are polynomials. Substitution into the TF gives:

CONTROL SYSTEM STABILITY CHARACTERISTIC EQUATION: The overall transfer function for a feedback control system is: TF = G / [1+GH]. The G and H functions can be put into the form: G(S) = A(S) / B(S) H(S)

### Frequency 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

### Nyquist Criterion For Stability of Closed Loop System

Nyquist Criterion For Stability of Closed Loop System Prof. N. Puri ECE Department, Rutgers University Nyquist Theorem Given a closed loop system: r(t) + KG(s) = K N(s) c(t) H(s) = KG(s) +KG(s) = KN(s)

### Topic # Feedback Control

Topic #4 16.31 Feedback Control Stability in the Frequency Domain Nyquist Stability Theorem Examples Appendix (details) This is the basis of future robustness tests. Fall 2007 16.31 4 2 Frequency Stability

### CHAPTER 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.

### Systems Analysis and Control

Systems Analysis and Control Matthew M. Peet Illinois Institute of Technology Lecture 22: The Nyquist Criterion Overview In this Lecture, you will learn: Complex Analysis The Argument Principle The Contour

### Dr 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 s-plane 4 Block Diagrams 5 Feedback System Characteristics Assign 1 Due 6 Root Locus 7 Root Locus 2 Assign

### ROOT LOCUS. Consider the system. Root locus presents the poles of the closed-loop 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 closed-loop system when the gain K changes from 0 to 1+ K G ( s)

### Unit 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

### I 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

### 1 (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

### r + - FINAL June 12, 2012 MAE 143B Linear Control Prof. M. Krstic

MAE 43B Linear Control Prof. M. Krstic FINAL June, One sheet of hand-written notes (two pages). Present your reasoning and calculations clearly. Inconsistent etchings will not be graded. Write answers

### MAS107 Control Theory Exam Solutions 2008

MAS07 CONTROL THEORY. HOVLAND: EXAM SOLUTION 2008 MAS07 Control Theory Exam Solutions 2008 Geir Hovland, Mechatronics Group, Grimstad, Norway June 30, 2008 C. Repeat question B, but plot the phase curve

### Lecture 6 Classical Control Overview IV. Dr. Radhakant Padhi Asst. Professor Dept. of Aerospace Engineering Indian Institute of Science - Bangalore

Lecture 6 Classical Control Overview IV Dr. Radhakant Padhi Asst. Professor Dept. of Aerospace Engineering Indian Institute of Science - Bangalore Lead Lag Compensator Design Dr. Radhakant Padhi Asst.

### DESIGN USING TRANSFORMATION TECHNIQUE CLASSICAL METHOD

206 Spring Semester ELEC733 Digital Control System LECTURE 7: DESIGN USING TRANSFORMATION TECHNIQUE CLASSICAL METHOD For a unit ramp input Tz Ez ( ) 2 ( z ) D( z) G( z) Tz e( ) lim( z) z 2 ( z ) D( z)

### ECE 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

### H(s) = s. a 2. H eq (z) = z z. G(s) a 2. G(s) A B. s 2 s(s + a) 2 s(s a) G(s) 1 a 1 a. } = (z s 1)( z. e ) ) (z. (z 1)(z e at )(z e at )

.7 Quiz Solutions Problem : a H(s) = s a a) Calculate the zero order hold equivalent H eq (z). H eq (z) = z z G(s) Z{ } s G(s) a Z{ } = Z{ s s(s a ) } G(s) A B Z{ } = Z{ + } s s(s + a) s(s a) G(s) a a

### Class 13 Frequency domain analysis

Class 13 Frequency domain analysis The frequency response is the output of the system in steady state when the input of the system is sinusoidal Methods of system analysis by the frequency response, as

### MEM 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

### Control Systems Engineering ( Chapter 8. Root Locus Techniques ) Prof. Kwang-Chun Ho Tel: Fax:

Control Systems Engineering ( Chapter 8. Root Locus Techniques ) Prof. Kwang-Chun Ho kwangho@hansung.ac.kr Tel: 02-760-4253 Fax:02-760-4435 Introduction In this lesson, you will learn the following : The

### Frequency methods for the analysis of feedback systems. Lecture 6. Loop analysis of feedback systems. Nyquist approach to study stability

