Lecture 13: Internal Model Principle and Repetitive Control

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Lecture 13: Internal Model Principle and Repetitive Control"

Transcription

1 ME 233, UC Berkeley, Spring 2014 Xu Chen Lecture 13: Internal Model Principle and Repetitive Control Big picture review of integral control in PID design example: 0 Es) C s) Ds) + + P s) Y s) where P s) = 1 ms + b, C s) = k p + k i 1 s + k ds, k p,k i,k d > 0 the integral action in PID control perfectly rejects asymptotically) constant disturbances D s) = d o /s): E s) = P s) 1 + P s)c s) D s) = d o m + k d )s 2 + k p + b)s + k i e t) 0 Lecture 13: Internal Model Principle and Repetitive Control ME

2 Big picture review of integral control in PID design 0 Es) C s) Ds) + + P s) Y s) the structure of the reference/disturbance is built into the integral controller: controller: 1 C s) = k p + k i s + k ds = 1 kd s 2 ) + k p s + k i s constant disturbance: d t) = d o L {d t)} = 1 s d o Lecture 13: Internal Model Principle and Repetitive Control ME General case: internal model principle IMP) Theorem Internal Model Principle) Rs) + Es) C s) = B cs) Ds)= B d s) A d s) + A c s) P s) = B ps) + A p s) Y s) Assume B p s) = 0 and A d s) = 0 do not have common roots. If the closed loop is asymptotically stable, and A c s) can be factoried as A c s) = A d s)a c s), then the disturbance is asymptotically rejected. Lecture 13: Internal Model Principle and Repetitive Control ME

3 General case: internal model principle IMP) Rs) + Es) C s) = B cs) Ds)= B d s) A d s) + A c s) P s) = B ps) + A p s) Y s) Proof: The steady-state error response to the disturbance is E s) = = P s) 1 + P s)c s) D s) = B p s)a c s) B d s) A p s)a c s) + B p s)b c s) A d s) B p s)a c s)b d s) A p s)a c s) + B p s)b c s) where all roots of A p s)a c s) + B p s)b c s) = 0 are on the left half plane. Hence e t) 0 Lecture 13: Internal Model Principle and Repetitive Control ME Internal model principle discrete-time case: Theorem Discrete-time IMP) R 1 ) + E 1 ) B c 1 ) D 1 )= B d 1 ) A d 1 ) + d B p 1 ) A c 1 ) + A p 1 ) Y 1 ) Assume B p 1 ) = 0 and A d 1 ) = 0 do not have common eros. If the closed loop is asymptotically stable, and A c 1 ) can be factoried as A c 1 ) = A d 1 ) A c 1 ), then the disturbance is asymptotically rejected. Proof: analogous to the continuous-time case. Lecture 13: Internal Model Principle and Repetitive Control ME

4 Internal model principle the disturbance structure: R 1 ) + E 1 ) B c 1 ) D 1 )= B d 1 ) A d 1 ) + d B p 1 ) A c 1 ) + A p 1 ) Y 1 ) example disturbance structures: d k) A d 1 ) constant d o 1 1 cosω 0 k) and sinω 0 k) cosω 0 ) + 2 shifted ramp signal d k) = αk + β periodic: d k) = d k N) 1 N Lecture 13: Internal Model Principle and Repetitive Control ME Internal model principle R 1 ) + E 1 ) B c 1 ) D 1 )= B d 1 ) A d 1 ) + d B p 1 ) A c 1 )A d 1 ) + A p 1 ) Y 1 ) observations: the controller contains a counter disturbance generator high-gain control: the open-loop frequency response P e jω) C e jω) = edjω B p e jω ) B c e jω ) A p e jω )A c e jω )A d e jω ) is large at frequencies where A d e jω ) = 0 D 1) = B d 1 ) /A d 1 ) means d k) is the impulse response of B d 1 ) /A d 1 ) : A d 1 ) d k) = B d 1 ) δ k) Lecture 13: Internal Model Principle and Repetitive Control ME

5 Outline 1. Big Picture review of integral control in PID design 2. Internal Model Principle theorems typical disturbance structures 3. Repetitive Control use of internal model principle design by pole placement design by stable pole-ero cancellation Lecture 13: Internal Model Principle and Repetitive Control ME Repetitive control Repetitive control focus on attenuating periodic disturbances with A d 1 ) = 1 N Control structure: R 1 ) + E 1 ) B c 1 ) D 1 )= B d 1 ) A d 1 ) + d B p 1 ) A c 1 )A d 1 ) + A p 1 ) Y 1 ) It remains to design B c 1 ) and A c 1 ). We discuss two methods: pole placement partial) cancellation of plant dynamics: prototype repetitive control Lecture 13: Internal Model Principle and Repetitive Control ME

