Intro to Frequency Domain Design


 Evelyn Webster
 2 years ago
 Views:
Transcription
1 Intro to Frequency Domain Design MEM 355 Performance Enhancement of Dynamical Systems Harry G. Kwatny Department of Mechanical Engineering & Mechanics Drexel University
2 Outline Closed Loop Transfer Functions The gang of four Sensitivity functions Traditional Performance Measures Time domain Frequency domain Stability & Robustness Introduction role of sensitivity functions Nyquist Traditional gain/phase margins
3 0/4/208 Closed Loop Transfer Functions
4 Closed Loop Transfer Functons error control d disturbance r  G c u G p y output d 2 reference p c ( ) ( ) Y = G U + D U = G R Y D E = R Y 2 noise
5 Transfer Functions Output GG G GG Ys () = Rs () D() s + D() s + GG + GG + GG Error p c p p c 2 p c p c p c G GG Es () = Rs () D() s + D() s + GG + GG + GG Control p p c 2 p c p c p c G GG G U() s = R() s D () s D () s + GG + GG + GG c p c c 2 p c p c p c
6 Gang of Four Outputs Inputs Output Error Control Reference GG p c + GG p c Gc + GG p c + GG p c Disturbance Noise Gp G GG p p c + GG + GG + GG p c GG p c GG p c Gc + GG + GG + GG p c p p c c p p c c
7 Sensitivity Functions [ ] [ ] [ ] E() s = I + L R() s I + L G D () s + I + L LD () s where L: = GG p [ ] 2 sensitivity function: S: = I + L command error complementary sensitivity c function: For SISO systems Bode derived: dt T dl L = dt dl { 2 [ L] L [ L] } [ L T = = + + [ L] L L] = + = change in command S p L [ + L] [ ] T : = I + L L with respect to change in L L command output transfer function (normalized) output
8 0/4/208 Traditional Performance Criteria
9 Traditional Performance ~ Time Domain ultimate error, limit of e(t) as t approaches infinity 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).
10 TTrr rise timettpp peak timettss settling time Traditional Performance ~ Time Domain Cont d y p.4 Im.2 α degree of stability, decay rate /α 2ε ideal region for closed loop poles θ damping ratio θ = sin Re ρ T r T T p s rise time peak time settling time t T r T p T s Time response parameters Ideal pole locations
11 Bode Plot Given any transfer function describing a SISO system Y s = GsU s ( ) ( ) ( ) Suppose the input is a sinusoid u t = Asin ωt, ( ) ( ) then the output is a sinusoid y t = Bsin ωt+ θ ( ) ( ) B = G jω A, θ = G jω ( ) ( )
12 Traditional Performance ~ Frequency Domain Sensitivity Complementary sensitivity Desired Altitude  2 s +.5s+ 0.5 K s disturbance ( 20s+ )( 0s+ )( 0.5s+ ) V22 Osprey altitude control K=280
13 Sensitivity Functions, Cont d For unity feed back systems: S is the error response to command transfer function T is the output response to command transfer function ( ) Suppose L s is strictly proper m< n, then lim S( jω ) = lim = ω + L j ω lim S ( jω ) ω 0 ω 0 ( jω ) ω 0 ω 0 ( ω) = lim = + c L( jω ) ( ω) L( jω ) ( ω) L( jω ) L j lim T( jω ) = lim = 0 ω + ω limt 0 type L << type L= 0 L j type L = lim = + c type L= 0 We would like S=0 We would like T= A system is of type p if the transfer function has p free integrators in the denomintor, i.e. m m s + am s + + a0 L( s) = k s p s n p b s n p b ( + n p + + 0) L
14 Traditional Performance ~ Frequency Domain Bandwidth Definitions Sensitivity Function (first crosses / 2=0.707~3db from below): { v S j 2 [0, v) } ω = max : ( ω) < ω BS v Complementary Sensitivity Function crosses ω BT 2 from above) = min{ v : T( jω) < 2 ω (, v ) } v Crossover frequency { v L j v } ω = max : ( ω) ω [0, ) c v (highest frequency where T
