Design Methods for Control Systems


 Sophie Ball
 1 years ago
 Views:
Transcription
1 Design Methods for Control Systems Maarten Steinbuch TU/e Gjerrit Meinsma UT Dutch Institute of Systems and Control Winter term
2 Schedule November 25 MSt December 2 MSt Homework # 1 December 9 MSt December 16 MSt Homework # 2 January 6 GM January 13 GM Homework # 3 January 20 GM January 27 GM Homework # 4 Some time in February: Review session
3 Overview Ch Introduction to to feedback control theory Ch Classical control system design Ch LQ, LQG and H 2 control system design Ch Design of of multivariable control systems Ch Uncertainty models and robustness Ch H optimization and µ synthesis
4 Scope and features Mature review of classical and modern control system design techniques Linear timeinvariant systems 70% SISO 30% MIMO Continuoustime MATLAB exercises Control toolbox MuTools and Robust Control toolboxes
5 Overview Ch Introduction to to feedback control theory Introduction Introduction Types Types of of control control systems systems Design Design issues issues Configurations Configurations Highgain Highgain feedback feedback Stability Stability Closedloop Closedloop characteristic characteristic polynomial polynomial Nyquist Nyquist criterion criterion Stability Stability margins margins Performance Performance System System functions functions Low Low and and high high frequencies frequencies Robustness Robustness Robustness Robustness functions functions Loop Loop shaping shaping Limits Limits of of performance performance Twodegreeoffreedom control control systems systems
6 Types of control systems Regulator systems Servo or positioning systems Tracking systems
7 Design issues Targets Closedloop stability Disturbance attenuation Good command response Robustness Limitations Plant capacity Measurement noise
8 Configurations Two degrees of freedom One degree of freedom
9 Highgain feedback1 Loop gain γ = ψ oφ Signal balance e = r γ () e Feedback equation e+ γ () e = r
10 Highgain feedback2 Feedback equation e+ γ () e = r High gain: γ ( e ) >> e Implies: γ () e r Hence: e << r r ψ ( y) So that y ψ 1 () r In case of unit feedback y r Good tracking
11 Highgain feedback3 Loop gain δ = ( φ) o ( ψ) Signal balance: z = d δ ( z) High gain: δ ( z ) >> z Hence: z << d Good disturbance reduction
12 Highgain feedback4 Need closedloop stability Good tracking and disturbance attentuation are retained as long as the closedloop system remains stable the gain remains high Under these conditions highgain feedback implies robustness with respect to loop uncertainty
13 Pitfalls of highgain feedback Highgain feedback has pitfalls: Naively making the gain large easily results in an unstable feedback system Even if the feedback system is stable overly large plant inputs may occur that exceed the plant capacity Measurement noise causes loss of performance
14 Stability1 State space representation: xt &() = Axt () + Brt () et () ut () = Cxt () + Drt () zt ()
15 Stability2 xt &() = Axt () + Brt () et () ut () = Cxt () + Drt () zt () The closedloop system is stable if its state space representation is asymptotically stable Equivalent statements: xt () 0 as t for every solution of xt &() = Axt () All eigenvalues of A have strictly negative real parts All roots of det(si A) have strictly negative real parts
16 Stability3 The control system is BIBO stable if every bounded input signal r results in bounded output signals e, u and z. BIBO = bounded input bounded output Asymptotic stability BIBO stability The converse is not true
17 Stability4 Internal stability Inject internal signals into each exposed interconnection of the system, and define additional internal output signals after each injection point Then the system is internally stable if it is BIBO stable with respect to all inputs (external and internal) and all (external and internal) outputs
18 Stability5 Example
