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Solving Systems with Gaussian Elimination

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Systems with Hysteresis Systems with Hysteresis Analysis, Identification and Control using the Bouc–Wen Model Fayçal Ikhouane Department of Applied Mathematics III School of Technical Industrial Engineering Technical University of Catalunya Barcelona, Spain José Rodellar Department of Applied Mathematics III School of Civil Engineering Technical University of Catalunya Barcelona, Spain Copyright © 2007 John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex PO19 8SQ, England Telephone +44 1243 779777 Email (for orders and customer service enquiries): cs-books@wiley.co.uk Visit our Home Page on www.wiley.com All Rights Reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except under the terms of the Copyright, Designs and Patents Act 1988 or under the terms of a licence issued by the Copyright Licensing Agency Ltd, 90 Tottenham Court Road, London W1T 4LP, UK, without the permission in writing of the Publisher. Requests to the Publisher should be addressed to the Permissions Department, John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex PO19 8SQ, England, or emailed to permreq@wiley.co.uk, or faxed to (+44) 1243 770620. This publication is designed to provide accurate and authoritative information in regard to the subject matter covered. It is sold on the understanding that the Publisher is not engaged in rendering professional services. If professional advice or other expert assistance is required, the services of a competent professional should be sought. Other Wiley Editorial Offices John Wiley & Sons Inc., 111 River Street, Hoboken, NJ 07030, USA Jossey-Bass, 989 Market Street, San Francisco, CA 94103-1741, USA Wiley-VCH Verlag GmbH, Boschstr. 12, D-69469 Weinheim, Germany John Wiley & Sons Australia Ltd, 42 McDougall Street, Milton, Queensland 4064, Australia John Wiley & Sons (Asia) Pte Ltd, 2 Clementi Loop #02-01, Jin Xing Distripark, Singapore 129809 John Wiley & Sons Canada Ltd, 6045 Freemont Blvd, Mississauga, ONT, L5R 4J3 Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic books. Anniversary Logo Design: Richard J. Pacifico Library of Congress Cataloging in Publication Data Ikhouane, Fayçal. Systems with hysteresis : analysis, identification and control using the Bouc-Wen model / Fayçal Ikhouane, José Rodellar. p. cm. Includes bibliographical references and index. ISBN 978-0-470-03236-7 (cloth) 1. Hysteresis—Mathematical models. I. Rodellar, José. II. Title. QC754.2.H9I34 2007 621—dc22 2007019894 British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN-13 978-0-470-03236-7 Typeset in 11/13pt Sabon by Integra Software Services Pvt. Ltd, Pondicherry, India Printed and bound in Great Britain by TJ International, Padstow, Cornwall This book is printed on acid-free paper responsibly manufactured from sustainable forestry in which at least two trees are planted for each one used for paper production. To my mother and brothers, Imad and Hicham F. Ikhouane To Anna, Laura and Silvia J. Rodellar Contents Preface xi List of Figures xv List of Tables xix 1 Introduction 1 1.1 Objective and Contents of the Book 1 1.2 The Bouc–Wen Model: Origin and Literature Review 5 2 Physical Consistency of the Bouc–Wen Model 13 2.1 Introduction 13 2.2 BIBO Stability of the Bouc–Wen Model 16 2.2.1 The Solving Systems with Gaussian Elimination Solving Systems with Gaussian Elimination By: OpenStaxCollege German mathematician Carl Friedrich Gauss (1777–1855) 1/29 Solving Systems with Gaussian Elimination Carl Friedrich Gauss lived during the late 18th century and early 19th century, but he is still considered one of the most prolific mathematicians in history His contributions to the science of mathematics and physics span fields such as algebra, number theory, analysis, differential geometry, astronomy, and optics, among others His discoveries regarding matrix theory changed the way mathematicians have