Lecture 2. FRTN10 Multivariable Control. Automatic Control LTH, 2018
|
|
- Sandra Hicks
- 5 years ago
- Views:
Transcription
1 Lecture 2 FRTN10 Multivariable Control
2 Course Outline L1 L5 Specifications, models and loop-shaping by hand 1 Introduction 2 Stability and robustness 3 Specifications and disturbance models 4 Control synthesis in frequency domain 5 Case study: DVD player L6 L8 Limitations on achievable performance L9 L11 Controller optimization: analytic approach L12 L14 Controller optimization: numerical approach L15 Course review
3 Lecture 2 Outline 1 Stability 2 Sensitivity and robustness 3 The Small Gain Theorem 4 Singular values
4 Lecture 2 Outline 1 Stability 2 Sensitivity and robustness 3 The Small Gain Theorem 4 Singular values
5 Stability is crucial Examples: bicycle JAS 39 Gripen Mercedes A-class ABS brakes
6 Input output stability u S y= S(u) Ageneral system S is calledinput output stable (or L 2 stable or BIBOstable orjust stable )if its L 2 gain is finite: S = sup u S(u) u <
7 Input output stability of LTI systems ForanLTIsystem S withimpulse response g(t)andtransfer function G(s), the following stability conditions are equivalent: S is bounded g(t) decays exponentially All polesof G(s)arein the left half-plane (LHP),i.e.,allpoles have negative real part
8 Internal stability The LTI system dx dt = Ax+Bu y= Cx+Du is called internally stable if the following equivalent conditions hold: Thestate x decaysexponentially when u=0 All eigenvaluesof Aare in the LHP
9 Internal vs input output stability If x=ax+bu y= Cx+Du is internally stable then G(s)=C(sI A) 1 B+D is input output stable. Warning The opposite is not always true! There may be unstable pole-zero cancellations(that also render the system uncontrollable and/or unobservable), and these may not be seen in the transfer function!
10 Stability of feedback loops Assumescalaropen-loopsystem G 0 (s) Σ G 0 (s) 1 Theclosed-loopsystemis stable if and only if all solutionsto the characteristic equation 1+G 0 (s)=0 are in the left half-plane.
11 Simplified Nyquist criterion If G 0 (s)isstable,thentheclosed-loopsystem[1+g 0 (s)] 1 isstable if and only if the Nyquist curve of G 0 (s)doesnot encircle 1. Imaginary Axis Real Axis (Note: Matlab gives a Nyquist plot for both positive and negative frequencies)
12 General Nyquist criterion Let P=numberof unstable(rhp) poles in G 0 (s) N=number ofclockwise encirclements of 1bythe Nyquist plot of G 0 (s) Thentheclosed-loopsystem[1+G 0 (s)] 1 has P+N unstablepoles
13 Lecture 2 Outline 1 Stability 2 Sensitivity and robustness 3 The Small Gain Theorem 4 Singular values
14 Sensitivity and robustness How sensitive is the closed-loop system to model errors and disturbances? How do we measure the distance to instability? Is it possible to guarantee stability for all systems within some distance from the ideal model?
