Chaos control of new Mathieu-van der Pol systems by fuzzy logic constant controllers

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ContentslistsavailableatSciVerseScienceDirect

Applied

Soft

Computing

jo u r n al h om epa g e :w w w . e l s e v i e r . c o m / lo c a t e / a s o c

Chaos

control

of

new

Mathieu–van

der

Pol

systems

by

fuzzy

logic

constant

controllers

Shih-Yu

Li

∗

DepartmentofMechanicalEngineering,NationalChiaoTungUniversity,1001TaHsuehRoad,Hsinchu300,Taiwan,ROC

a

r

t

i

c

l

e

i

n

f

o

Articlehistory:

Received9September2009

Receivedinrevisedform12February2011 Accepted14August2011

Availableonline22August2011 PACS: 05.45.Xt 05.45.Pq 05.45.Gg 05.45.Vx Keywords: Chaoscontrol

Fuzzylogicconstantcontroller FLCC

NewMathieu–vanderPolsystems Qisystem

a

b

s

t

r

a

c

t

Inthispaper,anewfuzzylogiccontroller—fuzzylogicconstantcontroller(FLCC)isintroducedtochaotic

signalscontrolling.ThemainideasoftheFLCCaredescribedasfollows:(1)provingthetwochaotic

systemsaregoingtoachieveasymptoticallystableviaLyapunovdirectmethod;(2)viadetectingthesign

oftheerrors,theappropriatefuzzylogiccontrolschemeisoperated;(3)choosingtheupperboundand

thelowerboundoftheerrorderivativesofthechaoticsignalstobetheconsequentparts(corresponding

controllers).

Duetocontrollersintraditionalmethod–derivedbyLyapunovdirectmethod,arealwayscomplicated,

nonlinearformorthefunctionsoferrors,anewsimplestcontroller—FLCCispresentedinthispaperto

synchronizetwochaoticsignals.ThroughtheFLCC,therearethreemaincontributionscanbeobtained:

(1)themathematicalmodelsofthenonlinearchaoticsystemscanbeunknown,allwehavetodois

capturingthesignalsoftheunknownsystems;(2)throughthefuzzylogicrules,thestrengthofcontrollers

canbeadjustedviathecorrespondingmembershipfunctions(whicharedecidedbythevaluesoferror

derivatives);(3)bytheFLCC,thechaoticsystemcanbemuchmoreexactlyandefﬁcientlycontrolledto

thetrajectoryofourgoalthantraditionalones.Threecases,originalpoint,regularfunctionandchaotic

Qisystem(withlargevaluesofinitialconditions),aregiventoillustratetheeffectivenessofournewFLC.

1. Introduction

SinceOtt et al.[1] gave thefamous OGY controlmethod in

1990,theapplicationsofthevariousmethodstocontrolachaotic

behaviorinnaturalsciencesandengineeringarewellknown.For

example,theadaptivecontrol[2–5],themethodofchaoscontrol

basedonsampleddata[6],themethodofpulsefeedbackof

system-aticvariable[7],theactivecontrol[8,9]andlinearerrorfeedback

control[10,11].However,when Lyapunovstabilityofzero

solu-tionofstates isstudied,thestability ofsolutions onthewhole

neighborhoodregionoftheoriginisdemanded.

Inrecentyears,somechaoscontrolbasedonfuzzysystemshas

beenproposedsincethefuzzysettheorywasinitiatedbyZadeh

[12], suchas fuzzy sliding mode controllingtechnique [13,14],

LMI-basedsynchronization[15]andextendedbackstepping

slid-ingmodecontrollingtechnique[16].Thefuzzylogiccontrol(FLC)

scheme has been widely developed for almost 40 years and

hasbeen successfully appliedto many applications[17]. Many

researchershaveworkedtoimprovetheperformanceoftheFLCs

andensuretheirstability.LiandGatlandinRefs.[18,19]proposed

∗ Tel.:+88635712121x55179;fax:+88635720634. E-mailaddress:agenghost@gmial.com

amoresystematicdesignmethodforPD-andPI-typeFLCs[20].

Choietal.[21]presentsasingleinputFLCensuringstability.Ying

[22]presentsapracticaldesignmethodfornonlinearfuzzy

con-trollers,andmanyotherresearchershaveresultsonthematterof

thestabilityofFLCs,inCastilloetal.[23]andCázarezetal.[24]was

presentedanextensionoftheMargaliotwork[25]tobuiltstable

type-2fuzzylogiccontrollersinLyapunovsense.

