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Industrial Electronics Handbook
Citation preview
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
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)
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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
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
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
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
xi
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
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:[email protected]:.www.mathworks.com
2011 by Taylor and Francis Group, LLC
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
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
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.
2011 by Taylor and Francis Group, LLC
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.
2011 by Taylor and Francis Group, LLC
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.
2011 by Taylor and Francis Group, LLC
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)
2011 by Taylor and Francis Group, LLC
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.
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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).
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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.
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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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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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..
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.
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)
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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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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.
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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.restored.
1.8.2 Period-Doubling Bifurcation
Considering.now.the.period-doubling.route,.let.us.assume.a.LC.as.starting.state.in.a.three-dimensional.system.(Figure.1.21a)..The.trajectory.crosses.the.Poincar.plane.at.point.P..As.a.result.of.changing.one.system.parameter,.the.eigenvalue.1.of.the.Jacobian.matrix.of.PMF.belonging.to.FP.P.is.moving.along.the.negative.real.axis.within.the.unit.circle.toward.point.1.and.crosses.it.as.1.becomes.1..The.eigenvalue.1.moves.outside.the.unit.circle..FP.P.belonging.to.the.first.iterate.loses.its.stability..Simultaneously,.two.new.stable.FPs.P1.and.P2.are.born.in.the.second.iterate.process.having.two.eigenvalues.2.=.3.(Figure.1.21b)..Their.value.is.equal.+1.at.the.bifurcation.point.and.it.is.getting.smaller.than.one.as.the.system.parameter.is.changing.further.in.the.same.direction..2.=.3.are.moving.along.the.positive.real.axis.toward.the.origin.
Continuing.the.parameter.change,.new.bifurcation.occurs.following.the.same.pattern.just.described.and.the.trajectory.will.cross.the.Poincar.plane.four.times.(2..2).in.one.period.instead.of.twice.from.the.new.bifurcation.point..The.period-doubling.process.keeps.going.on.but.the.difference.between.two.consecutive.parameter.values.belonging.to.bifurcations.is.getting.smaller.and.smaller.
1.8.3 Period-Doubling Scenario in General
The.period-doubling.scenario.is.shown.in.Figure.1.22.where.. is.the.system.parameter,.the.so-called.bifurcation parameter,.x.is.one.of.the.system.state.variables.belonging.to.the.FP,.or.more.generally.to.the.intersection.points.of.the.trajectory.with.the.Poincar.plane..The.intersection.points.belonging.to.the.transient.process.are.excluded..For.this.reason,.the.diagram.is.called.sometime.final state diagram..The.name.bifurcation.diagram.refers.to.the.bifurcation.points.shown.in.the.diagram.and.it.is.more.generally.used.
Feigenbaum.[7].has.shown.that.the.ratio.of.the.distances.between.successive.bifurcation.points.mea-sured.along.the.parameter.axis..approaches.to.a.constant.number.as.the.order.of.bifurcation,.labeled.by.k,.approaches.infinity:
2011 by Taylor and Francis Group, LLC
NonlinearDynamics 1-25
.
= = .
+
limk
k
k 14 6693
.(1.36)
where..is.the.so-called.Feigenbaum constant..It.is.found.that.the.ratio.of.the.distances.measured.in.the.axis.x.at.the.bifurcation.points.approaches.another.constant.number,.the.so-called.Feigenbaum.
.
= = .
+
limk
k
k 12 5029
.(1.37)
.is.universal.constant.in.the.theory.of.chaos.like.other.fundamental.numbers,.for.example,.e.=.2.718,.,.and.the.golden.mean.ratio.( )5 1 2 / .
(a)
(b)
Limitcycle
Newlimitcycle
Before bifurcation
After bifurcation
Parameterchange
Poincarplane
Poincarplane
P
P1
P2
P
FIGURE 1.21 Period-doubling.bifurcation..(a).Period-1.state.before,.and.(b).period-2.state.after.the.bifurcation.
x Final state ofstate variable
1 2 3S
Parameter
2
1
3
FIGURE 1.22 Final.state.or.bifurcation.diagram..S.=.Feigenbaum.point.
