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The Industrial Electronics HandbookS E c o n d E d I T I o n
control and mechatronIcs
2011 by Taylor and Francis Group, LLC
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The Industrial Electronics HandbookS E c o n d E d I T I o n
Fundamentals oF IndustrIal electronIcs
Power electronIcs and motor drIves
control and mechatronIcs
IndustrIal communIcatIon systems
IntellIgent systems
2011 by Taylor and Francis Group, LLC
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The Electrical Engineering Handbook Series
Series EditorRichard C. DorfUniversity of California, Davis
Titles Included in the Series
The Avionics Handbook, Second Edition, Cary R. SpitzerThe Biomedical Engineering Handbook, Third Edition, Joseph D. BronzinoThe Circuits and Filters Handbook, Third Edition, Wai-Kai ChenThe Communications Handbook, Second Edition, Jerry GibsonThe Computer Engineering Handbook, Vojin G. OklobdzijaThe Control Handbook, Second Edition, William S. Levine CRC Handbook of Engineering Tables, Richard C. DorfDigital Avionics Handbook, Second Edition, Cary R. SpitzerThe Digital Signal Processing Handbook, Vijay K. Madisetti and Douglas WilliamsThe Electric Power Engineering Handbook, Second Edition, Leonard L. GrigsbyThe Electrical Engineering Handbook, Third Edition, Richard C. DorfThe Electronics Handbook, Second Edition, Jerry C. WhitakerThe Engineering Handbook, Third Edition, Richard C. DorfThe Handbook of Ad Hoc Wireless Networks, Mohammad IlyasThe Handbook of Formulas and Tables for Signal Processing, Alexander D. PoularikasHandbook of Nanoscience, Engineering, and Technology, Second Edition, William A. Goddard, III, Donald W. Brenner, Sergey E. Lyshevski, and Gerald J. IafrateThe Handbook of Optical Communication Networks, Mohammad Ilyas and Hussein T. MouftahThe Industrial Electronics Handbook, Second Edition, Bogdan M. Wilamowski and J. David IrwinThe Measurement, Instrumentation, and Sensors Handbook, John G. WebsterThe Mechanical Systems Design Handbook, Osita D.I. Nwokah and Yidirim HurmuzluThe Mechatronics Handbook, Second Edition, Robert H. BishopThe Mobile Communications Handbook, Second Edition, Jerry D. GibsonThe Ocean Engineering Handbook, Ferial El-HawaryThe RF and Microwave Handbook, Second Edition, Mike GolioThe Technology Management Handbook, Richard C. DorfTransforms and Applications Handbook, Third Edition, Alexander D. PoularikasThe VLSI Handbook, Second Edition, Wai-Kai Chen
2011 by Taylor and Francis Group, LLC
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The Industrial Electronics HandbookS E c o n d E d I T I o n
control and mechatronIcs
Edited by
Bogdan M. WilamowskiJ. david Irwin
2011 by Taylor and Francis Group, LLC
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MATLAB is a trademark of The MathWorks, Inc. and is used with permission. The MathWorks does not warrant the accuracy of the text or exercises in this book. This books use or discussion of MATLAB software or related products does not constitute endorsement or sponsorship by The MathWorks of a particular pedagogical approach or particular use of the MATLAB software.
CRC PressTaylor & Francis Group6000 Broken Sound Parkway NW, Suite 300Boca Raton, FL 33487-2742
2011 by Taylor and Francis Group, LLCCRC Press is an imprint of Taylor & Francis Group, an Informa business
No claim to original U.S. Government works
Printed in the United States of America on acid-free paper10 9 8 7 6 5 4 3 2 1
International Standard Book Number: 978-1-4398-0287-8 (Hardback)
This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the valid-ity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint.
Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or uti-lized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopy-ing, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers.
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Library of Congress CataloginginPublication Data
Control and mechatronics / editors, Bogdan M. Wilamowski and J. David Irwin.p. cm.
A CRC title.Includes bibliographical references and index.ISBN 978-1-4398-0287-8 (alk. paper)1. Mechatronics. 2. Electronic control. 3. Servomechanisms. I. Wilamowski, Bogdan M. II. Irwin,
J. David. III. Title.
TJ163.12.C67 2010629.8043--dc22 2010020062
Visit the Taylor & Francis Web site athttp://www.taylorandfrancis.comand the CRC Press Web site athttp://www.crcpress.com
2011 by Taylor and Francis Group, LLC
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vii
Contents
Preface....................................................................................................................... xiAcknowledgments................................................................................................... xiiiEditorial.Board..........................................................................................................xvEditors..................................................................................................................... xviiContributors............................................................................................................ xxi
Part I Control System analysis
. 1. Nonlinear.Dynamics........................................................................................1-1Istvn Nagy and Zoltn Sto
. 2. Basic.Feedback.Concept.................................................................................. 2-1Tong Heng Lee, Kok Zuea Tang, and Kok Kiong Tan
. 3. Stability.Analysis............................................................................................. 3-1Naresh K. Sinha
. 4. Frequency-Domain.Analysis.of.Relay.Feedback.Systems.............................. 4-1Igor M. Boiko
. 5. Linear.Matrix.Inequalities.in.Automatic.Control......................................... 5-1Miguel Bernal and Thierry Marie Guerra
. 6. Motion.Control.Issues..................................................................................... 6-1Roberto Oboe, Makoto Iwasaki, Toshiyuki Murakami, and Seta Bogosyan
. 7. New.Methodology.for.Chatter.Stability.Analysis.in.Simultaneous.Machining........................................................................................................7-1Nejat Olgac and Rifat Sipahi
Part II Control System Design
. 8. Internal.Model.Control................................................................................... 8-1James C. Hung
. 9. Dynamic.Matrix.Control................................................................................ 9-1James C. Hung
2011 by Taylor and Francis Group, LLC
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viii Contents
.10. PID.Control....................................................................................................10-1James C. Hung and Joel David Hewlett
.11. Nyquist.Criterion........................................................................................... 11-1James R. Rowland
.12. Root.Locus.Method........................................................................................12-1Robert J. Veillette and J. Alexis De Abreu Garcia
.13. Variable.Structure.Control.Techniques.........................................................13-1Asif abanovic and Nadira abanovic-Behlilovic
.14. Digital.Control...............................................................................................14-1Timothy N. Chang and John Y. Hung
.15. Phase-Lock-Loop-Based.Control...................................................................15-1Guan-Chyun Hsieh
.16. Optimal.Control.............................................................................................16-1Victor M. Becerra
.17. Time-Delay.Systems....................................................................................... 17-1Emilia Fridman
.18. AC.Servo.Systems...........................................................................................18-1Yong Feng, Liuping Wang, and Xinghuo Yu
.19. Predictive.Repetitive.Control.with.Constraints...........................................19-1Liuping Wang, Shan Chai, and Eric Rogers
.20. Backstepping.Control.....................................................................................20-1Jing Zhou and Changyun Wen
.21. Sensors............................................................................................................ 21-1Tiantian Xie and Bogdan M. Wilamowski
.22. Soft.Computing.Methodologies.in.Sliding.Mode.Control............................22-1Xinghuo Yu and Okyay Kaynak
Part III Estimation, Observation, and Identification
.23. Adaptive.Estimation.......................................................................................23-1Seta Bogosyan, Metin Gokasan, and Fuat Gurleyen
.24. Observers.in.Dynamic.Engineering.Systems................................................24-1Christopher Edwards and Chee Pin Tan
.25. Disturbance.ObservationCancellation.Technique......................................25-1Kouhei Ohnishi
.26. Ultrasonic.Sensors.........................................................................................26-1Lindsay Kleeman
.27. Robust.Exact.Observation.and.Identification.via.High-Order.Sliding.Modes............................................................................................................. 27-1Leonid Fridman, Arie Levant, and Jorge Angel Davila Montoya
2011 by Taylor and Francis Group, LLC
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Contents ix
Part IV Modeling and Control
.28. Modeling.for.System.Control.........................................................................28-1A. John Boye
.29. Intelligent.Mechatronics.and.Robotics..........................................................29-1Satoshi Suzuki and Fumio Harashima
.30. State-Space.Approach.to.Simulating.Dynamic.Systems.in.SPICE................30-1Joel David Hewlett and Bogdan M. Wilamowski
.31. Iterative.Learning.Control.for.Torque.Ripple.Minimization.of.Switched.Reluctance.Motor.Drive................................................................................. 31-1Sanjib Kumar Sahoo, Sanjib Kumar Panda, and Jian-Xin Xu
.32. Precise.Position.Control.of.Piezo.Actuator...................................................32-1Jian-Xin Xu and Sanjib Kumar Panda
.33. Hardware-in-the-Loop.Simulation................................................................33-1Alain Bouscayrol
Part V Mechatronics and robotics
.34. Introduction.to.Mechatronic.Systems...........................................................34-1Ren C. Luo and Chin F. Lin
.35. Actuators.in.Robotics.andAutomation.Systems...........................................35-1Choon-Seng Yee and Marcelo H. Ang Jr.
.36. Robot.Qualities..............................................................................................36-1Raymond Jarvis
.37. Robot.Vision................................................................................................... 37-1Raymond Jarvis
.38. Robot.Path.Planning......................................................................................38-1Raymond Jarvis
.39. Mobile.Robots.................................................................................................39-1Miguel A. Salichs, Ramn Barber, and Mara Malfaz
Index.................................................................................................................. Index-1
2011 by Taylor and Francis Group, LLC
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Preface
The.field.of.industrial.electronics.covers.a.plethora.of.problems.that.must.be.solved.in.industrial.prac-tice..Electronic.systems.control.many.processes.that.begin.with.the.control.of.relatively.simple.devices.like.electric.motors,.through.more.complicated.devices.such.as.robots,.to.the.control.of.entire.fabrica-tion.processes..An.industrial.electronics.engineer.deals.with.many.physical.phenomena.as.well.as.the.sensors.that.are.used.to.measure.them..Thus,.the.knowledge.required.by.this.type.of.engineer.is.not.only.traditional.electronics.but.also.specialized.electronics,.for.example,.that.required.for.high-power.appli-cations..The.importance.of.electronic.circuits.extends.well.beyond.their.use.as.a.final.product.in.that.they.are.also.important.building.blocks.in.large.systems,.and.thus.the.industrial.electronics.engineer.must.also.possess.knowledge.of.the.areas.of.control.and.mechatronics..Since.most.fabrication.processes.are.relatively.complex,.there.is.an.inherent.requirement.for.the.use.of.communication.systems.that.not.only.link.the.various.elements.of.the.industrial.process.but.are.also.tailor-made.for.the.specific.indus-trial.environment..Finally,.the.efficient.control.and.supervision.of.factories.require.the.application.of.intelligent.systems.in.a.hierarchical.structure.to.address.the.needs.of.all.components.employed.in.the.production.process..This.need. is.accomplished. through. the.use.of. intelligent.systems.such.as.neural.networks,.fuzzy.systems,.and.evolutionary.methods..The.Industrial.Electronics.Handbook.addresses.all.these.issues.and.does.so.in.five.books.outlined.as.follows:
. 1.. Fundamentals of Industrial Electronics
. 2.. Power Electronics and Motor Drives
. 3.. Control and Mechatronics
. 4.. Industrial Communication Systems
. 5.. Intelligent Systems
The.editors.have.gone.to.great.lengths.to.ensure.that.this.handbook.is.as.current.and.up.to.date.as.pos-sible..Thus,.this.book.closely.follows.the.current.research.and.trends.in.applications.that.can.be.found.in.IEEE Transactions on Industrial Electronics..This.journal.is.not.only.one.of.the.largest.engineering.pub-lications.of.its.type.in.the.world.but.also.one.of.the.most.respected..In.all.technical.categories.in.which.this.journal.is.evaluated,.its.worldwide.ranking.is.either.number.1.or.number.2..As.a.result,.we.believe.that.this.handbook,.which.is.written.by.the.worlds.leading.researchers.in.the.field,.presents.the.global.trends.in.the.ubiquitous.area.commonly.known.as.industrial.electronics.