Recently,YauandShieh[26]proposedanamazingnewideato

designfuzzylogiccontrollers—constructingfuzzyrulessubjecttoa

commonLyapunovfunctionsuchthatthemaster-slavechaos

sys-temssatisfystabilityintheLyapunovsense.InRef.[26],thereare

twomaincontrollersintheirslavesystem.Oneisusedin

elim-inationofnonlinearterms andtheotheris builtbyfuzzyrules

subjecttoacommonLyapunovfunction.Therefore,theresulting

controllersarenonlinearform.Otherwise,inRef.[26],theregular

formisnecessary.Inordertocarryoutthenewmethod,theoriginal

systemmusttobetransformedintotheirregularform.

Inthis paper,wepropose anewstrategy whichis also

con-structingfuzzyrulessubjecttoaLyapunovdirectmethod.Error

dynamicsareusedtobeupperboundandlowerbound.Through

this new approach, a simplest controller, constant controller,

canbeobtainedand thedifﬁcultyin realizationof complicated

controllersinchaossynchronizationbyLyapunovdirectmethod

canbealsocoped.Unlikeconventionalapproaches,theresulting

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S.-Y.Li/AppliedSoftComputing11(2011)4474–4487 4475

Fig.1.Theconﬁgurationoffuzzylogiccontroller.

controllawhasless maximum magnitudeofthe instantaneous

controlcommandanditcanreducetheactuatorsaturation

phe-nomenoninrealphysicsystem.

Therestofthepaperisorganizedasfollows:inSection2,chaos

controlbyFLCC schemeispresented. InSection 3,newchaotic

Mathieu–vanderPolsystemisintroduced.InSection4,simulation

resultsareshown.InSection5conclusionsaregiven.

2. ChaoscontrolbyusingtheschemeofFLCC

2.1. Chaoscontrolscheme

Considerthefollowingchaoticsystem

˙x₌f(t,x)+u (2-1)

wherex= [x1,x2,... ,xn]T∈ Rnisathestatevector,f:R+×Rn→Rn

isavectorfunctionandu=[u1,u2,...,un]T∈Rnisthefuzzylogic

controllerneededtobedesigned.

Thegoalsystemwhichcanbeeitherchaoticorregular,is

˙y=g(t,y) (2-2)

wherey= [y1,y2,...,yn]T∈Rnisastatevector,g:R+×Rn→Rnis

avectorfunction.

Inordertomakethechaosstatexapproachingthegoalstatey,

deﬁnee=x−y=[e1,e3,e3,e4]asthestateerror.Thechaoscontrol

isaccomplishedinthesensethat[13–22]:

lim

t→∞e=t→∞lim(x−y)=0 (2-3)

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where

e=x−y (2-4)

FromEq.(2-4)wehavethefollowingerrordynamics:

˙e= ˙x− ˙y=f(t,x)−g(t,y)−u (2-5)

AccordingtoLyapunovdirectmethod,wehavethefollowing

Lyapunovfunctiontoderivethefuzzylogiccontrollerfor

synchro-nization: V=f(e1,...em,...en)=1 2(e 2 1+···+e 2 m+···e2n)>0 (2-6)

ThederivativeoftheLyapunovfunctioninEq.(2-5)is:

˙V=e1˙e1+···+em˙em+···en˙en (2-7)

If the controllers included in ˙e1... ˙em... ˙en can be suitably

designedtoachievethetarget: ˙V <0,thenthetwochaoticsystems

areasymptoticallystable.ThedesignprocessofFLCCisintroduced

inthefollowingsection.

2.2. Fuzzylogicconstantcontrollerdesignprocess

Thebasicconﬁgurationofthefuzzylogicsystemisshownin

Fig.1.Itiscomposedofﬁvefunctionblocks[27]:

1.Arulebasecontainsanumberoffuzzyif–thenrules.

2.Adatabasedeﬁnesthemembershipfunctionsofthefuzzysets

usedinfuzzyrules.

3.Adecision-makingunitperformstheinferenceoperationsonthe

rules.

4.Afuzziﬁcationinterfacetransformsthecrispinputsintodegrees

ofmatchwithlinguisticvalue.

5.Adefuzziﬁcationinterfacetransformsthefuzzyresultsofthe

inferenceintoacrispoutput.

Thefuzzyrulebaseconsistsofcollectionoffuzzyif–thenrules

expressedastheformifaisAthenbisB,whereaandbdenote

linguisticvariables,AandBrepresentlinguisticvalueswhichare

characterizedbymembershipfunctions.Allofthefuzzyrulescan

beusedtoconstructthefuzzyassociatedmemory.

