Robust Control 3 The Closed Loop


 Laureen Haynes
 9 months ago
 Views:
Transcription
1 Robust Control 3 The Closed Loop Harry G. Kwatny Department of Mechanical Engineering & Mechanics Drexel University /2/2002
2 Outline Closed Loop Transfer Functions Traditional Performance Measures Time domain Frequency domain Robustness Traditional gain/phase margins Multivariable Nyquist
3 Closed Loop Transfer Functons d r  K u G y d 2 ( ) ( ) Y = G U + D U = K R Y D E = R Y 2
4 Transfer Functions Con t Output [ ] [ ] [ I GsKs () ()] GsKsD () () () s [ ] [ ] [ I GsKs () ()] GsKsD () () () s [ ] [ ] Ys () = I+ GsKs () () GsKsRs () () () I+ GsKs () () GsD () () s Error Es () = I+ GsKs () () Rs () I+ GsKs () () GsD () () s Control U() s = K() s I + G() s K() s R() s K() s I + G() s K() s G() s D () s Ks () I+ GsKs () () D() s [ ]
5 Sensitivity Functions [ ] [ ] [ ] E() s = I + L R() s I + L GD () s + I + L LD () s where L: = GK sensitivity function: [ ] S: = I + L 2 { [ L] L [ L] } complementary sensitivity function: For SISO systems Bode derived: dt T dt L = dl L dl T = = [+ L] L + = [ + L] = S L [ + L] 2 [ ] T : = I + L L L
6 Traditional Performance ~ Time Domain rise time, T r, usually defined as the time to get from 0% to 90% of its ultimate (i.e., final) value. settling time, T s, the time at which the trajectory first enters an εtolerance of its ultimate value and remains there (ε is often taken as 2% of the ultimate value). peak time, T p, the time at which the trajectory attains its peak value. peak overshoot, OS, the peak or supreme value of the trajectory ordinarily expressed as a percentage of the ultimate value of the trajectory. An overshoot of more than 30% is often considered undesirable. A system without overshoot is overdamped and may be too slow (as measured by rise time and settling time).
7 Traditional Performance ~ Time Domain Cont d y p.4.2 2ε t T r T p T s
8 Traditional Performance ~ Frequency Domain db deg MAGNITUDE rad êsec PHASE rad êsec Complementary sensitivity 2 s +.5s+ 0.5 K s db deg MAGNITUDE rad êsec ( 20s+ )( 0s+ )( 0.5s+ ) PHASE rad êsec Sensitivity V22 Osprey altitude control K=280
9 Traditional Performance ~ Frequency Domain Sensitivity Function (first crosses / 2=0.707~3db from below): { v S j v } ω = max : ( ω) < 2 ω [0, ) BS v Complementary Sensitivity Function (highest frequency where T crosses 2 from above) { v T j v } ω = min : ( ω) < 2 ω (, ) BT v Crossover frequency { v L j v } ω = max : ( ω) ω [0, ) c v
10 Traditional Performance ~ Frequency Domain Sensitivity peaks are related to gain and phase margin. Sensitivity peaks are related to overshoot and damping ratio. Im L plane M M S T = max S( jω ) ω = max T ( jω ) ω S > S <  a Re S = a L( jω) S = + L
11 A Fundamental Tradeoff Note that Usefull identities: [ ] [ ] I + L + I + L L= I S + T = I [ ] [ ] T = I + L L= I + L = L I + L [ + ] = [ + ] [ + ] = [ + ] [ + ] = [ + ] = [ + ] = ( ) + 2 = ( ) ( ) ( ) ( ) G I KG I GK G K I GK I KG K GK I GK G I KG K I GK GK Es () SsRs () () SsGsD () () s TsD () () s { 2 } U() s K() s S s R() s G s D s D s
12 Bode Waterbed Formula systems with relative degree 2 or greater: (Waterbed effect) 0 0 ln S( jω) dω = π p ln T( jω) dω = π 2 ω ORHP poles ORHP zeros i q i
13 Cauchy Theorem Theorem (Cauchy): Let C be a simple closed curve in the splane. F s is a rational function, having neither poles nor zeros on C. If C is the ( ) image of C under the map F s, then Z = P N where N the number of counterclockwise encirclements of the origin by C as s traverses C once in the clockwise direction. ( ) ( ) Z the number of zeros of F s enclosed by C, counting multiplicities. P the number of poles of F s enclosed by C, counting multiplicities. ( )
14 Nyquist Take F(s)=+L(s) (return difference, F=S  ) Choose a C that encloses the entire RHP Map into Lplane instead of Fplane (shift by ) R
15 Nyquist Theorem Theorem (Nyquist): If the plot of L(s) (i.e., the image of the Nyquist contour in the L plane) encircles the point +j0 in the counterclockwise direction as many times as there are unstable open loop poles (poles of L(s) within the Nyquist contour) then the feedback system has no poles in the RHP. Z = P N closed loop poles in RHP = open loop poles in RHP  cc encirclements of 