2011 by Taylor and Francis Group, LLC
1-26 ControlandMechatronics
1.9 Chaotic State
1.9.1 Introduction
The.recently.discovered.new.state.is.the.chaotic.state..It.can.evolve.only.in.nonlinear.systems..The.condi-tions.required.for.chaotic.state.are.as.follows:.at.least.three.or.higher.dimensions.in.autonomous.systems.and.with.at.least.two.or.higher.dimensions.in.non-autonomous.systems.provided.that.they.are.described.by.CTM..On.the.other.hand,.when.DTM.is.used,.two.or.more.dimensions.suffice.for.invertible.iteration.functions.and.only.one.dimension.(e.g.,.logistic.map).or.more.for.non-invertible.iteration.functions.are.needed.for.developing.chaotic.state..In.addition.to.that,.some.other.universal.qualitative.features.com-mon.to.nonlinear.chaotic.systems.are.summarized.as.follows:
. The.systems.are.deterministic,.the.equations.describing.them.are.completely.known.. They.have.extreme.sensitive.dependence.on.initial.conditions.. Exponential.divergence.of.nearby.trajectories.is.one.of.the.signatures.of.chaos.. Even.though.the.systems.are.deterministic,.their.behavior.is.unpredictable.on.the.long.run.. The.trajectories. in.chaotic.state.are.non-periodic,.bounded,.cannot.be.reproduced,.and.do.not.
intersect.each.other.. The.motion.of.the.trajectories.is.random-like.with.underlying.order.and.structure.
As. it. was. mentioned,. the. trajectories. setting. off. from. initial. conditions. approach. after. the. transient.process.either.FPs.or.LC.or.Qu-P.curves.in.dissipative.systems..All.of.them.are.called.attractors.or.clas-sical attractors.since.the.system.is.attracted.to.one.of.the.above.three.states..When.a.chaotic.state.evolves.in.a.system,.its.trajectory.approaches.and.sooner.or.later.reaches.an.attractor.too,.the.so-called.strange attractor.
In.three.dimensions,.the.classical.attractors.are.associated.with.some.geometric.form,.the.station-ary.state.with.point,.the.LC.with.a.closed.curve,.and.the.quasi-periodic.state.with.surface..The.strange.attractor.is.associated.with.a.new.kind.of.geometric.object..It.is.called.a.fractal structure.
1.9.2 Lyapunov Exponent
Conceptually,.the.Lyapunov.exponent.is.a.quantitative.test.of.the.sensitive.dependence.on.initial.condi-tions.of.the.system..It.was.stated.earlier.that.one.of.the.properties.of.chaotic.systems.is.the.exponential.divergence.of.nearby.trajectories..The.calculation.methods.usually.apply.this.property.to.determine.the.Lyapunov.exponent..
After.the.transient.process,.the.trajectories.always.find.their.attractor.belonging.to.the.special.initial.condition.in.dissipative.systems..In.general,.the.attractor.as.reference.trajectory.is.used.to.calculate...Starting.two.trajectories.from.two.nearby.initial.points.placed.from.each.other.by.small.distance.d0,.the.distance.d.between.the.trajectories.is.given.by
. d t d et( ) = 0 . (1.38)
where..is.the.Lyapunov.exponent..One.of.the.initial.points.is.on.the.attractor..The.equation.can.hold.true.only.locally.because.the.chaotic.systems.are.bounded,.i.e.,.d(t).cannot.increase.to.infinity..The.value..may.depend.on.the.initial.point.on.the.attractor..To.characterize.the.attractor.by.a.Lyapunov.expo-nent,.the.calculation.described.above.has.to.be.repeated.for.a.large.number.of.n.of.initial.points.distrib-uted.along.the.attractor..Eventually,.the.average.Lyapunov.exponent..calculated.from.the.individual.n.values.will.characterize.the.state.of.the.system..The.criterion.for.chaos.is..>.0..When...0,.the.system.is.in.regular.state.(Figure.1.23).
2011 by Taylor and Francis Group, LLC
NonlinearDynamics 1-27
1.10 Examples from Power Electronics
1.10.1 Introduction
Power.electronic.systems.have.wide.applications.both.in.industry.and.at.home..The.structure.of.these.systems. keeps. changing. due. to. consecutive. turn-on. and. turn-off. processes. of. electronic. switches..They. are. variable structure,. piece-wise. linear,. nonlinear. dynamic. controlled. systems.. The. primary.source.of.nonlinearity.in.these.systems.is.that.the.switching.instants.depend.on.the.state.variables.[1]..Furthermore,.the.frequent.nonlinearities.can.also.be.found.in.the.systems..In.order.to.show.the.applica-tions.of.the.theory.of.nonlinear.dynamics.summarized.in.the.previous.sections,.five.examples.have.been.selected.from.the.field.of.power.electronics.[15,16]..Various.system.states.and.bifurcations.will.be.pre-sented.without.the.claim.of.completeness.in.this.section..Our.main.objective.is.to.direct.the.attention.of.the.readers,.engineers.to.the.possible.strange.phenomenon.that.can.be.encountered.in.power.electronics.and.explained.by.the.theory.of.nonlinear.dynamics.