6 1, Pole placement: prerequisite Theorem Consider G ) = β) α) = β 1 n1 +β 2 n2 + +β n. α ) and β ) are n +α 1 n1 + +α n coprime no common roots) iff S Sylvester matrix) is nonsingular: S = β α β 2 β α 1 1 β n α n1 α 1 β n β 1 α n β n... β 2 0 α n α n1.... β n β n α n β n 2n1) 2n1) Lecture 13: Internal Model Principle and Repetitive Control ME , Pole placement: prerequisite Example: G ) = β 1 n1 + β 2 n2 + + β n n + α 1 n1 + + α n = n1 + α 1 n2 + + α n1 n + α 1 n1 + + α n1 + 0 i.e. β 1 = 1 β i = α i1 i 2 α n = 0 α ) and β ) are not coprime, and S is clearly singular. Lecture 13: Internal Model Principle and Repetitive Control ME

7 1, Pole placement: big picture R 1 ) + E 1 ) B c 1 ) D 1 )= B d 1 ) A d 1 ) + d B p 1 ) A c 1 )A d 1 ) + A p 1 ) Y 1 ) Disturbance model: A d 1 ) = 1 N Pole placement assigns the closed-loop characteristic equation: d B p 1 ) B c 1 ) + A p 1 ) A c 1 ) A d 1 ) = 1 + η η η q q }{{} η 1 ) which is in the structure of a Diophantine equation. Design procedure: specify the desired closed-loop dynamics η 1) ; match coefficients of i on both sides to get B c 1 ) and A c 1 ). Lecture 13: Internal Model Principle and Repetitive Control ME , Pole placement: big picture R 1 ) + E 1 ) B c 1 ) D 1 )= B d 1 ) A d 1 ) + d B p 1 ) A c 1 )A d 1 ) + A p 1 ) Y 1 ) Diophantine equation in Pole placement: d B p 1 ) B c 1 ) + A p 1 ) A c 1 ) A d 1 ) = 1 + η η η q q }{{} η 1 ) Questions: what are the constraints for choosing η 1)? how to guarantee unique solution in Diophantine equation? Lecture 13: Internal Model Principle and Repetitive Control ME

8 Design and analysis procedure General procedure of control design: Problem definition Control design for solution current stage) Prove stability Prove stability robustness Case study or implementation results Lecture 13: Internal Model Principle and Repetitive Control ME , Pole placement: details Theorem Diophantine equation) Given η 1) = 1 + η η η q q α 1) = 1 + α α n n β 1) = β β β n n The Diophantine equation α 1) σ 1) + β 1) γ 1) = η 1) can be solved uniquely for σ 1) and γ 1) σ 1) = 1 + σ σ n1 n1) γ 1) = γ 0 + γ γ n1 n1) if the numerators of α 1) and β 1) are coprime and deg η 1)) = q 2n 1 Lecture 13: Internal Model Principle and Repetitive Control ME

9 1, Pole placement: details Proof of Diophantine equation Theorem key ideas): α 1) σ 1) +β 1) γ 1) = η 1) }{{}}{{} unknown unknown Matching the coefficients of i gives see one numerical example in course reader) σ 1 σ 2 S. σ n1 γ 0. γ n1 + α 1 α 2. α n 0. 0 = η 1 η 2. η n1 η n. η 2n1 The coprime condition assures S is invertible. deg η 1) 2n 1 assures the proper dimension on the right hand side of the equality. Lecture 13: Internal Model Principle and Repetitive Control ME , Prototype repetitive control: simple case D 1 )= B d 1 ) A d 1 ) R 1 ) + E 1 ) + C 1) d B p 1 ) + A p 1 ) Y 1 ) A d 1 ) = 1 N If all poles and eros of the plant are stable, then prototype repetitive control uses C 1) = k r N+d A p 1 ) ) 1 N B p 1 ) Theorem Prototype repetitive control) Under the assumptions above, the closed-loop system is asymptotically stable for 0 < k r < 2 Lecture 13: Internal Model Principle and Repetitive Control ME

10 2, Prototype repetitive control: stability Proof of Theorem on prototype repetitive control: From 1 + k r N+d A p 1 ) ) d B p 1 ) 1 N B p 1 ) A p 1 ) the closed-loop characteristic equation is = 0 A p 1 ) B p 1 )[ 1 1 k r ) N] = 0 roots of A p 1 ) B p 1 ) = 0 are all stable roots of 1 1 k r ) N = 0 are 1 k r 1 N e j 2πi N, i = 0,±1,..., for 0 < kr 1 1 k r 1 N e j 2πi N + π N ), i = 0,±1,..., for 1 < kr which are all inside the unit circle Lecture 13: Internal Model Principle and Repetitive Control ME , Prototype repetitive control: stability robustness Consider the case with plant uncertainty dk) rk)+ k r N+d A p 1 ) + d B p 1 ) 1 N )B p 1 ) + A p 1 ) )) yk) N open-loop poles on the unit circle Root locus example: N = 4, 1 + 1) = q/ p) Imaginary Axis Root Locus Real Axis k r > 0, the closedloop system is now unstable! Lecture 13: Internal Model Principle and Repetitive Control ME