The.successful.construction.of.industrial.systems.requires.an.understanding.of.the.various.aspects.of.control.theory..This.area.of.engineering,.like.that.of.power.electronics,.is.also.seldom.covered.in.depth.in.engineering.curricula.at.the.undergraduate.level..In.addition,.the.fact.that.much.of.the.research.in.control.theory.focuses.more.on.the.mathematical.aspects.of.control.than.on.its.practical.applications.makes.matters.worse..Therefore,.the.goal.of.Control and Mechatronics.is.to.present.many.of.the.concepts.of.control.theory.in.a.manner.that.facilitates.its.understanding.by.practicing.engineers.or.students.who.would.like.to.learn.about.the.applications.of.control.systems..Control and Mechatronics.is.divided.into.several.parts..Part.I.is.devoted.to.control.system.analysis.while.Part.II.deals.with.control.system.design..
2011 by Taylor and Francis Group, LLC
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xii Preface
Various.techniques.used.for.the.analysis.and.design.of.control.systems.are.described.and.compared.in.these.two.parts..Part.III.deals.with.estimation,.observation,.and.identification.and.is.dedicated.to.the.identification.of.the.objects.to.be.controlled..The.importance.of.this.part.stems.from.the.fact.that. in.order.to.efficiently.control.a.system,.it.must.first.be.clearly.identified..In.an.industrial.environment,.it.is.difficult.to.experiment.with.production.lines..As.a.result,.it.is.imperative.that.good.models.be.developed.to.represent.these.systems..This.modeling.aspect.of.control.is.covered.in.Part.IV..Many.modern.factories.have.more.robots.than.humans..Therefore,.the.importance.of.mechatronics.and.robotics.can.never.be.overemphasized..The.various.aspects.of.robotics.and.mechatronics.are.described.in.Part.V..In.all.the.material.that.has.been.presented,.the.underlying.central.theme.has.been.to.consciously.avoid.the.typical.theorems.and.proofs.and.use.plain.English.and.examples.instead,.which.can.be.easily.understood.by.students.and.practicing.engineers.alike.
For.MATLAB.and.Simulink.product.information,.please.contact
The.MathWorks,.Inc.3.Apple.Hill.DriveNatick,.MA,.01760-2098.USATel:.508-647-7000Fax:.508-647-7001E-mail:.info@mathworks.comWeb:.www.mathworks.com
2011 by Taylor and Francis Group, LLC
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xiii
Acknowledgments
The.editors.wish.to.express.their.heartfelt.thanks.to.their.wives.Barbara.Wilamowski.and.Edie.Irwin.for.their.help.and.support.during.the.execution.of.this.project.
2011 by Taylor and Francis Group, LLC
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xv
Editorial Board
Timothy N. ChangNew.Jersey.Institute.of.TechnologyNewark,.New.Jersey
Okyay KaynakBogazici.UniversityIstanbul,.Turkey
Ren C. LuoNational.Taiwan.UniversityTaipei,.Taiwan
Istvn NagyBudapest.University.of.Technology.
andEconomicsBudapest,.Hungary
Kouhei OhnishiKeio.UniversityYokohama,.Japan
James R. RowlandUniversity.of.KansasLawrence,.Kansas
Xinghuo Yu RMIT.UniversityMelbourne,.Victoria,.Australia
2011 by Taylor and Francis Group, LLC
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xvii
Editors
Bogdan M. Wilamowski.received.his.MS.in.computer.engineering.in.1966,.his.PhD.in.neural.computing.in.1970,.and.Dr..habil..in.integrated.circuit.design.in.1977..He.received.the.title.of.full.professor.from.the.president.of.Poland. in.1987..He.was. the.director.of. the. Institute.of.Electronics. (19791981). and. the. chair. of. the. solid. state. electronics.department. (19871989). at. the. Technical. University. of. Gdansk,.Poland..He.was.a.professor.at.the.University.of.Wyoming,.Laramie,.from. 1989. to. 2000.. From. 2000. to. 2003,. he. served. as. an. associate.director. at. the. Microelectronics. Research. and. Telecommunication.Institute,.University.of.Idaho,.Moscow,.and.as.a.professor.in.the.elec-trical.and.computer.engineering.department.and.in.the.computer.sci-ence.department.at.the.same.university..Currently,.he.is.the.director.
of.ANMSTCAlabama.Nano/Micro.Science.and.Technology.Center,.Auburn,.and.an.alumna.professor.in.the.electrical.and.computer.engineering.department.at.Auburn.University,.Alabama..Dr.Wilamowski.was.with.the.Communication.Institute.at.Tohoku.University,.Japan.(19681970),.and.spent.one.year.at.the.Semiconductor.Research.Institute,.Sendai,.Japan,.as.a.JSPS.fellow.(19751976)..He.was.also.a.visiting.scholar.at.Auburn.University.(19811982.and.19951996).and.a.visiting.professor.at.the.University.of.Arizona,.Tucson.(19821984)..He.is.the.author.of.4.textbooks,.more.than.300.refereed.publications,.and.has.27.patents..He.was.the.principal.professor.for.about.130.graduate.students..His.main.areas.of.interest.include.semiconductor.devices.and.sensors,.mixed.signal.and.analog.signal.processing,.and.computa-tional.intelligence.
Dr..Wilamowski.was.the.vice.president.of.the.IEEE.Computational.Intelligence.Society.(20002004).and.the.president.of.the.IEEE.Industrial.Electronics.Society.(20042005)..He.served.as.an.associate.edi-tor.of.IEEE Transactions on Neural Networks,.IEEE Transactions on Education,.IEEE Transactions on Industrial Electronics,.the.Journal of Intelligent and Fuzzy Systems,.the.Journal of Computing,.and.the.International Journal of Circuit Systems and IES Newsletter..He.is.currently.serving.as.the.editor.in.chief.of.IEEE Transactions on Industrial Electronics.
Professor.Wilamowski. is.an.IEEE.fellow.and.an.honorary.member.of. the.Hungarian.Academy.of.Science..In.2008,.he.was.awarded.the.Commander.Cross.of.the.Order.of.Merit.of.the.Republic.of.Poland.for.outstanding.service. in. the.proliferation.of. international.scientific.collaborations.and.for.achieve-ments.in.the.areas.of.microelectronics.and.computer.science.by.the.president.of.Poland.
2011 by Taylor and Francis Group, LLC
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xviii Editors
J. David Irwin.received.his.BEE.from.Auburn.University,.Alabama,.in. 1961,. and. his. MS. and. PhD. from. the. University. of. Tennessee,.Knoxville,.in.1962.and.1967,.respectively.
In.1967,.he.joined.Bell.Telephone.Laboratories,.Inc.,.Holmdel,.New.Jersey,.as.a.member.of.the.technical.staff.and.was.made.a.supervisor.in.1968..He. then. joined.Auburn.University. in.1969.as.an.assistant.professor.of.electrical.engineering..He.was.made.an.associate.profes-sor.in.1972,.associate.professor.and.head.of.department.in.1973,.and.professor.and.head.in.1976..He.served.as.head.of.the.Department.of.Electrical.and.Computer.Engineering.from.1973.to.2009..In1993,.he.was.named.Earle.C..Williams.Eminent.Scholar.and.Head..From.
1982.to.1984,.he.was.also.head.of.the.Department.of.Computer.Science.and.Engineering..He.is.currently.theEarle.C..Williams.Eminent.Scholar.in.Electrical.and.Computer.Engineering.at.Auburn.
Dr.. Irwin. has. served. the. Institute. of. Electrical. and. Electronic. Engineers,. Inc.. (IEEE). Computer.Society.as.a.member.of.the.Education.Committee.and.as.education.editor.of.Computer..He.has.served.as. chairman. of. the. Southeastern. Association. of. Electrical. Engineering. Department. Heads. and. the.National. Association. of. Electrical. Engineering. Department. Heads. and. is. past. president. of. both. the.IEEE.Industrial.Electronics.Society.and.the.IEEE.Education.Society..He.is.a.life.member.of.the.IEEE.Industrial.Electronics.Society.AdCom.and.has.served.as.a.member.of.the.Oceanic.Engineering.Society.AdCom..He.served.for.two.years.as.editor.of.IEEE Transactions on Industrial Electronics..He.has.served.on. the. Executive. Committee. of. the. Southeastern. Center. for. Electrical. Engineering. Education,. Inc.,.and.was.president.of.the.organization.in.19831984..He.has.served.as.an.IEEE.Adhoc.Visitor.for.ABET.Accreditation.teams..He.has.also.served.as.a.member.of. the.IEEE.Educational.Activities.Board,.and.was.the.accreditation.coordinator.for.IEEE.in.1989..He.has.served.as.a.member.of.numerous.IEEE.com-mittees,.including.the.Lamme.Medal.Award.Committee,.the.Fellow.Committee,.the.Nominations.and.Appointments.Committee,.and.the.Admission.and.Advancement.Committee..He.has.served.as.a.mem-ber.of.the.board.of.directors.of.IEEE.Press..He.has.also.served.as.a.member.of.the.Secretary.of.the.Armys.Advisory.Panel.for.ROTC.Affairs,.as.a.nominations.chairman.for.the.National.Electrical.Engineering.Department.Heads.Association,.and.as.a.member.of.the.IEEE.Education.Societys.McGraw-Hill/Jacob.Millman. Award. Committee.. He. has. also. served. as. chair. of. the. IEEE. Undergraduate. and. Graduate.Teaching.Award.Committee..He.is.a.member.of.the.board.of.governors.and.past.president.of.Eta.Kappa.Nu,.the.ECE.Honor.Society..He.has.been.and.continues.to.be.involved.in.the.management.of.several.international.conferences.sponsored.by.the.IEEE.Industrial.Electronics.Society,.and.served.as.general.cochair.for.IECON05.
Dr.. Irwin. is. the. author. and. coauthor. of. numerous. publications,. papers,. patent. applications,. and.presentations,. including. Basic Engineering Circuit Analysis,. 9th. edition,. published. by. John. Wiley. &.Sons,.which.is.one.among.his.16.textbooks..His.textbooks,.which.span.a.wide.spectrum.of.engineering.subjects,.have.been.published.by.Macmillan.Publishing.Company,.Prentice.Hall.Book.Company,.John.Wiley.&.Sons.Book.Company,.and.IEEE.Press..He.is.also.the.editor.in.chief.of.a.large.handbook.pub-lished.by.CRC.Press,.and.is.the.series.editor.for.Industrial.Electronics.Handbook.for.CRC.Press.
Dr..Irwin.is.a.fellow.of.the.American.Association.for.the.Advancement.of.Science,.the.American.Society. for. Engineering. Education,. and. the. Institute. of. Electrical. and. Electronic. Engineers.. He.received. an. IEEE. Centennial. Medal. in. 1984,. and. was. awarded. the. Bliss. Medal. by. the. Society. of.American.Military.Engineers.in.1985..He.received.the.IEEE.Industrial.Electronics.Societys.Anthony.J..Hornfeck.Outstanding.Service.Award.in.1986,.and.was.named.IEEE.Region.III.(U.S..Southeastern.Region). Outstanding. Engineering. Educator. in. 1989.. In. 1991,. he. received. a. Meritorious. Service.Citation. from. the. IEEE. Educational. Activities. Board,. the. 1991. Eugene. Mittelmann. Achievement.Award.from.the.IEEE.Industrial.Electronics.Society,.and.the.1991.Achievement.Award.from.the.IEEE.Education.Society..In.1992,.he.was.named.a.Distinguished.Auburn.Engineer..In.1993,.he.received.the.IEEE.Education.Societys.McGraw-Hill/Jacob.Millman.Award,.and.in.1998.he.was.the.recipient.of.the.