15 Amplitude and phase margins Amplitude margin A m : arg G 0 (iω 0 )= 180, G 0 (iω 0 ) = 1/A m Phasemarginϕ m : G 0 (iω c ) = 1, arg G 0 (iω c )=ϕ m /A m Gain A m 10 1 ω c G 0 (ω o ) ϕ m G 0 (ω c ) Phase ϕ m ω o
16 Mini-problem k(s+ 1) s 2 e + cs+ 1 sl 1 Nominally k= 1, c=1and L= 0. How muchmargin isthereineach parameter before the closed-loop system becomes unstable? Im Re Magnitude (abs) Phase (deg) Gm = Inf, Pm = 109 deg (at 1.41 rad/s) Frequency (rad/s)
17 Mini-problem
18 Sensitivity functions r C(s) P(s) y 1 S(s)= 1 1+ P(s)C(s) T(s)= P(s)C(s) 1+ P(s)C(s) sensitivity function complementary sensitivity function Notethat we alwayshave S(s)+ T(s)=1
19 Sensitivity towards changes in plant r C(s) P(s) y 1 Howsensitive is the closedloopto a(small) changein P? dt dp = C (1+ PC) 2= T P(1+ PC) Relativechange in T comparedto relative change in P: dt/t dp/p = 1 1+ PC = S
20 Sensitivity towards disturbances r d + e u C + P + v y 1 + n Open-loop response (C = 0) to process disturbances d, v: Closed-loop response: Y cl = Y ol = V+ PD 1 1+ PC V+ P 1+ PC D=S Y ol
21 Interpretation as stability margin The L 2 gain ofthe sensitivityfunctionmeasuresthe inverse ofthe distance between the Nyquist plot and the point 1: R 1 = sup ω 1 1+ P(iω)C(iω) = M s 1 Im R Re P(iω)C(iω)
22 Lecture 2 Outline 1 Stability 2 Sensitivity and robustness 3 The Small Gain Theorem 4 Singular values
23 Robustness analysis How large plant uncertainty can be tolerated without risking instability? Example(multiplicative uncertainty): v (s) w P(s) C(s)
24 The Small Gain Theorem r 1 e 1 S 1 S 2 e 2 r 2 Assumethat S 1 and S 2 are stable. If S 1 S 2 < 1, thenthe closed-loopsystem(from (r 1, r 2 )to (e 1, e 2 ))is stable. Note 1: The theorem applies also to nonlinear, time-varying, and multivariable systems Note 2: The stability condition is sufficient but not necessary, so the results may be conservative
25 The Small Gain Theorem r 1 e 1 S 1 S 2 e 2 r 2 Assumethat S 1 and S 2 are stable. If S 1 S 2 < 1, thenthe closed-loopsystem(from (r 1, r 2 )to (e 1, e 2 ))is stable. Note 1: The theorem applies also to nonlinear, time-varying, and multivariable systems Note 2: The stability condition is sufficient but not necessary, so the results may be conservative
26 Proof e 1 = r 1 +S 2 (r 2 +S 1 (e 1 )) ( ) e 1 r 1 + S 2 r 2 + S 1 e 1 e 1 r 1 + S 2 r 2 1 S 1 S 2 This showsboundedgain from (r 1, r 2 ) to e 1. Thegain to e 2 is boundedin the sameway.
27 Application to robustness analysis v (s) w P(s) C(s) Thediagram canbe redrawnas v w PC 1+PC
28 Application to robustness analysis v w PC 1+PC Assumingthat T= PC 1+PC guarantees stability if is stable,the SmallGain Theorem T < 1
29 Lecture 2 Outline 1 Stability 2 Sensitivity and robustness 3 The Small Gain Theorem 4 Singular values
30 Gain of multivariable systems Recall from Lecture 1 that forastable LTIsystem S. S = sup G(iω) = G ω How to calculate G(iω) for a multivariable system?
31 Vector norm and matrix gain Foravector x C n,we use the 2-norm x = x x= x x n 2 (A denotestheconjugate transpose of A) Foramatrix A C n m,we use the L 2 -induced norm Ax x A := sup = sup A Ax x x x x = λ(a x A) λ(a A) denotesthelargest eigenvalue of A A. The ratio Ax / x is maximized when x is a corresponding eigenvector.