11 2, Prototype repetitive control: stability robustness dk) rk)+ + C 1) d B p 1 ) + A p 1 ) )) yk) To make the controller robust to plant uncertainties, do instead C 1) = k rq, 1 ) N+d A p 1 ) 1 q, 1 ) N) B p 1 ) 1) q, 1 ) : low-pass filter. e.g. ero-phase low pass α α 0 + α 1 α 0 + 2α 1 which shifts the marginary stable open-loop poles to be inside the unit circle: A p 1 ) B p 1 )[ 1 1 k r )q, 1 ) N] = 0 Lecture 13: Internal Model Principle and Repetitive Control ME , Prototype repetitive control: extension D 1 )= B d 1 ) A d 1 ) R 1 ) + E 1 ) + C 1) d B p 1 ) + A p 1 ) Y 1 ) If poles of the plant are stable but NOT all eros are stable, let B p 1 ) = Bp 1 )B p + 1 ) [Bp 1 ) the uncancellable part] and C 1) = k r N+µ A p 1 )B p ) µ 1 N )B + p 1 ) d b, b > max ω [0,π] B p e jω ) 2 2) Similar as before, can show that the closed-loop system is stable in-class exercise). Lecture 13: Internal Model Principle and Repetitive Control ME

12 2, Prototype repetitive control: extension Exercise: analye the stability of D 1 )= B d 1 ) A d 1 ) R 1 ) + E 1 ) + C 1) d B p 1 ) + A p 1 ) Y 1 ) C 1) = k r N+µ A p 1 )Bp ) µ 1 N )B p + 1 ) d b, b > max ω [0,π] B p e jω ) 2 3) Key steps: B p e jω )Bp e jω ) b < 1; k r Bp e jω )Bp e jω ) b 1 < 1; all roots from [ kr N Bp )Bp 1 ) ] = 0 b are inside the unit circle. Lecture 13: Internal Model Principle and Repetitive Control ME , Prototype repetitive control: extension dk) rk)+ + C 1) d B p 1 ) + A p 1 ) )) yk) Robust version in the presence of plant uncertainties: C 1) = k r N+µ q, 1 )A p 1 )B p ) µ 1 q, 1 ) N )B + p 1 ) d b 4) where q, 1 ) : low-pass filter. e.g. ero-phase low pass α α 0 + α 1 α 0 + 2α 1 and µ is the order of B p ) Lecture 13: Internal Model Principle and Repetitive Control ME

13 Example dk) rk)+ + C 1) + Plant yk) disturbance period: N = 10; nominal plant: d B p 1 ) 1 A p 1 = ) ) ) C 1) = k r ) ) q, 1 ) q, 1 ) 10 ) Lecture 13: Internal Model Principle and Repetitive Control ME Additional reading ME233 course reader X. Chen and M. Tomiuka, An Enhanced Repetitive Control Algorithm using the Structure of Disturbance Observer, in Proceedings of 2012 IEEE/ASME International Conference on Advanced Intelligent Mechatronics, Taiwan, Jul , 2012, pp X. Chen and M. Tomiuka, New Repetitive Control with Improved Steady-state Performance and Accelerated Transient, IEEE Transactions on Control Systems Technology, vol. 22, no. 2, pp pages), Mar Lecture 13: Internal Model Principle and Repetitive Control ME

14 Summary 1. Big Picture review of integral control in PID design 2. Internal Model Principle theorems typical disturbance structures 3. Repetitive Control use of internal model principle design by pole placement design by stable pole-ero cancellation Lecture 13: Internal Model Principle and Repetitive Control ME

EECE 460 : Control System Design

EECE 460 : Control System Design EECE 460 : Control System Design SISO Pole Placement Guy A. Dumont UBC EECE January 2011 Guy A. Dumont (UBC EECE) EECE 460: Pole Placement January 2011 1 / 29 Contents 1 Preview 2 Polynomial Pole Placement

More information

Control System Design

Control System Design ELEC ENG 4CL4: Control System Design Notes for Lecture #14 Wednesday, February 5, 2003 Dr. Ian C. Bruce Room: CRL-229 Phone ext.: 26984 Email: ibruce@mail.ece.mcmaster.ca Chapter 7 Synthesis of SISO Controllers

More information

Control Systems Design

Control Systems Design ELEC4410 Control Systems Design Lecture 3, Part 2: Introduction to Affine Parametrisation School of Electrical Engineering and Computer Science Lecture 3, Part 2: Affine Parametrisation p. 1/29 Outline

More information

Analysis of SISO Control Loops

Analysis of SISO Control Loops Chapter 5 Analysis of SISO Control Loops Topics to be covered For a given controller and plant connected in feedback we ask and answer the following questions: Is the loop stable? What are the sensitivities

More information

Digital Control: Summary # 7

Digital 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 information

DESIGN USING TRANSFORMATION TECHNIQUE CLASSICAL METHOD

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

More information

Chapter 13 Digital Control

Chapter 13 Digital Control Chapter 13 Digital Control Chapter 12 was concerned with building models for systems acting under digital control. We next turn to the question of control itself. Topics to be covered include: why one