16 Example Hs + L Hs 2 +. s + L MAGNITUDE db rad êsec 0 4. PHASE deg rad êsec
17 Example Hs + L Hs L s Hs s + 2L Hs s + 4L.5 db MAGNITUDE rad êsec PHASE deg rad êsec
18 Gain & Phase Margin γ m Gain, db γ m φ m Phase, deg φ m L( jω ) Nyquist Bode
19 Guaranteed Gain/Phase Margin Theorem: Suppose M s is the sensitivity peak value, then α GM, PM ± 2 sin, α = M ± α 2 S R = S = a a GHplane  θ = 2sin 2a + a
20 Multivariable Nyquuist Recall that the closed loop poles are the roots of the polynomial ( ) = det I + L( s) ( s) [ ] ds () = ()det s I+ Ls () L If the open loop system is stable, then we need only be concerned with the zeros of F s i.e., SISO MIMO + L ( ) or L( s) ( s) det I + L s det I + L
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
More informationIntro 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
More informationMEM 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
More informationIntroduction. 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
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 informationSystems 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
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 informationTopic # 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
More informationSystems 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
More informationThe Nyquist criterion relates the stability of a closed system to the openloop frequency response and open loop pole location.
Introduction to the Nyquist criterion The Nyquist criterion relates the stability of a closed system to the openloop frequency response and open loop pole location. Mapping. If we take a complex number
More informationSystems 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
More informationLet the plant and controller be described as:
Summary of Fundamental Limitations in Feedback Design (LTI SISO Systems) From Chapter 6 of A FIRST GRADUATE COURSE IN FEEDBACK CONTROL By J. S. Freudenberg (Winter 2008) Prepared by: Hammad Munawar (Institute
More informationSystems Analysis and Control
Systems Analysis and Control Matthew M. Peet Arizona State University Lecture 24: Compensation in the Frequency Domain Overview In this Lecture, you will learn: Lead Compensators Performance Specs Altering
More informationFrequency 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
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 informationECSE 4962 Control Systems Design. A Brief Tutorial on Control Design
ECSE 4962 Control Systems Design A Brief Tutorial on Control Design Instructor: Professor John T. Wen TA: Ben Potsaid http://www.cat.rpi.edu/~wen/ecse4962s04/ Don t Wait Until The Last Minute! You got
More informationLinear Control Systems Lecture #3  Frequency Domain Analysis. Guillaume Drion Academic year
Linear Control Systems Lecture #3  Frequency Domain Analysis Guillaume Drion Academic year 20182019 1 Goal and Outline Goal: To be able to analyze the stability and robustness of a closedloop system
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 informationChapter 2. Classical Control System Design. Dutch Institute of Systems and Control
Chapter 2 Classical Control System Design Overview Ch. 2. 2. Classical control system design Introduction Introduction Steadystate Steadystate errors errors Type Type k k systems systems Integral Integral
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 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 information(Continued on next page)
(Continued on next page) 18.2 Roots of Stability Nyquist Criterion 87 e(s) 1 S(s) = =, r(s) 1 + P (s)c(s) where P (s) represents the plant transfer function, and C(s) the compensator. The closedloop characteristic
More informationRichiami 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
More informationFEL3210 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
More informationRobust and Optimal Control, Spring A: SISO Feedback Control A.1 Internal Stability and Youla Parameterization