32 Example: Different gains in different directions: [ ] y1 = y 2 [ ] [ ] 2 4 u1 0 3 u Input u= [ ] T, u = u u 1 y=gu = [ ] T, y = y (red):eigenvectors ; (blue): V ; (green): U A=U*S*V T
33 Singular Values Foramatrix A,its singular valuesσ i aredefined as σ i = λ i whereλ i arethe eigenvalues of A A. Let σ(a) denote the largest singular value and σ(a) the smallest singular value. Foralinear map y= Ax, it holds that σ(a) y x σ(a) The singular values are typically computed using singular value decomposition(svd): A=UΣV
34 Singular Values Foramatrix A,its singular valuesσ i aredefined as σ i = λ i whereλ i arethe eigenvalues of A A. Let σ(a) denote the largest singular value and σ(a) the smallest singular value. Foralinear map y= Ax, it holds that σ(a) y x σ(a) The singular values are typically computed using singular value decomposition(svd): A=UΣV
35 SVD example Matlab code for singular value decomposition of the matrix [ ] 2 4 A= 0 3 SVD: A=U S V whereboththematricesu andv areunitary(i.e. have orthonormal columns s.t. V V= I) and S is the diagonalmatrix with(sorteddecreasing) singular values σ i. Multiplying A with an input vector along the first column inv gives A V (:,1) =USV V (:,1) = [ 1 =US =U 0] (:,1) σ 1 That is, we get maximal gain σ 1 in the output direction U (:,1) if we use an input in directionv (:,1) (and minimal gain σ 2 if we use the second columnv (:,n) =V (:,2) ). >> A = [2 4; 0 3] A = >> [U,S,V] = svd(a) U = S = V = >> A*V(:,1) ans = >> U(:,1)*S(1,1) ans =
36 Example: Gain of multivariable system Consider the transfer function matrix 2 G(s)= s+ 1 s s s s+ 1 3 s+ 1 >> s=zpk('s') >> G=[ 2/(s+1) 4/(2*s+1); s/(s^2+0.1*s+1) 3/(s+1)]; >> sigma(g) % plot sigma values of G wrt freq >> grid on >> norm(g,inf) % infinity norm = system gain ans =
37 Singular Values System: G Frequency (rad/sec): Singular Value (abs): 5.26 Singular Values (abs) 10 0 System: G Frequency (rad/sec): Singular Value (abs): Frequency (rad/sec) The singular values of the tranfer function matrix(prev slide). Note that G(0)= [2 4;03] whichcorrespondsto Ain thesvd-exampleabove. G =
38 Lecture 2 summary Input output stability: S < Sensitivity function: S := dt/t dp/p = 1 1+PC SmallGain Theorem: Thefeedbackinterconnectionof S 1 and S 2 is stable if S 1 S 2 < 1 Conservative compared to the Nyquist criterion Useful for robustness analysis Thegain ofamultivariable system G(s)is givenby sup ω σ(g(iω)),whereσis the largest singular value
Stability 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 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 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 informationCDS 101/110a: Lecture 10-1 Robust Performance
CDS 11/11a: Lecture 1-1 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 informationControl System Design
ELEC ENG 4CL4: Control System Design Notes for Lecture #11 Wednesday, January 28, 2004 Dr. Ian C. Bruce Room: CRL-229 Phone ext.: 26984 Email: ibruce@mail.ece.mcmaster.ca Relative Stability: Stability
More informationLecture 7 (Weeks 13-14)
Lecture 7 (Weeks 13-14) Introduction to Multivariable Control (SP - Chapters 3 & 4) Eugenio Schuster schuster@lehigh.edu Mechanical Engineering and Mechanics Lehigh University Lecture 7 (Weeks 13-14) p.