More information

Control Systems. Root Locus & Pole Assignment. L. Lanari

Control 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 information

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

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

More information

Raktim Bhattacharya. . AERO 422: Active Controls for Aerospace Vehicles. Basic Feedback Analysis & Design

Raktim 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 information

Chapter 2. Classical Control System Design. Dutch Institute of Systems and Control

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 Steady-state Steady-state errors errors Type Type k k systems systems Integral Integral

More information

Controls Problems for Qualifying Exam - Spring 2014

Controls 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 information

Repetitive control : Power Electronics. Applications

Repetitive control : Power Electronics. Applications Repetitive control : Power Electronics Applications Ramon Costa Castelló Advanced Control of Energy Systems (ACES) Instituto de Organización y Control (IOC) Universitat Politècnica de Catalunya (UPC) Barcelona,

More information

Outline. Classical Control. Lecture 1

Outline. 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 information

Module 3F2: Systems and Control EXAMPLES PAPER 2 ROOT-LOCUS. Solutions

Module 3F2: Systems and Control EXAMPLES PAPER 2 ROOT-LOCUS. Solutions Cambridge University Engineering Dept. Third Year Module 3F: Systems and Control EXAMPLES PAPER ROOT-LOCUS Solutions. (a) For the system L(s) = (s + a)(s + b) (a, b both real) show that the root-locus

More information

Robust Internal Model Control for Impulse Elimination of Singular Systems

Robust Internal Model Control for Impulse Elimination of Singular Systems International Journal of Control Science and Engineering ; (): -7 DOI:.59/j.control.. Robust Internal Model Control for Impulse Elimination of Singular Systems M. M. Share Pasandand *, H. D. Taghirad Department

More information

Lecture 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 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 information

Alireza Mousavi Brunel University

Alireza Mousavi Brunel University Alireza Mousavi Brunel University 1 » Control Process» Control Systems Design & Analysis 2 Open-Loop Control: Is normally a simple switch on and switch off process, for example a light in a room is switched

More information

Internal Model Principle

Internal Model Principle Internal Model Principle If the reference signal, or disturbance d(t) satisfy some differential equation: e.g. d n d dt n d(t)+γ n d d 1 dt n d 1 d(t)+...γ d 1 dt d(t)+γ 0d(t) =0 d n d 1 then, taking Laplace

More information

Observer Based Friction Cancellation in Mechanical Systems

Observer Based Friction Cancellation in Mechanical Systems 2014 14th International Conference on Control, Automation and Systems (ICCAS 2014) Oct. 22 25, 2014 in KINTEX, Gyeonggi-do, Korea Observer Based Friction Cancellation in Mechanical Systems Caner Odabaş

More information

Root Locus Design Example #3

Root Locus Design Example #3 Root Locus Design Example #3 A. Introduction The system represents a linear model for vertical motion of an underwater vehicle at zero forward speed. The vehicle is assumed to have zero pitch and roll

More information

Control System Design

Control 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 information

Robust Control. 2nd class. Spring, 2018 Instructor: Prof. Masayuki Fujita (S5-303B) Tue., 17th April, 2018, 10:45~12:15, S423 Lecture Room

Robust Control. 2nd class. Spring, 2018 Instructor: Prof. Masayuki Fujita (S5-303B) Tue., 17th April, 2018, 10:45~12:15, S423 Lecture Room Robust Control Spring, 2018 Instructor: Prof. Masayuki Fujita (S5-303B) 2nd class Tue., 17th April, 2018, 10:45~12:15, S423 Lecture Room 2. Nominal Performance 2.1 Weighted Sensitivity [SP05, Sec. 2.8,

More information

Control Systems II. ETH, MAVT, IDSC, Lecture 4 17/03/2017. G. Ducard

Control 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 information

A NEW APPROACH TO MIXED H 2 /H OPTIMAL PI/PID CONTROLLER DESIGN

A NEW APPROACH TO MIXED H 2 /H OPTIMAL PI/PID CONTROLLER DESIGN Copyright 2002 IFAC 15th Triennial World Congress, Barcelona, Spain A NEW APPROACH TO MIXED H 2 /H OPTIMAL PI/PID CONTROLLER DESIGN Chyi Hwang,1 Chun-Yen Hsiao Department of Chemical Engineering National

More information

2.004 Dynamics and Control II Spring 2008

2.004 Dynamics and Control II Spring 2008 MT OpenCourseWare http://ocw.mit.edu 2.004 Dynamics and Control Spring 2008 or information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. Reading: ise: Chapter 8 Massachusetts

More information

Rational Implementation of Distributed Delay Using Extended Bilinear Transformations

Rational Implementation of Distributed Delay Using Extended Bilinear Transformations Rational Implementation of Distributed Delay Using Extended Bilinear Transformations Qing-Chang Zhong zhongqc@ieee.org, http://come.to/zhongqc School of Electronics University of Glamorgan United Kingdom

More information

Introduction to Feedback Control

Introduction to Feedback Control Introduction to Feedback Control Control System Design Why Control? Open-Loop vs Closed-Loop (Feedback) Why Use Feedback Control? Closed-Loop Control System Structure Elements of a Feedback Control System

