Optimal control of Takagi–Sugeno fuzzy-model-based systems representing dynamic ship positioning systems

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ContentslistsavailableatSciVerseScienceDirect

Applied

Soft

Computing

jo u r n al hom e p a g e :w w w . e l s e v i e r . c o m / l o c a t e / a s o c

Optimal

control

of

Takagi–Sugeno

fuzzy-model-based

systems

representing

dynamic

ship

positioning

systems

Wen-Hsien

Ho

a

_,

_Shinn-Horng

_Chen

b

_,

_Jyh-Horng

_Chou

b,c,∗

a_Department_of_Healthcare_{Administration}_and_Medical_Informatics,_Kaohsiung_Medical_University,₁₀₀_Shi-Chuan_1st_Road,_Kaohsiung_807,_Taiwan,_ROC b_Department_of_Mechanical_Engineering,_National_Kaohsiung_University_of_Applied_Sciences,₄₁₅_Chien-Kung_Road,_Kaohsiung_807,_Taiwan,_ROC c_Institute_of_System_Information_and_Control,_National_Kaohsiung_First_University_of_Science_and_Technology,₁_University_Road,_Yenchao,_Kaohsiung_824,

Taiwan,ROC

a

r

t

i

c

l

e

i

n

f

o

Articlehistory:

Received1February2012

Receivedinrevisedform13January2013 Accepted7February2013

Availableonline19March2013 Keywords:

Dynamicshippositioningsystems HybridTaguchi-geneticalgorithm Orthogonalfunctions

Quadraticﬁnite-horizonoptimalcontrol Takagi–Sugenofuzzymodel

a

b

s

t

r

a

c

t

Orthogonalfunctionapproach(OFA)andthehybridTaguchi-geneticalgorithm(HTGA)areusedtosolve quadraticfinite-horizonoptimalcontrollerdesignproblemsinbothafuzzyparalleldistributed compen-sation(PDC)controllerandanon-PDCcontroller(linearstatefeedbackcontroller)forTakagi–Sugeno(TS) fuzzy-model-basedcontrolsystemsfordynamicshippositioningsystems(TS-DSPS).BasedontheOFA,an algorithmrequiringonlyalgebraiccomputationisusedtosolvedynamicequationsfor TS-fuzzy-model-basedfeedbackandisthenintegratedwithHTGAtodesignquadraticfinite-horizonoptimalcontrollers forTS-DSPSunderthecriterionofminimizingaquadraticfinite-horizonintegralperformanceindex, whichisalsoconvertedtoalgebraicformbytheOFA.IntegrationofOFAandHTGAintheproposed approachenablesuseofsimplealgebraiccomputationandiswelladaptedtothecomputer implementa-tion.Therefore,itfacilitatesdesigntasksofquadraticfinite-horizonoptimalcontrollersfortheTS-DSPS. Theapplicabilityoftheproposedapproachisdemonstratedintheexampleofamooredtankerdesigned usingquadraticfinite-horizonoptimalcontrollers.

1. Introduction

Acurrentissueintheliteratureonfuzzy-model-basedcontrol systemsisthenonlinearcontrolproblemindynamicship position-ingsystems(DSPS)forcontrollingthethrustersandpropellersof marinesurfacevesselswhenmaneuveringatlowspeedorwhen maintainingastationaryposition[1–6].TheDSPSmaintainsa ﬂoat-ingvesselinthevicinityofareferencepoint(e.g.,verticallyabove awell) andstabilizestheheadingofa vessel.Thesemaneuvers requirecontrollersformultiplethrusters,includingthemain pro-pellersaftof theship,tunnel thrusters,and azimuth(rotatable) thrustersmountedunderthehull.Sincethemathematicalmodel usedforcontrolinthisstudydoesnotaccountforpitchandroll motion,Fig.1showsthatonly3degreesoffreedom(DOF)are con-sideredasdiscussedfurtherinSection2[7].Thatis,positionand headingarecontrolledonlyforthehorizontalplane.Thepractical implementationofthis techniqueisin maintainingthe station-arypositionofa shipbyformulatingthecontroloflongitudinal

∗ Correspondingauthorat:InstituteofSystemInformationandControl,National KaohsiungFirstUniversityofScienceandTechnology,1UniversityRoad,Yenchao, Kaohsiung824,Taiwan.Tel.:+88676011000;fax:+88676011066.