Robust and Optimal Control, Spring 2015 Instructor: Prof. Masayuki Fujita (S5303B) A: SISO Feedback Control A.1 Internal Stability and Youla Parameterization A.2 Sensitivity and Feedback Performance A.3
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 informationToday (10/23/01) Today. Reading Assignment: 6.3. Gain/phase margin lead/lag compensator Ref. 6.4, 6.7, 6.10
Today Today (10/23/01) Gain/phase margin lead/lag compensator Ref. 6.4, 6.7, 6.10 Reading Assignment: 6.3 Last Time In the last lecture, we discussed control design through shaping of the loop gain GK:
More informationDESIGN 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)
More informationLecture 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,
More informationOutline. Classical Control. Lecture 1
Outline Outline Outline 1 Introduction 2 Prerequisites Block diagram for system modeling Modeling Mechanical Electrical Outline Introduction Background Basic Systems Models/Transfers functions 1 Introduction
More informationData Based Design of 3Term Controllers. Data Based Design of 3 Term Controllers p. 1/10
Data Based Design of 3Term Controllers Data Based Design of 3 Term Controllers p. 1/10 Data Based Design of 3 Term Controllers p. 2/10 History Classical Control  single controller (PID, lead/lag) is designed
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 informationDesign Methods for Control Systems
Design Methods for Control Systems Maarten Steinbuch TU/e Gjerrit Meinsma UT Dutch Institute of Systems and Control Winter term 20022003 Schedule November 25 MSt December 2 MSt Homework # 1 December 9
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 informationNyquist 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
More informationSTABILITY ANALYSIS TECHNIQUES
ECE4540/5540: Digital Control Systems 4 1 STABILITY ANALYSIS TECHNIQUES 41: Bilinear transformation Three main aspects to controlsystem design: 1 Stability, 2 Steadystate response, 3 Transient response
More informationReturn Difference Function and ClosedLoop Roots SingleInput/SingleOutput Control Systems
Spectral Properties of Linear Quadratic Regulators Robert Stengel Optimal Control and Estimation MAE 546 Princeton University, 2018! Stability margins of singleinput/singleoutput (SISO) systems! Characterizations
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 informationDr Ian R. Manchester
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 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 informationStability of CL System
Stability of CL System Consider an open loop stable system that becomes unstable with large gain: At the point of instability, K( j) G( j) = 1 0dB K( j) G( j) K( j) G( j) K( j) G( j) =± 180 o 180 o Closed
More informationDigital 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
More informationModel Uncertainty and Robust Stability for Multivariable Systems
Model Uncertainty and Robust Stability for Multivariable Systems ELEC 571L Robust Multivariable Control prepared by: Greg Stewart Devron Profile Control Solutions Outline Representing model uncertainty.
More informationH(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
More informationActive Control? Contact : Website : Teaching
Active Control? Contact : bmokrani@ulb.ac.be Website : http://scmero.ulb.ac.be Teaching Active Control? Disturbances System Measurement Control Controler. Regulator.,,, Aims of an Active Control Disturbances
More informationFREQUENCYRESPONSE DESIGN
ECE45/55: Feedback Control Systems. 9 FREQUENCYRESPONSE DESIGN 9.: PD and lead compensation networks The frequencyresponse methods we have seen so far largely tell us about stability and stability margins
More informationCDS 101/110: Lecture 10.3 Final Exam Review
CDS 11/11: Lecture 1.3 Final Exam Review December 2, 216 Schedule: (1) Posted on the web Monday, Dec. 5 by noon. (2) Due Friday, Dec. 9, at 5: pm. (3) Determines 3% of your grade Instructions on Front
More informationLecture 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.