More information

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

Introduction. Performance and Robustness (Chapter 1) Advanced Control Systems Spring / 31 Introduction Classical Control Robust Control u(t) y(t) G u(t) G + y(t) G : nominal model G = G + : plant uncertainty Uncertainty sources : Structured : parametric uncertainty, multimodel uncertainty Unstructured

More information

An Analysis and Synthesis of Internal Model Principle Type Controllers

An Analysis and Synthesis of Internal Model Principle Type Controllers 29 American Control Conference Hyatt Regency Riverfront, St. Louis, MO, USA June -2, 29 WeA5.6 An Analysis and Synthesis of Internal Model Principle Type Controllers Yigang Wang, Kevin C. Chu and Tsu-Chin

More information

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

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

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

Classify a transfer function to see which order or ramp it can follow and with which expected error. Dr. J. Tani, Prof. Dr. E. Frazzoli 5-059-00 Control Systems I (Autumn 208) Exercise Set 0 Topic: Specifications for Feedback Systems Discussion: 30.. 208 Learning objectives: The student can grizzi@ethz.ch,

More information

Automatic Control (TSRT15): Lecture 7

Automatic 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: 13-282226 Office: B-house extrance 25-27 Outline 2 Feedforward

More information

Control of Manufacturing Processes

Control of Manufacturing Processes Control of Manufacturing Processes Subject 2.830 Spring 2004 Lecture #19 Position Control and Root Locus Analysis" April 22, 2004 The Position Servo Problem, reference position NC Control Robots Injection

More information

Control System Design

Control 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 information

PD, PI, PID Compensation. M. Sami Fadali Professor of Electrical Engineering University of Nevada

PD, PI, PID Compensation. M. Sami Fadali Professor of Electrical Engineering University of Nevada PD, PI, PID Compensation M. Sami Fadali Professor of Electrical Engineering University of Nevada 1 Outline PD compensation. PI compensation. PID compensation. 2 PD Control L= loop gain s cl = desired closed-loop

More information

IMC based automatic tuning method for PID controllers in a Smith predictor configuration

IMC based automatic tuning method for PID controllers in a Smith predictor configuration Computers and Chemical Engineering 28 (2004) 281 290 IMC based automatic tuning method for PID controllers in a Smith predictor configuration Ibrahim Kaya Department of Electrical and Electronics Engineering,

More information

Pole-Placement Design A Polynomial Approach

Pole-Placement Design A Polynomial Approach TU Berlin Discrete-Time Control Systems 1 Pole-Placement Design A Polynomial Approach Overview A Simple Design Problem The Diophantine Equation More Realistic Assumptions TU Berlin Discrete-Time Control

More information

Raktim Bhattacharya. . AERO 422: Active Controls for Aerospace Vehicles. Frequency Response-Design Method

Raktim Bhattacharya. . AERO 422: Active Controls for Aerospace Vehicles. Frequency Response-Design Method .. AERO 422: Active Controls for Aerospace Vehicles Frequency Response- Method Raktim Bhattacharya Laboratory For Uncertainty Quantification Aerospace Engineering, Texas A&M University. ... Response to

More information

Chapter 15 - Solved Problems

Chapter 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 information

Performance of Feedback Control Systems

Performance 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 Steady-state Error and Type 0, Type

More information

Automatic 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 Automatic Control 2 Loop shaping Prof. Alberto Bemporad University of Trento Academic year 21-211 Prof. Alberto Bemporad (University of Trento) Automatic Control 2 Academic year 21-211 1 / 39 Feedback

More information

2.004 Dynamics and Control II Spring 2008

2.004 Dynamics and Control II Spring 2008 MT OpenCourseWare http://ocw.mit.edu.004 Dynamics and Control Spring 008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. Massachusetts nstitute of Technology

More information

FEL3210 Multivariable Feedback Control

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

More information

Control of Electromechanical Systems

Control 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 information

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

Dr Ian R. Manchester Dr Ian R. Manchester AMME 3500 : Review Week Date Content Notes 1 6 Mar Introduction 2 13 Mar Frequency Domain Modelling 3 20 Mar Transient Performance and the s-plane 4 27 Mar Block Diagrams Assign 1 Due 5 3 Apr Feedback System Characteristics

More information

AN INTRODUCTION TO THE CONTROL THEORY

AN INTRODUCTION TO THE CONTROL THEORY Open-Loop controller An Open-Loop (OL) controller is characterized by no direct connection between the output of the system and its input; therefore external disturbance, non-linear dynamics and parameter

More information

Dr Ian R. Manchester

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

More information

Dr Ian R. Manchester Dr Ian R. Manchester AMME 3500 : Root Locus

Dr Ian R. Manchester Dr Ian R. Manchester AMME 3500 : Root Locus Week Content Notes 1 Introduction 2 Frequency Domain Modelling 3 Transient Performance and the s-plane 4 Block Diagrams 5 Feedback System Characteristics Assign 1 Due 6 Root Locus 7 Root Locus 2 Assign