Fig.3.Membershipfunction.

We use one signal, error derivatives ˙e(t)=

[ ˙e1, ˙e2,... ˙em,... ˙en]T, as the antecedent part of the proposed

FLCC to design the control input u that will be used in the

consequentpartoftheproposedFLCCasfollows:

u=[u1,u2...um,...un]T (2-8)

whereuisaconstantcolumnvectorandtheFLCCaccomplishesthe

objectivetostabilizetheerrordynamics(2-5).

ThestrategyoftheFLCCdesigningisproposedasfollowsand

theconﬁgurationofthestrategyisshowninFig.2.

Assumetheupperboundandlowerboundof ˙emareZmand−Zm,

thentheFLCCcanbedesignstepbystepasfollows:

(1)If em is detected as positive (em>0), we have to design a

controller for ˙em<0, then ˙V=em˙em<0 can be achieved.

Thereforewehavethefollowingithif–thenfuzzyrulesas:

Rule 1: if ˙em is Mmi then um1=Zm (2-9)

Rule 2: if ˙em is Mm2 then um2=Zm (2-10)

Rule 3: if ˙em is Mm3 then um3=em (2-11)

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Fig.5. TimehistoriesoferrorderivativesforCase1(withoutcontrollers).

(2) If em is detected as negative (em<0), we have to design a

controller for ˙em>0, then ˙V=em˙em<0 can be achieved.

Thereforewehavethefollowingithif–thenfuzzyrulesas:

Rule1: if ˙em is Mm1 then um1=−Zm (2-12)

Rule 2: if ˙em is Mm2 then um2=−Zm (2-13)

Rule 3: if ˙em is Mm3 then um3=em (2-14)

(3)Ifem approaches tozero,thenthesynchronization isnearly

achieved.Therefore wehave thefollowing ithif–thenfuzzy

rulesas:

Rule 1: if ˙em is Mm1 then um1=em≈0 (2-15)

whereMm1=

˙em

/Zm,Mm2=

˙em

/Zm andMm3=sgn((Zm−

˙em)/Zm)+sgn(( ˙em−Zm)/Zm),Mm1,Mm2andMm3 refertothe

membershipfunctionsofpositive (P),negative(N)and zero

(Z)separatelywhicharepresentedinFig.3.Foreachcase,umi,

i=1–3isthei-rdoutputof ˙emwhichisaconstantcontroller.The

centroiddefuzziﬁerevaluatestheoutputofallrulesasfollows:

um=

3 i=1Mmi×umi

3 i=1Mmi (2-18)

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Fig.7. TimehistoriesofstatesforCase1—theFLCCiscomingintoafter20s.

Table1

Rule-tableofFLCC.

Rule Antecedent Consequentpart

˙em umi

1 Negative(N) um1

2 Positive(P) um2

3 Zero(Z) um3

ThefuzzyrulebaseislistedinTable1,inwhichtheinput

vari-ablesintheantecedentpartoftherulesare ˙emandtheoutput

variableintheconsequentpartisumi.

Afterdesigningappropriatefuzzylogicconstantcontrollersand

beingsubstitutedintoEq.(2-7),anegativedeﬁniteofderivatives

ofLyapunovfunction ˙V canbeobtainedandtheasymptotically

stabilityofLyapunovtheoremcanbeachieved.

Consequently,theprocessesofFLCCdesigningtocontrola

sys-temfollowingthetrajectoryofagoalsystemcanbeconcludedas

follows:

1.ConstructaFLCCsysteminMATLAB(Simulink)followingFig.2

andEqs.(2-9)–(2-17).

2.Gettheupperboundandlowerboundoftheerrorderivativesof

thegoalandcontrolsystemswithoutanycontroller,i.e.−Zm≤

˙em≤Zm,whichareusedtobetheconstantcontrollers.

3.Design the membership functions of positive (P), negative

(N) and zero (Z) M1=

˙em

/Zm, M2=

˙em

/Zm and M3=

sgn((Zm− ˙em)/Zm)+sgn(( ˙em−Zm)/Zm).

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Fig.9. TimehistoriesofstatesforCase1—bynonlinearcontrollers(20s).

4.AnegativedeﬁniteofderivativesofLyapunovfunction ˙Vcanbe

obtainedandtheasymptoticallystabilityofLyapunovtheorem

canbeachieved.

3. NewchaoticMathieu–vanderPolsystem

ThissectionintroducesnewMathieu–vanderPolsystem.