E-mailaddresses:[email protected](W.-H.Ho),[email protected]

(J.-H.Chou).

movement (surge, earth-ﬁxed position on X-axis), transverse movement (sway, earth-ﬁxed position onY-axis), and heading (yaw)asa3DOFprobleminthehorizontalplane.Basedon3DOF errorsresultinginoffsetofthevessel,thecontrollerdeterminesthe thrustercommandsneededtoreturnthevesseltoitsinitial posi-tion.Thethrustercommandsneededforcorrectionareupdated periodically [1]. The recent literature shows a rapidly growing interest in applyingthe Takagi–Sugeno (TS)fuzzy-model-based approachtononlinearcontrolsystems[8–13].Thefuzzymodel pro-posedby[14]isdescribedbythefuzzyIF-THENrulesthatrepresent locallinearinput-outputrelationsofanonlinearcontrolsystem. ThemainfeatureofaTSfuzzymodelisitsuseofalinearsystem modeltoexpressthelocaldynamicsofeachfuzzyrule.Oncethe TSfuzzymodelsareobtained,thelinearcontrolmethodologycan beusedtodesignthelocalstatefeedbackcontrollersforeach lin-earmodel.However,recentapplicationsoftheTSfuzzymodelin

[4–6],inwhichnonlinearDSPSis representedusinglinear con-trolmethodology,haveonlydesignedthestabilizingcontrollersof theTS-fuzzy-model-basedcontrolsystemrepresentingtheDSPS (TS-DSPS),wherethecontrollersbelongtothetypeofparallel dis-tributedcompensation(PDC).Despitetheirsuccess,ithasbecome evidentthatmanyresearchissuesfortheTS-DSPSremaintobe addressed.Incontrolsystemdesign,thecontrolobjectiveof mini-mizingaquadraticﬁnite-horizonintegralperformancecriterionis oftenachievedbysynthesizingaquadraticﬁnite-horizonoptimal

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Fig.1. Earth-ﬁxedandvessel-ﬁxedcoordinateframes.

controller[15].Thus,akeyissueisthedesignofquadratic ﬁnite-horizonoptimalcontrollersfortheTS-DSPS.However,aliterature reviewshowsthatthisdesignissuehasnotbeenaddressed.

Recentstudiessuchas[16–21],however,haveapplied ortho-gonalfunctionapproach(OFA)[22,23]tosolvedynamicequations inTS-fuzzy-model-basedsystems.TheOFAischaracterizedbythe simplificationofTS-fuzzy-model-baseddynamicequationsby con-vertingthemtostraightforwardalgebraicequationsthatareeasily solvedbycomputer.Hence,theOFAhasarelativelyshorterand simplersolutionprocedure.TheOFAhasalsoshownhighly satis-factorysolutionresults.Therefore,theobjectivesofthisstudywere todesignbothaquadraticfinite-horizonoptimalfuzzyPDC con-trollerandaquadraticfinite-horizonoptimalnon-PDCcontroller (quadraticfinite-horizonoptimallinearstatefeedbackcontroller) fortheTS-DSPSbyintegratingtheOFAandthehybrid Taguchi-geneticalgorithm (HTGA) for direct minimization of a defined quadraticfinite-horizonintegralperformanceindexrepresenting thecontrolobjective.ThisstudyappliestheHTGAintroducedin

[24,25]notonlybecauseit effectivelyﬁndsoptimalor close-to-optimalsolutions,butalsobecauseitssolutionresultsaremore robustcomparedtoimprovedgeneticalgorithmsreportedinthe literature.

Therefore,toachievetheobjectiveusingtheelegantoperational propertiesoftheOFA,thisstudyfirstdevelopedacomputational algorithmforsolving TS-DSPSfeedbackdynamic equations.The novelapproach is theexpression ofstate variables in terms of orthogonalfunctions.Theproposedmethodsimplifiesthe proce-dureforsolvingTS-DSPSfeedbackdynamicequationsassuccessive solutionsforasystemofrecursiveformulaetakingonlythe expan-sioncoefficients.Basedonthepresentedrecursiveformulae,only astraightforwardalgebraiccomputationisneededtoperformthe computationalgorithm.Thedevelopedalgorithmisthenintegrated withtheHTGAin designs forboth the quadraticfinite-horizon optimal fuzzy PDC controller and the quadratic finite-horizon optimalnon-PDCcontroller(quadraticfinite-horizonoptimal lin-earstatefeedbackcontroller)oftheTS-DSPSunderthecriterion of minimizing a quadratic finite-horizon integral performance index,wheretheOFA alsoconvertsthequadraticfinite-horizon integral performance index to algebraic form. The proposed approachof integratingOFA andHTGAis non-differential, non-integral,efficient,andwell-suitedforcomputerimplementation.

