Optimal design of fractional order linear system with stochastic inputs/parametric uncertainties by hybrid spectral method

ContentslistsavailableatScienceDirect

Journal of Process Control

jo u r n al h om ep ag e :w w w . e l s e v i e r . c o m / l o c a t e / j p r o c o n t

Optimal design of fractional order linear system with stochastic inputs/parametric uncertainties by hybrid spectral method

Pham Luu Trung Duong, Moonyong Lee

SchoolofChemicalEngineering,YeungnamUniversity,Gyeongsan,RepublicofKorea

a r t i c l e i n f o

Articlehistory:

Received20November2013 Receivedinrevisedform14June2014 Accepted6August2014

Availableonline10September2014

Keywords:

Blockpulsefunction Fractionalcalculus Operationalmatrix Stochasticcollocation Polynomialchaos

a b s t r a c t

Thispaperreportsthedesignofafractionallinearsystemunderstochasticinputs/uncertainties.The designmethodswerebasedonthehybridspectralmethodforexpandingthesystemsignalsoverortho- gonalfunctions.Theuseofthehybridspectralmethodledtoalgebraicrelationshipsbetweenthefirstand secondorderstochasticmomentsoftheinputandoutputofasystem.Thespectralmethodcouldobtain ahighlyaccuratesolutionwithlesscomputationaldemandthanthetraditionalMonteCarlomethod.

Basedonthehybridspectralframework,theoptimaldesignwaselaboratedbyminimizingthesuitably definedconstrained-optimizationproblem.

©2014ElsevierLtd.Allrightsreserved.

1. Introduction

Inrecentyears,fractionalcalculushasattractedconsiderable attentionandisbecomingincreasinglypopularbecauseofitsprac- ticalapplicationsinarangeofscienceandengineeringfields[1–5].

Thisisbecausemathematicalmodelsbasedonfractionalderiva- tivescandescribeavarietyofnaturalphenomena,suchasflexible structures[6], anomaloussystem[7],and viscoelasticmaterials [8].Ontheotherhand,fractionalordersystemsareoftenstud- iedusingmodelswithfixeddeterministicparametersandinputs.

Moreover,theinput/parametersofthesemodelsareuncertaindue totheinherentvariabilityand/orincompleteknowledge.There- fore,thedevelopmentofmethodscapableofdesigningfractional ordersystemswithuncertaintiesisnecessary.

Thewell-knownMonteCarlo(MC)methodisatypicalapproach forsimulatingstochasticmodels[9,10].Thismethodinvolvesthe generationof independentrealizations ofrandom inputs based ontheirprescribedprobabilitydistribution.Foreachrealization, thedataisfixedandtheproblembecomesdeterministic.Solving the multiple deterministic realizations builds an ensemble of solutions, i.e. the realization of random solutions, from which

∗ Correspondingauthorat:SchoolofChemicalEngineering,YeungnamUniversity, Gyeongsan712-749,RepublicofKorea.Tel.:+82538103241;fax:+82538113262.

E-mailaddress:mynlee@yu.ac.kr(M.Lee).

statisticalinformationcanbeextracted,e.g.meanandvariance.

Thisapproachissimpletoapply,involvingonlyrepeateddetermi- nisticsimulations,buttheconvergenceisslowandlargenumbers ofcalculationsarerequired.Forexample,themeanvaluestypically convergeas1/√

M,whereMisthenumberofrealizations.

Generalizedpolynomialchaos(gPC)[11–13]isamorerecent approachforquantifyingtheuncertaintywithinsystemmodels.

Ontheotherhand,tosimulatestochasticsystemsusingthegPC method, the random inputs of many systems involve random processesapproximatedbytruncatedKarhunen–Loeve(KL)expan- sion,andthedimensionalityoftheinputdependsonthecorrelation lengths ofthese processes.For an inputwitha low correlation length(idealwhitenoise),thenumberofdimensionsrequiredfor anaccuraterepresentationcanbelarge,whichincreasesthecom- putationaldemandsubstantiallyusingthegPCmethod.

Arecent study[14]introduceda hybridspectralmethodfor quantifyingtheuncertaintiesinsingleinputsingleoutput(SISO) fractional order systems. Thispaper extends the framework in Ref.[14]fortheoptimaldesignofafractionalSISOsystemunder stochasticinput/parametricuncertainty.

Thispaperisorganizedasfollows:Section2brieflyintroducesa hybridspectralmethodforuncertaintyquantificationinfractional ordersystems.Section3definesthesuitableperformanceobjec- tivescoupledwiththespectralmethodforthedesignofastochastic linearfractionalsystem.Section4considersexamplesrangingfrom integertofractionalordertodemonstratetheproposedmethod.

http://dx.doi.org/10.1016/j.jprocont.2014.08.009 0959-1524/©2014ElsevierLtd.Allrightsreserved.

2. Fractionalordersystem

Thissection summarizesthe main concepts,definitions and basicresultsfromfractionalcalculus,whichareusefulforfurther developments.