Math-ieuequationandvanderPolequationaretwotypicalnonlinear

non-autonomoussystems:

˙x1=x2

˙x2=−(a+bsinωt)x1−(a+bsinωt)x31−cx2+dsinωt

(3-1)

˙x3=x4

˙x4=−ex3+f(1−x2₃)x4+g sin ωt

(3-2)

ExchangingsinωtinEq.(3-1)withx3andsinωtinEq.(3-2)with

x1,weobtaintheautonomousnewMathieu–vanderPolsystem:

⎧

⎪

⎨

⎪

⎩

˙x1=x2 ˙x2=−(a+bx3)x1−(a+bx3)x13−cx2+dx3 ˙x3=x4 ˙x4=−ex3+f(1−x₃2)x4+gx1 (3-3)

wherea,b,c,d,e,f,gareuncertainparameters.Thissystemexhibits

chaoswhentheparametersofsystemarea=10,b=3,c=0.4,d=70,

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Fig.11.TimehistoriesoferrorsforCase2—theFLCCiscomingintoafter20s.

Table2

Comparisonbetweenformsofcontrollers.

FLCC(components) Nonlinearcontrollers u1 ±Z1=±50ore1≈0 −x2−e1

u2 ±Z2=±500ore2≈0 (a+bx3)x1+(a+bx3)x3₁+cx2−dx3−e2

u3 ±Z3=±10ore3≈0 −x4−e3

u4 ±Z4=±10ore4≈0 ex3−f(1−x23)x4−gx1−e4

e=1,f=5,g=0.1andtheinitialstatesofsystemare(x10,x20,x30,

x40)=(1,5,1,5).Theprojectionsofphaseportraitsareshownin

Fig.4.

4. Simulationresults

Inordertolead(x1,x2,x3,x4)inEq.(3-3)tothegoal,weadd

controltermsu1,u2,u3andu4toeachequationofEq.(3-3),

respec-tively.

⎧

⎪

⎨

⎪

⎩

˙x1=x2+u1 ˙x2=−(a+bx3)x1−(a+bx3)x31−cx2+dx3+u2 ˙x3=x4+u3 ˙x4=−ex3+f(1−x23)x4+gx1+u4 (4-1)

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Fig.13.TimehistoriesoferrorsforCase2—bynonlinearcontrollers(20s).

Table3

Comparisonbetweenerrordataat99.96s,99.97s,99.98s,99.99sand100.00safter theactionofcontrollers.

Time FLCC Nonlinearcontrollers

e1 e1 99.96s −3.5694e−035 6.8001e−016 99.97s −3.5339e−035 6.8001e−016 99.98s −3.4987e−035 6.8001e−016 99.99s −3.4639e−035 6.8001e−016 100.00s −3.4294e−035 6.8001e−016

e2 e2 99.96s 5.5438e−048 3.0531e−015 99.97s 5.4885e−048 3.0531e−015 99.98s 5.4340e−048 3.0531e−015 99.99s 5.3799e−048 3.0531e−015 100.00s 5.3264e−048 3.0531e−015

Therearethreecasestoshowtheeffectivenessandfeasibility

ofthenewapproach—FLCC.CaseI:Controlthechaoticmotionto

zero,CaseII:Controlthechaoticmotiontoaregularfunctionand

CaseIII:ControlthechaoticmotionofthenewMathieu–vanderPol

systemtoanotherchaoticmotionoftheQisystem.Furthermore,

chaoscontrolviatraditionalnonlinearcontrollersinCasesI–IIIis

alsogivenintablesandﬁguresforcomparison.

4.1. CaseI:controlthechaoticmotiontozeroby(1)FLCCand(2)

traditionalcontrollers

In this case, we will control thechaotic motionof the new

Mathieu–vanderPolsystem(4-1)tozero.Thegoalisy=0.Thestate

errorisei=xi−yi=xi(i=1,2,3,4)anderrordynamicsbecomes

⎧

⎪

⎨

⎪

⎩

˙e1= ˙x1=x2+u1 ˙e2= ˙x2=−(a+bx3)x1−(a+bx3)x31−cx2+dx3+u2 ˙e3= ˙x3=x4+u3 ˙e4= ˙x4=−ex3+f(1−x32)x4+gx1+u4 (4-2)

ChoosingLyapunovfunctionas:

V= 1₂(e2₁+e₂2+e2₃+e2₄) (4-3)

Itstimederivativeis:

˙V₌e1˙e1+e2˙e2+e3˙e3+e4˙e4=e1(x2+u1)