19 Stability6 If each component system is stabilizable and detectable ( has no hidden unstable modes ) then Stability Internal stability When inputoutput descriptions are used (such as transfer functions) internal stability is often easier to check than asymptotic stability
20 Closedloop characteristic polynomial1 State space representation of the openloop system: Characteristic polynomial: State space representation of the closedloop system: Characteristic polynomial: xt &() = Axt () + Bet () yt () = Cxt () + Det () χ () s = det( si A) xt &() = Axt (), cl 1 Acl = A B( I + D) C χ () s = det( si A ) cl cl
21 Closedloop characteristic polynomial2 P(s) C(s) L(s)=P(s)C(s) plant transfer matrix compensator transfer matrix loop gain transfer matrix Then χ cl det( I + L( s)) () s = χ() s det( I + L ( ))
22 Closedloop characteristic polynomial3 χ cl det( I + L( s)) () s = χ() s det( I + L ( )) SISO case: N() s Y() s Ls () = PsCs () () = Ds () X() s Then (within a nonzero constant factor) χ () s = D() s X() s χ cl () s = D() s X() s + N() s Y() s
23 Nyquist criterion Consider the SISO case The locus of L( jω), ω R in the complex plane is called the Nyquist plot of the loop gain The number of unstable closedloop poles = The number of times the Nyquist plot encircles the point 1 + The number of unstable openloop poles
24 Generalized Nyquist criterion Consider the MIMO case The number of unstable closedloop poles = The number of times the locus of det( I + L( jω)), ω R encircles the origin + The number of unstable openloop poles (Principle of the argument)
25 Stability margins1 In the SISO case, the point 1 is a critical point for the Nyquist plot of the closedloop system. If the Nyquist plot is changed so that it crosses the point 1 then the system becomes unstable If the closedloop system is stable but the Nyquist plot passes closely by 1 then the system is nearunstable, that is, has an oscillatory response the system may become unstable by small perturbations of the plant, that is, the system is not robust
26 Stability margins2 There exist various stability margins. They measure how close the Nyquist plot gets to 1 Gain margin k m Phase margin φ m Modulus margin τ m (Landau)
27 System functions: L and S Loop gain L Sensitivity function S L = PC z 1 = v 1{ + L S
28 System functions: R Input disturbance ( proces ) sensitivity function R 1 z = Pv = SPr { 1+ L R
29 System functions: H and T Closedloop transfer function H Complementary sensitivity function T z L L = Fr H = L {1 + L H T F
30 System functions: U Input ( control ) sensitivity function U C u = ( Fr m v) CP U
31 Measurement noise 1 PC PC z = v+ Fr m PC PC PC S T T
32 S, R, U and T 1 P z1 1+ PC 1+ PC v1 S R v1 z = = C PC v v U T 1+ PC 1+ PC 2 2 2
33 Design interrelations S T 1 = 1 + L L = 1 + L U= T / P R= SP H = T F S and T are suitable objects for manipulation
34 Low and high frequencies1 Typical shapes for S and T
35 Low and high frequencies2 Loop gain L large at low frequencies: small at high frequencies: L( jω ) >> 1, S 1/ L, T 1 L( jω ) << 1, S 1, T L Crossover region: L( jω) 1
36 Low and high frequencies3 input sensitivity C 1/ P for low frequencies U = T / P = 1+ PC C for high frequencies input disturbance sensitivity R P 1/ C for low frequencies = SP = 1+ PC P for high frequencies closedloop transfer function H = TF F corrects T
37 Robustness functions1 Sufficient condition for stability under perturbation: L( jω) L ( jω) < 1 + L ( jω), o ω R o
38 Robustness functions2 Equivalently, L( jω) Lo( jω) 1 + Lo( jω) <, ω L ( jω) L ( jω) o o R or L( jω) Lo ( jω) 1 <, ω L ( jω) T ( jω) o o R
39 Robustness functions3 Bound on the relative size of perturbations: L( jω) L ( jω) L o o ( jω ) W 1 ( jω), ω R Sufficient and necessary condition for stability under all perturbations that satisfy the bound: W 1 1 ( jω) <, ω R To ( jω )
40 Robustness functions4 Size of the smallest perturbation that may destabilize the system: W 1 1 ( jω) =, ω R T ( jω ) o
41 Robustness functions5 The preceding discussion focuses on preventing the Nyquist plot of the loop gain L from crossing the point 1. Preventing the inverse Nyquist plot that is, the Nyquist plot of 1/L from crossing the point 1 also guarantees stability. Sufficient condition: < 1 +, ω R L( jω) L ( jω) L ( jω) o o