2011 by Taylor and Francis Group, LLC
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Editors xix
IEEE.Undergraduate.Teaching.Award.. In.2000,.he.received.an.IEEE.Third.Millennium.Medal.and.the.IEEE.Richard.M..Emberson.Award..In.2001,.he.received.the.American.Society.for.Engineering.Educations. (ASEE). ECE. Distinguished. Educator. Award.. Dr.. Irwin. was. made. an. honorary. profes-sor,.Institute.for.Semiconductors,.Chinese.Academy.of.Science,.Beijing,.China,.in.2004..In.2005,.he.received.the.IEEE.Education.Societys.Meritorious.Service.Award,.and.in.2006,.he.received.the.IEEE.Educational.Activities.Board.Vice.Presidents.Recognition.Award..He.received.the.Diplome.of.Honor.from.the.University.of.Patras,.Greece,.in.2007,.and.in.2008.he.was.awarded.the.IEEE.IES.Technical.Committee.on.Factory.Automations.Lifetime.Achievement.Award..In.2010,.he.was.awarded.the.elec-trical.and.computer.engineering.department.heads.Robert.M..Janowiak.Outstanding.Leadership.and.Service.Award..In.addition,.he.is.a.member.of.the.following.honor.societies:.Sigma.Xi,.Phi.Kappa.Phi,.Tau.Beta.Pi,.Eta.Kappa.Nu,.Pi.Mu.Epsilon,.and.Omicron.Delta.Kappa.
2011 by Taylor and Francis Group, LLC
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xxi
Contributors
Marcelo H. Ang Jr.Department.of.Mechanical.EngineeringNational.University.of.SingaporeSingapore
Ramn BarberDepartment.of.System.Engineering.
andAutomationUniversity.Carlos.IIIMadrid,.Spain
Victor M. BecerraSchool.of.Systems.EngineeringUniversity.of.ReadingReading,.United.Kingdom
Miguel BernalCentro.Universitavio.de.los.VallesUniversity.of.GuadalajaraJalisco,.Mexico
Seta BogosyanElectrical.and.Computer.Engineering.
DepartmentUniversity.of.Alaska,.FairbanksFairbanks,.Alaska
Igor M. BoikoDepartment.of.Electrical.and.Computer.
EngineeringUniversity.of.CalgaryCalgary,.Alberta,.Canada
Alain BouscayrolLaboratoire.dElectrotechnique.et.dElectronique.
de.Puissance.de.LilleUniversity.of.Lille.1Lille,.France
A. John BoyeDepartment.of.Electrical.EngineeringUniversity.of.NebraskaandNeurintel,.LLCLincoln,.Nebraska
Shan ChaiSchool.of.Electrical.and.Computer.EngineeringRMIT.UniversityMelbourne,.Victoria,.Australia
Timothy N. ChangDepartment.of.Electrical.and.Computer.
EngineeringNew.Jersey.Institute.of.TechnologyNewark,.New.Jersey
J. Alexis De Abreu GarciaDepartment.of.Electrical.and.Computer.
EngineeringThe.University.of.AkronAkron,.Ohio
Christopher EdwardsDepartment.of.EngineeringUniversity.of.LeicesterLeicester,.United.Kingdom
Yong FengSchool.of.Electrical.and.Computer.EngineeringRMIT.UniversityMelbourne,.Victoria,.Australia
Emilia FridmanDepartment.of.Electrical.EngineeringSystemsTel.Aviv.UniversityTel.Aviv,.Israel
2011 by Taylor and Francis Group, LLC
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xxii Contributors
Leonid FridmanControl.and.Advanced.
RoboticsDepartmentNational.Autonomus.University.of.MexicoMexico.City,.Mexico
Metin GokasanControl.Engineering.DepartmentIstanbul.Technical.UniversityIstanbul,.Turkey
Thierry Marie GuerraLaboratory.of.Industrial.and.Human.
AutomationControlMechanical.Engineering.and.
ComputerScienceUniversity.of.Valenciennes.and.
Hainaut-CambresisValenciennes,.France
Fuat GurleyenControl.Engineering.DepartmentIstanbul.Technical.UniversityIstanbul,.Turkey
Fumio HarashimaTokyo.Metropolitan.UniversityTokyo,.Japan
Joel David HewlettDepartment.of.Electrical.and.Computer.
EngineeringAuburn.UniversityAuburn,.Alabama
Guan-Chyun HsiehDepartment.of.Electrical.EngineeringChung.Yuan.Christian.UniversityChung-Li,.Taiwan
James C. HungDepartment.of.Electrical.Engineering.
andComputer.ScienceThe.University.of.Tennessee,.KnoxvilleKnoxville,.Tennessee
John Y. HungDepartment.of.Electrical.and.Computer.
EngineeringAuburn.UniversityAuburn,.Alabama
Makoto IwasakiDepartment.of.Computer.Science.
andEngineeringNagoya.Institute.of.TechnologyNagoya,.Japan
Raymond JarvisIntelligent.Robotics.Research.CentreMonash.UniversityMelbourne,.Victoria,.Australia
Okyay KaynakDepartment.of.Electrical.and.Electronic.
EngineeringBogazici.UniversityIstanbul,.Turkey
Lindsay KleemanDepartment.of.Electrical.and.Computer.Systems.
EngineeringMonash.UniversityMelbourne,.Victoria,.Australia
Tong Heng LeeDepartment.of.Electrical.and.Computer.
EngineeringNational.University.of.SingaporeSingapore
Arie LevantApplied.Mathematics.DepartmentTel.Aviv.UniversityTel.Aviv,.Israel
Chin F. LinDepartment.of.Electrical.EngineeringNational.Chung.Cheng.UniversityChia-Yi,.Taiwan
Ren C. LuoDepartment.of.Electrical.EngineeringNational.Taiwan.UniversityTaipei,.Taiwan
Mara MalfazDepartment.of.System.Engineering.
andAutomationUniversity.Carlos.IIIMadrid,.Spain
2011 by Taylor and Francis Group, LLC
-
Contributors xxiii
Jorge Angel Davila MontoyaAeronautic.Engineering.DepartmentNational.Polytechnic.InstituteMexico.City,.Mexico
Toshiyuki MurakamiDepartment.of.System.Design.EngineeringKeio.UniversityYokohama,.Japan
Istvn NagyDepartment.of.Automation.and.Applied.
InformaticsBudapest.University.of.Technology.
andEconomicsBudapest,.Hungary
Roberto OboeDepartment.of.Management.and.EngineeringUniversity.of.PadovaVicenza,.Italy
Kouhei OhnishiDepartment.of.System.Design.EngineeringKeio.UniversityYokohama,.Japan
Nejat OlgacDepartment.of.Mechanical.EngineeringUniversity.of.ConnecticutStorrs,.Connecticut
Sanjib Kumar PandaDepartment.of.Electrical.and.Computer.
EngineeringNational.University.of.SingaporeSingapore
Eric RogersSchool.of.Electronics.and.Computer.ScienceUniversity.of.SouthamptonSouthampton,.United.Kingdom
James R. RowlandDepartment.of.Electrical.Engineering.
andComputer.ScienceUniversity.of.KansasLawrence,.Kansas
Asif abanoviFaculty.of.Engineering.and.Natural.SciencesSabanci.UniversityIstanbul,.Turkey
Nadira abanovi-BehliloviFaculty.of.Engineering.and.Natural.SciencesSabanci.UniversityIstanbul,.Turkey
Sanjib Kumar SahooDepartment.of.Electrical.and.Computer.
EngineeringNational.University.of.SingaporeSingapore
Miguel A. SalichsSystems.Engineering.and.Automation.
DepartmentUniversity.Carlos.IIIMadrid,.Spain
Naresh K. SinhaDepartment.of.Electrical.and.Computer.
EngineeringMcMaster.UniversityHamilton,.Ontario,.Canada
Rifat SipahiDepartment.of.Mechanical.and.Industrial.
EngineeringNortheastern.UniversityBoston,.Massachusetts
Zoltn StoDepartment.of.Automation.and.Applied.
InformaticsBudapest.University.of.Technology.
andEconomicsBudapest,.Hungary
Satoshi SuzukiDepartment.of.Robotics.and.MechatronicsSchool.of.Science.and.Technology.for.Future.LifeTokyo.Denki.UniversityTokyo,.Japan
2011 by Taylor and Francis Group, LLC
-
xxiv Contributors
Kok Kiong TanDepartment.of.Electrical.and.Computer.
EngineeringNational.University.of.SingaporeSingapore
Chee Pin TanSchool.of.EngineeringMonash.UniversityMalaysia
Kok Zuea TangDepartment.of.Electrical.and.Computer.
EngineeringNational.University.of.SingaporeSingapore,.Singapore
Robert J. VeilletteDepartment.of.Electrical.and.Computer.
EngineeringThe.University.of.AkronAkron,.Ohio
Liuping WangSchool.of.Electrical.and.Computer.EngineeringRMIT.UniversityMelbourne,.Victoria,.Australia
Changyun WenSchool.of.Electrical.and.Electronic.EngineeringNanyang.Technological.UniversitySingapore
Bogdan M. WilamowskiDepartment.of.Electrical.and.Computer.
EngineeringAuburn.UniversityAuburn,.Alabama
Tiantian XieDepartment.of.Electrical.and.Computer.
EngineeringAuburn.UniversityAuburn,.Alabama
Jian-Xin XuDepartment.of.Electrical.and.Computer.
EngineeringNational.University.of.SingaporeSingapore
Choon-Seng YeeDepartment.of.Mechanical.EngineeringNational.University.of.SingaporeSingapore
Xinghuo YuSchool.of.Electrical.and.Computer.EngineeringRMIT.UniversityMelbourne,.Victoria,.Australia
Jing ZhouPetroleum.DepartmentInternational.Research.Institute.of.StavangerBergen,.Norway
2011 by Taylor and Francis Group, LLC
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I-1
IControl System Analysis 1 Nonlinear Dynamics Istvn Nagy and Zoltn Sto...........................................................1-1
Introduction. . Basics. . Equilibrium.Points. . Limit.Cycle. . Quasi-Periodic.and.Frequency-Locked.State. . Dynamical.Systems.Described.by.Discrete-Time.Variables:.Maps. . Invariant.Manifolds:.Homoclinic.and.Heteroclinic.Orbits. . Transitions.to.Chaos. . Chaotic.State. . Examples.from.Power.Electronics. . Acknowledgments. . References
2 Basic Feedback Concept Tong Heng Lee, Kok Zuea Tang, and Kok Kiong Tan.............2-1Basic.Feedback.Concept. . Bibliography
3 Stability Analysis Naresh K. Sinha.......................................................................................3-1Introduction. . States.of.Equilibrium. . Stability.of.Linear.Time-Invariant.Systems. . Stability.of.Linear.Discrete-Time.Systems. . Stability.of.Nonlinear.Systems. . References
4 Frequency-Domain Analysis of Relay Feedback Systems Igor M. Boiko......................4-1RelayFeedback.Systems. . Locus.of.a.Perturbed.Relay.System.Theory. . Design.ofCompensating.Filters.in.Relay.Feedback.Systems. . References
5 Linear Matrix Inequalities in Automatic Control Miguel Bernal and Thierry Marie Guerra................................................................................................................................5-1What.Are.LMIs?. . What.Are.LMIs.Good.For?. . References
6 Motion Control Issues Roberto Oboe, Makoto Iwasaki, Toshiyuki Murakami, and Seta Bogosyan........................................................................................................................6-1Introduction. . High-Accuracy.Motion.Control. . Motion.Control.and.Interaction.withthe.Environment. . Remote.Motion.Control. . Conclusions. . References
7 New Methodology for Chatter Stability Analysis in Simultaneous Machining Nejat Olgac and Rifat Sipahi..............................................................................7-1Introduction.and.a.Review.of.Single.Tool.Chatter. . Regenerative.Chatter.in.Simultaneous.Machining. . CTCR.Methodology. . Example.Case.Studies. . Optimization.of.the.Process. . Conclusion. . Acknowledgments. . References
2011 by Taylor and Francis Group, LLC
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1-1
1.1 Introduction
A.new.class.of.phenomena.has.recently.been.discovered.three.centuries.after.the.publication.of.Newtons Principia.(1687).in.nonlinear.dynamics..New.concepts.and.terms.have.entered.the.vocabulary.to.replace.time. functions. and. frequency. spectra. in. describing. their. behavior,. e.g.,. chaos,. bifurcation,. fractal,.Lyapunov.exponent,.period.doubling,.Poincar.map,.and.strange.attractor.