+e2(−(a+bx3)x1−(a+bx3)x31−cx2+dx3+u2)

+e3(x4+u3)+e4(−ex3+f(1−x32)x4+gx1+u4) (4-4)

(1)ByFLCC

InordertodesignFLCC,wedivideEq.(4-3)intofourparts:

V= 1₂(e2₁+e2₂+e2₃+e2₄)=V1+V2+V3+V4 (4-5)

thentheerrorderivativeis

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where

Part1: ˙V1=e1˙e1=e1(x2+u1)

Part2: ˙V2=e2˙e2=e2(−(a+bx3)x1−(a+bx3)x31−cx2+dx3+u2)

Part3: ˙V3=e3˙e3=e3(x4+u3)

Part4: ˙V4=e4˙e4=e4(−ex3+f(1−x2₃)x4+gx1+u4)

AccordingtotheprocessoftheFLCCdesigning,theerror

deriva-tivesofCaseIwithoutanycontrollerareshowninFig.5,andthe

valuesoftheupperboundandlowerboundcanbeobtainedas

follows:

Z1=50, Z2=500, Z3=10, Z4=10 (4-7)

AftergettingZ1∼Z4,alltheconstantcontrollers,membership

func-tionsandthecorrespondingFLCCcanbedecided.Therefore, ˙V1=

e1˙e1<0, ˙V2=e2˙e2<0, ˙V3=e3˙e3<0 and ˙V4=e4˙e4<0 can

beachieved,thenwehave ˙V= ˙V1+ ˙V2+ ˙V3+ ˙V4<0.Itisclearthat

thederivativeofLyapunovfunctionisnegativedeﬁniteandthe

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Fig.16.TimehistoriesoferrorderivativesforCase3(withoutcontrollers).

errordynamicssystemisgoingtoachieveasymptoticallystable.

ThesimulationresultsareshowninFigs.6and7.

(2) Bytraditionalnonlinearcontrollers

InordertoleadtheerrordynamicsinEq.(4-4)toachieve

asymp-toticallystable, we havethe following nonlinear controllersby

traditionalmethodas:

⎧

⎪

⎨

⎪

⎩

u1=−x2−e1 u2=(a+bx3)x1+(a+bx3)x31+cx2−dx3−e2 u3=−x4−e3 u4=+ex3−f(1−x23)x4−gx1−e4 (4-8)

substitutingallthenonlinearcontrollersinEq.(4-8)intoEq.(4-4):

˙V =−e2

1−e22−e23−e24<0 (4-9)

whichisanegativedeﬁnitefunctionandtheerrordynamicssystem

isgoingtoachieveasymptoticallystable.Thesimulationresultsare

showninFigs.8and9.

4.1.1. ComparisonwithFLCCandtraditionalmethod

In this section, numerical simulation results by FLCC and

traditionalcontrollersarelistedinTables2and3forcomparison.

ComparingthetwosimulationresultsinTables2and3,itisclear

toﬁndoutthat(1)theperformance(accuracyandspeedof

con-vergence)ofthestatesoferrorsconvergingtotheoriginalpointby

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Fig.18.TimehistoriesofstatesforCase3—theFLCCiscomingintoafter20s.

Table4

FLCC(components) Nonlinearcontrollers u1 ±Z1=±50ore1≈0 −x2+ω×F1cosωt−e1 u2 ±Z2=±500ore2≈0 +(a+bx3)x1+(a+bx3)x13+cx2+dx3+ω× F2cosωt−e2 u3 ±Z3=±20ore3≈0 −x4+ω×F3cosωt−e3 u4 ±Z4=±100ore4≈0 ex3−f(1−x2₃)x4−gx1+ω×F4cosωt−e4 Table5

e1 e1 29.96s 0.00031450 0.0065468 29.97s 0.00031388 0.0065308 29.98s 0.00031325 0.0065146 29.99s 0.00031262 0.0064984 30.00s 0.00031197 0.0064820

e2 e2 29.96s 0.00064831 0.0064258 29.97s 0.00064702 0.0064110 29.98s 0.00064572 0.0063961 29.99s 0.00064440 0.0063810 30.00s 0.00064306 0.0063658

e3 e3 29.96s 0.00037285 0.0112380 29.97s 0.00037216 0.0111750 29.98s 0.00037146 0.0111130 29.99s 0.00037075 0.0110510 30.00s 0.00037003 0.0109890

e4 e4 29.96s 0.00142410 0.0066563 29.97s 0.00142150 0.0066392 29.98s 0.00141900 0.0066220 29.99s 0.00141630 0.0066047 30.00s 0.00141370 0.0065873

FLCCismuchbetterthantheperformancebytraditionalones;(2)

thecontrollersinFLCCdesigningaremuchsimplerthantraditional

ones.