Therefore, computational complexity is substantially reduced. Thus, the approach facilitatesdesign tasksfor quadratic ﬁnite-horizonoptimalcontrollersforTS-DSPS.

This paper is organized as follows. Section 2 describes the problem statement of the TS-DSPS. Section 3 presents a new approachfor integratingOFA andHTGAin thedesignsfor both a quadratic finite-horizon optimal fuzzy PDC controller and a quadratic finite-horizon optimal non-PDC controller for a TS-DSPSwhileminimizingadefinedquadraticfinite-horizonintegral performance index. Section 4 gives an illustrative example to demonstratetheapplicabilityoftheproposedapproach.Finally, Section5concludesthestudy.

2. ProblemstatementofTS-DSPS

InthenormalizedmodelofhorizontalmotioninaDSPS,the motioncomponentsaresurge,swayandyaw[2].Thisstudy evalu-atedthecaseofaDSPScontrolledexclusivelybythrusters.Anchors can also be included in the model of ship motion to analyze thruster-assistedmooring.Dampingforcesareassumedtobe lin-earsince thespeed of theship is alwaysslowduring dynamic positioning.TheDSPSmodelalsoassumesahomogeneousmass distributionandxz-planesymmetry.Theoriginofthecoordinates issetatthecenterlineoftheship.

Wheretheposition(x,y)andheading ofthevesselrelativeto anearth-ﬁxedframeareexpressedinvectorformas=[x,y, ]T_,

letthevelocitiesbedecomposedinavessel-ﬁxedreferenceframe as=[u,

v

,r]T.Thesethreemodesarereferredtoasthesurge, sway,andyawmodesofaship.Theoriginofavessel-fixedframe locatedatthevesselcenterlineisexpressedasthedistancefrom thecenterofgravity.Sincethemeanrollandpitchanglesarezero, DSPSapplicationstypicallyassumethatmotionsinducedbyrolling andpitchingoftheshiparenegligible.Therefore,therigid-body dynamicequationsformotionofthevessel(surge,swayandyaw) obtainthevessel-fixedvelocityvectorsrelativetotheearth-fixed velocity(seeFig.1,[7])are

˙(t)=J((t))(t), (1)

and

B˙(t)+D(t)+G(t)=(t), (2)

where (t)=[x(t), y(t), (t)]T _denotes _the _earth-ﬁxed

orienta-tion vector describing thesurge, sway and yaw modes, (t)= [u(t),

v

(t),r(t)]Tdenotesthebody-ﬁxedlinearandangularvelocity vectordescribingthesurge,swayandyawmodes,(t)=[1(t),2(t),

3(t)]Tdenotestheinputvectorforthecontrolforcesandmoment

providedbythethrustersystem(mainpropellersaftoftheship 1(t)tunnelthrusters2(t)andazimuth(rotatable)thrusters3(t)),

andTmeanstransposeofamatrix.InEqs.(1)and(2),

J((t))=

⎡

⎣

cos( (t))_{sin( (t))} −_{cos( (t))}sin( (t)) 00

0 0 1

⎤

⎦

(3)

denotesthetransformationmatrixinyaw(assumingthatJ((t))is non-singular),Bdenotestheinertiamatrixincludingthe hydro-dynamic added inertia, D denotes the damping matrix, and G=diag{g11,g22,g33}denotesthesystemstatematrixofthesurge,

swayandyawmodesfortheanchorforcesandmoment.Eqs.(1) and(2)arepresentedindetailin[2].Thestarboard-portsymmetry ofshipsimpliesthatBandDhavethefollowingstructure:

B=

⎡

⎢

⎣

b11 0 0 0 b22 b23 0 b32 b33

⎤

⎥

⎦

(4a)

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Fig.2.BlockdiagramofatypicalDSPScontrolsystem. and D=

⎡

⎢

⎣

d11 0 0 0 d22 d23 0 d32 d33

⎤

⎥

⎦

, (4b)

thatis,nocouplingoccursbetweenthesurgeandthesway–yaw subsystems.