42 Robustness functions6 Equivalently, 1 1 L( jω) Lo ( jω) 1 <, ω R 1 So ( jω ) L ( jω ) o
43 Robustness functions7 Consider perturbations such that 1 1 L( jω) Lo ( jω) 1 L ( jω ) o W 2 ( jω), ω R Sufficient and necessary condition for robust stability: W 2 1 ( jω) <, So ( j ) ω R ω
44 Robustness functions8 Size of the smallest perturbation that may destabilize the system: W 2 1 ( jω) =, ω R So ( jω )
45 Combined robustness test1 Define δ L ( jω) = L( jω) L ( jω) L o o ( jω ) δ L 1 ( jω) = 1 1 L( jω) Lo ( jω) 1 L ( jω ) o
46 Combined robustness test2 Then the perturbed closedloop system is stable if ω R δ L ( j ) 1 ω < 1 S( jω ) or δ L ( jω) < 1 T( jω ) typically satisfied at low frequencies typically satisfied at high frequencies
47 Combined robustness test3 Critical frequency region: crossover area
48 Loop shaping Low frequencies: large loop gain High frequencies: small loop gain In the crossover region the phase is constrained because of stability
49 Bode s gainphase relationship1 Between break frequencies the loop gain behaves as Hence L( jω) c( jω) n L( jω) cω n π arg L( jω ) n 2 Phase and magnitude do not behave independently Bode s gainphase relationship describes the relation more accurately
50 Bode s gainphase relationship2 The limitations imposed by stability on the phase in the crossover region by Bode s gainphase relationship limit the rate at which the loop gain decreases: If, say, π arg L( jω) in the crossover region 2 then 1 L( jω) cω in the crossover region
51 Limits of performance Bode s integral The FreudenbergLooze equalities Limitations are imposed by causality the polezero configuration
52 Bode s integral1 If L has at least two more poles than zeros then 0 log S( jω) dω = π Re pi 0 The p i are the righthalf plane poles of the loop gain. i Proof: Use the Poisson integral from complex function theory
53 Bode s integral1
54 Bode s integral1
55 Bode s integral2 Dual result: Suppose that the loop has integrating action of at least order 2. Then 0 1 log T(1/ jω) dω = π Re 0 z The z i are the righthalf plane zeros of the loop gain. i i
56 FreudenbergLooze equality1 Let z be any righthalf plane zero of the loop gain. Poisson s formula of complex function theory leads to the equality 0 log( S( jω) ) dw ( ω) = log B ( z) 0 z 1 poles Strengthens Bode s integral W z 1 ω Imz 1 ω + Im ( ω) = arctan + arctan π Re z π Re z z Increasing function. Rises most steeply at z. B () poles s = i p p i i + s s Blaschke product
57 FreudenbergLooze equality2 W z for different values of arg z (a) arg z = 0 (b) arg z is almost π/2 Frequencies where W z rises most steeply contribute most to the integral
58 FreudenbergLooze equality3 The bounds for S hold provided µ 1 ε Wz ( ω1 ) 1 W ( ω ) z 1 B ( 1 ) poles ( z) Wz ω The dependence of the righthand side on the various parameters may be analyzed
59 FreudenbergLooze equality4 Effects of righthalf plane zeros on S S may be made small up to the frequency min i z i. Attempting to make S small beyond this frequency makes S peak Righthalf plane poles further impair the achievable reduction of S (in particular, nearlycancelling righthalf plane polezero pairs)
60 FreudenbergLooze equality5 Rederivation of the FreudenbergLooze equality while 1 1 L replacing L with 1/L, so that S = = = 1+ L L L interchanging the roles of the poles and the zeros T leads to 0 log( T( jω) ) dw ( ω) = log B ( p) 0 p 1 zeros
61 FreudenbergLooze equality6 0 log( T( jω) ) dw ( ω) = log B ( p) 0 p 1 zeros p is any righthalf plane pole of the loop gain, and B () zeros s = i z z i i + s s In the application of the equality, interchange the roles of low and high frequencies
62 FreudenbergLooze equality7 Effects of righthalf plane poles on T T may be made small above the frequency max i p i. Attempting to make T small below this frequency makes T peak Righthalf plane zeros further impair the achievable reduction of T (in particular, nearlycancelling righthalf plane polezero pairs)