Consequently,eventhesystemissocomplicatedandthe

con-trollers provided in FLCC are such a constant ones, the high

performanceandexact numericalsimulationresultscanbestill

remained.

4.2. CaseII:controlthechaoticmotiontoaregularfunctionby

(1)FLCCand(2)traditionalcontrollers

In this case we will control the chaotic motion of thenew

Mathieu–vanderPolsystem(4-1)toregularfunctionoftime.The

goalisyi=Fisinωt(i=1,2,3,4).Theerrorequation

ei=xi−yi=xi−Fisinωt, (i=1,2,3,4) (4-10)

lim

t→∞ei=t→∞lim(xi−Fisinωt)=0, (i=1,2,3,4)

whereF1=0.5,F2=1,F3=0.5,F4=2andω=0.5

Theerrordynamicsis

⎧

⎪

⎨

⎪

⎩

˙e1=x2−ω×F1cosωt+u1

˙e2=−(a+bx3)x1−(a+bx3)x13−cx2+dx3−ω×F2cosωt+u2

˙e3=x4−ω×F3cosωt+u3

˙e4=−ex3+f(1−x23)x4+gx1−ω×F4cosωt+u4

(4-11)

V= 1₂(e2₁+e₂2+e2₃+e2₄) (4-12)

˙V =e1˙e1+e2˙e2+e3˙e3+e4˙e4=e1(x2−ω×F1cosωt+u1)

+e2(−(a+bx3)x1−(a+bx3)x13−cx2+dx3−ω

×F2 cosωt+u2)+e3(x4−ω×F3cosωt+u3)

+e4(−ex3+f(1−x23)x4+gx1−ω×F4 cosωt+u4) (4-13)

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Fig.19.TimehistoriesoferrorsforCase3—bynonlinearcontrollers(20s).

InordertodesignFLCC,wedivideEq.(4-13)intofourpartsas

follows:

Assume V=(1/2)(e2

1+e22+e23+e24)=V1+V2+V3+V4,

then ˙V =e1˙e1+e2˙e2+e3˙e3+e4˙e4= ˙V1+ ˙V2+ ˙V3+ ˙V4, where

V1=(1/2)e21, V2=(1/2)e22, V3=(1/2)e23andV4=(1/2)e24.

Part1: ˙V1=e1˙e1=e1(x2−ω×F1cosωt+u1)

Part 2: ˙V2=e2˙e2=e2(−(a+bx3)x1−(a+bx3)x31−cx2+dx3−

ω×F2cosωt+u2)

Part3: ˙V3=e3˙e3=e3(x4−ω×F3cosωt+u3)

Part 4: ˙V4=e4˙e4=e4(−ex3+f(1−x23)x4+gx1−ω×F4cosωt+

u4)

deriva-tivesofCaseIIwithoutanycontrollerareshowninFig.10,andthe

follows:

Z1=50, Z2=500, Z3=20, Z4=100 (4-14)

(2)Bytraditionalnonlinearcontrollers

In ordertolead theerrordynamicsin Eq.(4-11) toachieve

asymptoticallystable,wehavethefollowingnonlinearcontrollers

bytraditionalmethodas:

⎧

⎪

⎨

⎪

⎩

u1=−x2+ω×F1cosωt−e1 u2=+(a+bx3)x1+(a+bx3)x13+cx2+dx3+ω×F2cosωt−e2 u3=−x4+ω×F3cosωt−e3 u4=ex3−f(1−x23)x4−gx1+ω×F4cosωt−e4 (4-15)

substitutingallthenonlinearcontrollersinEq.(4-15)intoEq.

(4-13):

˙V =−e2

1−e22−e23−e24<0 (4-16)

showninFigs.13and14.

NumericalsimulationresultsbyFLCCandtraditionalcontrollers

arelistedinTables4and5forcomparison.Itcanbefoundoutthat

(1)theperformance(accuracyandspeedofconvergence)ofthe

statesoferrorsconvergingtotheoriginalpointbyFLCCismuch

betterthantheperformancebytraditionalones;(2)thecontrollers

inFLCCdesigningaremuchsimplerthantraditionalones.

Consequently, even thesystem is socomplicated or regular

form,andthecontrollersprovidedinFLCCaresuchaconstantones,

thehighperformanceandexactnumericalsimulationresultscan

bestillremained.