More informationLecture 5: Frequency domain analysis: Nyquist, Bode Diagrams, second order systems, system types
Lecture 5: Frequency domain analysis: Nyquist, Bode Diagrams, second order systems, system types Venkata Sonti Department of Mechanical Engineering Indian Institute of Science Bangalore, India, 562 This
More informationRobust Control 9 Design Summary & Examples
Robust Control 9 Design Summary & Examples Harry G. Kwatny Department of Mechanical Engineering & Mechanics Drexel University 5/6/003 Outline he H Problem Solution via γiteration Robust stability via
More informationMAE143a: Signals & Systems (& Control) Final Exam (2011) solutions
MAE143a: Signals & Systems (& Control) Final Exam (2011) solutions Question 1. SIGNALS: Design of a noisecancelling headphone system. 1a. Based on the lowpass filter given, design a highpass filter,
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 informationMAS107 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
More informationCDS 101/110a: Lecture 81 Frequency Domain Design
CDS 11/11a: Lecture 81 Frequency Domain Design Richard M. Murray 17 November 28 Goals: Describe canonical control design problem and standard performance measures Show how to use loop shaping to achieve
More informationELECTRONICS & COMMUNICATIONS DEP. 3rd YEAR, 2010/2011 CONTROL ENGINEERING SHEET 5 LeadLag Compensation Techniques
CAIRO UNIVERSITY FACULTY OF ENGINEERING ELECTRONICS & COMMUNICATIONS DEP. 3rd YEAR, 00/0 CONTROL ENGINEERING SHEET 5 LeadLag Compensation Techniques [] For the following system, Design a compensator such
More informationNyquist 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)
More informationLecture 6. Chapter 8: Robust Stability and Performance Analysis for MIMO Systems. Eugenio Schuster.
Lecture 6 Chapter 8: Robust Stability and Performance Analysis for MIMO Systems Eugenio Schuster schuster@lehigh.edu Mechanical Engineering and Mechanics Lehigh University Lecture 6 p. 1/73 6.1 General
More informationx(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 TimeDelay 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
More informationCDS 101/110a: Lecture 101 Robust Performance
CDS 11/11a: Lecture 11 Robust Performance Richard M. Murray 1 December 28 Goals: Describe how to represent uncertainty in process dynamics Describe how to analyze a system in the presence of uncertainty
More informationThe Frequencyresponse Design Method
Chapter 6 The Frequencyresponse 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)
More informationSystems Analysis and Control
Systems Analysis and Control Matthew M. Peet Arizona State University Lecture 21: Stability Margins and Closing the Loop Overview In this Lecture, you will learn: Closing the Loop Effect on Bode Plot Effect
More informationDesign and Tuning of Fractionalorder PID Controllers for Timedelayed Processes
Design and Tuning of Fractionalorder PID Controllers for Timedelayed Processes Emmanuel Edet Technology and Innovation Centre University of Strathclyde 99 George Street Glasgow, United Kingdom emmanuel.edet@strath.ac.uk
More informationLABORATORY INSTRUCTION MANUAL CONTROL SYSTEM I LAB EE 593
LABORATORY INSTRUCTION MANUAL CONTROL SYSTEM I LAB EE 593 ELECTRICAL ENGINEERING DEPARTMENT JIS COLLEGE OF ENGINEERING (AN AUTONOMOUS INSTITUTE) KALYANI, NADIA CONTROL SYSTEM I LAB. MANUAL EE 593 EXPERIMENT
More informationTable of Laplacetransform
Appendix Table of Laplacetransform pairs 1(t) f(s) oct), unit impulse at t = 0 a, a constant or step of magnitude a at t = 0 a s t, a ramp function e at, an exponential function s + a sin wt, a sine fun
More informationDefinition of Stability
Definition of Stability Transfer function of a linear timeinvariant (LTI) system Fs () = b 2 1 0+ b1s+ b2s + + b m m m 1s   + bms a0 + a1s+ a2s2 + + an1sn 1+ ansn Characteristic equation and poles
More informationControl 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 DMAVT ETH Zürich November 24, 2017 E. Frazzoli (ETH) Lecture
More informationMCE693/793: Analysis and Control of Nonlinear Systems
MCE693/793: Analysis and Control of Nonlinear Systems Introduction to Describing Functions Hanz Richter Mechanical Engineering Department Cleveland State University Introduction Frequency domain methods
More informationAutomatic Control 2. Loop shaping. Prof. Alberto Bemporad. University of Trento. Academic year
Automatic Control 2 Loop shaping Prof. Alberto Bemporad University of Trento Academic year 21211 Prof. Alberto Bemporad (University of Trento) Automatic Control 2 Academic year 21211 1 / 39 Feedback
More informationClassify a transfer function to see which order or ramp it can follow and with which expected error.