Forthissystem,Eq.(2)canbere-writtenas

˙(t)=A1(t)+A2(t)+A3(t), (5) where A1=−B−1G=

⎡

⎢

⎣

a111 a112 a113 a121 a122 a123 a131 a132 a133

⎤

⎥

⎦

, (6a) A2=−B−1D=

⎡

⎢

⎣

a211 a212 a213 a221 a222 a223 a231 a232 a233

⎤

⎥

⎦

, (6b) and A3=B−1=

⎡

⎢

⎣

a311 a312 a313 a321 a322 a323 a331 a332 a333

⎤

⎥

⎦

. (6c)

CombiningEqs.(1)and(5)inthespaceofstatevariablesobtains thefollowingstateequations:

˙1(t)=cos(3(t))4(t)−sin(3(t))5(t), (7a)

˙2(t)=sin(3(t))4(t)+cos(3(t))5(t), (7b) ˙3(t)=6(t), (7c) ˙4(t)=a1111(t)+a1122(t)+a1133(t)+a2114(t)+a2125(t) +a2136(t)+a3111(t)+a3122(t)+a3133(t), (7d) ˙5(t) =a1211(t)+a1222(t)+a1233(t)+a2214(t)+a2225(t) +a2236(t)+a3211(t)+a3222(t)+a3233(t), (7e) and ˙6(t)=a1311(t)+a1322(t)+a1333(t)+a2314(t)+a2325(t) +a2336(t)+a3311(t)+a3322(t)+a3333(t), (7f) where (t)=[1(t),2(t),3(t),4(t),5(t),6(t)]T =[x(t),y(t), (t),u(t),

v

(t),r(t)]T, (8) and (t)=[1(t),2(t),3(t)]T. (9)

Fig.2showsanoverallblockdiagramofatypicalDSPScontrol system[26].Thesetpointforthecontrolsystemistheshipposition setbytheoperator.Comparisonofthesetpointwiththeactual vesselpositionproducesanerrorsignalthatthecontrollerusesto deriveathrustdemandforrestoringthepositionoftheship.Thrust demandsobtainedforeachofthreeaxes,fore/aft(X),port/starboard (Y)andheading( )arecombinedandallocatedtoeachthruster oftheship.Theoutputofthisalgorithm,whichisdesignatedthe thrustertransform,is(t)demandtothethrusters.Byresponding tothesecommands,thethrusters(t)correctthepositionofthe vessel,whichthenclosesthecontrolloop.

Additionally,theproposedapproachofusingsector nonlinear-ityinthefuzzymodelconstructioneasilyderivesboththefuzzy setsofthepremisepartandthelineardynamicmodelofthe con-sequentpartoftheTSfuzzymodel fromthephysicalmodelof thegivennonlineardynamicsystem[27].Thisapproachensures anexactfuzzymodelconstructionforagivennonlineardynamic model[27].Theadvantageofthesectornonlinearityapproachisthe eliminationofapproximationerrorbetweentheoriginalnonlinear systemanditsTS-fuzzy-model-basedsystem[27].Therefore,this studyappliessectornonlinearityapproach[27]sothatthe non-linearequationfortheDSPSmotioncanbeexactlyrepresented asaTS-fuzzy-model-baseddynamicsystem.Assumingyawangle 3(t)= (t) varies between−/2 and /2, the exact TS-DSPSis

obtainedbyapplyingthefollowingrules: Rule1: IF z1(t) is M11 and z1(t) is M12, THEN ˙(t)= Ã1(t)+ ˜B1(t), (10a) Rule2: IF z1(t) is M21 and z2(t) is M22, THEN ˙(t)= Ã2(t)+ ˜B2(t), (10b) Rule3: IF z1(t) is M31 and z2(t) is M32, THEN ˙(t)= Ã3(t)+ ˜B3(t), (10c) Rule4: IF z1(t)isM41 and z2(t)isM42, THEN ˙(t)= Ã4(t)+ ˜B4(t), (10d) wherez1(t)=sin3(t)andz2(t)=cos3(t)arethepremisevariables;

(t)=[1(t),2(t),3(t),4(t),5(t),6(t)]Tdenotesthestatevector;

1(t)and4(t)denotetheearth-ﬁxedpositionontheX-axisand

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2(t)and5(t)denotetheearth-ﬁxedpositionontheY-axisandthe

body-ﬁxedvelocityonthey-axis,respectively;5(t)= ˙2(t),3(t)

and6(t)denotetheyawangleandyawangularvelocity,

respec-tively;and6(t)= ˙3(t),(t)=[1(t),2(t),3(t)]Tdenotestheinput

vectorwiththecontrolforcesandthemomentprovidedbythe thrustersystem,3(t)∈[−/2,/2], ˜ A1=