63 FreudenbergLooze equality8 Consequences for S and T
64 Twodegreeoffreedom systems1 N Y Let P =, C = D X Then the closedloop transfer function is PC NY H = F = F 1+ PC D cl Dcl = DX + NY with D cl the closedloop characteristic polynomial
65 Twodegreeoffreedom systems2 Other twodegreeoffreedom configuration: H = NX D cl F has zeros at the roots of N and X H NY = F has zeros at the Dcl roots of N and Y Can the zeros of H be made independent of the feedback compensator?
66 Twodegreeoffreedom systems3 Further twodegreeoffreedom configuration N Y Y P =, C = C = D X X Need XX 1 2 = X, YY 1 2 = Y to achieve the same loop gain as in the two previous cases Have H PC = = 1+ PC NX Y D cl
67 Twodegreeoffreedom systems4 H is independent of the compensator if we let so that H = NX Y D 2 1 cl Y = X = 1 Y = Y, X = X N C1 =, C2 = Y, H = X D cl
68 Twodegreeoffreedom systems5 Resulting feedback system Equivalent configuration
69 Twodegreeoffreedom systems6 Extension May choose F polynomial so that we obtain a 1½degreeoffreedom system
70 Twodegreeoffreedom systems7 Further extension: F polynomial, F o rational: 2½degreeoffreedom system F =1, F o rational: H = NF D cl F o 2degreeoffreedom system
Chapter 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 informationChapter 3. LQ, LQG and Control System Design. Dutch Institute of Systems and Control
Chapter 3 LQ, LQG and Control System H 2 Design Overview LQ optimization state feedback LQG optimization output feedback H 2 optimization nonstochastic version of LQG Application to feedback system design
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 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 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 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 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 informationControl Systems I. Lecture 7: Feedback and the Root Locus method. Readings: Jacopo Tani. Institute for Dynamic Systems and Control DMAVT ETH Zürich
Control Systems I Lecture 7: Feedback and the Root Locus method Readings: Jacopo Tani Institute for Dynamic Systems and Control DMAVT ETH Zürich November 2, 2018 J. Tani, E. Frazzoli (ETH) Lecture 7:
More 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 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 informationControl Systems I. Lecture 6: Poles and Zeros. Readings: Emilio Frazzoli. Institute for Dynamic Systems and Control DMAVT ETH Zürich
Control Systems I Lecture 6: Poles and Zeros Readings: Emilio Frazzoli Institute for Dynamic Systems and Control DMAVT ETH Zürich October 27, 2017 E. Frazzoli (ETH) Lecture 6: Control Systems I 27/10/2017
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 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 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 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 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 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 information3 Stabilization of MIMO Feedback Systems
3 Stabilization of MIMO Feedback Systems 3.1 Notation The sets R and S are as before. We will use the notation M (R) to denote the set of matrices with elements in R. The dimensions are not explicitly
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 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 informationMTNS 06, Kyoto (July, 2006) Shinji Hara The University of Tokyo, Japan
MTNS 06, Kyoto (July, 2006) Shinji Hara The University of Tokyo, Japan Outline Motivation & Background: H2 Tracking Performance Limits: new paradigm Explicit analytical solutions with examples H2 Regulation
More informationEE Control Systems LECTURE 9
Updated: Sunday, February, 999 EE  Control Systems LECTURE 9 Copyright FL Lewis 998 All rights reserved STABILITY OF LINEAR SYSTEMS We discuss the stability of input/output systems and of statespace
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 informationSTABILITY OF CLOSEDLOOP CONTOL SYSTEMS