Lecture 6. Loop analysis of feedback systems 1. Motivation 2. Graphical representation of frequency response: Bode and Nyquist curves 3. Nyquist stability theorem 4. Stability margins Frequency methods

### Course Outline. Higher Order Poles: Example. Higher Order Poles. Amme 3500 : System Dynamics & Control. State Space Design. 1 G(s) = s(s + 2)(s +10)

Amme 35 : System Dynamics Control State Space Design Course Outline Week Date Content Assignment Notes 1 1 Mar Introduction 2 8 Mar Frequency Domain Modelling 3 15 Mar Transient Performance and the s-plane

### FREQUENCY RESPONSE ANALYSIS Closed Loop Frequency Response

Closed Loop Frequency Response The Bode plot is generally constructed for an open loop transfer function of a system. In order to draw the Bode plot for a closed loop system, the transfer function has

### Intro to Frequency Domain Design

Intro to Frequency Domain Design MEM 355 Performance Enhancement of Dynamical Systems Harry G. Kwatny Department of Mechanical Engineering & Mechanics Drexel University Outline Closed Loop Transfer Functions

### FEL3210 Multivariable Feedback Control

FEL3210 Multivariable Feedback Control Lecture 5: Uncertainty and Robustness in SISO Systems [Ch.7-(8)] Elling W. Jacobsen, Automatic Control Lab, KTH Lecture 5:Uncertainty and Robustness () FEL3210 MIMO

### STABILITY ANALYSIS. Asystemmaybe stable, neutrallyormarginallystable, or unstable. This can be illustrated using cones: Stable Neutral Unstable

ECE4510/5510: Feedback Control Systems. 5 1 STABILITY ANALYSIS 5.1: Bounded-input bounded-output (BIBO) stability Asystemmaybe stable, neutrallyormarginallystable, or unstable. This can be illustrated

### Homework 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

### Digital Control Systems

Digital Control Systems Lecture Summary #4 This summary discussed some graphical methods their use to determine the stability the stability margins of closed loop systems. A. Nyquist criterion Nyquist

### a. Closed-loop 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

### CHAPTER # 9 ROOT LOCUS ANALYSES

F K א CHAPTER # 9 ROOT LOCUS ANALYSES 1. Introduction The basic characteristic of the transient response of a closed-loop system is closely related to the location of the closed-loop poles. If the system

### Robust Control 3 The Closed Loop

Robust Control 3 The Closed Loop Harry G. Kwatny Department of Mechanical Engineering & Mechanics Drexel University /2/2002 Outline Closed Loop Transfer Functions Traditional Performance Measures Time

### ECE 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

### Analysis 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

### MASSACHUSETTS 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:

### Lecture 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

### The Frequency-Response

6 The Frequency-Response Design Method A Perspective on the Frequency-Response Design Method The design of feedback control systems in industry is probably accomplished using frequency-response methods

### School 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

### Module 07 Control Systems Design & Analysis via Root-Locus Method

Module 07 Control Systems Design & Analysis via Root-Locus Method Ahmad F. Taha EE 3413: Analysis and Desgin of Control Systems Email: ahmad.taha@utsa.edu Webpage: http://engineering.utsa.edu/ taha March

### The Frequency-response Design Method

Chapter 6 The Frequency-response Design Method Problems and Solutions for Section 6.. (a) Show that α 0 in Eq. (6.2) is given by α 0 = G(s) U 0ω = U 0 G( jω) s jω s= jω 2j and α 0 = G(s) U 0ω = U 0 G(jω)

### x(t) = x(t h), x(t) 2 R ), where is the time delay, the transfer function for such a e s Figure 1: Simple Time Delay Block Diagram e i! =1 \e i!t =!