worked for the last two centuries We first encountered Gaussian elimination in Systems of Linear Equations: Two Variables In this section, we will revisit this technique for solving systems, this time using matrices Writing the Augmented Matrix of a System of Equations A matrix can serve as a device for representing and solving a system of equations To express a system in matrix form, we extract the coefficients of the variables and the constants, and these become the entries of the matrix We use a vertical line to separate the coefficient entries from the constants, essentially replacing the equal signs When a system is written in this form, we call it an augmented matrix For example, consider the following × system of equations 3x + 4y = 4x−2y = We can write this system as an augmented matrix: [ 4 −2 | ] We can also write a matrix containing just the coefficients This is called the coefficient matrix [ ] 4 −2 A three-by-three system of equations such as 3x − y − z = x+y=5 2x−3z = 2/29 Solving Systems with Gaussian Elimination has a coefficient matrix [ −1 −1 1 −3 ] and is represented by the augmented matrix [ −1 −1 1 −3 |] Notice that the matrix is written so that the variables line up in their own columns: x-terms go in the first column, y-terms in the second column, and z-terms in the third column It is very important that each equation is written in standard form ax + by + cz = d so that the variables line up When there is a missing variable term in an equation, the coefficient is How To Given a system of equations, write an augmented matrix Write the coefficients of the x-terms as the numbers down the first column Write the coefficients of the y-terms as the numbers down the second column If there are z-terms, write the coefficients as the numbers down the third column Draw a vertical line and write the constants to the right of the line Writing the Augmented Matrix for a System of Equations Write the augmented matrix for the given system of equations x + 2y − z = 2x − y + 2z = x − 3y + 3z = The augmented matrix displays the coefficients of the variables, and an additional column for the constants 3/29 Solving Systems with Gaussian Elimination [ |] −1 −1 −3 3 Try It Write the augmented matrix of the given system of equations 4x−3y = 11 3x + 2y = [ | ] −3 11 Writing a System of Equations from an Augmented Matrix We can use augmented matrices to help us solve systems of equations because they simplify operations when the systems are not encumbered by the variables However, it is important to understand how to move back and forth between formats in order to make finding solutions smoother and more intuitive Here, we will use the information in an augmented matrix to write the system of equations in standard form Writing a System of Equations from an Augmented Matrix Form Find the system of equations from the augmented matrix [ −3 −5 −5 −4 −3 | ] −2 When the columns represent the variables x, y, and z, [ −3 −5 −5 −4 −3 | ] −2 x − 3y − 5z = − → 2x − 5y − 4z = −3x + 5y + 4z = Try It 4/29 Solving Systems with Gaussian Elimination Write the system of equations from the augmented matrix [ −1 −1 x − y + | ] −9 z=5 2x − y + 3z = y + z = −9 Performing Row Operations on a Matrix Now that we can write systems of equations in augmented matrix form, we will examine the various row operations that can be performed on a matrix, such as addition, multiplication by a constant, and interchanging rows Performing row operations on a matrix is the method we use for solving a system of equations In order to solve the system of equations, we want to convert the matrix to row-echelon form, in which there are ones down the main diagonal from the upper left corner to the lower right corner, and zeros in every position below the main diagonal as shown Row-echelon form [ ] a b d 0 We use row operations corresponding to equation operations to obtain a new matrix that is row-equivalent in a simpler form Here are the guidelines to obtaining row-echelon form In any nonzero row, the