More information

INTRODUCTION TO DIGITAL CONTROL

INTRODUCTION TO DIGITAL CONTROL ECE4540/5540: Digital Control Systems INTRODUCTION TO DIGITAL CONTROL.: Introduction In ECE450/ECE550 Feedback Control Systems, welearnedhow to make an analog controller D(s) to control a linear-time-invariant

More information

(Continued on next page)

(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 information

Lecture: Sampling. Automatic Control 2. Sampling. Prof. Alberto Bemporad. University of Trento. Academic year

Lecture: Sampling. Automatic Control 2. Sampling. Prof. Alberto Bemporad. University of Trento. Academic year Automatic Control 2 Sampling Prof. Alberto Bemporad University of rento Academic year 2010-2011 Prof. Alberto Bemporad (University of rento) Automatic Control 2 Academic year 2010-2011 1 / 31 ime-discretization

More information

Locus 6. More Root S 0 L U T I 0 N S. Note: All references to Figures and Equations whose numbers are not preceded by an "S"refer to the textbook.

Locus 6. More Root S 0 L U T I 0 N S. Note: All references to Figures and Equations whose numbers are not preceded by an Srefer to the textbook. S 0 L U T I 0 N S More Root Locus 6 Note: All references to Figures and Equations whose numbers are not preceded by an "S"refer to the textbook. For the first transfer function a(s), the root locus is

More information

Root Locus. Motivation Sketching Root Locus Examples. School of Mechanical Engineering Purdue University. ME375 Root Locus - 1

Root Locus. Motivation Sketching Root Locus Examples. School of Mechanical Engineering Purdue University. ME375 Root Locus - 1 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: D 0.09 Position

More information

Step input, ramp input, parabolic input and impulse input signals. 2. What is the initial slope of a step response of a first order system?

Step input, ramp input, parabolic input and impulse input signals. 2. What is the initial slope of a step response of a first order system? IC6501 CONTROL SYSTEM UNIT-II TIME RESPONSE PART-A 1. What are the standard test signals employed for time domain studies?(or) List the standard test signals used in analysis of control systems? (April

More information

Stability of Parameter Adaptation Algorithms. Big picture

Stability of Parameter Adaptation Algorithms. Big picture ME5895, UConn, Fall 215 Prof. Xu Chen Big picture For ˆθ (k + 1) = ˆθ (k) + [correction term] we haven t talked about whether ˆθ(k) will converge to the true value θ if k. We haven t even talked about

More information

ECE 486 Control Systems

ECE 486 Control Systems ECE 486 Control Systems Spring 208 Midterm #2 Information Issued: April 5, 208 Updated: April 8, 208 ˆ This document is an info sheet about the second exam of ECE 486, Spring 208. ˆ Please read the following

More information

Design and Tuning of Fractional-order PID Controllers for Time-delayed Processes

Design and Tuning of Fractional-order PID Controllers for Time-delayed Processes Design and Tuning of Fractional-order PID Controllers for Time-delayed Processes Emmanuel Edet Technology and Innovation Centre University of Strathclyde 99 George Street Glasgow, United Kingdom emmanuel.edet@strath.ac.uk

More information

School of Mechanical Engineering Purdue University. DC Motor Position Control The block diagram for position control of the servo table is given by:

School of Mechanical Engineering Purdue University. DC Motor Position Control The block diagram for position control of the servo table is given by: Root Locus Motivation Sketching Root Locus Examples ME375 Root Locus - 1 Servo Table Example DC Motor Position Control The block diagram for position control of the servo table is given by: θ D 0.09 See

More information

Linear State Feedback Controller Design

Linear 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 information

The 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 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 information

ME 304 CONTROL SYSTEMS Spring 2016 MIDTERM EXAMINATION II

ME 304 CONTROL SYSTEMS Spring 2016 MIDTERM EXAMINATION II ME 30 CONTROL SYSTEMS Spring 06 Course Instructors Dr. Tuna Balkan, Dr. Kıvanç Azgın, Dr. Ali Emre Turgut, Dr. Yiğit Yazıcıoğlu MIDTERM EXAMINATION II May, 06 Time Allowed: 00 minutes Closed Notes and

More information

Laplace Transform Analysis of Signals and Systems

Laplace Transform Analysis of Signals and Systems Laplace Transform Analysis of Signals and Systems Transfer Functions Transfer functions of CT systems can be found from analysis of Differential Equations Block Diagrams Circuit Diagrams 5/10/04 M. J.