⎡

⎢

⎣

0 0 0 1 −1 0 0 0 0 1 1 0 0 0 0 0 0 1 a111 a112 a113 a211 a212 a213 a121 a122 a123 a221 a222 a223 a131 a132 a133 a231 a232 a233

⎤

⎥

⎦

, _A˜₂₌

⎡

⎢

⎣

0 0 0 0 −1 0 0 0 0 1 0 0 0 0 0 0 0 1 a111 a112 a113 a211 a212 a213 a121 a122 a123 a221 a222 a223 a131 a132 a133 a231 a232 a233

⎤

⎥

⎦

, ˜ A3=

⎡

⎢

⎣

0 0 0 1 1 0 0 0 0 −1 1 0 0 0 0 0 0 1 a111 a112 a113 a211 a212 a213 a121 a122 a123 a221 a222 a223 a131 a132 a133 a231 a232 a233

⎤

⎥

⎦

, A˜4=

⎡

⎢

⎣

0 0 0 0 1 0 0 0 0 −1 0 0 0 0 0 0 0 1 a111 a112 a113 a211 a212 a213 a121 a122 a123 a221 a222 a223 a131 a132 a133 a231 a232 a233

⎤

⎥

⎦

, ˜ B1= ˜B2= ˜B3= ˜B4= ˜B =

⎡

⎢

⎣

0 0 0 a311 a321 a331 0 0 0 a312 a322 a332 0 0 0 a313 a323 a333

⎤

⎥

⎦

T , M11(z1(t))=M21(z1(t))= z1(t)+ 1 2 , (11a) M31(z1(t))=M41(z1(t))= 1−z1(t) 2 , (11b) M12(z2(t))=M32(z2(t))=z2(t), (11c) and M22(z2(t))=M42(z2(t))=1−z2(t). (11d)

TheresultingTS-DSPSinferredfromEq.(10)isrepresentedas ˙(t)=

4

i=1

hi(z(t))( ˜Ai(t)+ ˜Bi(t)), (12)

inwhichz(t)=[z1(t),z2(t)]denotesthepremisevector,

h1(z(t))=M11(z1(t))M12(z2(t)), (13a)

h2(z(t))=M21(z1(t))M22(z2(t)), (13b)

h3(z(t))=M31(z1(t))M32(z2(t)), (13c)

and

h4(z(t))=M41(z1(t))M42(z2(t)). (13d)

Therefore,forallt,hi(z(t))≥0and

4

i=1hi(z(t))=1.

Theconsideredproblemisfindingboth thequadratic finite-horizon optimalfuzzy PDC controller and the quadratic finite-horizon optimal non-PDC controller (quadratic finite-horizon optimallinearstatefeedbackcontroller)fortheTS-DSPSinEq.(12)

suchthatthequadraticﬁnite-horizonintegralperformanceindex

˜J =

qt_f 0 [T(t)Q(t)+T(t)R(t)]dt

= q−1

k=0

(k+1)t_f kt_f [T(t)Q(t)+T(t)R(t)]dt (14)

isminimizedtorepresentthecontrolobjectiveofmaintainingthe systemstate,(t)isminimizedascloseaspossibletothedesired statevaluesofzerowithaminimumexpenditureofcontroleffort, tfdenotesasmalltimeintervalchosenforindependentvariablet,

qisapositiveintegerspeciﬁedbythedesigner,Qisasymmetric

positive-semideﬁnitematrix,andRisasymmetricpositive-deﬁnite matrix.

3. Quadraticﬁnite-horizonoptimalcontrollersdesignfor theTS-DSPS

FortheTS-DSPSinEq.(12),thissectiondescribeshowboththe OFAandtheHTGAareusedtofindboththequadraticfinite-horizon optimal fuzzy PDC controller and the quadratic finite-horizon optimalnon-PDC controller for the TS-DSPSthat minimizethe quadraticfinite-horizonintegralperformanceindexinEq.(14). 3.1. Quadraticfinite-horizonoptimalfuzzyPDCcontroller

ConsiderthefollowingfuzzyPDCcontrollerfortheTS-DSPS: (t)=−

4

i=1

hi(z(t))Fi(t), (15)

whereFi(i=1,2,3,4)denotethe3×6localfeedbackgainmatrices.