CHBE320 LECTURE X STABILITY OF CLOSEDLOOP CONTOL SYSTEMS Professor Dae Ryook Yang Spring 2018 Dept. of Chemical and Biological Engineering 101 Road Map of the Lecture X Stability of closedloop control
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 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 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 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 informationTopic # Feedback Control Systems
Topic #17 16.31 Feedback Control Systems Deterministic LQR Optimal control and the Riccati equation Weight Selection Fall 2007 16.31 17 1 Linear Quadratic Regulator (LQR) Have seen the solutions to the
More informationControl Systems Design
ELEC4410 Control Systems Design Lecture 18: State Feedback Tracking and State Estimation Julio H. Braslavsky julio@ee.newcastle.edu.au School of Electrical Engineering and Computer Science Lecture 18:
More informationTopic # Feedback Control. StateSpace Systems Closedloop control using estimators and regulators. Dynamics output feedback
Topic #17 16.31 Feedback Control StateSpace Systems Closedloop control using estimators and regulators. Dynamics output feedback Back to reality Copyright 21 by Jonathan How. All Rights reserved 1 Fall
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 informationExam. 135 minutes, 15 minutes reading time
Exam August 15, 2017 Control Systems I (151059100L) Prof Emilio Frazzoli Exam Exam Duration: 135 minutes, 15 minutes reading time Number of Problems: 44 Number of Points: 52 Permitted aids: Important:
More informationChapter Stability Robustness Introduction Last chapter showed how the Nyquist stability criterion provides conditions for the stability robustness of
Lectures on Dynamic Systems and Control Mohammed Dahleh Munther A Dahleh George Verghese Department of Electrical Engineering and Computer Science Massachuasetts Institute of Technology c Chapter Stability
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 informationThe loop shaping paradigm. Lecture 7. Loop analysis of feedback systems (2) Essential specifications (2)
Lecture 7. Loop analysis of feedback systems (2). Loop shaping 2. Performance limitations The loop shaping paradigm. Estimate performance and robustness of the feedback system from the loop transfer L(jω)
More informationProblem Set 4 Solution 1
Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.245: MULTIVARIABLE CONTROL SYSTEMS by A. Megretski Problem Set 4 Solution Problem 4. For the SISO feedback
More informationControl Systems. Root Locus & Pole Assignment. L. Lanari
Control Systems Root Locus & Pole Assignment L. Lanari Outline rootlocus definition main rules for hand plotting root locus as a design tool other use of the root locus pole assignment Lanari: CS  Root
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 informationExam. 135 minutes, 15 minutes reading time
Exam August 6, 208 Control Systems II (5059000) Dr. Jacopo Tani Exam Exam Duration: 35 minutes, 5 minutes reading time Number of Problems: 35 Number of Points: 47 Permitted aids: 0 pages (5 sheets) A4.
More informationCONTROL DESIGN FOR SET POINT TRACKING
Chapter 5 CONTROL DESIGN FOR SET POINT TRACKING In this chapter, we extend the pole placement, observerbased output feedback design to solve tracking problems. By tracking we mean that the output is commanded
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 informationECE317 : Feedback and Control
ECE317 : Feedback and Control Lecture : RouthHurwitz stability criterion Examples Dr. Richard Tymerski Dept. of Electrical and Computer Engineering Portland State University 1 Course roadmap Modeling
More 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 informationFall 線性系統 Linear Systems. Chapter 08 State Feedback & State Estimators (SISO) FengLi Lian. NTUEE Sep07 Jan08
Fall 2007 線性系統 Linear Systems Chapter 08 State Feedback & State Estimators (SISO) FengLi Lian NTUEE Sep07 Jan08 Materials used in these lecture notes are adopted from Linear System Theory & Design, 3rd.