1 Time-Delay Systems 1.1 Introduction Recitation Notes: Time Delays and Nyquist Plots Review In control systems a challenging area is operating in the presence of delays. Delays can be attributed to acquiring

### IC6501 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

### Course 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: Steady-State Error Unit 7: Root Locus Techniques

### Lecture 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

### MEM 355 Performance Enhancement of Dynamical Systems

MEM 355 Performance Enhancement of Dynamical Systems Frequency Domain Design Intro Harry G. Kwatny Department of Mechanical Engineering & Mechanics Drexel University /5/27 Outline Closed Loop Transfer

### Root 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

### Software 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

### Analysis of Discrete-Time Systems

TU Berlin Discrete-Time Control Systems 1 Analysis of Discrete-Time Systems Overview Stability Sensitivity and Robustness Controllability, Reachability, Observability, and Detectabiliy TU Berlin Discrete-Time

### Definition of Stability

Definition of Stability Transfer function of a linear time-invariant (LTI) system Fs () = b 2 1 0+ b1s+ b2s + + b m m m 1s - - + bms a0 + a1s+ a2s2 + + an-1sn- 1+ ansn Characteristic equation and poles

### Root 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

### Dr Ian R. Manchester

Week Content Notes 1 Introduction 2 Frequency Domain Modelling 3 Transient Performance and the s-plane 4 Block Diagrams 5 Feedback System Characteristics Assign 1 Due 6 Root Locus 7 Root Locus 2 Assign

### Control Systems I. Lecture 6: Poles and Zeros. Readings: Emilio Frazzoli. Institute for Dynamic Systems and Control D-MAVT ETH Zürich

Control Systems I Lecture 6: Poles and Zeros Readings: Emilio Frazzoli Institute for Dynamic Systems and Control D-MAVT ETH Zürich October 27, 2017 E. Frazzoli (ETH) Lecture 6: Control Systems I 27/10/2017

### Control Systems I Lecture 10: System Specifications

Control Systems I Lecture 10: System Specifications Readings: Guzzella, Chapter 10 Emilio Frazzoli Institute for Dynamic Systems and Control D-MAVT ETH Zürich November 24, 2017 E. Frazzoli (ETH) Lecture

### Control 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

### Discrete Systems. Step response and pole locations. Mark Cannon. Hilary Term Lecture

Discrete Systems Mark Cannon Hilary Term 22 - Lecture 4 Step response and pole locations 4 - Review Definition of -transform: U() = Z{u k } = u k k k= Discrete transfer function: Y () U() = G() = Z{g k},

### The Nyquist Stability Test

Handout X: EE24 Fall 2002 The Nyquist Stability Test.0 Introduction With negative feedback, the closed-loop transfer function A(s) approaches the reciprocal of the feedback gain, f, as the magnitude of

### Radar 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

### Root 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

### Analysis of Discrete-Time Systems

TU Berlin Discrete-Time Control Systems TU Berlin Discrete-Time Control Systems 2 Stability Definitions We define stability first with respect to changes in the initial conditions Analysis of Discrete-Time

### 2.004 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

### CONTROL * ~ SYSTEMS ENGINEERING

CONTROL * ~ SYSTEMS ENGINEERING H Fourth Edition NormanS. Nise California State Polytechnic University, Pomona JOHN WILEY& SONS, INC. Contents 1. Introduction 1 1.1 Introduction, 2 1.2 A History of Control

### Stability of Feedback Control Systems: Absolute and Relative

Stability of Feedback Control Systems: Absolute and Relative Dr. Kevin Craig Greenheck Chair in Engineering Design & Professor of Mechanical Engineering Marquette University Stability: Absolute and Relative

### AA/EE/ME 548: Problem Session Notes #5

AA/EE/ME 548: Problem Session Notes #5 Review of Nyquist and Bode Plots. Nyquist Stability Criterion. LQG/LTR Method Tuesday, March 2, 203 Outline:. A review of Bode plots. 2. A review of Nyquist plots

### AMME3500: System Dynamics & Control

Stefan B. Williams May, 211 AMME35: System Dynamics & Control Assignment 4 Note: This assignment contributes 15% towards your final mark. This assignment is due at 4pm on Monday, May 3 th during Week 13

### Classify a transfer function to see which order or ramp it can follow and with which expected error.

Dr. J. Tani, Prof. Dr. E. Frazzoli 5-059-00 Control Systems I (Autumn 208) Exercise Set 0 Topic: Specifications for Feedback Systems Discussion: 30.. 208 Learning objectives: The student can grizzi@ethz.ch,

### ESE319 Introduction to Microelectronics. Feedback Basics

Feedback Basics Stability Feedback concept Feedback in emitter follower One-pole feedback and root locus Frequency dependent feedback and root locus Gain and phase margins Conditions for closed loop stability