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 2018-2019 1 Goal and Outline Goal: To be able to analyze the stability and robustness of a closed-loop system
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 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 informationECEN 605 LINEAR SYSTEMS. Lecture 20 Characteristics of Feedback Control Systems II Feedback and Stability 1/27
1/27 ECEN 605 LINEAR SYSTEMS Lecture 20 Characteristics of Feedback Control Systems II Feedback and Stability Feedback System Consider the feedback system u + G ol (s) y Figure 1: A unity feedback system
More 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 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 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 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 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 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 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 informationCDS 101/110a: Lecture 8-1 Frequency Domain Design
CDS 11/11a: Lecture 8-1 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 System Design
ELEC ENG 4CL4: Control System Design Notes for Lecture #22 Dr. Ian C. Bruce Room: CRL-229 Phone ext.: 26984 Email: ibruce@mail.ece.mcmaster.ca Friday, March 5, 24 More General Effects of Open Loop Poles
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 2002-2003 Schedule November 25 MSt December 2 MSt Homework # 1 December 9
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 informationFRTN10 Multivariable Control, Lecture 13. Course outline. The Q-parametrization (Youla) Example: Spring-mass System
FRTN Multivariable Control, Lecture 3 Anders Robertsson Automatic Control LTH, Lund University Course outline The Q-parametrization (Youla) L-L5 Purpose, models and loop-shaping by hand L6-L8 Limitations
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 D-MAVT ETH Zürich November 24, 2017 E. Frazzoli (ETH) Lecture
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 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 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 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 informationControl Systems Engineering ( Chapter 6. Stability ) Prof. Kwang-Chun Ho Tel: Fax:
Control Systems Engineering ( Chapter 6. Stability ) Prof. Kwang-Chun Ho kwangho@hansung.ac.kr Tel: 02-760-4253 Fax:02-760-4435 Introduction In this lesson, you will learn the following : How to determine
More information6.245: MULTIVARIABLE CONTROL SYSTEMS by A. Megretski. Solutions to Problem Set 1 1. Massachusetts Institute of Technology
Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.245: MULTIVARIABLE CONTROL SYSTEMS by A. Megretski Solutions to Problem Set 1 1 Problem 1.1T Consider the
More informationNonlinear Control. Nonlinear Control Lecture # 18 Stability of Feedback Systems
Nonlinear Control Lecture # 18 Stability of Feedback Systems Absolute Stability + r = 0 u y G(s) ψ( ) Definition 7.1 The system is absolutely stable if the origin is globally uniformly asymptotically stable
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 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 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 informationAn Internal Stability Example
An Internal Stability Example Roy Smith 26 April 2015 To illustrate the concept of internal stability we will look at an example where there are several pole-zero cancellations between the controller and
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 informationMAE 143B - Homework 9
MAE 143B - Homework 9 7.1 a) We have stable first-order poles at p 1 = 1 and p 2 = 1. For small values of ω, we recover the DC gain K = lim ω G(jω) = 1 1 = 2dB. Having this finite limit, our straight-line
More informationCourse Outline. FRTN10 Multivariable Control, Lecture 13. General idea for Lectures Lecture 13 Outline. Example 1 (Doyle Stein, 1979)
Course Outline FRTN Multivariable Control, Lecture Automatic Control LTH, 6 L-L Specifications, models and loop-shaping by hand L6-L8 Limitations on achievable performance L9-L Controller optimization:
More informationControl Systems I. Lecture 7: Feedback and the Root Locus method. Readings: Jacopo Tani. Institute for Dynamic Systems and Control D-MAVT ETH Zürich
Control Systems I Lecture 7: Feedback and the Root Locus method Readings: Jacopo Tani Institute for Dynamic Systems and Control D-MAVT ETH Zürich November 2, 2018 J. Tani, E. Frazzoli (ETH) Lecture 7:
More informationProfessor Fearing EE C128 / ME C134 Problem Set 7 Solution Fall 2010 Jansen Sheng and Wenjie Chen, UC Berkeley
Professor Fearing EE C8 / ME C34 Problem Set 7 Solution Fall Jansen Sheng and Wenjie Chen, UC Berkeley. 35 pts Lag compensation. For open loop plant Gs ss+5s+8 a Find compensator gain Ds k such that the
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 5-059-00 Control Systems I (Autumn 208) Exercise Set 0 Topic: Specifications for Feedback Systems Discussion: 30.. 208 Learning objectives: The student can grizzi@ethz.ch,
More informationRobust Multivariable Control
Lecture 1 Anders Helmersson anders.helmersson@liu.se ISY/Reglerteknik Linköpings universitet Addresses email: anders.helmerson@liu.se mobile: 0734278419 http://users.isy.liu.se/rt/andersh/teaching/robkurs.html
More informationEECS C128/ ME C134 Final Thu. May 14, pm. Closed book. One page, 2 sides of formula sheets. No calculators.