Until.recently,.chaos.and.order.have.been.viewed.as.mutually.exclusive..Maxwells.equations.gov-ern.the.electromagnetic.phenomena;.Newtons.laws.describe.the.processes.in.classical.mechanics,.etc..
1Nonlinear Dynamics
1.1. Introduction....................................................................................... 1-11.2. Basics................................................................................................... 1-2
Classification. . Restrictions. . Mathematical.Description1.3. Equilibrium.Points............................................................................ 1-5
Introduction. . Basin.of.Attraction. . Linearizing.around.theEP. . Stability. . Classification.of.EPs,.Three-Dimensional.StateSpace.(N.=.3). . No-Intersection.Theorem
1.4. Limit.Cycle.......................................................................................... 1-9Introduction. . Poincar.Map.Function.(PMF). . Stability
1.5. Quasi-Periodic.and.Frequency-Locked.State.............................. 1-12Introduction. . Nonlinear.Systems.with.Two.Frequencies. . .Geometrical.Interpretation. . N-Frequency.Quasi-Periodicity
1.6. Dynamical.Systems.Described.by.Discrete-Time.Variables:Maps.................................................................................1-14Introduction. . Fixed.Points. . MathematicalApproach. . .Graphical.Approach. . Study.of.Logistic.Map. .Stability.of.Cycles
1.7. Invariant.Manifolds:.Homoclinic.and.Heteroclinic.Orbits...... 1-21Introduction. . Invariant.Manifolds,.CTM. . Invariant.Manifolds,.DTM. . Homoclinic.and.Heteroclinic.Orbits,.CTM
1.8. Transitions.to.Chaos....................................................................... 1-24Introduction. . Period-Doubling.Bifurcation. . Period-Doubling.Scenario.in.General
1.9. Chaotic.State..................................................................................... 1-26Introduction. . Lyapunov.Exponent
1.10. Examples.from.Power.Electronics................................................ 1-27Introduction. . High-Frequency.Time-Sharing.Inverter. . Dual.Channel.Resonant.DCDC.Converter. . Hysteresis.Current-Controlled.Three-Phase.VSC. . Space.Vector.Modulated.VSC.withDiscrete-Time.Current.Control. . Direct.Torque.Control
Acknowledgments....................................................................................... 1-41References..................................................................................................... 1-41
Istvn NagyBudapestUniversityofTechnologyandEconomics
Zoltn StoBudapestUniversityofTechnologyandEconomics
2011 by Taylor and Francis Group, LLC
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1-2 ControlandMechatronics
They.represent.the.world.of.order,.which.is.predictable..Processes.were.called.chaotic.when.they.failed.to.obey.laws.and.they.were.unpredictable..Although.chaos.and.order.have.been.believed.to.be.quite.distinct.faces.of.our.world,.there.were.tricky.questions.to.be.answered..For.example,.knowing.all.the.laws.governing.our.global.weather,.we.are.unable.to.predict.it,.or.a.fluid.system.can.turn.easily.from.order.to.chaos,.from.laminar.flow.into.turbulent.flow.
It.came.as.an.unexpected.discovery.that.deterministic.systems.obeying.simple.laws.belonging.undoubt-edly.to.the.world.of.order.and.believed.to.be.completely.predictable.can.turn.chaotic..In.mathematics,.the.study.of.the.quadratic.iterator.(logistic.equation.or.population.growth.model).[xn+1.=.axn(1..xn),.n.=.0,.1,.2,.].revealed.the.close.link.between.chaos.and.order.[5]..Another.very.early.example.came.from.the.atmospheric.science.in.1963;.Lorenzs.three.differential.equations.derived.from.the.NavierStokes.equa-tions.of.fluid.mechanics.describing.the.thermally.induced.fluid.convection.in.the.atmosphere,.Peitgen.etal..[9]..They.can.be.viewed.as.the.two.principal.paradigms.of.the.theory.of.chaos..One.of.the.first.cha-otic.processes.discovered.in.electronics.can.be.shown.in.diode.resonator.consisting.of.a.series.connection.of.a.pn.junction.diode.and.a.10100.mH.inductor.driven.by.a.sine.wave.generator.of.50100.kHz.
The.chaos.theory,.although.admittedly.still.young,.has.spread.like.wild.fire.into.all.branches.of.sci-ence..In.physics,.it.has.overturned.the.classic.view.held.since.Newton.and.Laplace,.stating.that.our.uni-verse.is.predictable,.governed.by.simple.laws..This.illusion.has.been.fueled.by.the.breathtaking.advances.in.computers,.promising.ever-increasing.computing.power.in.information.processing..Instead,.just.the.opposite.has.happened..Researchers.on.the.frontier.of.natural.science.have.recently.proclaimed.that.this.hope. is.unjustified.because.a. large.number.of.phenomena.in.nature.governed.by.known.simple. laws.are.or.can.be.chaotic..One.of.their.principle.properties.is.their.sensitive.dependence.on.initial.condi-tions..Although.the.most.precise.measurement.indicates.that.two.paths.have.been.launched.from.the.same.initial.condition,.there.are.always.some.tiny,.impossible-to-measure.discrepancies.that.shift.the.paths.along.very.different.trajectories..The.uncertainty. in.the. initial.measurements.will.be.amplified.and.become.overwhelming.after.a.short.time..Therefore,.our.ability.to.predict.accurately.future.develop-ments.is.unreasonable..The.irony.of.fate.is.that.without.the.aid.of.computers,.the.modern.theory.of.chaos.and.its.geometry,.the.fractals,.could.have.never.been.developed.
The.theory.of.nonlinear.dynamics.is.strongly.associated.with.the.bifurcation.theory..Modifying.the.parameters.of.a.nonlinear.system,.the.location.and.the.number.of.equilibrium.points.can.change..The.study.of.these.problems.is.the.subject.of.bifurcation.theory.
The.existence.of.well-defined.routes.leading.from.order.to.chaos.was.the.second.great.discovery.and.again.a.big.surprise.like.the.first.one.showing.that.a.deterministic.system.can.be.chaotic.
The.overview.of.nonlinear.dynamics.here.has. two.parts..The.main.objective. in. the.first.part. is. to.summarize. the. state. of. the. art. in. the. advanced. theory. of. nonlinear. dynamical. systems.. Within. the.overview,.five.basic.states.or.scenario.of.nonlinear.systems.are.treated:.equilibrium.point,.limit.cycle,.quasi-periodic. (frequency-locked).state,. routes. to.chaos,.and.chaotic.state..There.will.be.some.words.about.the.connection.between.the.chaotic.state.and.fractal.geometry.
In.the.second.part,.the.application.of.the.theory.is.illustrated.in.five.examples.from.the.field.of.power.elec-tronics..They.are.as.follows:.high-frequency.time-sharing.inverter,.voltage.control.of.a.dual-channel.resonant.DCDC.converter,.and.three.different.control.methods.of.the.three-phase.full.bridge.voltage.source.DCAC/ACDC.converter,.a.sophisticated.hysteresis.current.control,.a.discrete-time.current.control.equipped.with.space.vector.modulation.(SVM).and.the.direct.torque.control.(DTC).applied.widely.in.AC.drives.
1.2 Basics
1.2.1 Classification
The. nonlinear. dynamical. systems. have. two. broad. classes:. (1). autonomous systems. and. (2). non-autonomous systems..Both.are.described.by.a.set.of.first-order.nonlinear.differential.equations.and.can.be.represented.in.state.(phase).space..The.number.of.differential.equations.equals.the.degree of freedom.
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NonlinearDynamics 1-3
(ordimension).of.the.system,.which.is.the.number.of.independent.state.variables.needed.to.determine.uniquely.the.dynamical.state.of.the.system.
1.2.1.1 autonomous Systems
There.are.no.external.input.or.forcing.functions.applied.to.the.system..The.set.of.nonlinear.differential.equations.describing.the.system.is
.ddtx v f x= = ,( )m
.(1.1)
wherexvT
NT
N
x x xv v v
=
=
[ , , , ][ , , ]
1 2
1 2
is the state vectoris the velocity vvectoris the nonlinear vector functionf T N
Tf f f=
=
[ , , ][ ,
1 2
1 2
m ,, ]NTt
is the parameter vectordenotes the transpose of a vectoris tthe timeis the dimension of the systemN
The.time.t.does.not.appear.explicitly.
1.2.1.2 Non-autonomous Systems
Time-dependent.external.inputs.or.forcing.functions.u(t).are.applied.to.the.system..It.is.a.set.of.nonlin-ear.differential.equations:
.ddt
tx v f x u= = , ,( ( ) )m.
(1.2)
Time.t.explicitly.appears.in.u(t)..(1.1).and.(1.2).can.be.solved.analytically.or.numerically.for.a.given.ini-tial.condition.x0.and.parameter.vector...The.solution.describes.the.state.of.the.system.as.a.function.of.time..The.solution.can.be.visualized.in.a.reference.frame.where.the.state.variables.are.the.coordinates..It.is.called.the.state space.or.phase space..At.any.instant,.a.point.in.the.state.space.represents.the.state.of.the.system..As.the.time.evolves,.the.state.point.is.moving.along.a.path.called.trajectory.or.orbit.starting.from.the.initial.condition.
1.2.2 restrictions
The.non-autonomous.system.can.always.be.transformed.to.autonomous.systems.by.introducing.a.new.state.variable.xN+1.=.t..Now.the.last.differential.equation.in.(1.1).is
.dxdt
dtdt
N += =
1 1.
(1.3)
The.number.of.dimensions.of.the.state.space.was.enlarged.by.one.by.including.the.time.as.a.state.vari-able..From.now.on,.we.confine.our.consideration.to.autonomous.systems.unless.it.is.stated.otherwise..By.this.restriction,.there.is.no.loss.of.generality.
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1-4 ControlandMechatronics
The.discussion.is.confined.to.real,.dissipative.systems..As.time.evolves,.the.state.variables.will.head.for.some.final.point,.curve,.area,.or.whatever.geometric.object.in.the.state.space..They.are.called.the.attractor.for.the.particular.system.since.certain.dedicated.trajectories.will.be.attracted.to.these.objects..We.focus.our.considerations.on.the.long-term.behavior.of.the.system.rather.than.analyzing.the.start-up.and.transient.processes.
The.trajectories.are.assumed.to.be.bounded.as.most.physical.systems.are..They.cannot.go.to.infinity.
1.2.3 Mathematical Description
Basically,.two.different.concepts.applying.different.approaches.are.used..The.first.concept.considers.all.the.state.variables.as.continuous.quantities.applying.continuous-time.model.(CTM)..As.time.evolves,.the.system.behavior.is.described.by.a.moving.point. in.state.space.resulting.in.a.trajectory.(or.flow).obtained.by.the.solution.of.the.set.of.differential.equations.(1.1).[or.(1.2)]..Figure.1.1.shows.the.continu-ous.trajectory.of.function.(x0,.t,.).for.three-dimensional.system,.where.x0.is.the.initial.point..The.second.concept.takes.samples.from.the.continuously.changing.state.variables.and.describes.the.system.behavior.by.discrete.vector.function.applying.the.Poincar concept..Figure.1.1.shows.the.way.how.the.samples.are.taken.for.a.three-dimensional.autonomous.system..A.so-called.Poincar plane,.in.general.Poincar.section,.is.chosen.and.the.intersection.points.cut.by.the.trajectory.are.recorded.as.samples..The.selection.of.the.Poincar.plane.is.not.crucial.as.long.as.the.trajectory.cuts.the.surface.transversely..The.relation.between.xn.and.xn+1,.i.e.,.between.subsequent.intersection.points.generated.always.from.the.same.direction.are.described.by.the.so-called.Poincar.map.function.(PMF)
. x P xn n+ =1 ( ) . (1.4)
Pay.attention,.the.subscript.of.x.denotes.the.time.instant,.not.a.component.of.vector.x..The.Poincar.sec-tion.is.a.hyperplane.for.systems.with.dimension.higher.than.three,.while.it.is.a.point.and.a.straight.line.for.a.one-.and.a.two-dimensional.system,.respectively.