15 Example: Osprey Bandwidth Complementary sensitivity Open loop Sensitivity
16 Interpretation of Bode Plot G( s) Yresponse to input U: E( s) = GsU ( ) ( s) ( ω) = ( ω) ( ω) ( ω) = ( ω) ( ω), ( ω) = ( ω) + U ( jω) Any transfer function: Output Y j G j U j Y j G j U j Y j G j e sin ( ωt) 0.00 t 0. t e MAGNITUDE MAGNITUDE db 0 db rad sec input frequency distribution rad sec
17 A Fundamental Tradeoff [ ] [ ] Note that + L + + L L= S + T = Making S small improves tracking & disturbance rejection but makes system susceptible to noise S T : ( ) Es () = SsRs () () SsG () s D() s+ TsD () () s Typical design specifications S T ( jω ) < < ω [ 0, ω ] ( jω) << ω [ ω, ] And, there are other limitations ω > ω 2
18 Cauchy Theorem splane F(s) Fplane Any function of a complex number can be viewed as a map of points in one complex plane to another 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 ( ) 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. C ( ) ( ) C =Image(C) is the 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. ( )
19 Nyquist return difference Sensitivity function Take F(s)=+L(s) (note: F=S  ) Choose a C that encloses the entire RHP Map into Lplane instead of Fplane (shift by ) X splane X splane L( s) Lplane X R X R X X Typical Nyquist contours Image of Nyquist contour
20 Nyquist Theorem Theorem (Nyquist): If the plot of L(s) (i.e., the image of the Nyquist contour in the Lplane) 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 
21 Example X I splane R II L( s) Lplane X III ( ) L s = 2 2 ( s+ p)( s + 2ζω0s+ ω0 ) jθ ( ρ ) I: s = jω L jω =, 0 < ω < ( ) 2 2 ( jω + p)( ω + 2ζω0jω + ω0 ) jθ π π II: s = ρe, ρ, = 2 2 L e = 0 III: s = jω, III I jθ 2 j2θ jθ 2 ( ρ e + p )( ρ e + 2ζωρ e + ω ) * X ρ
22 Example 2 ( ) L s = 2 2 ( + 2ζω0s+ ω0 ) s s X X X I IV III 2 2 ( + j ) ( ) * ( ) splane R I: s = jω L( jω) =, 0 < ω < jω ω ζω ω ω jθ π π II: s = ρe, ρ, θ = 2 2 jθ L( ρ e ) = 0 jθ j θ jθ ρ ρ e ρ e 2ζω ρ e ω III: s = jω, III I jθ π π IV : s = εe, ε, θ = 2 2 jθ L( ε e ) = jθ j θ jθ ε e ε e 2ζω ε e ω εω II ε e L( s) jθ Lplane
23 Example 3 ( ) G s = ( ) ( s+ ) s s >> s=tf( s ); >> G=/(s*(s^2+2*0.*s+)); >> nyquist(g) >> s=tf( s ); >> G=/((s+0.0)*(s^2+2*0.*s+)); >> nyquist(g)
24 Gain & Phase Margin Gain margin 0 LL jjjj γ m Gain, db 00 γ m frequency 0 Phase margin φ m Phase, deg 80 φ m LL jjjj Nyquist plot L( jω ) Assumption: the nominal system is stable. Bode plot
25 Robustness From Sensitivity Functions M M Sensitivity peaks are related to gain and phase margin. Sensitivity peaks are related to overshoot and damping ratio. S T = max S( jω) ω = max T( jω) ω Unit circle centered at  Im L plane S > S < = ρ = + L= + ρ ( ) jθ S e L e jθ S  = a a L( jω) Re constant S prescribes circle S = + L
26 Example: Osprey Sensitivity function plots for K=280, 500, 000 MAGNITUDE db rad sec
27 Example: Osprey K=280 5 Increased gain yields higher bandwidth (and reduced settling time) but higher sensitivity peak (lower damping ratio and more overshoot) K=000
28 V22 with PID disturbance Desired Altitude  K s 2 +.5s+ 0.5 s ( 20s+ )( 0s+ )( 0.5s+ )
29 V22 with PID db MAGNITUDE rad sec db MAGNITUDE rad sec db MAGNITUDE rad sec db MAGNITUDE rad sec Complementary sensitivity sensitivity
30 V22 with PID >> s=tf('s'); >> G=280*((s^2+.5*s+0.5)/s) /((20*s+)*(0*s+)*(0.5*s+)); >> margin(g) Notice that gain margin is negative because system loses stability when gain drops.