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 informationAnalysis and Synthesis of SingleInput SingleOutput Control Systems
Lino Guzzella Analysis and Synthesis of SingleInput SingleOutput Control Systems l+kja» \Uja>)W2(ja»\ um Contents 1 Definitions and Problem Formulations 1 1.1 Introduction 1 1.2 Definitions 1 1.2.1 Systems
More informationChapter 7 Interconnected Systems and Feedback: WellPosedness, Stability, and Performance 7. Introduction Feedback control is a powerful approach to o
Lectures on Dynamic Systems and Control Mohammed Dahleh Munther A. Dahleh George Verghese Department of Electrical Engineering and Computer Science Massachuasetts Institute of Technology c Chapter 7 Interconnected
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 informationTopic # Feedback Control Systems
Topic #20 16.31 Feedback Control Systems Closedloop system analysis Bounded Gain Theorem Robust Stability Fall 2007 16.31 20 1 SISO Performance Objectives Basic setup: d i d o r u y G c (s) G(s) n control
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 informationUnit 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
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 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 informationExam. 135 minutes + 15 minutes reading time
Exam January 23, 27 Control Systems I (559L) Prof. Emilio Frazzoli Exam Exam Duration: 35 minutes + 5 minutes reading time Number of Problems: 45 Number of Points: 53 Permitted aids: Important: 4 pages
More informationThe stability of linear timeinvariant feedback systems
The stability of linear timeinvariant feedbac systems A. Theory The system is atrictly stable if The degree of the numerator of H(s) (H(z)) the degree of the denominator of H(s) (H(z)) and/or The poles
More informationÜbersetzungshilfe / Translation aid (English) To be returned at the end of the exam!
Prüfung Regelungstechnik I (Control Systems I) Prof. Dr. Lino Guzzella 3. 8. 24 Übersetzungshilfe / Translation aid (English) To be returned at the end of the exam! Do not mark up this translation aid
More informationChapter 9 Robust Stability in SISO Systems 9. Introduction There are many reasons to use feedback control. As we have seen earlier, with the help of a
Lectures on Dynamic Systems and Control Mohammed Dahleh Munther A. Dahleh George Verghese Department of Electrical Engineering and Computer Science Massachuasetts Institute of Technology c Chapter 9 Robust
More informationLinear State Feedback Controller Design
Assignment For EE5101  Linear Systems Sem I AY2010/2011 Linear State Feedback Controller Design Phang Swee King A0033585A Email: king@nus.edu.sg NGS/ECE Dept. Faculty of Engineering National University
More informationẋ n = f n (x 1,...,x n,u 1,...,u m ) (5) y 1 = g 1 (x 1,...,x n,u 1,...,u m ) (6) y p = g p (x 1,...,x n,u 1,...,u m ) (7)
EEE582 Topical Outline A.A. Rodriguez Fall 2007 GWC 352, 9653712 The following represents a detailed topical outline of the course. It attempts to highlight most of the key concepts to be covered and
More informationControl Systems 2. Lecture 4: Sensitivity function limits. Roy Smith
Control Systems 2 Lecture 4: Sensitivity function limits Roy Smith 2017314 4.1 Inputoutput controllability Control design questions: 1. How well can the plant be controlled? 2. What control structure
More informationCDS 101/110a: Lecture 102 Control Systems Implementation
CDS 101/110a: Lecture 102 Control Systems Implementation Richard M. Murray 5 December 2012 Goals Provide an overview of the key principles, concepts and tools from control theory  Classical control 
More informationAircraft Stability & Control
Aircraft Stability & Control Textbook Automatic control of Aircraft and missiles 2 nd Edition by John H Blakelock References Aircraft Dynamics and Automatic Control  McRuler & Ashkenas Aerodynamics, Aeronautics