1.10.2 High-Frequency time-Sharing Inverter
1.10.2.1 the System
It. is.applied. for. induction.heating.where.high-frequency.power.supply. is. required.with. frequency.of.several.thousand.hertz.Figure.1.24.shows.the.inverter.configuration.in.its.simplest.form..I+.and.I,.the.so-called.positive.and.negative.sub-inverters,.are.encircled.by.dotted.lines.
The. parallel. oscillatory. circuit. LpCpRp. represents. the. load.. The. supply. is. provided. by. a. center-tapped.DC.voltage.source..In.order.to.explain.the.mode.of.action.of.the.inverter,.ideal.components.are.assumed..The.basic.operation.of.the.inverter.can.be.understood.by.the.time.functions.of.Figure.1.25..In.Figure.1.25a,.the.high.frequency.approximately.sinusoidal.output.voltage.v0,.the.inverter.output.current.pulses.i0,.and.the.condenser.voltage.vc.can.be.seen..Thyristors.T1.and.T2.are.alternately.fired.at.instants.located.at.every.sixth.zero.crossings.on.the.positive.and.on.the.negative.slope.of.the.output.voltage,.respectively..After.firing.a.thyristor,.an.output.current.pulse.i0.is.flowing.into.the.load,.which.changes.the.polarity.of.the.series.condenser.voltage.vc,.for.instance,.from.Vcm.to.Vcm..The.thyristor.turn-off.time.can.be.a.little.bit.longer.than.two.and.half.cycles.of.the.output.voltage.(Figure.1.25b)..Figure.1.26.presents.a.configuration.with.three.positive.and.three.negative.sub-inverters..Here,.the.respective.input.and. output. terminals. of. the. sub-inverters. are. paralleled.. The. numbering. of. the. sub-inverters. corre-sponds.to.the.firing.order..Each.sub-inverter.pair.works.in.the.same.way.as.it.was.previously.described.
x2
x3
x1
d0
d(t) =d0et
>0Chaotic
x2
x3
x1
d0
d(t) =d0et
1-28 ControlandMechatronics
1.10.2.2 results
One.of.the.most.interesting.results.is.that.by.using.an.approximate.model.assuming.sinusoidal.output.voltage,.no.steady-state.solution.can.be.found.for.the.current.pulse.and.other.variables.in.certain.opera-tion.region.and.its. theoretical.explanation.can.be.found.in.Ref..[6]..The.laboratory.tests.verified.this.theoretical.conclusion..To.describe.the.phenomena.in.the.region.in.quantitative.form,.a.more.accurate.model.was.used..The.independent.energy.storage.elements.are.six.Ls.series.inductances,.three.Cs.series.capacitances,.one.Lp.inductance,.and.one.Cp.capacitance,.altogether.11.elements,.with.11.state.variables..The.accurate.analysis.was.performed.by.simulation.in.MATLAB.environment.for.open-.and.for.closed-loop.control.
Ls
Ls
vL
vT1Cs/2
Cs/2vT2
vC
I+
I
Loadiip
iin
io
vi
+
vivo
iCp
iLp
Cp
Lp
RpT1
T2
FIGURE 1.24 The.basic.configuration.of.the.inverter.
io vc
vT1
Vom
t
(a)
(b)
t
vo
Vcm
vi
vcritic=(vi+Vom) Vcm
t1 Vcm+ viVom
Vcm> (vi+Vom)
Vcm1 2 1
2
To
FIGURE 1.25 Time.functions.of.(a).output.voltage.vo,.output.current.io, condenser.voltage.vc, and.(b).thyristor.voltage.vT1.in.basic.operation.
