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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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 Cramer's Rule Solving Systems with Cramer's Rule By: OpenStaxCollege We have learned how to solve systems of equations in two variables and three variables, and by multiple methods: substitution, addition, Gaussian elimination, using the inverse of a matrix, and graphing Some of these methods are easier to apply than others and are more appropriate in certain situations In this section, we will study two more strategies for solving systems of equations Evaluating the Determinant of a 2×2 Matrix A determinant is a real number that can be very useful in mathematics because it has multiple applications, such as calculating area, volume, and other quantities Here, we will use determinants to reveal whether a matrix is invertible by using the entries of a square matrix to determine whether there is a solution to the system of equations Perhaps one of the more interesting applications, however, is their use in cryptography Secure signals or messages are sometimes sent encoded in a matrix The data can only be decrypted with an invertible matrix and the determinant For our purposes, we focus on the determinant as an indication of the invertibility of the matrix Calculating the determinant of a matrix involves following the specific patterns that are outlined in this section Find the Determinant of a × Matrix The determinant of a × matrix, given A= [ ] a b c d is defined as 1/42 Solving Systems with Cramer's Rule Notice the change in notation There are several ways to indicate the determinant, including det(A) and replacing the brackets in a matrix with straight lines, |A| Finding the Determinant of a × Matrix Find the determinant of the given matrix A= [ −6 det(A) = | ] −6 | = 5(3) − (−6)(2) = 27 Using Cramer’s Rule to Solve a System of Two Equations in Two Variables We will now introduce a final method for solving systems of equations that uses determinants Known as Cramer’s Rule, this technique dates back to the middle of the 18th century and is named for its innovator, the Swiss mathematician Gabriel Cramer (1704-1752), who introduced it in 1750 in Introduction l'Analyse des lignes Courbes algébriques Cramer’s Rule is a viable and efficient method for finding solutions to systems with an arbitrary number of unknowns, provided that we have the same number of equations as unknowns Cramer’s Rule will give us the unique solution to a system of equations, if it exists However, if the system has no solution or an infinite number of solutions, this will be indicated by a determinant of zero To find out if the system is inconsistent or dependent, another method, such as elimination, will have to be used To understand Cramer’s Rule, let’s look closely at how we solve systems of linear equations using basic row operations Consider a system of two equations in two variables a1x + b1y = c1 (1) a2x + b2y = c2 (2) 2/42 Solving Systems with Cramer's Rule We eliminate one variable using row operations and solve for the other Say that we wish to solve for x If equation (2) is multiplied by the opposite of the coefficient of y in equation (1), equation (1) is multiplied by the coefficient of y in equation (2), and we add the two equations, the variable y will be eliminated b2a1x + b2b1y = b2c1 Multiply R1 by b2 − b1a2x − b1b2y = − b1c2 Multiply R2 by − b1 b2a1x − b1a2x = b2c1 − b1c2 Now, solve for x b2a1x − b1a2x = b2c1 − b1c2 x(b2a1 − b1a2) = b2c1 − b1c2 [ ] [ ] c1 b1 b2c1 − b1c2 x= = b2a1 − b1a2 c2 b2 a1 b1 a2 b2 Similarly, to solve for y, we will eliminate x a2a1x + a2b1y = a2c1 − a1a2x − a1b2y = − a1c2 Multiply R1 by a2 Multiply R2 by − a1 a2b1y − a1b2y = a2c1 − a1c2 Solving for y gives 3/42 Solving Systems with Cramer's Rule a2b1y − a1b2y = a2c1 − a1c2 y(a2b1 − a1b2) = a2c1 − a1c2 | | | | a1 c1 y= a2 c2 a2c1 − a1c2 a1c2 − a2c1 = = a2b1 − a1b2 a1b2 − a2b1 a1 b1 a2 b2 Notice that the denominator for both x and y is the determinant of the coefficient matrix We can use these formulas to solve for x and y, but Cramer’s Rule also introduces new notation: • D : determinant of the coefficient matrix • Dx : determinant of the numerator in the solution ofx Dx x= D • Dy : determinant of the numerator in the solution of y y= Dy D The key to Cramer’s Rule is replacing the variable column of interest with the constant column and calculating the determinants We can then express x and y as a quotient of two determinants Cramer’s Rule for 2×2 Systems Cramer’s Rule is a method that uses determinants to solve systems of equations that have the same number of equations as variables Consider a system of two linear equations in two variables a1x + b1y = c1 a2x + b2y = c2 The solution using Cramer’s Rule is given as 4/42 Solving Systems with Cramer's Rule | | | | | | | | c1 b1 x= a1 c1 c2 b2 a2 c2 Dx Dy = , D ≠ 0; y = = , D ≠ D D a1 b1 a1 b1 a2 b2 a2 b2 If we are solving for x, the x column is replaced with the constant column If we are solving for y, the y column is ... 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 ... instruction and practice with Cramer s Rule • Solve a System of Two Equations Using Cramer' s Rule • Solve a Systems of Three Equations using Cramer' s Rule 14/42 Solving Systems with Cramer' s Rule Key... answer Solving a × System Using Cramer s Rule Find the solution to the given × system using Cramer s Rule x+y−z=6 3x − 2y + z = −5 x + 3y − 2z = 14 Use Cramer s Rule 8/42 Solving Systems with Cramer' s... 16/42 Solving Systems with Cramer' s Rule | | −3 −2 | −3 3.1 4, 000 | − 7, 990.7 | | − 1.1 0.6 7.2 − 0.5 −1 0 0 −3 | | | | | | −1 0 0 −3 1 1 0 −1 | −3 −4 −5 | 17/42 Solving Systems with Cramer' s