4.3. CaseIII:controlthechaoticmotionofthenewMathieu–van

derPolsystemtoanotherchaoticmotionoftheQisystemvia(1)

FLCCand(2)traditionalcontrollers

In Case III, we will control chaotic motion of the new

Mathieu–vanderPolsystem(4-1)tothatoftheQisystem.The

goalsystemforcontrolisQisystemshownasfollows:

⎧

⎪

⎨

⎪

⎩

˙y1=a1(y2−y1)+y2y3y4 ˙y2=b1(y1+y2)−y1y3y4 ˙y3=−c1y3+y1y2y4 ˙y4=−d1y4+y1y2y3 (4-17)

wherey1,y2,y3andy4 arethestatevariablesofthesystemand

a1,b1,c1andd1areallpositiverealparameters.ThisQisystemin

Eq.(4-17)wasrecentlyintroducedbyQietal.[28]andithasbeen

shownexhibitcomplexdynamicalbehaviorincludingthefamiliar

(13)

systemparameters:a1=30,b1=10,c1=1,d1=10andinitial

con-ditions(y10,y20,y30,y40)=(20,50,20,50),theQimodelexhibits

chaoticmotionwhichisshowninFig.15.

Theerrorequation

ei=xi−yi, (i=1,2,3,4) (4-18)

lim

t→∞ei=tlim→∞(xi−yi)=0, (i=1,2,3,4)

Theerrordynamicsis

⎧

⎪

⎨

⎪

⎩

˙e1= ˙x1=x2−(a1(y2−y1)+y2y3y4)+u1 ˙e2= ˙x2=−(a+bx3)x1−(a+bx3)x31−cx2+dx3−(b1(y1+y2)−y1y3y4)+u2 ˙e3= ˙x3=x4−(−c1y3+y1y2y4)+u3 ˙e4= ˙x4=−ex3+f(1−x23)x4+gx1−(−d1y4+y1y2y3)+u4 (4-19)

V=1₂(e2₁+e₂2+e2₃+e2₄) (4-20)

˙V =e1˙e1+e2˙e2+e3˙e3+e4˙e4=e1(x2−(a1(y2−y1)

+y2y3y4)+u1)+e2(−(a+bx3)x1−(a+bx3)x31−cx2+dx3

−(b1(y1+y2)−y1y3y4)+u2)+e3(x4−(−c1y3+y1y2y4)

+u3)+e4(−ex3+f(1−x23)x4+gx1−(−d1y4+y1y2y3)+u4)

(4-21)

(1)ByFLCC

InordertodesignFLCC,wedivideEq.(4-12)intofourpartsas

follows: Assume V₌(1/2)(e2 1+e 2 2+e 2 3+e 2 4)=V1+V2+V3+V4,

then ˙V₌e1˙e1+e2˙e2+e3˙e3+e4˙e4= ˙V1+ ˙V2+ ˙V3+ ˙V4, where

V1=(1/2)e21, V2=(1/2)e22, V3=(1/2)e23andV4=(1/2)e24.

Part1: ˙V1=e1˙e1=e1(x2−(a1(y2−y1)+y2y3y4)+u1)

Part 2: ˙V2=e2˙e2=e2(−(a+bx3)x1−(a+bx3)x3₁−cx2+dx3−

(b1(y1+y2)−y1y3y4)+u2)

Part3: ˙V3=e3˙e3=e3(x4−(−c1y3+y1y2y4)+u3)

Part 4: ˙V4=e4˙e4=e4(−ex3+f(1−x₃2)x4+gx1−(−d1y4+

y1y2y3)+u4)

deriva-tivesofCaseIIIwithoutanycontrollerareshowninFig.16,andthe

follows:

Z1=2000, Z2=2000, Z3=1000, Z4=1000 (4-22)

(2)Bytraditionalnonlinearcontrollers

In ordertolead theerrordynamicsin Eq.(4-19) toachieve

asymptoticallystable,wehavethefollowingnonlinearcontrollers

bytraditionalmethodas:

⎧

⎪

⎨

⎪

⎩

u1=−x2+(a1(y2−y1)+y2y3y4)−e1 u2=(a+bx3)x1+(a+bx3)x31+cx2−dx3+(b1(y1+y2)−y1y3y4)−e2 u3=−x4+(−c1y3+y1y2y4)−e3 u4=+ex3−f(1−x23)x4−gx1+(−d1y4+y1y2y3)−e4 (4-23)

(14)