first nonzero number is a It is called a leading Any all-zero rows are placed at the bottom on the matrix Any leading is below and to the right of a previous leading Any column containing a leading has zeros in all other positions in the column 5/29 Solving Systems with Gaussian Elimination To solve a system of equations we can perform the following row ... plc wiring - 2.12 2.1.5 Ladder Logic Outputs In ladder logic there are multiple types of outputs, but these are not consistently available on all PLCs. Some of the outputs will be externally connected to devices outside the PLC, but it is also possible to use internal memory locations in the PLC. Six types of outputs are shown in Figure 2.12. The first is a normal output, when energized the output will turn on, and energize an output. The circle with a diagonal line through is a normally on output. When energized the output will turn off. This type of output is not available on all PLC types. When initially energized the OSR (One Shot Relay) instruction will turn on for one scan, but then be off for all scans after, until it is turned off. The L (latch) and U (unlatch) instructions can be used to lock outputs on. When an L output is energized the output will turn on indefinitely, even when the output coil is deenergized. The output can only be turned off using a U output. The last instruction is the IOT (Immediate OutpuT) that will allow outputs to be updated without having to wait for the ladder logic scan to be completed. When power is applied (on) the output x is activated for the left output, but turned An input transition on will cause the output x to go on for one scan xx OSR x (this is also known as a one shot relay) off for the output on the right. plc wiring - 2.13 Figure 2.12 Ladder Logic Outputs 2.2 A CASE STUDY Problem: Try to develop (without looking at the solution) a relay based controller that will allow three switches in a room to control a single light. When the L coil is energized, x will be toggled on, it will stay on until the U coil Some PLCs will allow immediate outputs that do not wait for the program scan to L U IOT end before setting an output. (Note: This instruction will only update the outputs using is energized. This is like a flip-flop and stays set even when the PLC is turned off. x xx the output table, other instruction must change the individual outputs.) Note: Outputs are also commonly shown using parentheses -( )- instead of the circle. This is because many of the programming systems are text based and circles cannot be drawn. plc wiring - 2.14 2.3 SUMMARY • Normally open and closed contacts. • Relays and their relationship to ladder logic. • PLC outputs can be inputs, as shown by the seal in circuit. • Programming can be done with ladder logic, mnemonics, SFCs, and structured text. • There are multiple ways to write a PLC program. Solution: There are two possible approaches to this problem. The first assumes that any one of the switches on will turn on the light, but all three switches must be off for the light to be off. switch 1 switch 2 switch 3 light The second solution assumes that each switch can turn the light on or off, regardless of the states of the other switches. This method is more complex and involves thinking through all of the possible combinations of switch positions. You might recognize this problem as an exclusive or problem. switch 1 switch 1 switch 1 light switch 2 switch 2 switch 2 switch 3 switch 3 switch 3 switch 1 switch 2 switch 3 Note: It is important to get a clear understanding of how the controls are expected to work. In this example two radically different solutions were obtained based upon a simple difference in the operation. plc wiring - 2.15 2.4 PRACTICE PROBLEMS 1. Give an example of where a PLC could be used. 2. Why would relays be used in place of PLCs? 3. Give a concise description of a PLC. 4. List the advantages of a PLC over relays. 