SubstitutingEq.(15)intoEq.(12)obtains ˙(t)= 4

i=1 4

j=1 hi(z(t))hj(z(t))( ˜Ai− ˜BiFj)(t). (16)

Thesufﬁcientconditionforquadraticstabilityis

Theorem1. When ˜B1= ˜B2= ˜B3= ˜B4= ˜B,theequilibriumofafuzzy

controlsystem(16)isgloballyandasymptoticallystableifthereexists acommonpositivedeﬁnitematrixPsuchthat

{˜Ai− ˜BiFi} T

P+P{ ˜Ai− ˜BiFi}<0, i=1,2,3,4. (17) Proof. Thedetailedprocedureofthisproofisshownin[27].

Now,assumethatallelementsof␰(t)areabsolutelyintegrable withinktf≤t≤(k+1)tf,anddeﬁne

t=ktf+ε, (18)

and

k=(ktf), (19)

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Withinktf≤t≤(k+1)tf,statevector␰(t)canthenbe

approxi-matedasthetruncatedorthogonalfunctionrepresentation

(t)=

m−1

s=0

s(k)Ts(t)= ˜(k)T(t), (20)

wheremisthenumberoftermsrequiredfortheorthogonal functions,T(t)=[T0(t),T1(t),...,Tm−1(t)]Tdenotesthem×1

ortho-gonalbasisvector,Ti(t)(i=0,1,...,m−1)denotetheorthogonal

functions,(k)s (s=0,1,...,m−1)isthe6×1coefﬁcientvector,and

˜(k)₌_[(k) 0 ,

(k) 1 ,...,

(k)

m−1]isthe6×mcoefﬁcientmatrix.

Theoret-ically,theaccuracyofapproximatesolutionsshouldincreasewith thevalueofm.Basedontheobservationsreportedin[22]and[23], m∈[4,8]generallyenablestheshiftedChebyshevfunctionandthe shiftedLegendrefunctiontoobtainsatisfactorilyaccurateresults.

AftersubstitutingEq.(15)andthetruncatedorthogonalfunction representationof␰(t)inEq.(20)intothequadraticﬁnite-horizon integralperformanceindexinEq.(14),thequadraticﬁnite-horizon integralperformanceindex ˜Jbecomes

˜J = q−1

k=0 trace

⎡

⎣

W(˜(k))T

⎛

⎝

Q+ 4

i=1 4

j=1 hi(zk)hj(zk)FiTRFj

⎞

⎠

(˜(k))

⎤

⎦

. (21)

Sincethedegreeoffulﬁllmentoftheantecedentmustbe com-putedbeforetheconsequentoutputcanbeinferred,andsincetf

isasmalltimeinterval,thevalueofhi(z(t))withinktf≤t≤(k+1)tf,

ishi(z(ktf))andisassumedlyconstantwherehi(zk)=hi(z(ktf)),and

Wdenotestheproduct-integrationmatrixoftwoorthogonalbasis vectors. Theconstantmatrix Wdepends ontheselected ortho-gonalfunctionvectorT(t).TheconstantmatricesWfortheshifted Chebyshevfunctionweregivenin[17].

Integrating Eq.(12)fromt=ktf tot=t withinktf≤t≤(k+1)tf

obtains (t)−(ktf)= 4

i=1 hi(zk)

˜ Ai

t kt_f (t)dt+ ˜Bi

t kt_f (t)dt

, (22)

wherethevalueofhi(z(t))isassumedtobehi(z(ktf))withink

tf≤t≤(k+1)tfbecausethedegreeoffulﬁllmentoftheantecedent

mustbecomputedbeforetheconsequentoutputcanbeinferred andbecausetfisasmalltimeinterval.