For.a.non-autonomous.system.having.a.periodic.forcing.function,.the.samples.are.taken.at.a.definite.phase.of.the.forcing.function,.e.g.,.at.the.beginning.of.the.period..It.is.a.stroboscopic.sampling,.the.state.variables.for.a.mechanical.system.are.recorded.with.a.flash.lamp.fired.once.in.every.period.of.the.forc-ing.function.[8]..Again,.the.PMF.describes.the.relation.between.sampled.values.of.the.state.variables.
Knowing.that.the.trajectories.are.the.solution.of.differential.equation.system,.which.are.unique.and.deterministic,.it.implies.the.existence.of.a.mathematical.relation.between.xn.and.xn+1,.i.e.,.the.existence.
x3
x2
xn
x0 x1
xn+1=P(xn)
Poincarplane
Trajectory (x0, t, )
FIGURE 1.1 Trajectory.described.by.(x0,. t,.)..xn,.xn+1. are. the. intersection.points.of. the. trajectory.with. the.Poincar.plane.
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NonlinearDynamics 1-5
of.PMF..However,.the.discrete-time.equation.(1.4).can.be.solved.analytically.or.numerically.indepen-dent.of.the.differential.equation..In.the.second.concept,.the.discrete-time.model.(DTM).is.used.
The.Poincar.section.reduces.the.dimensionality.of.the.system.by.one.and.describes.it.by.an.iterative,.finite-size.time.step.function.rather.than.a.differential,.infinitesimal.time.step..On.the.other.hand,.PMF.retains.the.essential.information.of.the.system.dynamics.
Even.though.the.state.variables.are.changing.continuously.in.many.systems.like.in.power.electron-ics,.they.can.advantageously.be.modeled.by.discrete-iteration.function.(1.4)..In.some.other.cases,.the.system.is.inherently.discrete,.their.state.variables.are.not.changing.continuously.like.in.digital.systems.or.models.describing.the.evolution.of.population.of.species.
1.3 Equilibrium Points
1.3.1 Introduction
The.nonlinear.world.is.much.more.colorful.than.the.linear.one..The.nonlinear.systems.can.be.in.various.states,.one.of.them.is.the.equilibrium point.(EP)..It.is.a.point.in.the.state.space.approached.by.the.trajec-tory.of.a.continuous,.nonlinear.dynamical.system.as.its.transients.decay..The.velocity.of.state.variables.v.=.x.is.zero.in.the.EP:
.ddtx v f x= = , =( )m 0
.(1.5)
The. solution. of. the. nonlinear. algebraic. function. (1.5). can. result. in. more. than. one. EPs.. They. are.x x x x1 2* * * *, , , , k n, ..The.stable.EPs.are.attractors.
1.3.2 Basin of attraction
The.natural.consequence.of.the.existence.of.multiple.attractors.is.the.partition.of.state.space.into.dif-ferent. regions. called. basins of attractions.. Any. of. the. initial. conditions. within. a. basin. of. attraction.launches.a.trajectory.that.is.finally.attracted.by.the.particular.EP.belonging.to.the.basin.of.attraction.(Figure.1.2)..The.border.between.two.neighboring.basins.of.attraction.is.called.separatrix..They.organize.the.state.space.in.the.sense.that.a.trajectory.born.in.a.basin.of.attraction.will.never.leave.it.
x2
x1
x3
x*2
x*1
x*i
x*2 (0)
x*k (0)
x*k
Basins of attractions
Linearizedregion byJacobianmatrix Jk
FIGURE 1.2 Basins. of. attraction. and. their. EP. (N. =. 3).. x1,. x2,. and. x3. are. coordinates. of. the. state. space..vectorx..xk*.is.a.particular.value.of.state.space.vector.x,.xk*.denotes.an.EP,.and.xk(0).is.an.initial.condition.in.the.basin.of.attraction.of.xk*.
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1-6 ControlandMechatronics
1.3.3 Linearizing around the EP
Introducing. the.small.perturbation. = x x xk*,. (1.1).can.be. linearized. in. the.close.neighborhood.of.the.EP.xk* ..Now. f x x f x J x( * ) ( * )k k k+ = + +, ,m m ..Neglecting.the.terms.of.higher.order.than.x.and.substituting.it.back.to.(1.1):
.ddt k
= x v J x.
(1.6)
where
.
Jk
N
N
N N
fx
fx
fx
fx
fx
fx
fx
fx
f
=
1
1
1
2
1
2
1
2
2
2
1 2
NN
Nx
.
(1.7)
is.the.Jacobian.matrix..The.partial.derivatives.have.to.be.evaluated.at. xk* ..f(x*,.).=.0.was.observed.in.(1.6)..Jk.is.a.real,.time-independent.N..N.matrix..Seeking.the.solution.of.(1.7).in.the.form
. =x erte . (1.8)
and.substituting.it.back.to.(1.6):
. J e ek r r= . (1.9)
Its.nontrivial.solution.for..must.satisfy.the.Nth.order.polynomial.equation
. det( )J Ik = 0 . (1.10)
where.I.is.the.N..N.identity.matrix..From.(1.6),.(1.8),.and.(1.9),
. = =v J e ek rt
rte e . (1.11)
Selecting.the.direction.of.vector.x.in.the.special.way.given.by.(1.8),.i.e.,.its.change.in.time.depends.only.on.one.constant.,.it.has.two.important.consequences:
. 1.. The.product.Jker.(=.er).only.results.in.the.expansion.or.contraction.of.er.by...The.direction.of.er.is.not.changed.
. 2.. The.direction.of.the.perturbation.of.the.velocity.vector.v.will.be.the.same.as.that.of.vector.er .
The.direction.of.er.is.called.characteristic.direction.and.er.is.the.right-hand.side.eigenvector.of.Jk.as.Jk.is.multiplied.from.the.right.by.er...is.the.eigenvalue.(or.characteristic.exponent).of.Jk.
We.confine.our.consideration.of.N.distinct.roots.of.(1.10).(multiple.roots.are.excluded)..They.are.1,.2,.m,,.N..Correspondingly,.we.have.N.distinct.eigenvectors.as.well..The.general.solution.of.(1.6):
. = =
= =
x e( ) ( )t e tm
N
mt
m
N
mm
1 1
.
(1.12)
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-
NonlinearDynamics 1-7
Here.we.assumed.that.the.initial.condition.was. = ==
x e( )t mN m0 1 ..The.roots.of.(1.10).are.either.real.or.complex.conjugate.ones.since.the.coefficients.are.real.in.(1.10).
When. we. have. complex. conjugate. pairs. of. eigenvalues. m. =. m+1. =. m. . jm,. the. corresponding.eigenvectors.are.em.=.m+1.=.em,R.+.jem,I,.where.the..denotes.complex.conjugate.and.where.m,.m,.and.em,R,.em,I.are.all.real.and.real-valued.vectors,.respectively..From.the.two.complex.solutions.m(t).=.em.exp(mt).and.m+1(t).=.em+1.exp(m+1t),.two.linearly.independent.real.solutions.sm(t).and.sm+1(t).can.be.composed:
.s e em m m t m R m m I mt t t e t tm( ) ( ) ( ) cos sin= + = + + , ,12 1
.
(1.13)
.s e em m m t m I m m R mt j
t t e t tm+ + , ,= = 1 112( ) ( ) ( ) cos sin
.(1.14)
When.the.eigenvalue.is.real.m.=.m.and.the.solution.belonging.to.m,
. s em m mtt t e m( ) ( )= = . (1.15)
Note.that.the.time.function.belonging.to.an.eigenvalue.or.a.pair.of.eigenvalues.is.the.same.for.all.state.variables.
1.3.4 Stability
The.EP.is.stable.if.and.only.if.the.real.part.m.of.all.eigenvalues.belonging.to.the.EP.is.negative..Otherwise,.one.or.more.solutions.sm(t).goes.to.infinity..m.=.0.is.considered.as.unstable.case..When.m.0
Unstablem>0
em
em,I
em,RP
PStablem
-
1-8 ControlandMechatronics
1.3.5 Classification of EPs, three-Dimensional State Space (N = 3)
Depending.on.the.location.of.the.three.eigenvalues.in.the.complex.plane,.eight.types.of.the.EPs.are.distin-guished.(Figure.1.4)..Three.eigenvectors.are.used.for.reference.frame..The.origin.is.the.EP..Eigenvectors.e1,.e2,.and.e3.are.used.when.all.eigenvalues.are.real.(Figure.1.4a,c,e,g).and.e1,.e2,R,.and.e2,I.are.used.when.we.have.a.pair.of.conjugate.complex.eigenvalues.(Figure.1.4b,d,f,h)..The.orbits.(trajectories).are.moving.exponentially.in.time.along.the.eigenvectors.e1,.e2,.or.e3.when.the.initial.condition.is.on.them..The.orbits.are.spiraling.in.the.plane.spanned.by.e2,R.and.e2,I.with.initial.condition.in.the.plane..The.operation.points.are.attracted.(repelled).by.stable.(unstable).EP.
All.three.eigenvalues.are.on.the.left-hand.side.of.the.complex.plane.for.node.and.spiral node.(Figure.1.4a.and.b)..The.spiral.node.is.also.called.attracting focus..All.three.eigenvalues.are.on.the.right-hand.side.for.repeller.and.spiral repeller.(Figure.1.4c.and.d)..The.spiral.repeller.is.also.called.repelling focus..For.saddle points,.either.one.(Index.1).(Figure.1.4e.and.f).or.two.(Index.2).(Figure.1.4g.and.h).eigenvalues.are.on.the.right-hand.side.
Saddle.points.play.very.important.role.in.organizing.the.trajectories.in.state.space..A.trajectory.asso-ciated. to.an.eigenvector.or.a.pair.of.eigenvectors.can.be.stable.when. its.eigenvalue(s). is. (are).on. the.left-hand.side.of.the.complex.plane.(Figure.1.4a.and.b).or.unstable.when.they.are.on.the.right-hand.side.(Figure.1.4c.through.f)..Trajectories.heading.directly.to.and.directly.away.from.a.saddle.point.are.called.
e2
e2
e1
e1
Im
Im
Re
Re
e2,R
e2,R
e2,I
e2,I
e1
e1
Im
Im
Re
Re
Hopfbifurcation
e3
e3
(a) (b)
(c) (d)
e2
e2
e1
e1
Im
Im
Re
Re
e2,R
e2,R
e2,I
e2,I
e1
e1
Im
Im
Re
Re
e3
e3
(e) (f )
(g) (h)
Stablemanifold
Stablemanifold
Unstablemanifold
Unstablemanifold
FIGURE 1.4 Classifications.of.EPs. (N.=.3).. (a).Node,. (b). spiral.node,. (c). repeller,. (d). spiral. repeller,. (e). saddle.pointindex.1,.(f).spiral.saddle.pointindex.1,.(g).saddle.point2,.and.(h).spiral.saddle.pointindex.2.
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-
NonlinearDynamics 1-9
stable.and.unstable invariant manifold.or.shortly.manifold..Sometimes,.the.stable.(unstable).manifolds.are.called.insets.(outsets)..The.operation.point.either.on.the.stable.or.on.the.unstable.manifold.cannot.leave. the.manifold..The.manifolds.of. a. saddle.point. in. its.neighborhood. in.a. two-dimensional. state.space.divide.it.into.four.regions..A.trajectory.born.in.a.region.is.confined.to.the.region..The.manifolds.are.part.of.the.separatrices.separating.the.basins.of.attractions..In.this.sense,.the.manifolds.organize.the.state.space.