5. A PLC can effectively replace a number of components. Give examples and discuss some good and bad applications of PLCs. 6. Explain the trade-offs between relays and PLCs for control applications. 7. Explain why ladder logic outputs are coils? 8. In the figure below, will the power for the output on the first rung normally be on or off? Would the output on the second rung normally page 0 A C + B C * B B T1 ST2 A⋅= T1 T2 T3 T4 ST1 ST2 ST3 FS = first scan ST1 ST1 T1+()T2⋅ FS+= ST2 ST2 T2 T3++()T1 T4⋅⋅= ST3 ST3 T4 T1⋅+()T3⋅= T2 ST1 B⋅= T3 ST3 CB⋅()⋅= T4 ST2 CB+()⋅= ST2 A ST1 B ST3 C B T1 T2 T3 T4 ST2 C B ST1 T2 ST1 T1 first scan ST2 T1 ST2 T2 T3 ST3 T3 ST3 T4 T4 T1 Automating Manufacturing Systems with PLCs (Version 4.2, April 3, 2003) Hugh Jack page i Copyright (c) 1993-2003 Hugh Jack (jackh@gvsu.edu). Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled "GNU Free Documentation License". This document is provided as-is with no warranty, implied or otherwise. There have been attempts to eliminate errors from this document, but there is no doubt that errors remain. As a result, the author does not assume any responsibility for errors and omissions, or damages resulting from the use of the information pro- vided. Additional materials and updates for this work will be available at http://clay- more.engineer.gvsu.edu/~jackh/books.html page ii 1.1 TODO LIST 1.4 2. PROGRAMMABLE LOGIC CONTROLLERS . . . . . . . . . . . . . 2.1 2.1 INTRODUCTION 2.1 2.1.1 Ladder Logic 2.1 2.1.2 Programming 2.6 2.1.3 PLC Connections 2.10 2.1.4 Ladder Logic Inputs 2.11 2.1.5 Ladder Logic Outputs 2.12 2.2 A CASE STUDY 2.13 2.3 SUMMARY 2.14 2.4 PRACTICE PROBLEMS 2.15 2.5 PRACTICE PROBLEM SOLUTIONS 2.15 2.6 ASSIGNMENT PROBLEMS 2.16 3. PLC HARDWARE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 3.1 INTRODUCTION 3.1 3.2 INPUTS AND OUTPUTS 3.2 3.2.1 Inputs 3.3 3.2.2 Output Modules 3.7 3.3 RELAYS 3.13 3.4 A CASE STUDY 3.14 3.5 ELECTRICAL WIRING DIAGRAMS 3.15 3.5.1 JIC Wiring Symbols 3.17 3.6 SUMMARY 3.21 3.7 PRACTICE PROBLEMS 3.21 3.8 PRACTICE PROBLEM SOLUTIONS 3.24 3.9 ASSIGNMENT PROBLEMS 3.27 4. LOGICAL SENSORS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 4.1 INTRODUCTION 4.1 4.2 SENSOR WIRING 4.1 4.2.1 Switches 4.2 4.2.2 Transistor Transistor Logic (TTL) 4.3 4.2.3 Sinking/Sourcing 4.3 4.2.4 Solid State Relays 4.10 4.3 PRESENCE DETECTION 4.11 4.3.1 Contact Switches 4.11 4.3.2 Reed Switches 4.11 4.3.3 Optical (Photoelectric) Sensors 4.12 4.3.4 Capacitive Sensors 4.19 4.3.5 Inductive Sensors 4.23 4.3.6 Ultrasonic 4.25 4.3.7 Hall Effect 4.25 page iii 4.3.8 Fluid Flow 4.26 4.4 SUMMARY 4.26 4.5 PRACTICE PROBLEMS 4.27 4.6 PRACTICE PROBLEM SOLUTIONS 4.30 4.7 ASSIGNMENT PROBLEMS 4.36 5. LOGICAL ACTUATORS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 5.1 INTRODUCTION 5.1 5.2 SOLENOIDS 5.1 5.3 VALVES 5.2 5.4 CYLINDERS 5.4 5.5 HYDRAULICS 5.6 5.6 PNEUMATICS 5.8 5.7 MOTORS 5.9 5.8 OTHERS 5.10 5.9 SUMMARY 5.10 5.10 PRACTICE PROBLEMS 5.10 5.11 PRACTICE PROBLEM SOLUTIONS 5.10 5.12 ASSIGNMENT PROBLEMS 5.11 6. BOOLEAN LOGIC DESIGN . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 6.1 INTRODUCTION 6.1 6.2 BOOLEAN ALGEBRA 6.1 6.3 LOGIC DESIGN 6.6 6.3.1 Boolean Algebra Techniques 6.13 6.4 COMMON LOGIC FORMS 6.14 6.4.1 Complex Gate Forms 6.14 6.4.2 Multiplexers 6.15 6.5 SIMPLE DESIGN CASES 6.17 6.5.1 Basic Logic Functions 6.17 6.5.2 Car Safety System 6.18 6.5.3 Motor Forward/Reverse 6.18 6.5.4 A Burglar Alarm 6.19 6.6 SUMMARY 6.23 6.7 PRACTICE PROBLEMS 6.24 6.8 PRACTICE PROBLEM SOLUTIONS 6.27 6.9 ASSIGNMENT PROBLEMS 6.37 7. KARNAUGH MAPS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 7.1 INTRODUCTION 7.1 7.2 SUMMARY 7.4 7.3 PRACTICE PROBLEMS 7.4 7.4 PRACTICE PROBLEM SOLUTIONS 7.10 page iv 7.5 ASSIGNMENT PROBLEMS 7.16 8. PLC OPERATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 8.1 INTRODUCTION 8.1 8.2 OPERATION SEQUENCE 8.3 8.2.1 The Input and Output Scans 8.4 8.2.2 The Logic Scan 8.4 page 0 A C + B C * B B T1 ST2 A⋅= T1 T2 T3 T4 ST1 ST2 ST3 FS = first scan ST1 ST1 T1+()T2⋅ FS+= ST2 ST2 T2 T3++()T1 T4⋅⋅= ST3 ST3 T4 T1⋅+()T3⋅= T2 ST1 B⋅= T3 ST3 CB⋅()⋅= T4 ST2 CB+()⋅= ST2 A ST1 B ST3 C B T1 T2 T3 T4 ST2 C B ST1 T2 ST1 T1 first scan ST2 T1 ST2 T2 T3 ST3 T3 ST3 T4 T4 T1 Automating Manufacturing Systems with PLCs (Version 4.2, April 3, 2003) Hugh Jack page i Copyright (c) 1993-2003 Hugh Jack (jackh@gvsu.edu). Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled "GNU Free Documentation License". This document is provided as-is with no warranty, implied or otherwise. There have been attempts to eliminate errors from this document, but there is no doubt that errors remain. As a result, the author does not assume any responsibility for errors and omissions, or damages resulting from the use of the information pro- vided. Additional materials and updates for this work will be available at http://clay- more.engineer.gvsu.edu/~jackh/books.html page ii 1.1 TODO LIST 1.4 2. PROGRAMMABLE LOGIC CONTROLLERS . . . . . . . . . . . . . 2.1 2.1 INTRODUCTION 2.1 2.1.1 Ladder Logic 2.1 2.1.2 Programming 2.6 2.1.3 PLC Connections 2.10 2.1.4 Ladder Logic Inputs 2.11 2.1.5 Ladder Logic Outputs 2.12 2.2 A CASE STUDY 2.13 2.3 SUMMARY 2.14 2.4 PRACTICE PROBLEMS 2.15 2.5 PRACTICE PROBLEM SOLUTIONS 2.15 2.6 ASSIGNMENT PROBLEMS 2.16 3. PLC HARDWARE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 3.1 INTRODUCTION 3.1 3.2 INPUTS AND OUTPUTS 3.2 3.2.1 Inputs 3.3 3.2.2 Output Modules 3.7 3.3 RELAYS 3.13 3.4 A CASE STUDY 3.14 3.5 ELECTRICAL WIRING DIAGRAMS 3.15 3.5.1 JIC Wiring Symbols 3.17 3.6 SUMMARY 3.21 3.7 PRACTICE PROBLEMS 3.21 3.8 PRACTICE PROBLEM SOLUTIONS 3.24 3.9 ASSIGNMENT PROBLEMS 3.27 4. LOGICAL SENSORS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 4.1 INTRODUCTION 4.1 4.2 SENSOR WIRING 4.1 4.2.1 Switches 4.2 4.2.2 Transistor Transistor Logic (TTL) 4.3 4.2.3 Sinking/Sourcing 4.3 4.2.4 Solid State Relays 4.10 4.3 PRESENCE DETECTION 4.11 4.3.1 Contact Switches 4.11 4.3.2 Reed Switches 4.11 4.3.3 Optical (Photoelectric) Sensors 4.12 4.3.4 Capacitive Sensors 4.19 4.3.5 Inductive Sensors 4.23 4.3.6 Ultrasonic 4.25 4.3.7 Hall Effect 4.25 page iii 4.3.8 Fluid Flow 4.26 4.4 SUMMARY 4.26 4.5 PRACTICE PROBLEMS 4.27 4.6 PRACTICE PROBLEM SOLUTIONS 4.30 4.7 ASSIGNMENT PROBLEMS 4.36 5. LOGICAL ACTUATORS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 5.1 INTRODUCTION 5.1 5.2 SOLENOIDS 5.1 5.3 VALVES 5.2 5.4 CYLINDERS 5.4 5.5 HYDRAULICS 5.6 5.6 PNEUMATICS 5.8 5.7 MOTORS 5.9 5.8 OTHERS 5.10 5.9 SUMMARY 5.10 5.10 PRACTICE PROBLEMS 5.10 5.11 PRACTICE PROBLEM SOLUTIONS 5.10 5.12 ASSIGNMENT PROBLEMS 5.11 6. BOOLEAN LOGIC DESIGN . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 6.1 INTRODUCTION 6.1 6.2 BOOLEAN ALGEBRA 6.1 6.3 LOGIC DESIGN 6.6 6.3.1 Boolean Algebra Techniques 6.13 6.4 COMMON LOGIC FORMS 6.14 6.4.1 Complex Gate Forms 6.14 6.4.2 Multiplexers 6.15 6.5 SIMPLE DESIGN CASES 6.17 6.5.1 Basic Logic Functions 6.17 6.5.2 Car Safety System 6.18 6.5.3 Motor Forward/Reverse 6.18 6.5.4 A Burglar Alarm 6.19 6.6 SUMMARY 6.23 6.7 PRACTICE PROBLEMS 6.24 6.8 PRACTICE PROBLEM SOLUTIONS 6.27 6.9 ASSIGNMENT PROBLEMS 6.37 7. KARNAUGH MAPS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 7.1 INTRODUCTION 7.1 7.2 SUMMARY 7.4 7.3 PRACTICE PROBLEMS 7.4 7.4 PRACTICE PROBLEM SOLUTIONS 7.10 page iv 7.5 ASSIGNMENT PROBLEMS 7.16 8. PLC OPERATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 8.1 INTRODUCTION 8.1 8.2 OPERATION SEQUENCE 8.3 8.2.1 The Input and Output Scans 8.4 8.2.2 The Logic Scan 8.4 ... below 6/29 Solving Systems with Gaussian Elimination If any rows contain all zeros, place them at the bottom Solving a × System by Gaussian Elimination Solve the given system by Gaussian elimination. .. equations be solved by Gaussian elimination? Yes, a system of linear equations of any size can be solved by Gaussian elimination How To 14/29 Solving Systems with Gaussian Elimination Given a system... [ 1 0.05 0.08 0.09 −1 | ] 10, 000 770 Now, we perform Gaussian elimination to achieve row-echelon form 17/29 Solving Systems with Gaussian Elimination −0.05R1 + R2 = R2 → −2R1 + R3 = R3 → [ [

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