Solving Systems with Inverses

Solving Systems with Inverses tài liệu, giáo án, bài giảng , luận văn, luận án, đồ án, bài tập lớn về tất cả các lĩnh vự...

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. 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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 Inverses Solving Systems with Inverses By: OpenStaxCollege Nancy plans to invest $10,500 into two different bonds to spread out her risk The first bond has an annual return of 10%, and the second bond has an annual return of 6% In order to receive an 8.5% return from the two bonds, how much should Nancy invest in each bond? What is the best method to solve this problem? There are several ways we can solve this problem As we have seen in previous sections, systems of equations and matrices are useful in solving real-world problems involving finance After studying this section, we will have the tools to solve the bond problem using the inverse of a matrix Finding the Inverse of a Matrix We know that the multiplicative inverse of a real number a is a−1, and 1 aa−1 = a−1a = a a = For example, 2−1 = and 2 = The multiplicative inverse () () of a matrix is similar in concept, except that the product of matrixA and its inverse A−1 equals the identity matrix The identity matrix is a square matrix containing ones down the main diagonal and zeros everywhere else We identify identity matrices by In where n represents the dimension of the matrix [link] and [link] are the identity matrices for a × matrix and a × matrix, respectively I2 = [ I3 = [ ] 0 1 0 0 ] The identity matrix acts as a in matrix algebra For example, AI = IA = A 1/32 Solving Systems with Inverses A matrix that has a multiplicative inverse has the properties AA−1 = I A−1A = I A matrix that has a multiplicative inverse is called an invertible matrix Only a square matrix may have a multiplicative inverse, as the reversibility, AA−1 = A−1A = I, is a requirement Not all square matrices have an inverse, but if A is invertible, then A−1 is unique We will look at two methods for finding the inverse of a × matrix and a third method that can be used on both × and3 × matrices A General Note The Identity Matrix and Multiplicative Inverse The identity matrix, In, is a square matrix containing ones down the main diagonal and zeros everywhere else I2 = [ ] 0 [ ] I3 = × × 0 1 0 If A is an n × n matrix and B is an n × n matrix such that AB = BA = In, then B = A−1, the multiplicative inverse of a matrix A Showing That the Identity Matrix Acts as a Given matrix A, show that AI = IA = A A= [ ] −2 Use matrix multiplication to show that the product of A and the identity is equal to the product of the identity and A 2/32 Solving Systems with Inverses AI = AI = [ [ ] [ ] [ −2 −2 0 ][ ][ = = 3⋅1+4⋅0 3⋅0+4⋅1 −2 ⋅ + ⋅ −2 ⋅ + ⋅ 1 ⋅ + ⋅ (−2) 1⋅4+0⋅5 ⋅ + ⋅ (−2) 0⋅4+1⋅5 ][ ][ = = −2 −2 ] ] How To Given two matrices, show that one is the multiplicative inverse of the other Given matrix A of order n × n and matrix B of order n × n multiply AB If AB = I, then find the product BA If BA = I, then B = A−1 and A = B−1 Showing That Matrix A Is the Multiplicative Inverse of Matrix B Show that the given matrices are multiplicative inverses of each other A= [ −2 −9 ] [ ,B= −9 −5 ] Multiply AB and BA If both products equal the identity, then the two matrices are inverses of each other AB = [ = = BA = [ [ [ = = [ [ −2 −9 ][ · −9 −5 ] 1(−9) + 5(2) 1(−5) + 5(1) −2(−9)−9(2) −2(−5)−9(1) 0 ] ] −9 −5 ][ · −2 −9 ] −9(1)−5(−2) −9(5)−5(−9) 2(1) + 1(−2) 2(−5) + 1(−9) 0 ] ] 3/32 Solving Systems with Inverses A andBare inverses of each other Try It Show that the following two matrices are inverses of each other A= [ AB = BA = −1 −3 [ [ ] [ ][ ][ ,B= −3 −4 1 −3 −4 −1 −3 1 −3 −4 1 −1 −3 ] ][ ][ = = 1(−3) + 4(1) 1(−4) + 4(1) −1(−3) + −3(1) −1(−4) + −3(1) −3(1) + −4(−1) −3(4) + −4(−3) 1(1) + 1(−1) 1(4) + 1(−3) ][ ][ = = 0 1 0 ] ] Finding the Multiplicative Inverse Using Matrix Multiplication We can now determine whether two matrices are inverses, but how would we find the inverse of a given matrix? Since we know that the product of a matrix and its inverse is the identity matrix, we can find the inverse of a matrix by setting up an equation using matrix multiplication Finding the Multiplicative Inverse Using Matrix Multiplication Use matrix multiplication to find the inverse of the given matrix A= [ −2 −3 ] For this method, we multiply A by a matrix containing unknown constants and set it equal to the identity [ ][ ] [ ] −2 a b −3 c d = 0 Find the product of the two matrices on the left side of the equal sign [ ][ ] [ −2 a b −3 c d = 1a−2c 1b−2d 2a−3c 2b−3d ] 4/32 Solving Systems with Inverses Next, set up a system of equations with the entry in row 1, column of the new matrix equal to the first entry of the identity, Set the entry in row 2, column of the new matrix equal to the corresponding entry of the identity, which is 1a−2c = R1 2a−3c = R2 Using row operations, multiply and add as follows: (−2)R1 + R2 → R2 Add the equations, and solve for c 1a − 2c = + 1c = − c= −2 Back-substitute to solve for a a−2(−2) = a+4=1 a = −3 Write another system of equations setting the entry in row 1, column of the new matrix equal to the corresponding entry of the identity, Set the entry in row 2, column ... 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 ... 24/32 Solving Systems with Inverses [ ] 1 2 There is no inverse [ ] [ ] [ ] 1 0.5 1.5 −0.5 0.5 1.5 −0.5 [ ] [ ] [ ] [ ] −2 −3 1 −5 17 −3 20 −3 12 −1 −1 −3 −2 −4 −5 25/32 Solving Systems with Inverses. .. 29/32 Solving Systems with Inverses [ [ [ 39 −1 −1 1 −1 −1 1 −1 −1 0 2 −1 −3 1 −2 0 −2 −5 1 [ ] ] ] −7 18 − 53 32 10 24 − 36 21 −9 46 − 16 −5 ] [ ] 2 0 0 0 0 30/32 Solving Systems with Inverses. .. matrix A= [ −17 11 −1 11 −7 −2 ] 12/32 Solving Systems with Inverses [ ] 1 A−1 = 2 −3 −5 Solving a System of Linear Equations Using the Inverse of a Matrix Solving a system of linear equations