Finally,.when.the.real.part.m.is.zero.in.the.pair.of.the.conjugate.complex.eigenvalue.and.the.dimen-sion.is.N.=.2,.the.EP.is.called.center..The.trajectories.in.the.reference.frame.eReI.are.circles.(Figure.1.5)..Their.radius.is.determined.by.the.initial.condition.
1.3.6 No-Intersection theorem
Trajectories. in.state.space.cannot. intersect.each.other..The.theorem.is. the.direct.consequence.of. the.deterministic.system..The.state.of.the.system.is.unambiguously.determined.by.the.location.of.its.opera-tion.point.in.the.state.space..As.the.system.is.determined.by.(1.1),.all.of.the.derivatives.are.determined.by.the.instantaneous.values.of.the.state.variables..Consequently,.there.is.only.one.possible.direction.for.a.trajectory.to.continue.its.journey.
1.4 Limit Cycle
1.4.1 Introduction
Two-.or.higher-dimensional.nonlinear.systems.can.exhibit.periodic.(cyclic).motion.without.external.periodical.excitation..This.behavior.is.represented.by.closed-loop.trajectory.called.Limit Cycle.(LC).in.the.state.space..There.are.stable.(attracting).and.unstable.(repelling).LC..The.basic.difference.between.the.stable.LC.and.the.center.(see.Figure.1.5).having.closed.trajectory.is.that.the.trajectories.starting.from.nearby.initial.points.are.attracted.by.stable.limit.cycle.and.sooner.or.later.they.end.up.in.the.LC,.while.the.trajectories.starting.from.different.initial.conditions.will.stay.forever.in.different.tracks.determined.by.the.initial.conditions.in.the.case.of.center.
Figure.1.6.shows.a.stable.LC.together.with.the.Poincar.plane..Here,.the.dimension.is.3..The.LC.inter-sects.the.Poincar.plane.at.point.Pk.called.Fixed Point.(FP)..It.plays.a.crucial.role.in.nonlinear.dynamics..Instead.of.investigating.the.behavior.of.the.LC,.the.FP.is.studied.
1.4.2 Poincar Map Function (PMF)
After.moving.the.trajectory.from.the.LC.by.a.small.deviation,.the.discrete.Poinear.Map.Function.(PMF).relates.the.coordinates.of.intersection.point.Pn.to.those.of.the.previous.point.Pn1..All.points.are. the. intersection.points. in. the.Poincar.plane.generated.by. the. trajectory..The.PMF.in.a. three-dimensional.state.space.is.(Figure.1.6)
Initialconditions
e2
e1
Im
Re
FIGURE 1.5 Center..Eigenvalues.are.complex.conjugate,.N.=.2.
2011 by Taylor and Francis Group, LLC
-
1-10 ControlandMechatronics
.
u P u v
v P u v
n n n
n n n
= ,( )= ,( )
1 1 1
2 1 1 .
(1.16)
whereP1.and.P2.are.the.PMFun.and.vn.are.the.coordinates.of.the.intersection.point.Pn.in.the.Poincar.plane
Introducing.vector.zn.by
. znT
n nu v= , . (1.17)
where.zn.points.to.intersection.Pn.from.the.origin,.the.PMF.is
. z P zn n= ( )1 . (1.18)
At.fixed.point.Pk
. z P zk k= ( ) . (1.19)
The.first.benefit.of.applying.the.PMF.is.the.reduction.of.the.dimension.by.one.as.the.stability.of.FP.Pk.is.studied.now.in.the.two-dimensional.state.space.rather.than.studying.the.stability.of.the.LC.in.the.three-dimensional.state.space,.and.the.second.one.is.the.substitution.of.the.differential.equation.by.difference.equation.
1.4.3 Stability
The.stability.can.be.investigated.on.the.basis.of.the.PMF.(1.18)..First,.the.nonlinear.function.P(zn1).has.to.be.linearized.by.its.Jacobian.matrix.Jk.evaluated.at.its.FP.zk..Knowing.Jk,.(1.18).can.be.rewritten.for.small.perturbation.around.the.FP.zk.as
Poincarplane
Trajectory
Attracting limit cyclex2
x1
x3
Pk is fixed point
znPk Pn
Pn1
v
u
FIGURE 1.6 Stable.limit.cycle.and.the.Poincar.plane.(N.=.3).
2011 by Taylor and Francis Group, LLC
-
NonlinearDynamics 1-11
. = = z J J zn k n knz 1 0 . (1.20)
where.z0.is.the.initial.small.deviation.from.FP.Pk.Substituting.Jk.by.its.eigenvalues.k.and.right.emr.and.left.eml.eigenvectors,
. =
=
z e e zn mN
mn
mr mlT
1
1
0.
(1.21)
Due.to.(1.21),.the.LC.is.stable.if.and.only.if.all.eigenvalues.are.within.the.circle.with.unit.radius.in.the.complex.plane.
In.the.stability.analysis,.both.in.continuous-time.model.(CTM).and.in.discrete-time.model.(DTM),.the.Jacobian.matrix.is.operated.on.the.small.perturbation.of.the.state.vector.[see.(1.6).and.(1.20)]..The.essential. difference. is. that. it. determines. the. velocity. vector. for. CTM. and. the. next. iterate. for. DTM,.respectively.
Assume. that. the. very. first. point. at. the. beginning. of. iteration. is. placed. on. eigenvector. em. of. the.Jacobian. matrix.. Figure. 1.7. shows. four. different. iteration. processes. corresponding. to. the. particular.value.of.eigenvalue.m.associated.to.em..In.Figure.1.7a.and.b,.m.is.real,.but.its.value.is.0.
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1-12 ControlandMechatronics
1.5 Quasi-Periodic and Frequency-Locked State
1.5.1 Introduction
Beside.the.EP.and.LC,.another.possible.state.or.motion.is.the.quasi-periodic.(Qu-P).motion.in.CTM..In.Qu-P.state,.the.motionin.theorynever.exactly.repeats.itself..Other.terms.used.in.literature.are.conditionally periodic.or.almost periodic..Qu-P.state.is.possible.in.inherently.discrete.systems.as.well..The.frequency-locked.(F-L).state.is.a.special.case.of.Qu-P.state..Qu-P.state.is.not.possible.in.N.=.1.or.2.dimensional.systems,.i.e.,.N.must.be.N..3..Similarly.to.chaos,.Qu-P.state.is.aperiodic..In.chaotic.state,.two.points.in.state.space,.which.are.arbitrarily.close,.will.diverge..In.other.words,.the.chaotic.system.is.extremely.sensitive.to.initial.conditions.and.to.changes.in.control.parameters..In.contrast.to.chaotic.state,.two.points.that.are.initially.close.will.remain.close.over.time.in.Qu-P.state.
The. Qu-P. motion. is. a. mixture. of. periodic. motions. of. several. different. angular. frequencies. 1,.2,,.m..Depending.on.the.value.of.their.linear.combination.L,
. L k k km m= + + +1 1 2 2 . (1.22)
the.motion.can.be.Qu-P,.i.e.,.aperiodic.when.the.sum.L..0.or.it.can.be.F-L,.i.e.,.periodic.state.when.the.sum.L.=.0..Here.k1,.k2,,.km.are.any.positive.(or.negative).integer.(k1.=.k2.=..=.km.=.0.is.excluded).
The.EP,.LC,.Qu-P,.and.F-L.states.are.regular attractors.while.in.chaotic.state,.the.system.has.strange attractor. (see. later. in.Section.1.9)..The.Qu-P.motion.plays.central. role. in.Hamiltonian. systems,. e.g.,.in. mechanical. systems. modeled. without. friction,. which. are. non-dissipative. ones.. They. do. not. have.attractors.
1.5.2 Nonlinear Systems with two Frequencies
Qu-P.and.F-L.motions.occur.frequently.in.practice.in.systems.having.a.natural.oscillation.frequency.and.a.different.external.forcing.frequency.or.two.different.natural.oscillation.frequencies..Because.of.nonlinearity,.the.superposition.of.the.independent.frequencies.is.not.valid.
Starting.from.(1.22),.assume.that
.
1
2
2
1= =TT
pq .
(1.23)
where.T1.=.2/1.and.T2.=.2/2.are.the.periods.of.respective.harmonic.oscillations.and.p.and.q.are.positive.integers..Here.we.assume.that.any.common.factors.in.the.frequency.ratio.have.been.removed,.e.g.,.if.f1/f2.=.2/6,.the.common.factor.of.2.will.be.removed.and.f1/f2.=.p/q.=.1/3.can.be.written..When.the.fre-quency.ratio.is.the.ratio.of.two.integers,.then.the.ratio.is.called.rational.in.mathematical.sense,.i.e.,.the.two.frequencies.are.commensurate,.the.behavior.of.the.system.is.periodic..It.is.in.F-L.state.
On.the.other.hand,.when.the.frequency.ratio.is.irrational,.the.two.frequencies.are.incommensurate,.and.the.behavior.is.Qu-P..The.last.two.statements.can.easily.be.understood.in.the.geometrical.interpre-tation.of.the.system.trajectory.
1.5.3 Geometrical Interpretation
A.two-frequency.Qu-P.trajectory.on.a.toroidal.surface.in.the.three-dimensional.state.space.is.shown.in.Figure.1.8..Introducing.two.angles.r.=.rt.=.1t.and.R.=.Rt.=.2t,.they.determine.point.P.of.the.trajectory.on.the.surface.of.the.torus.provided.that.the.initial.condition.point.P0.belonging.to.t.=.0.is.known..The.center.of.the.torus.is.at.the.origin..R.is.the.large.radius.of.the.torus.whose.cross-sectional.radius.is.r..The.two.angles.R.and.r.are.increasing.as.time.evolves.and.therefore.point.P.is.moving.on.
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-
NonlinearDynamics 1-13
the.surface.of.the.torus.tracking.the.trajectory.of.state.vector.xT.=.[x1,.x2,.x3]..The.three.components.of.the.state.vector.x.are.given.by.the.equations.as.follows:
.
x R r
x r
x R r
r R
r
r R
1
2
3
=
=
=
( )cos
sin
( )sin
+ cos
+ cos
.
(1.24)
The.trajectory. is.winding.on.the. torus.around.the.cross.section.with.minor.radius.r,.making.R/2.rotations.per.unit.time..As.r/R.=.p/q.[see.(1.23)].and.assuming.that.p.and.q.are.integers,.the.number.of.rotations.around.circle.r.and.circle.R.in.per.unit.is.p.and.q,.respectively..For.example,.if.p.=.1.and.q=3,.point.P.makes.three.rotations.around.circle.R.as.long.as.it.makes.only.one.rotation.around.circle.r..Figure.1.9.shows.the.torus.and.the.Poincar.plane.intersecting.the.torus.together.with.the.trajectory.on.the.surface.of.the.torus..The.Poincar.plane.illustrates.its.intersection.points.with.the.trajectory.
When.the.frequency.ratio.p/q.=.1/3,.it.is.rational..As.long.as.point.P.rotates.once.around.circle.R,.it.rotates.120.around.circle.r..Starting.from.point.0.on.the.Poincar.plane.(Figure.1.9a).after.123.rotations.around.circle.R,.point.P.intersects.the.Ponicar.plane.successively.at.point.123..Point.3.
Torus
x2
x1
x3
P0
R
r
P
R
2r
FIGURE 1.8 Two-frequency.trajectory.on.a.toroidal.surface.in.state.space.(N.=.3).
r/R=p/q=1/3
Poincarplane(a)
1
230
reeintersection
points
r/R=p/q=2pi
(b)Poincarplane
After long time thenumber of intersection
points approaches innity
FIGURE 1.9 Example.for.(a).frequency-locked.and.(b).quasi-periodic.state.
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-
1-14 ControlandMechatronics
coincides.with.the.starting.point.0.as.three.times.120.is.360..The.process.is.periodic,.the.trajectory.closes.on.itself,.it.is.the.F-L.state.