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 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 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 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 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 informationState Regulator. Advanced Control. design of controllers using pole placement and LQ design rules
Advanced Control State Regulator Scope design of controllers using pole placement and LQ design rules Keywords pole placement, optimal control, LQ regulator, weighting matrixes Prerequisites Contact state
More informationAutonomous Mobile Robot Design
Autonomous Mobile Robot Design Topic: Guidance and Control Introduction and PID Loops Dr. Kostas Alexis (CSE) Autonomous Robot Challenges How do I control where to go? Autonomous Mobile Robot Design Topic:
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 informationSchool of Mechanical Engineering Purdue University. DC Motor Position Control The block diagram for position control of the servo table is given by:
Root Locus Motivation Sketching Root Locus Examples ME375 Root Locus  1 Servo Table Example DC Motor Position Control The block diagram for position control of the servo table is given by: θ D 0.09 See
More informationStability and Robustness 1
Lecture 2 Stability and Robustness This lecture discusses the role of stability in feedback design. The emphasis is notonyes/notestsforstability,butratheronhowtomeasurethedistanceto instability. The small
More informationA brief introduction to robust H control
A brief introduction to robust H control JeanMarc Biannic System Control and Flight Dynamics Department ONERA, Toulouse. http://www.onera.fr/staff/jeanmarcbiannic/ http://jm.biannic.free.fr/ European
More informationFull State Feedback for State Space Approach
Full State Feedback for State Space Approach State Space Equations Using Cramer s rule it can be shown that the characteristic equation of the system is : det[ si A] 0 Roots (for s) of the resulting polynomial
More informationMIMO analysis: loopatatime
MIMO robustness MIMO analysis: loopatatime y 1 y 2 P (s) + + K 2 (s) r 1 r 2 K 1 (s) Plant: P (s) = 1 s 2 + α 2 s α 2 α(s + 1) α(s + 1) s α 2. (take α = 10 in the following numerical analysis) Controller:
More informationControl Systems I. Lecture 7: Feedback and the Root Locus method. Readings: Guzzella 9.13, Emilio Frazzoli
Control Systems I Lecture 7: Feedback and the Root Locus method Readings: Guzzella 9.13, 13.3 Emilio Frazzoli Institute for Dynamic Systems and Control DMAVT ETH Zürich November 3, 2017 E. Frazzoli (ETH)
More informationUncertainty and Robustness for SISO Systems
Uncertainty and Robustness for SISO Systems ELEC 571L Robust Multivariable Control prepared by: Greg Stewart Outline Nature of uncertainty (models and signals). Physical sources of model uncertainty. Mathematical
More informationControl Design. Lecture 9: State Feedback and Observers. Two Classes of Control Problems. State Feedback: Problem Formulation
Lecture 9: State Feedback and s [IFAC PB Ch 9] State Feedback s Disturbance Estimation & Integral Action Control Design Many factors to consider, for example: Attenuation of load disturbances Reduction
More informationLecture 4 Stabilization
Lecture 4 Stabilization This lecture follows Chapter 5 of DoyleFrancisTannenbaum, with proofs and Section 5.3 omitted 17013 IOCUPC, Lecture 4, November 2nd 2005 p. 1/23 Stable plants (I) We assume that
More informationSingular Value Decomposition Analysis
Singular Value Decomposition Analysis Singular Value Decomposition Analysis Introduction Introduce a linear algebra tool: singular values of a matrix Motivation Why do we need singular values in MIMO control
More informationEL2520 Control Theory and Practice
EL2520 Control Theory and Practice Lecture 8: Linear quadratic control Mikael Johansson School of Electrical Engineering KTH, Stockholm, Sweden Linear quadratic control Allows to compute the controller
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 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 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 informationLecture 5: Linear Systems. Transfer functions. Frequency Domain Analysis. Basic Control Design.