31 V22 PID+Lead disturbance Desired Altitude  K 2 s s s s s+ ( 20s+ )( 0s+ )( 0.5s+ )
32 V22 PID+Lead Notice that sensitivity peak is reduced from 20 db to 0 db Complementary sensitivity Sensitivity MAGNITUDE MAGNITUDE db rad sec db rad sec 60 MAGNITUDE MAGNITUDE db db rad sec rad sec Disturbance to output K=50 Command to control
33 V22 PID+Lead Gain margin is infinite because the system is never unstable for any K>0.
34 Summary Need to consider 24 transfer functions to fully evaluate performance Bandwidth is inversely related to settling time Sensitivity function peak is related to overshoot and inversely to damping ratio Gain and phase margins can be determined from Nyquist or Bode plots Sensitivity peak is inversely related to stability margin Design tools: Root locus helps place poles Bode and/or Nyquist helps establish robustness (margins) & performance (sensitivity peaks)
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
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 informationRobust 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
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 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 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 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 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 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. 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 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 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 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 informationDiscrete 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},
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 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 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 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 23: Drawing The Nyquist Plot Overview In this Lecture, you will learn: Review of Nyquist Drawing the Nyquist Plot Using the
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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 informationControls Problems for Qualifying Exam  Spring 2014
Controls Problems for Qualifying Exam  Spring 2014 Problem 1 Consider the system block diagram given in Figure 1. Find the overall transfer function T(s) = C(s)/R(s). Note that this transfer function
More 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 informationAN INTRODUCTION TO THE CONTROL THEORY
OpenLoop controller An OpenLoop (OL) controller is characterized by no direct connection between the output of the system and its input; therefore external disturbance, nonlinear dynamics and parameter
More informationDigital Control: Summary # 7
Digital Control: Summary # 7 Proportional, integral and derivative control where K i is controller parameter (gain). It defines the ratio of the control change to the control error. Note that e(k) 0 u(k)
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 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 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 informationTradeoffs and Limits of Performance
Chapter 9 Tradeoffs and Limits of Performance 9. Introduction Fundamental limits of feedback systems will be investigated in this chapter. We begin in Section 9.2 by discussing the basic feedback loop
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 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 informationControl of Manufacturing Processes
Control of Manufacturing Processes Subject 2.830 Spring 2004 Lecture #18 Basic Control Loop Analysis" April 15, 2004 Revisit Temperature Control Problem τ dy dt + y = u τ = time constant = gain y ss =
More informationRoot Locus Design. MEM 355 Performance Enhancement of Dynamical Systems
Root Locus Design MEM 355 Performance Enhancement of Dynamical Systems Harry G. Kwatny Department of Mechanical Engineering & Mechanics Drexel University Outline The root locus design method is an iterative,
More informationChapter 15  Solved Problems
Chapter 5  Solved Problems Solved Problem 5.. Contributed by  Alvaro Liendo, Universidad Tecnica Federico Santa Maria, Consider a plant having a nominal model given by G o (s) = s + 2 The aim of the
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 informationAnalysis of DiscreteTime Systems
TU Berlin DiscreteTime Control Systems TU Berlin DiscreteTime Control Systems 2 Stability Definitions We define stability first with respect to changes in the initial conditions Analysis of DiscreteTime
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 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 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 informationFEEDBACK CONTROL SYSTEMS
FEEDBAC CONTROL SYSTEMS. Control System Design. Open and ClosedLoop Control Systems 3. Why ClosedLoop Control? 4. Case Study  Speed Control of a DC Motor 5. SteadyState Errors in Unity Feedback Control
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 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 informationChapter 7  Solved Problems