On.the.other.hand,.when.the.frequency.ratio.is.irrational,.e.g.,.p/q.=.2.as.long.as.the.point.P.makes.one.rotation.around.circle.R,.it.completes.2.rotations.around.circle.r..The.phase.shift.of.the.first.inter-section.point.on.the.Poincar.plane.from.the.initial.point.P0.is.given.by.an.irrational.angle.=.360..2(mod.1).where.(mod.1).is.the.modulus.operator.that.takes.the.fraction.of.a.number.(e.g.,.6.28(mod.1).=.0.28)..After.any.further.full.rotations.around.circle.R,.the.phase.shifts.of.the.intersection.points.from.P0.on.the.Poincar.plane.remain.irrational;.therefore,.they.will.never.coincide.with.P0,.and.the.trajectory.will.never.close.on.itself..All.intersection.points.will.be.different..As.t..,.the.number.of.intersection.points.will.be.infinite.and.a.circle.of.radius.r.will.be.visible.on.the.Poincar.plane.consisting.of.infinite.number.of.distinct.points.(Figure.1.9b)..The.system.is.in.Qu-P.state.
1.5.4 N-Frequency Quasi-Periodicity
We.have.treated.up.to.now.the.two-frequency.quasi-periodicity.a.little.bit.in.detail..It.has.to.be.stressed.that.in.mathematical.sense,.the.N-frequency.quasi-periodicity.is.essentially.the.same..The.N.frequen-cies.define.N.angles.1.=.1t,.2.=.2t,.N.=.Nt.determining.uniquely.the.position.and.movement.of.the.operation.point.P.on.the.surface.of.the.N-dimensional.torus..Now.again,.the.trajectory.fills.up.the.surface.of.the.N-dimensional.torus.in.the.state.space.
1.6 Dynamical Systems Described by Discrete-time Variables: Maps
1.6.1 Introduction
The.dynamical.systems.can.be.described.by.difference.equation.systems.with.discrete-time.variables..The.relation.in.vector.form.is
. x f xn n+ =1 ( ) . (1.25)
where.xn.is.K-dimensional.state.variable.xnT n n nKx x x= , ,[ ]( ) ( ) ( )1 2 ,.f.is.nonlinear.vector.function..State.vec-tor.xn.is.obtained.at.discrete.time.n.=.1.by.x1.=.f(x0),.where.x0.is.the.initial.condition..From.x1,.the.value.x2.=.f(x1).can.be.calculated,.etc..Knowing.x0,.the.orbit.of.discrete-time.system.x0,.x1,.x2,.is.generated.
We.can.consider.that.vector.function.f.maps.xn.into.xn+1..In.this.sense,.f.is.a.map function..The.number.of.state.variables.determines.the.dimension.of.the.map..
Examples:One-dimensional.map.(K.=.1):
. Logistic.map:. xn+1.=.axn(1..xn).
Tentmap: =
ifif
xax x
a x xnn n
n n+
>
10 5
1 0 5.
( ) ..
where.a.is.constant.Two-dimensional.map.(K.=.2):
.Henonmap:
x ax bxx cx
n n n
n n
+
+
= +
=
11 2 1
12 1
12( ) ( ) ( )
( ) ( )
.
where.a,.b,.and.c.are.constants.
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NonlinearDynamics 1-15
K-dimensional.map:
Poincar.map.of.an.N.=.K.+.1.dimensional.state.space.
Maps.can.give.useful.insight.for.the.behavior.of.complex.dynamic.systems.
The.map.function.(1.25).can.be.invertible.or.non-invertible..It.is.invertible.when.the.discrete.function
. x f xn n=
+1
1( ) . (1.26)
can. be. solved. uniquely. for. xn.. f 1. is. the. inverse. of. f.. Two. examples. are. given. next.. First,. the. invert-ible.Henon.map.and.after,.the.non-invertible.quadratic.(logistic).map.are.discussed..Henon map.is.fre-quently.cited.example.in.nonlinear.systems..It.has.two.dimensions.and.maps.the.point.with.coordinate.xn( )1 .and.xn( )2 . in. the.plane.to.a.new.point. xn+11( ) .and. xn+12( ) .. It. is. invertible,.because. xn+11( ) .and. xn+12( ) .uniquely.determine.the.value.xn( )1 .and.xn( )2 ,.since
.
x xc
x x bxa
nn
nn n
( )( )
( )( ) ( )
1 12
2 11 2
121
=
=
+
+
+ +
Here,.a..0.and.c..0.must.hold..(For.some.values.of.a,.b,.and.c,.the.Henon.map.can.exhibit.chaotic.behavior.)
Turning.now. to. the.non-invertible.maps,. the.quadratic or logistic map. is. taken.as. example.. It.was.developed.originally.as.a.demography.model..It.is.a.very.simple.system,.but.its.response.can.be.surpris-ingly.colorful.. It. is.non-invertible,.as.Figure.1.10.shows..We.cannot.uniquely.determine.xn. from.xn+1..Aswe.see.later,.an.invertible.map.can.be.chaotic.only.if.its.dimension.is.two.or.more.(Henon.map)..On.the.other.hand,.the.non-invertible.map.can.be.chaotic.even.in.one-dimensional.cases,.e.g.,.the.logistic.map.
1.6.2 Fixed Points
The.concept.of.FP.was.introduced.earlier.(see.Figure.1.6)..The.system.stays.in.steady.state.at.FP.where.xn+1.=.xn.=.x*..When.the.discrete.function.(1.25).is.nonlinear,.it.can.have.more.than.one.FP.
1.6.2.1 One-Dimensional Iterated Maps
For.the.sake.of.simplicity,.the.one-dimensional.maps.are.discussed.from.now.on..The.one-dimensional.iterated.maps.can.describe.the.dynamics.of.large.number.of.systems.of.higher.dimension..To.throw.light.
xn+1
xn
FIGURE 1.10 The.quadratic.or.logistic.iterated.map.
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1-16 ControlandMechatronics
to.the.statement,.consider.a.three-dimensional.dynamical.system.with.PMF. z PnT n n n n nu v u v+ + += , = ,1 1 1[ ] ( ) .[see. (1.18)].. In. certain. cases,. we. have. a. relation. between. the. two. coordinates. un. and. vn:. vn. =. Fv(un)..Substituting.it.into.the.map.function.un+1.=.Pu(un,.vn),.we.end.up.with
. u P u F u f un u n v n n+ = =1 ( ( )) ( ), . (1.27)
one-dimensional.map.function.More.arguments.can.be.found.in.the.literature.(see.chapter.5.2.in.Ref..[4].and.page.66.of.Ref..[5]).
on.the.wide.scope.of.applications.on.the.one-dimensional.discrete.map.functions..If.the.dissipation.in.the.system.is.high.enough,.then.even.systems.with.dimension.more.than.three.can.be.analyzed.by.one-dimensional.map.
1.6.2.2 return Map or Cobweb
The.iteration.in.the.one-dimensional.discrete.map.function.or.difference.equation
. x f xn n+ =1 ( ) . (1.28)
can.be.done.with.numerical.or.graphical.method..The.return.map.or.cobweb.is.a.graphical.method..To.illustrate.the.return.map.method,.we.take.as.example.the.equation
.x ax
bxnn
nc+ = +
1 1 ( ) .(1.29)
where.a,.b,.and.c.are.constants..The.iteration.has.to.be.performed.as.follows.(Figure.1.11):
. 1.. Plot.the.function.xn+1(xn).in.plane.xn+1.versus.xn.
. 2.. Select.x0.as.initial.condition.
. 3.. Draw.a.straight.line.starting.from.origin.with.slope.1.called.mirror line.or.diagonal.
. 4.. Read.the.value.x1.from.the.graph.and.draw.a.line.in.parallel.with.the.horizontal.axis.from.x1.to.the.mirror.line.(line.11).
. 5.. Read.the.value.x2.by.vertical.line.12.
. 6.. Repeat.the.graphical.process.by.drawing.the.horizontal.line.22.and.finally.determining.x3.
In.order.to.find.the.FP.x*,.we.have.to.repeat.the.graphical.process.
FP
xn+1
x2
x1
x1 x3 x2 x0
xnx*
2
1
2
1
FIGURE 1.11 Iteration.in.return.map.
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NonlinearDynamics 1-17
1.6.2.3 kth return Map
Periodic.steady.state.with.period.T.is.represented.by.a.single.FP.x*.in.the.mapping,.i.e.,.x*.=.f(x*)..kth-order.subharmonic. solutions. with. period. kT. correspond. to. FPs. { * *}x xk1 ,. where. x f x xn2 1 1* ( * ) *= =+f x x f xn k( *) * ( *) 1 = ..The.kth.iterate.of.f(x).is.defined.as.the.function.that.results.from.applying.f k.times,.its.notation.is.f (k)(x).=.f(f(f(x))),.and.its.mapping.is.called.kth.return.map.and.the.process.is.called.period-k.
1.6.2.4 Stability of FP in One-Dimensional Map
The.FP.is.locally stable.if.subsequent.iterates.starting.from.a.sufficiently.near.initial.point.to.the.FP.are.eventually.getting.closer.and.closer.the.FP..The.expressions.attracting.FP.or.asymptotically.stable.FP.are.also.used..On.the.other.hand,.if.the.subsequent.iterates.move.away.from.x*,.then.the.name.unstable.or.repelling.FP.is.used.
1.6.3 Mathematical approach
By.knowing.the.nonlinear.function.f(x).and.one.of.its.FPs.x*,.we.can.express.the.first.iterate.x1.by.apply-ing.a.Taylor.series.expansion.near.x*:
.x f x f x df
dxx f x df
dxx
x x1 0 0 0= = + + + ( ) ( *) ( *)
* *
.
(1.30)
where.x0.=.x0..x*.and.the.initial.condition.x0.is.sufficiently.near.x*..The.derivative..=.df/dx.has.to.be.evaluated.at.x*...is.the.eigenvalue.of.f(x).at.x*..The.nth.iterate.is
. = = =
x x x x dfdx
xn n nn
x
( *)*
0 0 0
.(1.31)
It.is.obvious.that.x*.is.stable.FP.if. | / | df dx x* 1..In.general,.when.the.dimension.is.more.than.one,..must.be.within.the.unit.circle.drawn.around.the.origin.of.the.complex.plane.for.stable.FP.
The.nonlinear.f(x).has.multiple.FPs..The.initial.conditions.leading.to.a.particular.x*.constitute.the.basin of attraction.of.x*..As.there.are.more.basins.of.attraction,.none.of.FPs.can.be.globally stable..They.can.be.only.locally.stable.
1.6.4 Graphical approach
Graphical approach.is.explained.in.Figure.1.12..Figure.1.12a,c,e,.and.g.presents.the.return.map.with.FP.x*.and.with.the.initial.condition.(IC)..The.thick.straight.line.with.slope..at.x*.approximates.the.function.f(x).at.x*..Figure.1.12b,d,f,.and.h.depicts.the.discrete-time.evolution.xn(n)..||.1..The.subsequent.iterates.explode,.FP.is.unstable..Note.that.the.cobweb.and.time.evolution.is.oscillating.when..
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1-18 ControlandMechatronics
In.general,.it.has.two.FPs:. x a1 1 1* = / .and.x2 0* = ..The.respective.eigenvalues.are.1.=.2..a.and.2.=.a..Figure.1.13.shows.the.return.map.(left).and.the.time.evolution.(right).at.different.value.a.changing.in.range.0.
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NonlinearDynamics 1-19
stability.of.cycles)..Increasing.a.over.a.=.3.570,.we.enter.the.chaotic.range.(Figure.1.13k.and.l),.the.itera-tion.is.a.periodic.with.narrow.ranges.of.a.producing.periodic.solutions.
As.a.is.increased,.first.we.have.period-1.in.steady.state,.later.period-2,.then.period-4.emerge,.etc..The.scenario.is.called.period doubling cascade.