Byapplyingthefollowingintegralpropertyoftheorthogonal functions

t

kt_f

T(t)dt=HT(t), (23)

andapplyingEqs.(15),(19)and(20),Eq.(22)canbereformulated as ˜(k)₋_[ k,0,0,...,0]= 4

i=1 4

j=1 hi(zk)hj(zk)

_˜ Ai− ˜BiFj

˜(k)_H, (24) whereHistheoperationalmatrixofintegrationfortheorthogonal functions[17].Forexample,theconstantmatrixHobtainedbythe shiftedChebyshevfunctionis[17]

H=tf

⎡

⎢

⎣

1 2 − 1 2 0 ··· 0 0 0 1 8 0 − 1 8 ··· 0 0 0 −1 6 1 4 0 ··· 0 0 0 −1 16 0 1 8 ··· 0 0 0 . . . . . . . . . . . . . . . . . . . . . −1 2(m−1)(m−3) 0 0 ··· 1 4(m−3) 0 −1 4(m−1) −1 2m(m−2) 0 0 ··· 0 1 4(m−2) 0

⎤

⎥

⎦

. (25)

Eq.(24)canberewrittenas ˜(k)₋ 4

i=1 4

j=1 hi(zk)hj(zk)( ˜Ai− ˜BiFj)˜(k)H= ˜Q(k), (26) where ˜Q(k)₌_[ k,0,0,...,0]isa6×mmatrix.

ByapplyingtheKroneckerproduct,Eq.(26)directlyderivesthe explicitformforthecoefﬁcientmatrix ˜(k)_as

ˆ(k)₌

⎡

⎣

I6m− 4

i=1 4

j=1 hi(zk)hj(zk)(HT⊗( ˜Ai− ˜BiFj))

⎤

⎦

−1 ˆ Q(k), (27)

where I6m denotes the 6m×6m identity matrix, ˆ(k)=

0(k)T, (k)T 1 ,..., (k)T m−1

T , Qˆ(k)₌_[T k,0T,0T,...,0T] T , and ⊗

denotestheKroneckerproduct[28].Thisimpliesthat ˜(k)_can_be

obtainedfromEq.(27).

Ifonesetoflocalfeedbackgainmatrices{F1,F2,F3,F4}isgiven,

˜(k)_(k₌_0,_1,._._.,_q₋₁₎_can_then_be_calculated_from_the_following

algorithm,whichrequiresonlyalgebraiccomputation. Algebraicalgorithm. DetailedSteps

Step1: Forasmalltimeintervaltf,aspeciﬁedpositiveintegerq,

andaninitialstatevector␰(0),setk=0 Step2:Calculatehi(z(ktf))fori=1,2,3,4.

Step3:Calculate ˆ(k)_from_Eq.₍₂₇₎_.

Step4:Compute k + 1 by using k+1=((k+1)tf)= ˜(k)T((k+

1)tf).

Step5:Setk=k+1Ifk>q−1thenstop;otherwise,gotoStep2. Theabovealgorithmclearlyshowsthat ˜(k)_(k₌_0,_1,._._.,_q₋₁₎_can

bedeterminedbyspecifyingonesetoflocalfeedbackgainmatrices

{F1,F2,F3,F4};thevalueoftheperformanceindexinEq.(21)

corre-spondingtoset{F1,F2,F3,F4}canthenbecalculated.Givenanother

setoflocalfeedbackgainmatrices{F1,F2,F3,F4}Eq.(21)obtains

anotherperformanceindexvalue.Thatis,theperformanceindex valueinEq.(21)isactuallydependentonthesetoflocalfeedback gainmatrices{F1,F2,F3,F4},i.e.,

˜J =G(f111,f112,...,f436), (28)

wherefijk(i=1,2,3,4,j=1,2,3andk=1,2,3,4,5,6)denotethe

elementsofthelocalgainmatricesFi(i=1,2,3,4).Therefore,the

designproblemofthequadraticﬁnite-horizonoptimalfuzzyPDC controllerfortheTS-DSPSistooptimizefijksuchthatthe

perfor-manceindexinEq.(21)isminimized.Theproblemcanbeexpressed as

(6)

Fig.10.Controlinput2(t)responsesfortheTS-DSPSwithquadraticﬁnite-horizon

optimalfuzzyPDCcontroller(solidline)andtheTS-DSPSwithquadratic ﬁnite-horizonoptimalnon-PDCcontroller(dashedline).

= q−1

k=0

(k+1)t_f kt_f [T(t) Q(t)+ T(t)R (t)]dt, (47)

inwhichq=1000,tf=0.01,Q=diag{1,1},andR=1.