Name: SID: EECS C28/ ME C34 Final Thu. May 4, 25 5-8 pm Closed book. One page, 2 sides of formula sheets. No calculators. There are 8 problems worth points total. Problem Points Score 4 2 4 3 6 4 8 5 3
More informationThe Q-parametrization (Youla) Lecture 13: Synthesis by Convex Optimization. Lecture 13: Synthesis by Convex Optimization. Example: Spring-mass System
The Q-parametrization (Youla) Lecture 3: Synthesis by Convex Optimization controlled variables z Plant distubances w Example: Spring-mass system measurements y Controller control inputs u Idea for lecture
More informationStability Margin Based Design of Multivariable Controllers
Stability Margin Based Design of Multivariable Controllers Iván D. Díaz-Rodríguez Sangjin Han Shankar P. Bhattacharyya Dept. of Electrical and Computer Engineering Texas A&M University College Station,
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 information6.241 Dynamic Systems and Control
6.241 Dynamic Systems and Control Lecture 17: Robust Stability Readings: DDV, Chapters 19, 20 Emilio Frazzoli Aeronautics and Astronautics Massachusetts Institute of Technology April 6, 2011 E. Frazzoli
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 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 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 informationMEM Chapter 2. Sensitivity Function Matrices
Applied Robust Control, Chap 2, 2012 Spring 1 MEM800-007 Chapter 2 Sensitivity Function Matrices r e K u d y G Loop transfer function matrix: L GK Sensitivity function matrix: S ( I L) Complementary Sensitivity
More informationControl Systems. Root Locus & Pole Assignment. L. Lanari
Control Systems Root Locus & Pole Assignment L. Lanari Outline root-locus definition main rules for hand plotting root locus as a design tool other use of the root locus pole assignment Lanari: CS - Root
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 informationReturn Difference Function and Closed-Loop Roots Single-Input/Single-Output Control Systems
Spectral Properties of Linear- Quadratic Regulators Robert Stengel Optimal Control and Estimation MAE 546 Princeton University, 2018! Stability margins of single-input/singleoutput (SISO) systems! Characterizations
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 informationOutline. Control systems. Lecture-4 Stability. V. Sankaranarayanan. V. Sankaranarayanan Control system
Outline Control systems Lecture-4 Stability V. Sankaranarayanan Outline Outline 1 Outline Outline 1 2 Concept of Stability Zero State Response: The zero-state response is due to the input only; all the
More informationSoftware Engineering/Mechatronics 3DX4. Slides 6: Stability
Software Engineering/Mechatronics 3DX4 Slides 6: Stability Dr. Ryan Leduc Department of Computing and Software McMaster University Material based on lecture notes by P. Taylor and M. Lawford, and Control
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 informationTopic # Feedback Control. State-Space Systems Closed-loop control using estimators and regulators. Dynamics output feedback
Topic #17 16.31 Feedback Control State-Space Systems Closed-loop control using estimators and regulators. Dynamics output feedback Back to reality Copyright 21 by Jonathan How. All Rights reserved 1 Fall
More informationCDS 101/110a: Lecture 10-2 Control Systems Implementation
CDS 101/110a: Lecture 10-2 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 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 informationAnalysis of Discrete-Time Systems
TU Berlin Discrete-Time Control Systems 1 Analysis of Discrete-Time Systems Overview Stability Sensitivity and Robustness Controllability, Reachability, Observability, and Detectabiliy TU Berlin Discrete-Time
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 informationThe Tuning of Robust Controllers for Stable Systems in the CD-Algebra: The Case of Sinusoidal and Polynomial Signals