2011 by Taylor and Francis Group, LLC
NonlinearDynamics 1-29
1.10.2.3 Open-Loop Control
Bifurcation.diagram.was.generated.here,.the.peak.values.of.the.output.voltage.Vom.were.sampled,.stored,.and.plotted.as.a.function.of.the.control.parameter.To,.where.To.is.the.period.between.two.consecutive.fir-ing.pulses.in.the.positive.or.in.the.negative.sub-inverters.(Figure.1.27)..Having.just.one.single.value.Vom.for.a.given.To,.the.output.voltage.vo.repeats.itself.in.each.period.To..This.state.is.called.period-1..Similarly,.period-5.state.develops,.for.example,.at.firing.period.To.=.277.s..Now,.there.are.five.consecutive.distinct.Vom.values..vo.is.still.periodic,.it.repeats.itself.after.5To.has.elapsed.(Figure.1.28).
1.10.2.4 Closed-Loop Control
A.self-control.structure.is.obtained.by.applying.a.feedback.control.loop..Now,.the.approximately.sinu-soidal.output.voltage.vo.is.compared.with.a.DC.control.voltage.VDC.and.the.thyristors.are.alternatively.fired.at.the.crossing.points.of.the.two.curves..The.study.is.concerned.with.the.effect.of.the.variation.of.the.DC.control.voltage.level.VDC.on.the.behavior.of.the.feedback-controlled.inverter..Again,.the.bifur-cation.diagram.is.used.for.the.presentation.of.the.results.(Figure.1.29)..The.peak.values.of.the.output.
Ls
T1 T3 T5 Csio iL
iC
Lp
Rp
Cp
T4 T6 T2
vT1
Ls Ls
Ls Ls Ls
iip
iin
vovC1 vi
vi
0
+
io5+ io2
io3+ io6
io1+ io4
FIGURE 1.26 The.power.circuit.of.the.time-sharing.inverter..(Reprinted.from.Bajnok,.M..et.al.,.Surprises.stem-ming.from.using.linear.models.for.nonlinear.systems:.Review..In.Proceedings of the 29th Annual Conference on Industrial Electronics, Control and Instrumentation (IECON03),.Roanoke,.VA,.vol..I,.pp..961971,.November.26,.2003..CD.Rom.ISBN:0-7803-7907-1...[2003].IEEE..With.permission.)
500
450
400
V om [V
]
350
300265 270 275 280
Period-5
To [s]285 290
FIGURE 1.27 Bifurcation.diagram.of.inverter.with.open-loop.control.
2011 by Taylor and Francis Group, LLC
1-30 ControlandMechatronics
voltage.Vom.were.sampled,.stored,.and.plotted.as.a.function.of.the.control.parameter.VDC..The.results.are.basically.similar.to.those.obtained.for.open-loop.control..As.in.the.previous.study,.the.feedback-controlled.inverter.generates.subharmonics.as.the.DC.voltage.level.is.varied.
1.10.3 Dual Channel resonant DCDC Converter
1.10.3.1 the System
The.dual.channel.resonant.DCDC.converter.family.was.introduced.earlier.[3]..The.family.has.12.members..A.common.feature.of.the.different.entities.is.that.they.transmit.power.from.input.to.output.through.two.channels,.the.so-called.positive.and.negative.ones,.coupled.by.a.resonating.capacitor..The.converter.can.operate.both.in.symmetrical.and.in.asymmetrical.mode..The.respective.variables.in.the.two.channels.
800
600
400
200
0v o [V
], i o1
[A]
200
400
0.092 0.094 0.096 0.098
3T5
To
voio1T5= 5To
t [s]0.1
FIGURE 1.28 vo(t).and.io1(t)..To.=.277.s..Period-5.state..(Reprinted.from.Bajnok,.M..et.al.,.Surprises.stemming.from.using.linear.models.for.nonlinear.systems:.Review..In.Proceedings of the 29th Annual Conference on Industrial Electronics, Control and Instrumentation (IECON03),.Roanoke,.VA,.vol..I,.pp..961971,.November.26,.2003..CD.Rom.ISBN:0-7803-7907-1...[2003].IEEE..With.permission.)
500
450
400
350
30020 0 20
VDC [V]
V om [V
]
40 60 80
FIGURE 1.29 Bifurcation. diagram. of. inverter. with. feedback. control. loop.. (Reprinted. from. Bajnok,. M.. et. al.,.Surprises.stemming.from.using. linear.models. for.nonlinear.systems:.Review..In.Proceedings of the 29th Annual Conference on Industrial Electronics, Control and Instrumentation (IECON03),.Roanoke,.VA,.vol..I,.pp..961971,.November.26,.2003..CD.Rom.ISBN:0-7803-7907-1...[2003].IEEE..With.permission.)