### FREQUENCY-RESPONSE ANALYSIS

ECE450/550: Feedback Control Systems. 8 FREQUENCY-RESPONSE ANALYSIS 8.: Motivation to study frequency-response methods Advantages and disadvantages to root-locus design approach: ADVANTAGES: Good indicator

### MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Mechanical Engineering Dynamics and Control II Fall 2007

MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Mechanical Engineering.4 Dynamics and Control II Fall 7 Problem Set #9 Solution Posted: Sunday, Dec., 7. The.4 Tower system. The system parameters are

### Solutions to Skill-Assessment Exercises

Solutions to Skill-Assessment Exercises To Accompany Control Systems Engineering 4 th Edition By Norman S. Nise John Wiley & Sons Copyright 2004 by John Wiley & Sons, Inc. All rights reserved. No part

### Unit 11 - Week 7: Quantitative feedback theory (Part 1/2)

X reviewer3@nptel.iitm.ac.in Courses» Control System Design Announcements Course Ask a Question Progress Mentor FAQ Unit 11 - Week 7: Quantitative feedback theory (Part 1/2) Course outline How to access

### Introduction. Performance and Robustness (Chapter 1) Advanced Control Systems Spring / 31

Introduction Classical Control Robust Control u(t) y(t) G u(t) G + y(t) G : nominal model G = G + : plant uncertainty Uncertainty sources : Structured : parametric uncertainty, multimodel uncertainty Unstructured

### Richiami di Controlli Automatici

Richiami di Controlli Automatici Gianmaria De Tommasi 1 1 Università degli Studi di Napoli Federico II detommas@unina.it Ottobre 2012 Corsi AnsaldoBreda G. De Tommasi (UNINA) Richiami di Controlli Automatici

### FATIMA MICHAEL COLLEGE OF ENGINEERING & TECHNOLOGY

FATIMA MICHAEL COLLEGE OF ENGINEERING & TECHNOLOGY Senkottai Village, Madurai Sivagangai Main Road, Madurai - 625 020. An ISO 9001:2008 Certified Institution DEPARTMENT OF ELECTRONICS AND COMMUNICATION

### 7.4 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)

### Design of a Lead Compensator

Design of a Lead Compensator Dr. Bishakh Bhattacharya Professor, Department of Mechanical Engineering IIT Kanpur Joint Initiative of IITs and IISc - Funded by MHRD The Lecture Contains Standard Forms of

### Dr. Ian R. Manchester

Dr Ian R. Manchester Week Content Notes 1 Introduction 2 Frequency Domain Modelling 3 Transient Performance and the s-plane 4 Block Diagrams 5 Feedback System Characteristics Assign 1 Due 6 Root Locus

### Nyquist Plots / Nyquist Stability Criterion

Nyquist Plots / Nyquist Stability Criterion Given Nyquist plot is a polar plot for vs using the Nyquist contour (K=1 is assumed) Applying the Nyquist criterion to the Nyquist plot we can determine the

### Stability & Compensation

Advanced Analog Building Blocks Stability & Compensation Wei SHEN (KIP) 1 Bode Plot real zeros zeros with complex conjugates real poles poles with complex conjugates http://lpsa.swarthmore.edu/bode/bode.html

### The Nyquist Feedback Stability Criterion

ECE137B notes; copyright 2018 The Nyquist Feedback Stability Criterion Mark Rodwell, University of California, Santa Barbara Feedback loop stability A () s AOL ( s) AOL ( s) 1 A ( s) ( s) 1 T ( s) Ns ()

### Homework 6 Solutions and Rubric

Homework 6 Solutions and Rubric EE 140/40A 1. K-W Tube Amplifier b) Load Resistor e) Common-cathode a) Input Diff Pair f) Cathode-Follower h) Positive Feedback c) Tail Resistor g) Cc d) Av,cm = 1/ Figure

### INTRODUCTION 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 linear-time-invariant

### STABILITY ANALYSIS TECHNIQUES

ECE4540/5540: Digital Control Systems 4 1 STABILITY ANALYSIS TECHNIQUES 41: Bilinear transformation Three main aspects to control-system design: 1 Stability, 2 Steady-state response, 3 Transient response

### ECE317 : Feedback and Control

ECE317 : Feedback and Control Lecture : Routh-Hurwitz stability criterion Examples Dr. Richard Tymerski Dept. of Electrical and Computer Engineering Portland State University 1 Course roadmap Modeling