Table6

FLCC(components) Nonlinearcontrollers u1 ±Z1=±2000ore1≈0 −x2+(a1(y2−y1)+y2y3y4)−e1 u2 ±Z2=±2000ore2≈0 (a+bx3)x1+(a+bx3)x31+cx2−dx3+(b1(y1+ y2)−y1y3y4)−e2 u3 ±Z3=±1000ore3≈0 −x4+(−c1y3+y1y2y4)−e3 u4 ±Z4=±1000ore4≈0 ex3−f(1−x32)x4−gx1+(−dy4+y1y2y3)−e4 Table7

e1 e1 59.96s −7.0533e−011 −1.5015e−004 59.97s −6.9883e−011 −1.7386e−004 59.98s −6.9228e−011 −1.9902e−004 59.99s −6.8573e−011 −2.2194e−004 60.00s −6.7920e−011 −2.3330e−004

e2 e2 59.96s 4.1470e−012 −1.3996e−004 59.97s 4.1160e−012 −1.3715e−004 59.98s 4.0852e−012 −1.2018e−004 59.99s 4.0544e−012 −7.9564e−005 60.00s 4.0239e−012 −2.2182e−006

e3 e3 59.96s 0 9.1480e−007 59.97s 0 −1.5572e−005 59.98s 0 −4.0608e−005 59.99s 0 −7.6908e−005 60.00s 0 −1.2420e−004

e4 e4 59.96s −3.0018e−011 −9.9919e−005 59.97s −2.9837e−011 −1.3461e−004 59.98s −2.9644e−011 −1.7607e−004 59.99s −2.9441e−011 −2.2263e−004 60.00s −2.9231e−011 −2.6721e−004

substitutingallthenonlinearcontrollersinEq.(4-23)intoEq.

(4-21):

˙V=−e2

1−e22−e23−e24<0 (4-24)

showninFigs.19and20.

Inthiscase,thenewMathieu–vanderPolsystemiscontrolled

toanotherchaoticmotion—theQisystemwithlargeinitial

con-ditions(y10,y20,y30,y40)=(20,50,20,50).Thiscaseisillustrated

toinvestigatetheeffectivenessandfeasibilityoftheFLCCevenif

thetwochaotictrajectoriesarefarfromeachother.Accordingto

Figs.17and19,thespeedofcontrollingtheerrorstatestothe

orig-inalpointsviaFLCC(aboutat30s)ismuchfasterthanthespeed

vianonlinearcontroller(aboutat60s).Asaresults,thenumerical

datain59.96–60.00sarefurtherproposedforcomparison.

NumericalsimulationresultsbyFLCCandtraditionalcontrollers

arelistedinTables6and7forcomparison.Thetwomain

supe-riorities still exist—(1)theperformance (accuracyand speed of

convergence)oftheconvergenceoferrorstatesbyFLCCismuch

better than by traditional method; (2) the controllers in FLCC

designingaremuchsimplerthantraditionalones.

5. Conclusions

Inthispaper,asimplestcontroller—fuzzylogicconstant

con-troller (FLCC) is introduced to chaos control. The illustrations

mentionedabovedemonstratethebetterperformanceand

accu-racyofnumericalsimulationresultsofthesynchronizationviaFLCC

clearlyeventhefuzzycontrollersareonlyasimpleformofconstant

numbers.

Asaresult,threemaincontributionscanbeconcluded—(1)high

performance oftheconvergence oferror statesin

synchroniza-tion;(2)thestrengthofthefuzzycontrollerscanbeadjustedvia

membershipfunctions;(3)fuzzylogiccontrollersareeasyto

pro-duce.Furthermore,duetothecharactersofFLCC,themathematical

modelsofdomainsystemscanbeunknown,allwehavetodois

cap-turingtheoutputsignals,constructingthefuzzylogicsystemand

calculatingtheareaoftheerrorderivatives,thenwecancontrol

anyoutputsignaltoanotherone.Hence,FLCCis sucha

poten-tialtoolandcanbeappliedtovariouskindsofﬁeldswithlotsof

unknownfunctions—suchasneuroscience,un-modelbio-systems,

andcomplicatedbrainnetwork.

Acknowledgment

ThisworkwassupportedinpartbytheUST-UCSDInternational

CenterofExcellenceinAdvancedBio-engineeringsponsoredbythe

TaiwanNationalScienceCouncilI-RiCEProgramunderGrant

Num-ber:NSC-99-2911-I-009-101.Thisresearchwassupportedbythe

NationalScienceCouncil,RepublicofChina,underGrantNumber

NSC99-2221-E-009-019.

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