In theTS-fuzzy-model-based controlsystem in Eq.(46),the proposedapproach,whichintegratestheOFA,theHTGA,andthe presented LMI-based stabilizability condition, is applied in the designofboththequadraticfinite-horizonoptimalfuzzyPDC con-trollerandthequadraticfinite-horizonoptimalnon-PDCcontroller suchthat thereexistsasymmetric positivedefinitematrix Pto maketheLMIsinEqs.(17)and(32)hold,respectively,andsuch thatthequadraticintegralperformanceindexinEq.(47)is mini-mized.Here,theconditionsoftheevolutionaryenvironmentofthe HTGAareapopulationsizeof100,acrossoverrateof0.8,amutation rateof0.1,andagenerationnumberof10.

AfterusingtheproposedintegrativeapproachandtheLMI tool-box[44] withm=4, |fijk|≤3.5(i=1, 2;j=1and k=1, 2)andthe

Fig.11.Controlinput3(t)responsesfortheTS-DSPSwithquadraticﬁnite-horizon

optimalfuzzyPDCcontroller(solidline)andtheTS-DSPSwithquadratic ﬁnite-horizonoptimalnon-PDCcontroller(dashedline).

orthogonalarrayL8(27)fordesigningthefuzzyPDCcontroller,in

whichfijk aretheelementsofthelocalfeedbackgainmatricesFi

(i=1,2),theresultsareperformanceindex ˜J =2.8124,local feed-backgainmatricesF1=[0.45650.8698]andF2=[0.9353−0.1302],

andsymmetricpositivedeﬁnitematrixP=

10.8909 7.5414

7.5414 29.0336

. Using the existing LMI-based approach [27,34] to design the fuzzy PDC controller then obtains performance index ˜J = 4.3202, local feedback gain matrices F1=[−0.14320.0825] and

F2=[−0.52710.0767]andsymmetricpositivedeﬁnitematrixP=

0.0154 0.0044 0.0044 0.0159

.

Next,theproposedintegrativeapproachandtheLMItoolbox

[44]withm=4,|fij|≤3.5(i=1andj=1,2)andtheorthogonalarray

L4(23) areused to design the non-PDCcontroller where fij are

the elementsof the feedback gain matrix F, we obtain perfor-manceindex ˜J=2.8125,feedbackgainmatrixF=[0.64570.3823] andsymmetricpositivedeﬁnitematrixP=

13.3892 8.5140

8.5140 34.6249

. IftheexistingLMI-basedapproach[27,34]isthenusedtodesignthe non-PDCcontroller,performanceindex ˜J =3.7442,feedbackgain matrixF=[0.1653−0.0511]andsymmetricpositivedeﬁnitematrix P=

0.0283 0.0089 0.0089 0.0231

.

Theresultsoftheperformancecomparisonsbetweenthe pro-posedintegrativeapproachandtheLMI-basedapproachshowthat theproposedintegrativeapproachobtainsasmallerperformance indexcomparedtotheLMI-basedapproach.Therefore,the pro-posedintegrativeapproachismoreeffectiveforﬁndingboththe fuzzyPDCcontrollerandnon-PDCcontroller.Additionally,unlike theLMI-basedapproach,theproposedintegrativeapproachcan solvetheproblemoffeedbackgainmatrixunderconstraints. 5. Conclusions

ThisstudycombinedOFAandHTGAtosolvequadratic finite-horizon optimal controller design problems in both fuzzy PDC controllers and non-PDC controllers for the TS-DSPS. An OFA requiringonlyalgebraiccomputationwasusedtosolve TS-fuzzy-model-based feedback dynamic equations. The OFA was then integratedwithHTGAtodesignquadraticfinite-horizonoptimal controllers for a TS-DSPS under the criterion of minimizing a quadratic finite-horizonintegral performance index. Theuseof standardalgebraiccomputationintheproposedapproachof inte-gratingOFA and HTGA enableseasy computerimplementation. Accordingly,theprocedurefordesigningquadraticfinite-horizon optimal controllers for the TS-DSPS is much shorter and sim-pler.Thus,theproposedapproachfacilitatesthedesigntasksof quadraticfinite-horizonoptimalcontrollersforTS-DSPS.The illus-trativeexampleofamooredtankerconfirmedthattheproposed approachfordesigningquadraticfinite-horizonoptimalcontrollers foraTS-DSPSismoreeffectivethantheLMI-basedapproach pre-sentedby[27,34].

Acknowledgements

ThisworkwasinpartsupportedbytheNationalScienceCouncil, Taiwan,RepublicofChina,underGrantnumbersNSC 98-2221-E-037-006,NSC99-2320-B-037-026-MY2,NSC101-2221-E-151-031 andNSC101-2320-B-037-022.

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