1.6.6 Stability of Cycles
We.have.already.introduced.the.notation.f (k)(x).=.f(f(f(x)).for.the.kth.iterate.that.results.from.apply-ing.f k-times..If.we.start.at.x1*.and.after.applying.f k-times.we.end.up.with.x xk* *= 1 ,.then.we.say.we.have.period.or.cycle-k.with.k.separate.FPs:.x x f x x f xk k1 2 1 1* * ( * ) , * ( * ), ,= = .
In.the.simplest.case.of.period-2,.the.two.FPs.are:.x f x2 1* ( *)= .and.x f x f f x1 2 1* ( * ) ( ( *))= = ..Referring.to.(1.30),.we.know.that.the.stability.of.FP.x1*.depends.on.the.value.of.the.derivative
.( ) ( ( ))
*2
1
=
df f xdx x .
(1.33)
(a)
1
1
xn+1
xn+1
0
0
0 1
1
IC
0 IC
xn
xn
xn
xna
a4 1
00
1
00
3
2
1
0
4
3
2
1
0
n
n
1xn+1
010 IC
xn
xna 1
00
4
3
2
1
0 n
(b)
(c) (d)
(e) (f )
FIGURE 1.13 Return.map.(left).and.time.evolution.(right).of.the.logistic.map.(continued)
2011 by Taylor and Francis Group, LLC
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1-20 ControlandMechatronics
Using.the.chain.rule.for.derivatives,
.
( )
( )
( )
*
( ( ))* * * *
2
1 1 1 1 2
df xdx
df f xdx
dfdx
dfdx
dfdx
dfd
x x f x x x
= = =
xx x1* .(1.34)
Consequently,
. x x
d f xdx
d f xdx
1 2
2 2
*
( ) ( )
*
( ) ( )
=
.
(1.35)
(1.35). states. that. the.derivatives.or.eigenvalues.of. the.second. iterate.of. f(x).are. the.same.at.both.FPs.belonging.to.period-2.
As. an. example,. Figure. 1.14. shows. the. return. map. for. the. first. (Figure. 1.14a). and. for. the. second.(Figure. 1.14b). iterate. when. a. =. 3.2.. Both. FPs. in. the. first. iterate. map. are. unstable. as. we. have. just.
1
1
xn+1
xn+1
0
0
0 1
1
IC
0 IC
xn
xn
xn
xna
a4 1
00
1
00
3
2
1
0
4
3
2
1
0
n
n
1xn+1
010 IC
xn
xna 1
00
4
3
2
1
0n
(g) (h)
(i) (j)
(k) (l)
FIGURE 1.13 (continued)
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NonlinearDynamics 1-21
discussed..We.have.four.FPs.in.the.second.iterate.map..Two.of.them,. x1*.and.x2*,.are.stable.FPs.(point.S1.and.S2).and.the.other.two.(zero.and.x*).are.unstable.(point.U1.and.U2)..The.FP.of.the.first.iterate.must.be.the.FP.of.the.second.iterate.as.well:.x*.=.f(x*).and.x*.=.f( f(x*)).
1.7 Invariant Manifolds: Homoclinic and Heteroclinic Orbits
1.7.1 Introduction
To. obtain. complete. understanding. of. the. global. dynamics. of.nonlinear.systems,.the.knowledge.of.invariant.manifolds.is.abso-lutely. crucial.. The. invariant. manifolds. or. briefly. the. manifolds.are.borders.in.state.space.separating.regions..A.trajectory.born.in.one.region.must.remain.in.the.same.region.as.time.evolves..The..manifolds. organize. the. state. space.. There. are. stable. and. unsta-ble. manifolds.. They. originate. from. saddle. points.. If. the. .initial..condition.is.on.the.manifold.or.subspace,.the.trajectory.stays.on.the. manifold.. Homoclinic. orbit. is. established. when. stable. and.unstable.manifolds.of.a.saddle.point.intersect..Heteroclinic.orbit.is.established.when.stable.and.unstable.manifolds.from.different.saddle.points.intersect.
1.7.2 Invariant Manifolds, CtM
The.CTM.is.applied.for.describing.the.system..Invariant.mani-fold. is. a. curve. (trajectory). in. plane. (N. =. 2). (Figure. 1.15),. or.curve.or.surface. in.space.(N.=.3).(Figure.1.16),.or. in.general.a.subspace.(hypersurface).of.the.state.space.(N.>.3)..The.manifolds.are.always.associated.with.saddle.point.denoted.here.by.x*..Any.initial.condition.in.the.manifold.results.in.movement.of.the.operation.point. in. the. manifold. under. the. action. of. the. relevant. differential. equations.. There. are. two. kinds.
Unstablemanifold
Stablemanifold
W u(x *)
W s(x*)x*
es
eu
t
t
FIGURE 1.15 Stable.Ws.and.unstable.Wu.manifold.(N.=.2)..The.CTM.is.used..es.and.eu.are.eigenvectors.at.saddle.point.x*.belonging.to.Ws.and.Wu,.respectively.
(a)
xn+1
xn+2
xnU1
S1
S2U2
x*1 x*2x*
xnx*
(b)
FIGURE 1.14 Cycle. of. period-2. in.the.logistic.map.for.a.=.3.3..(a).Return.map.for.first.iterate..(b).Return.map.for.second.iterate.xn+2.versus.xn.
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1-22 ControlandMechatronics
of. manifolds:. stable. manifold. denoted. by. Ws. and. unstable. manifold. denoted. by. Wu.. If. the. initial.points.are.on.Ws.or.on.Wu,.the.operation.points.remain.on.Ws.or.Wu.forever,.but.the.points.on.Ws.are.attracted.by.x*.and.the.points.on.Wu.are.repelled.from.x*..By.considering.t ,.every.movement.along.the.manifolds.is.reversed.
If.the.initial.conditions.(points.P1,.P2,.P3,.P4.in.Figure.1.17).are.not.on.the.manifolds,.their.trajectories.will.not.cross.any.of.the.manifolds,.they.remain.in.the.space.bounded.by.the.manifolds..The.trajecto-ries.are.repelled.from.Ws(x*).and.attracted.by.Wu(x*)..Any.of.these.trajectories.(orbits).must.remain.in.the.space.where.it.was.born..Wu(x*).(or.Ws(x*)).are.boundaries..Consequently,.the.invariant.manifolds.organize.the.state.space.
1.7.3 Invariant Manifolds, DtM
Applying. DTM,. i.e.,. difference. equations. describe. the. system,. then. mostly. the. PMF. is. used.. The.fixed.point.x*.must.be.a.saddle.point.of.PMF.to.have.manifolds..Figure.1.18.presents.the.Ponicar.surface.or.plane.with.the.stable.Ws(x*).and.unstable.Wu(x*).manifold..s0.and.u0.is.the.intersection.point.of.the.trajectory.with.the.Poincar.surface,.respectively..The.next.intersection.point.of.the.same.
Stablemanifold W s(x*)
Unstablemanifold W u(x*)
eu
Es
x*t
t
t
FIGURE 1.16 Stable.Ws.and.unstable.Wu.manifold.(N.=.3)..The.CTM.is.used..Es.=.span[es1,.es2].stable.subspace.is.tangent.of.Ws(x*).at.x*..eu.is.an.unstable.eigenvector.at.x*..x*.is.saddle.point.
W u(x*)P4
P3
P2P1
x*
W s(x*)
FIGURE 1.17 Initial.conditions.(P1,.P2,.P3,.P4).are.not.placed.on.any.of.the.invariant.manifolds..The.trajectories.are.repelled.from.Ws(x*).and.attracted.by.Wu(x*)..Any.of.the.trajectories.(orbits).must.remain.in.the.region.where.it.was.born..Wu(x*).(or.Ws(x*)).are.boundaries.
2011 by Taylor and Francis Group, LLC
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NonlinearDynamics 1-23
trajectory.with.the.Poincar.surface.is.s1.and.u1.the.following.s2.and.u2,.etc..If.Ws,.Wu.are.manifolds.and.s0,.u0.are.on.the.mani-folds,.all. subsequent. intersection.point.will.be.on. the.respective.manifold.. Starting. from. infinitely. large. number. of. initial. point.s0(u0). on. Ws. (or. on. Wu),. infinitely. large. number. of. intersection.points.are.obtained.along.W s.(or.Wu)..Curve.Ws(Wu).is.determined.by.using.infinitely.large.number.of.intersection.points.
1.7.4 Homoclinic and Heteroclinic Orbits, CtM
In.homoclinic connection,.the.stable.Ws.and.unstable.Wu.manifold.of.the.same.saddle.point.x*.intersect.each.other.(Figure.1.19)..The.two.manifolds,.Ws.and.Wu,.constitute.a.homolinic.orbit..The.opera-tion.point.on.the.homoclinic.orbits.approach.x*.both. in. forward.and.in.backward.time.under.the.action.of.the.relevant.differential.equation..In.heteroclinic connection,.the.stable.manifold.W xs( *)1 .of.saddle.point.x1*.is.connected.to.the.unstable.manifold.W xu( *)2 .of.saddle.point.x2*,.and.vice.versa.(Figure.1.20)..The.two.manifolds.W xs( *)1 .and.W xu( *)2 .[similarly.W xs( *)2 .and.W xu( *)1 ].constitute.a.heteroclinic.orbit.
Poincarsurface
eu
s0s1
s2 s3 s4 u0x*
u1u2
u3u4
es
W s (x*)
W u (x*)
Points denoted by s (by u) approach(diverge from) x* as n
FIGURE 1.18 Stable.Ws.and.unstable.Wu.manifold..DTM.is.used..es.and.eu.are.eigenvectors.at.saddle.point.x*.belonging.to.Ws.and.Wu,.respectively.
W u (x*)
W s (x*)
Homoclinic connection
Homoclinicorbit
x*
FIGURE 1.19 Homoclinic. con-nection.and.orbit.(CTM).
Wu (x1*)
W s (x2*)
W s (x1*)
Wu (x1*)
x11*x2*Heteroclinic
orbit
Heteroclinic connection
FIGURE 1.20 Heteroclinic.connection.and.orbit.(CTM).
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1-24 ControlandMechatronics
1.8 transitions to Chaos
1.8.1 Introduction
One.of. the.great.achievements.of. the. theory.of. chaos. is. the.discovery.of. several. typical. routes. from.regular.states.to.chaos..Quite.different.systems.in.their.physical.appearance.exhibit.the.same.route..The.main.thing.is.the.universality..There.are.two.broad.classes.of.transitions.to.chaos:.the local and global bifurcations..In.the.first.case,.for.example,.one.EP.or.one.LC.loses.its.stability.as.a.system.parameter.is.changed..The.local.bifurcation.has.three.subclasses:.period.doubling,.quasi-periodicity,.and.intermit-tency..The.most.frequent.route.is.the.period.doubling.
In.the.second.case,. the.global.bifurcation.involves. larger.scale.behavior. in.state.space,.more.EPs,.and/or.more.LCs.lose.their.stability..It.has.two.subclasses,.the.chaotic.transient.and.the.crisis..In.this.section,.only.the.local.bifurcations.and.the.period-doubling.route.is.treated..Only.one.short.comment.is.made.both.on.the.quasi-periodic.route.and.on.the.intermittency.
In.quasi-periodic route,.as.a.result.of.alteration.in.a.parameter,.the.system.state.changes.first.from.EP.to.LC.through.bifurcation..Later,.in.addition,.another.frequency.develops.by.a.new.bifurcation.and.the.system.exhibits.quasi-periodic.state..In.other.words,.there.are.two.complex.conjugate.eigenvalues.within.the.unit.circle.in.this.state..By.changing.further.the.parameter,.eventually.the.chaotic.state.is.reached.from.the.quasi-periodic.one.
In.the. intermittency route. to.chaos,.apparently.periodic.and.chaotic.states.alternately.develop..The.system.state.seems.to.be.periodic.in.certain.intervals.and.suddenly.it.turns.into.a.burst.of.chaotic.state..The.irregular.motion.calms.down.and.everything.starts.again..Changing.the.system.parameter.further,.the.length.of.chaotic.states.becomes.longer.and.finally.the.periodic.states.are.not
top related