Chapter 7  Solved Problems Solved Problem 7.1. A continuous time system has transfer function G o (s) given by G o (s) = B o(s) A o (s) = 2 (s 1)(s + 2) = 2 s 2 + s 2 (1) Find a controller of minimal
More informationAutomatic Control (TSRT15): Lecture 7
Automatic Control (TSRT15): Lecture 7 Tianshi Chen Division of Automatic Control Dept. of Electrical Engineering Email: tschen@isy.liu.se Phone: 13282226 Office: Bhouse extrance 2527 Outline 2 Feedforward
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 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 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 informationFeedback Control of Linear SISO systems. Process Dynamics and Control
Feedback Control of Linear SISO systems Process Dynamics and Control 1 OpenLoop Process The study of dynamics was limited to openloop systems Observe process behavior as a result of specific input signals
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 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 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 informationLecture 1: Feedback Control Loop
Lecture : Feedback Control Loop Loop Transfer function The standard feedback control system structure is depicted in Figure. This represend(t) n(t) r(t) e(t) u(t) v(t) η(t) y(t) F (s) C(s) P (s) Figure
More informationUltimate State. MEM 355 Performance Enhancement of Dynamical Systems
Ultimate State MEM 355 Performance Enhancement of Dnamical Sstems Harr G. Kwatn Department of Mechanical Engineering & Mechanics Drexel Universit Outline Design Criteria two step process Ultimate state
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 informationIC6501 CONTROL SYSTEMS
DHANALAKSHMI COLLEGE OF ENGINEERING CHENNAI DEPARTMENT OF ELECTRICAL AND ELECTRONICS ENGINEERING YEAR/SEMESTER: II/IV IC6501 CONTROL SYSTEMS UNIT I SYSTEMS AND THEIR REPRESENTATION 1. What is the mathematical
More informationIntroduction to Feedback Control
Introduction to Feedback Control Control System Design Why Control? OpenLoop vs ClosedLoop (Feedback) Why Use Feedback Control? ClosedLoop Control System Structure Elements of a Feedback Control System
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 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 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 informationChapter 9: Controller design
Chapter 9. Controller Design 9.1. Introduction 9.2. Effect of negative feedback on the network transfer functions 9.2.1. Feedback reduces the transfer function from disturbances to the output 9.2.2. Feedback
More informationPerformance of Feedback Control Systems
Performance of Feedback Control Systems Design of a PID Controller Transient Response of a Closed Loop System Damping Coefficient, Natural frequency, Settling time and Steadystate Error and Type 0, Type
More informationDr 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 splane 4 27 Mar Block Diagrams Assign 1 Due 5 3 Apr Feedback System Characteristics
More informationTransient response via gain adjustment. Consider a unity feedback system, where G(s) = 2. The closed loop transfer function is. s 2 + 2ζωs + ω 2 n
Design via frequency response Transient response via gain adjustment Consider a unity feedback system, where G(s) = ωn 2. The closed loop transfer function is s(s+2ζω n ) T(s) = ω 2 n s 2 + 2ζωs + ω 2
More 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 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 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 information100 (s + 10) (s + 100) e 0.5s. s 100 (s + 10) (s + 100). G(s) =
1 AME 3315; Spring 215; Midterm 2 Review (not graded) Problems: 9.3 9.8 9.9 9.12 except parts 5 and 6. 9.13 except parts 4 and 5 9.28 9.34 You are given the transfer function: G(s) = 1) Plot the bode plot
More informationCourse roadmap. Step response for 2ndorder system. Step response for 2ndorder system
ME45: Control Systems Lecture Time response of ndorder systems Prof. Clar Radcliffe and Prof. Jongeun Choi Department of Mechanical Engineering Michigan State University Modeling Laplace transform Transfer
More informationState Space Design. MEM 355 Performance Enhancement of Dynamical Systems
State Space Design MEM 355 Performance Enhancement of Dynamical Systems Harry G. Kwatny Department of Mechanical Engineering & Mechanics Drexel University Outline State space techniques emerged around
More informationMEM 255 Introduction to Control Systems: Analyzing Dynamic Response
MEM 55 Itroductio to Cotrol Systems: Aalyzig Dyamic Respose Harry G. Kwaty Departmet of Mechaical Egieerig & Mechaics Drexel Uiversity Outlie Time domai ad frequecy domai A secod order system Via partial
More informationStep Response Analysis. Frequency Response, Relation Between Model Descriptions
Step Response Analysis. Frequency Response, Relation Between Model Descriptions Automatic Control, Basic Course, Lecture 3 November 9, 27 Lund University, Department of Automatic Control Content. Step
More informationRobust stability and Performance
122 c Perry Y.Li Chapter 5 Robust stability and Performance Topics: ([ author ] is supplementary source) Sensitivities and internal stability (Goodwin 5.15.4) Modeling Error and Model Uncertainty (Goodwin
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 information