The Tuning of Robust Controllers for Stable Systems in the CD-Algebra: The Case of Sinusoidal and Polynomial Signals Timo Hämäläinen Seppo Pohjolainen Tampere University of Technology Department of Mathematics
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 informationr + - FINAL June 12, 2012 MAE 143B Linear Control Prof. M. Krstic
MAE 43B Linear Control Prof. M. Krstic FINAL June, One sheet of hand-written notes (two pages). Present your reasoning and calculations clearly. Inconsistent etchings will not be graded. Write answers
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, 965-3712 The following represents a detailed topical outline of the course. It attempts to highlight most of the key concepts to be covered and
More informationContents lecture 5. Automatic Control III. Summary of lecture 4 (II/II) Summary of lecture 4 (I/II) u y F r. Lecture 5 H 2 and H loop shaping
Contents lecture 5 Automatic Control III Lecture 5 H 2 and H loop shaping Thomas Schön Division of Systems and Control Department of Information Technology Uppsala University. Email: thomas.schon@it.uu.se,
More informationControl Systems I. Lecture 7: Feedback and the Root Locus method. Readings: Guzzella 9.1-3, Emilio Frazzoli
Control Systems I Lecture 7: Feedback and the Root Locus method Readings: Guzzella 9.1-3, 13.3 Emilio Frazzoli Institute for Dynamic Systems and Control D-MAVT ETH Zürich November 3, 2017 E. Frazzoli (ETH)
More informationModule 5: Design of Sampled Data Control Systems Lecture Note 8
Module 5: Design of Sampled Data Control Systems Lecture Note 8 Lag-lead Compensator When a single lead or lag compensator cannot guarantee the specified design criteria, a laglead compensator is used.
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 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 informationMAE143a: Signals & Systems (& Control) Final Exam (2011) solutions
MAE143a: Signals & Systems (& Control) Final Exam (2011) solutions Question 1. SIGNALS: Design of a noise-cancelling headphone system. 1a. Based on the low-pass filter given, design a high-pass filter,
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 informationD(s) G(s) A control system design definition
R E Compensation D(s) U Plant G(s) Y Figure 7. A control system design definition x x x 2 x 2 U 2 s s 7 2 Y Figure 7.2 A block diagram representing Eq. (7.) in control form z U 2 s z Y 4 z 2 s z 2 3 Figure
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 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 informationChapter 5. Standard LTI Feedback Optimization Setup. 5.1 The Canonical Setup
Chapter 5 Standard LTI Feedback Optimization Setup Efficient LTI feedback optimization algorithms comprise a major component of modern feedback design approach: application problems involving complex models
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 informationTheory of Robust Control
Theory of Robust Control Carsten Scherer Mathematical Systems Theory Department of Mathematics University of Stuttgart Germany Contents 1 Introduction to Basic Concepts 6 1.1 Systems and Signals..............................
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 informationExercise 1 (A Non-minimum Phase System)
Prof. Dr. E. Frazzoli 5-59- Control Systems I (Autumn 27) Solution Exercise Set 2 Loop Shaping clruch@ethz.ch, 8th December 27 Exercise (A Non-minimum Phase System) To decrease the rise time of the system,
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-RESPONSE DESIGN
ECE45/55: Feedback Control Systems. 9 FREQUENCY-RESPONSE DESIGN 9.: PD and lead compensation networks The frequency-response methods we have seen so far largely tell us about stability and stability margins
More informationECE382/ME482 Spring 2005 Homework 7 Solution April 17, K(s + 0.2) s 2 (s + 2)(s + 5) G(s) =
ECE382/ME482 Spring 25 Homework 7 Solution April 17, 25 1 Solution to HW7 AP9.5 We are given a system with open loop transfer function G(s) = K(s +.2) s 2 (s + 2)(s + 5) (1) and unity negative feedback.