More information

6.241 Dynamic Systems and Control

6.241 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 information

Robust Performance Example #1

Robust 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 information

Systems Analysis and Control

Systems Analysis and Control Systems Analysis and Control Matthew M. Peet Illinois Institute of Technology Lecture : Different Types of Control Overview In this Lecture, you will learn: Limits of Proportional Feedback Performance

More information

Disturbance Rejection in Parameter-varying Web-winding Systems

Disturbance Rejection in Parameter-varying Web-winding Systems Proceedings of the 17th World Congress The International Federation of Automatic Control Disturbance Rejection in Parameter-varying Web-winding Systems Hua Zhong Lucy Y. Pao Electrical and Computer Engineering

More information

H-infinity Model Reference Controller Design for Magnetic Levitation System

H-infinity Model Reference Controller Design for Magnetic Levitation System H.I. Ali Control and Systems Engineering Department, University of Technology Baghdad, Iraq 6043@uotechnology.edu.iq H-infinity Model Reference Controller Design for Magnetic Levitation System Abstract-

More information

Time Response of Systems

Time Response of Systems Chapter 0 Time Response of Systems 0. Some Standard Time Responses Let us try to get some impulse time responses just by inspection: Poles F (s) f(t) s-plane Time response p =0 s p =0,p 2 =0 s 2 t p =

More information

Lecture 10: Proportional, Integral and Derivative Actions

Lecture 10: Proportional, Integral and Derivative Actions MCE441: Intr. Linear Control Systems Lecture 10: Proportional, Integral and Derivative Actions Stability Concepts BIBO Stability and The Routh-Hurwitz Criterion Dorf, Sections 6.1, 6.2, 7.6 Cleveland State

More information

EE3CL4: Introduction to Linear Control Systems

EE3CL4: Introduction to Linear Control Systems 1 / 17 EE3CL4: Introduction to Linear Control Systems Section 7: McMaster University Winter 2018 2 / 17 Outline 1 4 / 17 Cascade compensation Throughout this lecture we consider the case of H(s) = 1. We

More information

Chapter 6 - Solved Problems

Chapter 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 Z-N tuning strategy for the nominal

More information

Robust Tuning of Power System Stabilizers Using Coefficient Diagram Method

Robust Tuning of Power System Stabilizers Using Coefficient Diagram Method International Journal of Electrical Engineering. ISSN 0974-2158 Volume 7, Number 2 (2014), pp. 257-270 International Research Publication House http://www.irphouse.com Robust Tuning of Power System Stabilizers

More information

Digital Control Systems

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

More information

CDS 101/110a: Lecture 8-1 Frequency Domain Design

CDS 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 information

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

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

More information

ECEN 605 LINEAR SYSTEMS. Lecture 20 Characteristics of Feedback Control Systems II Feedback and Stability 1/27

ECEN 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 information

Fundamental of Control Systems Steady State Error Lecturer: Dr. Wahidin Wahab M.Sc. Aries Subiantoro, ST. MSc.

Fundamental of Control Systems Steady State Error Lecturer: Dr. Wahidin Wahab M.Sc. Aries Subiantoro, ST. MSc. Fundamental of Control Systems Steady State Error Lecturer: Dr. Wahidin Wahab M.Sc. Aries Subiantoro, ST. MSc. Electrical Engineering Department University of Indonesia 2 Steady State Error How well can

More information

Systems Analysis and Control

Systems Analysis and Control Systems Analysis and Control Matthew M. Peet Illinois Institute of Technology Lecture 6: Generalized and Controlled Design Overview In this Lecture, you will learn: Generalized? What about changing OTHER

More information

Systems Analysis and Control

Systems Analysis and Control Systems Analysis and Control Matthew M. Peet Arizona State University Lecture 6: Generalized and Controller Design Overview In this Lecture, you will learn: Generalized? What about changing OTHER parameters

More information

Robust Loop Shaping Controller Design for Spectral Models by Quadratic Programming

Robust Loop Shaping Controller Design for Spectral Models by Quadratic Programming Robust Loop Shaping Controller Design for Spectral Models by Quadratic Programming Gorka Galdos, Alireza Karimi and Roland Longchamp Abstract A quadratic programming approach is proposed to tune fixed-order

More information

Automatic Control (TSRT15): Lecture 4

Automatic Control (TSRT15): Lecture 4 Automatic Control (TSRT15): Lecture 4 Tianshi Chen Division of Automatic Control Dept. of Electrical Engineering Email: tschen@isy.liu.se Phone: 13-282226 Office: B-house extrance 25-27 Review of the last

More information

Control Systems I. Lecture 9: The Nyquist condition

Control Systems I. Lecture 9: The Nyquist condition Control Systems I Lecture 9: The Nyquist condition Readings: Åstrom and Murray, Chapter 9.1 4 www.cds.caltech.edu/~murray/amwiki/index.php/first_edition Jacopo Tani Institute for Dynamic Systems and Control

More information

The requirements of a plant may be expressed in terms of (a) settling time (b) damping ratio (c) peak overshoot --- in time domain

The requirements of a plant may be expressed in terms of (a) settling time (b) damping ratio (c) peak overshoot --- in time domain Compensators To improve the performance of a given plant or system G f(s) it may be necessary to use a compensator or controller G c(s). Compensator Plant G c (s) G f (s) The requirements of a plant may

More information

Lecture 2: Discrete-time Linear Quadratic Optimal Control

Lecture 2: Discrete-time Linear Quadratic Optimal Control ME 33, U Berkeley, Spring 04 Xu hen Lecture : Discrete-time Linear Quadratic Optimal ontrol Big picture Example onvergence of finite-time LQ solutions Big picture previously: dynamic programming and finite-horizon