2011 by Taylor and Francis Group, LLC
NonlinearDynamics 1-31
vary.symmetrically.or.asymmetrically.in.the.two.modes..The.energy.change.between.the.two.channels.is.accomplished.by.a.capacitor. in.asymmetrical.operation..There. is.no.energy.exchange.between. the.positive.and.the.negative.channels.in.the.symmetrical.case..Our.description.is.restricted.to.the.buck.configuration.(Figure.1.30).in.symmetrical.operation..Suffix.i.and.o.refer.to.input.and.output.while.suf-fix.p.and.n.refer.to.positive.and.negative.channel,.respectively..Two.different.converter.versions.can.be.derived.from.the.configuration..The.first.version.contains.diodes.in.place.of.clamping.switches.Scp.and.Scn..The.second.one.applies.controlled.switches.conducting.current.in.the.direction.of.arrow..To.sim-plify,.the.configuration.capacitance.C.is.replaced.by.short.circuit.and.Scp,.Scn.are.applied..The.controlled.switches.within.one.channel.are.always.in.complementary.states.(i.e.,.when.Sp.is.on,.Scp.is.off,.and.vice.versa)..By.turning.on.switch.Sp,.a.sinusoidal.current.pulse.iip.is.developed.from.t.=.0.to.p.( = /1 LC .in.circuitSp,.L,.vop,.C,.and.vip.(Figure.1.31))..The.currents.are.iip.=.iop.=.icp.in.interval.0.
1-32 ControlandMechatronics
current..In.DCM,.the.current.is.zero.between.ep.and.Ts..In.the.continuous.current-conduction.mode.(CCM).of.operation,.the.inductor.current.flows.continuously.(iop.>.0)..The.inductor.current.iop.decreases.in.both.cases.in.a.linear.fashion..After.turning.on.Scp,.the.capacitor.voltage.vc.stops.changing..It.keeps.its.value.Vcp.
The.same.process.takes.place.at.the.negative.side,.resulting.in.a.negative.condenser.current.pulse.and.voltage.swing.after.turning.on.Sn.
1.10.3.2 the PWM Control
For.controlling.the.output.voltage.vo.=.vop.+.von.by.PWM.switching,.a.feedback.control. loop.is.applied.(Figure.1.32a)..The.control.signal.vcon.is.compared.to.the.repetitive.sawtooth.waveform.(Figure.1.32b)..The.control.voltage.signal.vcon. is.obtained.by.amplifying.the.error,.the.difference.between.the.actual.output.voltage.vo,.and.its.desired.value.Vref..When.the.amplified.error.signal.vcon.is.greater.than.the.sawtooth.wave-form,.the.switch.control.signal.(Figure.1.32c).becomes.high.and.the.selected.switch.turns.on..Otherwise,.the.switch.is.off..The.controlled.switches.are.Sp.and.Sn.(the.switches.within.one.channel,.e.g.,.Sp.and.Scp.are.in.complementary.states),.and.they.are.controlled.alternatively,.i.e.,.the.switch.control.signal.is.generated.for.the.switch.in.one.channel.in.one.period.of.the.sawtooth.wave.and.in.the.next.period,.the.signal.is.gener-ated.for.the.switch.in.the.other.channel..The.switching.frequency.of.these.switches.is.half.of.the.frequency.of.the.sawtooth.wave.
Vref
(a)
Kvvcon
vramp
voComparator ConverterSwitchcontrol
VU
(b)
VL T t
vconvramp
(c)
t
Switch controlHigh
Low
FIGURE 1.32 PWM.control:.(a).Block.diagram,.(b).input.and.(c).output.signals.of.the.comparator.
49.8
49.7
49.673.4 73.6 73.8
Kv
v ok (
V)
74 74.2
FIGURE 1.33 Enlarged.part.of.the.bifurcation.diagram.
2011 by Taylor and Francis Group, LLC
NonlinearDynamics 1-33
1.10.3.3 results
The. objective. is. the. calculation. of. the. controlled. variable. vo. in. steady. state. in.order.to.discover.the.various.possible.states.of.this.nonlinear.variable.structure.dynamical. system.. The. analysis. was. performed. by. simulation. in. MATLAB.environment..The.effect.of.variation.Kv.was.studied..A.representative.bifurcation.diagram.is.shown.in.Figure