Dr. J. Tani, Prof. Dr. E. Frazzoli 505900 Control Systems I (Autumn 208) Exercise Set 0 Topic: Specifications for Feedback Systems Discussion: 30.. 208 Learning objectives: The student can grizzi@ethz.ch,
More informationExercise 1 (A Nonminimum Phase System)
Prof. Dr. E. Frazzoli 559 Control Systems I (HS 25) Solution Exercise Set Loop Shaping Noele Norris, 9th December 26 Exercise (A Nonminimum Phase System) To increase the rise time of the system, we
More informationMASSACHUSETTS 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
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 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 System Design
ELEC ENG 4CL4: Control System Design Notes for Lecture #11 Wednesday, January 28, 2004 Dr. Ian C. Bruce Room: CRL229 Phone ext.: 26984 Email: ibruce@mail.ece.mcmaster.ca Relative Stability: Stability
More informationECEN 326 Electronic Circuits
ECEN 326 Electronic Circuits Stability Dr. Aydın İlker Karşılayan Texas A&M University Department of Electrical and Computer Engineering Ideal Configuration V i Σ V ε a(s) V o V fb f a(s) = V o V ε (s)
More informationFrequency domain analysis
Automatic Control 2 Frequency domain analysis Prof. Alberto Bemporad University of Trento Academic year 20102011 Prof. Alberto Bemporad (University of Trento) Automatic Control 2 Academic year 20102011
More informationSTABILITY 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: Boundedinput boundedoutput (BIBO) stability Asystemmaybe stable, neutrallyormarginallystable, or unstable. This can be illustrated
More informationAMME3500: 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
More informationAdditional ClosedLoop Frequency Response Material (Second edition, Chapter 14)
Appendix J Additional ClosedLoop Frequency Response Material (Second edition, Chapter 4) APPENDIX CONTENTS J. ClosedLoop Behavior J.2 Bode Stability Criterion J.3 Nyquist Stability Criterion J.4 Gain
More informationTheory of Machines and Automatic Control Winter 2018/2019
Theory of Machines and Automatic Control Winter 2018/2019 Lecturer: Sebastian Korczak, PhD, Eng. Institute of Machine Design Fundamentals  Department of Mechanics http://www.ipbm.simr.pw.edu.pl/ Lecture
More informationChapter 2 SDOF Vibration Control 2.1 Transfer Function
Chapter SDOF Vibration Control.1 Transfer Function mx ɺɺ( t) + cxɺ ( t) + kx( t) = F( t) Defines the transfer function as output over input X ( s) 1 = G( s) = (1.39) F( s) ms + cs + k s is a complex number:
More informationEngraving Machine Example
Engraving Machine Example MCE44  Fall 8 Dr. Richter November 24, 28 Basic Design The Xaxis of the engraving machine has the transfer function G(s) = s(s + )(s + 2) In this basic example, we use a proportional
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 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 informationClosedloop system 2/1/2016. Generally MIMO case. Twodegreesoffreedom (2 DOF) control structure. (2 DOF structure) The closed loop equations become
Closedloop system enerally MIMO case Twodegreesoffreedom (2 DOF) control structure (2 DOF structure) 2 The closed loop equations become solving for z gives where is the closed loop transfer function
More informationAnalysis of DiscreteTime Systems
TU Berlin DiscreteTime Control Systems 1 Analysis of DiscreteTime Systems Overview Stability Sensitivity and Robustness Controllability, Reachability, Observability, and Detectabiliy TU Berlin DiscreteTime
More informationLecture 7 (Weeks 1314)
Lecture 7 (Weeks 1314) Introduction to Multivariable Control (SP  Chapters 3 & 4) Eugenio Schuster schuster@lehigh.edu Mechanical Engineering and Mechanics Lehigh University Lecture 7 (Weeks 1314) p.