More information

Control Systems I Lecture 10: System Specifications

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

More information

Professor Fearing EE C128 / ME C134 Problem Set 4 Solution Fall 2010 Jansen Sheng and Wenjie Chen, UC Berkeley. control input. error Controller D(s)

Professor Fearing EE C128 / ME C134 Problem Set 4 Solution Fall 2010 Jansen Sheng and Wenjie Chen, UC Berkeley. control input. error Controller D(s) Professor Fearing EE C18 / ME C13 Problem Set Solution Fall 1 Jansen Sheng and Wenjie Chen, UC Berkeley reference input r(t) + Σ error e(t) Controller D(s) grid 8 pixels control input u(t) plant G(s) output

More information

Example on Root Locus Sketching and Control Design

Example on Root Locus Sketching and Control Design Example on Root Locus Sketching and Control Design MCE44 - Spring 5 Dr. Richter April 25, 25 The following figure represents the system used for controlling the robotic manipulator of a Mars Rover. We

More information

The loop shaping paradigm. Lecture 7. Loop analysis of feedback systems (2) Essential specifications (2)

The 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 information

Control System Design

Control System Design ELEC ENG 4CL4: Control System Design Notes for Lecture #36 Dr. Ian C. Bruce Room: CRL-229 Phone ext.: 26984 Email: ibruce@mail.ece.mcmaster.ca Friday, April 4, 2003 3. Cascade Control Next we turn to an

More information

Control Systems Design

Control Systems Design ELEC4410 Control Systems Design Lecture 13: Stability Julio H. Braslavsky julio@ee.newcastle.edu.au School of Electrical Engineering and Computer Science Lecture 13: Stability p.1/20 Outline Input-Output

More information

Lecture 9: Input Disturbance A Design Example Dr.-Ing. Sudchai Boonto

Lecture 9: Input Disturbance A Design Example Dr.-Ing. Sudchai Boonto Dr-Ing Sudchai Boonto Department of Control System and Instrumentation Engineering King Mongkuts Unniversity of Technology Thonburi Thailand d u g r e u K G y The sensitivity S is the transfer function

More information

TRACKING AND DISTURBANCE REJECTION

TRACKING AND DISTURBANCE REJECTION TRACKING AND DISTURBANCE REJECTION Sadegh Bolouki Lecture slides for ECE 515 University of Illinois, Urbana-Champaign Fall 2016 S. Bolouki (UIUC) 1 / 13 General objective: The output to track a reference

More information

Radar Dish. Armature controlled dc motor. Inside. θ r input. Outside. θ D output. θ m. Gearbox. Control Transmitter. Control. θ D.

Radar Dish. Armature controlled dc motor. Inside. θ r input. Outside. θ D output. θ m. Gearbox. Control Transmitter. Control. θ D. Radar Dish ME 304 CONTROL SYSTEMS Mechanical Engineering Department, Middle East Technical University Armature controlled dc motor Outside θ D output Inside θ r input r θ m Gearbox Control Transmitter

More information

Compensator Design to Improve Transient Performance Using Root Locus

Compensator Design to Improve Transient Performance Using Root Locus 1 Compensator Design to Improve Transient Performance Using Root Locus Prof. Guy Beale Electrical and Computer Engineering Department George Mason University Fairfax, Virginia Correspondence concerning

More information

Analysis and Synthesis of Single-Input Single-Output Control Systems

Analysis and Synthesis of Single-Input Single-Output Control Systems Lino Guzzella Analysis and Synthesis of Single-Input Single-Output 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 information

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

ROOT LOCUS. Consider the system. Root locus presents the poles of the closed-loop system when the gain K changes from 0 to. H(s) H ( s) = ( s) C1 ROOT LOCUS Consider the system R(s) E(s) C(s) + K G(s) - H(s) C(s) R(s) = K G(s) 1 + K G(s) H(s) Root locus presents the poles of the closed-loop system when the gain K changes from 0 to 1+ K G ( s)

More information

Outline. Classical Control. Lecture 5

Outline. Classical Control. Lecture 5 Outline Outline Outline 1 What is 2 Outline What is Why use? Sketching a 1 What is Why use? Sketching a 2 Gain Controller Lead Compensation Lag Compensation What is Properties of a General System Why use?

More information

9. Two-Degrees-of-Freedom Design

9. Two-Degrees-of-Freedom Design 9. Two-Degrees-of-Freedom Design In some feedback schemes we have additional degrees-offreedom outside the feedback path. For example, feed forwarding known disturbance signals or reference signals. In

More information

Automatic Control Systems (FCS) Lecture- 8 Steady State Error

Automatic Control Systems (FCS) Lecture- 8 Steady State Error Automatic Control Systems (FCS) Lecture- 8 Steady State Error Introduction Any physical control system inherently suffers steady-state error in response to certain types of inputs. A system may have no

More information