ISS0031 Modeling and Identification Lecture 5: Linear Systems. Transfer functions. Frequency Domain Analysis. Basic Control Design. Aleksei Tepljakov, Ph.D. September 30, 2015 Linear Dynamic Systems Definition
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 informationControl of Electromechanical Systems
Control of Electromechanical Systems November 3, 27 Exercise Consider the feedback control scheme of the motor speed ω in Fig., where the torque actuation includes a time constant τ A =. s and a disturbance
More informationDiscussion on: Measurable signal decoupling with dynamic feedforward compensation and unknowninput observation for systems with direct feedthrough
Discussion on: Measurable signal decoupling with dynamic feedforward compensation and unknowninput observation for systems with direct feedthrough H.L. Trentelman 1 The geometric approach In the last
More informationControl Systems I. Lecture 1: Introduction. Suggested Readings: Åström & Murray Ch. 1, Guzzella Ch. 1. Emilio Frazzoli
Control Systems I Lecture 1: Introduction Suggested Readings: Åström & Murray Ch. 1, Guzzella Ch. 1 Emilio Frazzoli Institute for Dynamic Systems and Control DMAVT ETH Zürich September 22, 2017 E. Frazzoli
More informationChapter 6  Solved Problems
Chapter 6  Solved Problems Solved Problem 6.. Contributed by  James Welsh, University of Newcastle, Australia. Find suitable values for the PID parameters using the ZN tuning strategy for the nominal
More informationControl Systems I. Lecture 2: Modeling. Suggested Readings: Åström & Murray Ch. 23, Guzzella Ch Emilio Frazzoli
Control Systems I Lecture 2: Modeling Suggested Readings: Åström & Murray Ch. 23, Guzzella Ch. 23 Emilio Frazzoli Institute for Dynamic Systems and Control DMAVT ETH Zürich September 29, 2017 E. Frazzoli
More informationECE 388 Automatic Control
Controllability and State Feedback Control Associate Prof. Dr. of Mechatronics Engineeering Çankaya University Compulsory Course in Electronic and Communication Engineering Credits (2/2/3) Course Webpage:
More informationCDS Solutions to Final Exam
CDS 22  Solutions to Final Exam Instructor: Danielle C Tarraf Fall 27 Problem (a) We will compute the H 2 norm of G using statespace methods (see Section 26 in DFT) We begin by finding a minimal statespace
More informationLimiting Performance of Optimal Linear Filters
Limiting Performance of Optimal Linear Filters J.H. Braslavsky M.M. Seron D.Q. Mayne P.V. Kokotović Technical Report CCEC 97414 Center for Control Engineering and Computation University of California
More informationSome solutions of the written exam of January 27th, 2014
TEORIA DEI SISTEMI Systems Theory) Prof. C. Manes, Prof. A. Germani Some solutions of the written exam of January 7th, 0 Problem. Consider a feedback control system with unit feedback gain, with the following
More informationLecture plan: Control Systems II, IDSC, 2017
Control Systems II MAVT, IDSC, Lecture 8 28/04/2017 G. Ducard Lecture plan: Control Systems II, IDSC, 2017 SISO Control Design 24.02 Lecture 1 Recalls, Introductory case study 03.03 Lecture 2 Cascaded
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 informationControl Systems I. Lecture 2: Modeling and Linearization. Suggested Readings: Åström & Murray Ch Jacopo Tani
Control Systems I Lecture 2: Modeling and Linearization Suggested Readings: Åström & Murray Ch. 23 Jacopo Tani Institute for Dynamic Systems and Control DMAVT ETH Zürich September 28, 2018 J. Tani, E.
More informationProblem Set 5 Solutions 1
Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.245: MULTIVARIABLE CONTROL SYSTEMS by A. Megretski Problem Set 5 Solutions The problem set deals with Hankel
More information