More informationRobust Control. 2nd class. Spring, 2018 Instructor: Prof. Masayuki Fujita (S5303B) Tue., 17th April, 2018, 10:45~12:15, S423 Lecture Room
Robust Control Spring, 2018 Instructor: Prof. Masayuki Fujita (S5303B) 2nd class Tue., 17th April, 2018, 10:45~12:15, S423 Lecture Room 2. Nominal Performance 2.1 Weighted Sensitivity [SP05, Sec. 2.8,
More informationStability analysis of feedback systems
Chapter 1 Stability analysis of feedback systems 1.1 Introduction A control system must be asymptotically stable. A method for determination of the stability property of a control system will be described
More informationRobust Performance Example #1
Robust Performance Example # The transfer function for a nominal system (plant) is given, along with the transfer function for one extreme system. These two transfer functions define a family of plants
More informationExercise 1 (A Nonminimum Phase System)
Prof. Dr. E. Frazzoli 559 Control Systems I (Autumn 27) Solution Exercise Set 2 Loop Shaping clruch@ethz.ch, 8th December 27 Exercise (A Nonminimum Phase System) To decrease the rise time of the system,
More information(a) Find the transfer function of the amplifier. Ans.: G(s) =
126 INTRDUCTIN T CNTR ENGINEERING 10( s 1) (a) Find the transfer function of the amplifier. Ans.: (. 02s 1)(. 001s 1) (b) Find the expected percent overshoot for a step input for the closedloop system
More informationCONTROL 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)
More informationTopic # Feedback Control Systems
Topic #19 16.31 Feedback Control Systems Stengel Chapter 6 Question: how well do the large gain and phase margins discussed for LQR map over to DOFB using LQR and LQE (called LQG)? Fall 2010 16.30/31 19
More informationControl Systems II. ETH, MAVT, IDSC, Lecture 4 17/03/2017. G. Ducard
Control Systems II ETH, MAVT, IDSC, Lecture 4 17/03/2017 Lecture plan: Control Systems II, IDSC, 2017 SISO Control Design 24.02 Lecture 1 Recalls, Introductory case study 03.03 Lecture 2 Cascaded Control
More informationRaktim Bhattacharya. . AERO 422: Active Controls for Aerospace Vehicles. Frequency ResponseDesign Method
.. AERO 422: Active Controls for Aerospace Vehicles Frequency Response Method Raktim Bhattacharya Laboratory For Uncertainty Quantification Aerospace Engineering, Texas A&M University. ... Response to
More informationControl for. Maarten Steinbuch Dept. Mechanical Engineering Control Systems Technology Group TU/e
Control for Maarten Steinbuch Dept. Mechanical Engineering Control Systems Technology Group TU/e Motion Systems m F Introduction Timedomain tuning Frequency domain & stability Filters Feedforward Servooriented
More informationMEM 355 Performance Enhancement of Dynamical Systems
MEM 355 Performance Enhancement of Dynamical Systems State Space Design Harry G. Kwatny Department of Mechanical Engineering & Mechanics Drexel University 11/8/2016 Outline State space techniques emerged
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 information