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 informationSAMPLE SOLUTION TO EXAM in MAS501 Control Systems 2 Autumn 2015
FACULTY OF ENGINEERING AND SCIENCE SAMPLE SOLUTION TO EXAM in MAS501 Control Systems 2 Autumn 2015 Lecturer: Michael Ruderman Problem 1: Frequency-domain analysis and control design (15 pt) Given is a
More informationChapter 20 Analysis of MIMO Control Loops
Chapter 20 Analysis of MIMO Control Loops Motivational Examples All real-world systems comprise multiple interacting variables. For example, one tries to increase the flow of water in a shower by turning
More informationEL2520 Control Theory and Practice
So far EL2520 Control Theory and Practice r Fr wu u G w z n Lecture 5: Multivariable systems -Fy Mikael Johansson School of Electrical Engineering KTH, Stockholm, Sweden SISO control revisited: Signal
More information5. Observer-based Controller Design
EE635 - Control System Theory 5. Observer-based Controller Design Jitkomut Songsiri state feedback pole-placement design regulation and tracking state observer feedback observer design LQR and LQG 5-1
More informationAnalysis of Discrete-Time Systems
TU Berlin Discrete-Time Control Systems TU Berlin Discrete-Time Control Systems 2 Stability Definitions We define stability first with respect to changes in the initial conditions Analysis of Discrete-Time
More informationRaktim Bhattacharya. . AERO 422: Active Controls for Aerospace Vehicles. Basic Feedback Analysis & Design
AERO 422: Active Controls for Aerospace Vehicles Basic Feedback Analysis & Design Raktim Bhattacharya Laboratory For Uncertainty Quantification Aerospace Engineering, Texas A&M University Routh s Stability
More informationThe Nyquist criterion relates the stability of a closed system to the open-loop frequency response and open loop pole location.
Introduction to the Nyquist criterion The Nyquist criterion relates the stability of a closed system to the open-loop frequency response and open loop pole location. Mapping. If we take a complex number
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 Steady-state Steady-state errors errors Type Type k k systems systems Integral Integral
More informationControl Systems I. Lecture 6: Poles and Zeros. Readings: Emilio Frazzoli. Institute for Dynamic Systems and Control D-MAVT ETH Zürich
Control Systems I Lecture 6: Poles and Zeros Readings: Emilio Frazzoli Institute for Dynamic Systems and Control D-MAVT ETH Zürich October 27, 2017 E. Frazzoli (ETH) Lecture 6: Control Systems I 27/10/2017
More informationPM diagram of the Transfer Function and its use in the Design of Controllers
PM diagram of the Transfer Function and its use in the Design of Controllers Santiago Garrido, Luis Moreno Abstract This paper presents the graphical chromatic representation of the phase and the magnitude
More informationRobust fixed-order H Controller Design for Spectral Models by Convex Optimization
Robust fixed-order H Controller Design for Spectral Models by Convex Optimization Alireza Karimi, Gorka Galdos and Roland Longchamp Abstract A new approach for robust fixed-order H controller design by
More informationECE 345 / ME 380 Introduction to Control Systems Lecture Notes 8
Learning Objectives ECE 345 / ME 380 Introduction to Control Systems Lecture Notes 8 Dr. Oishi oishi@unm.edu November 2, 203 State the phase and gain properties of a root locus Sketch a root locus, by
More informationÜbersetzungshilfe / Translation aid (English) To be returned at the end of the exam!
Prüfung Regelungstechnik I (Control Systems I) Prof. Dr. Lino Guzzella 3.. 24 Übersetzungshilfe / Translation aid (English) To be returned at the end of the exam! Do not mark up this translation aid -
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 informationÜbersetzungshilfe / Translation aid (English) To be returned at the end of the exam!
Prüfung Regelungstechnik I (Control Systems I) Prof. Dr. Lino Guzzella 9. 8. 2 Übersetzungshilfe / Translation aid (English) To be returned at the end of the exam! Do not mark up this translation aid -
More information