2.1. Governingequationforsystemdynamics

Amongthe many formulationsof thegeneralized derivative withnon-integerorder,theRiemann–Liouvilledefinitionisused mostcommonly[15]

D^˛₀f(t)= 1

(m−˛)



dt





0

f() (t−)^1−(m−˛)

d, (1)

where(x)denotesthegammafunction;mistheintegersatisfying m−1<˛<m.

TheRiemann–Liouvillefractionalintegral ofafunctionf(t) is definedas

I₀^˛f(t)= 1

(˛)



f()

(t−)^1−˛d. (2)

TheLaplacetransformforafractionalorderderivativeunder zeroinitialconditionsisdefinedas

L{D^˛0f(t)}=s^˛F(s), (3)

whereF(s)istheLaplacetransformoff(t).

Therefore,afractionalordersingleinputsingleoutput(SISO) systemcanbedescribedbythefollowingfractionalorderdifferen- tialequation

a0D^˛₀⁰y(t)+a1D^˛₀¹y(t)+···+a_lD^˛₀^ly(t)=b0D^ˇ₀⁰u(t)

+b₁D^ˇ₀¹u(t)+···+bmD^ˇ₀^mu(t), (4) orbythetransferfunction,

G(s)= Y (s)

U(s)=bms^ˇm+···+b₀s^ˇ0

a_ls^˛l+···+a₀s^˛0 , (5) where˛_iandˇ_iarethearbitraryrealpositivenumbers,andu(t) andy(t)arethesystem’sinputandoutput,respectively.

2.2. Operationalmatricesofblockpulsefunctionforanalysisof fractionalordersystem

Blockpulsefunctionsareacompletesetoforthogonalfunctions thataredefinedoverthetimeinterval,[0,],

i=

⎧ ⎨

⎩

1 i−1

N ≤t≤ i N 0 elsewhere

. (6)

whereNisthenumberofblockpulsefunctions.

Therefore,anyfunctionthatcanbeabsolutelyintegratedonthe timeinterval[0,]canbeexpandedintoaseriesfromtheblock pulsebasis:

f(t)=␺^TN(t)C_f=



c_{fi i}(t), (7)

where ␺^TN(t)=[ ₁(t),..., _N(t)]constitutes of theblockpulse basis.Fromhere,thesubscript,N,of␺^TN(t)isdroppedoutforthe convenienceofnotation.

Theexpansioncoefficients(orspectralcharacteristics)canbe calculatedasfollows

c_fi= N



(i/N)



[(i−1)/N]

f(t) _i(t)dt. (8)

Furthermore,anyfunctiong(t1,t2)absolutelyintegrableover thetimeinterval,[0,]×[0,],canbeexpandedas

g(t₁,t₂)=





c_{ij i}(t₁) _j(t₂)=␺^T(t₁)Cg␺(t2). (9) withexpansioncoefficients(orspectralcharacteristics)of

c_ij=









[(i−1)/N]

(i/N)



[(i−1)/N]

g(t1,t2) _i(t1) _j(t2)dt1dt2. (10)

Eq.(2)canbeexpressedintermsoftheoperationalmatrix[16],

I^˛₀f(t)=␺(t)^TA˛C_f. (11)

wherethegeneralizedoperationalmatrixintegrationoftheblock pulsefunction,A˛,is

A˛=P_˛^T=



N



(˛+2)

⎛

⎜ ⎜

⎜ ⎜

⎝

f1 f2 f3 ··· fN

0 f₁ f₂ ··· fN−1

..

. . .. . .. . .. ... 0 ... ... ... f₁

⎞

⎟ ⎟

⎟ ⎟

⎠

T

. (12)

Theelementsofthegeneralizedoperationalmatrixintegration canbeexpressedas

f₁=1; fp=p^˛+1−2(p−1)^˛+1+(p−2)^˛+1 for p=2,3... (13)

Thegeneralizedoperationalmatrixofaderivativeoforder˛is

B_˛A_˛=I, (14)

whereIistheidentitymatrix.

Thegeneralizedoperationalmatrixofthederivativecanbeused toapproximateEq.(1)asfollows:

D₀^˛f(t)=␺(t)^TB˛C_f. (15)

Usingtheoperationalmatrixofthefractionalorderderivative, Eq.(4)canberewritteninthefollowingform:

A_G=(a_lD˛_l+···+a₀D˛₀)⁻¹(bmD_ˇm+···+boD_ˇo). (16) Theinputandoutputofthesystemisthuslinkedby

CY=AGCU; Y (t)=(CY)^T␺(t); U(t)=(CU)^T␺(t). (17) Closed-loop control systems normally comprise severalele- ments,suchasthecontrollerandplantinFigs.1and2,interms ofthetransferfunctionandoperationalmatrix,respectively.The operationalmatrixofaclosed-loopsystemcanbefoundusingblock diagramalgebrasimilartotheblockalgebrausedforthetransfer function[10].

Moreondetailontheoperationalmatrixwithrespecttothe differentpolynomial functionscanbefound in[16,17] andthe referencestherein.

+ +

+ - C(s) G(s)

R(s) ^Y(s)

N(s)

U(s)

Fig.1.Closed-loopcontrolsystem.

+ - CR

CY

CU pi d

A AG

Fig.2. Closed-loopcontrolsystemintermsofoperationalmatrices.

2.3. Stochasticanalysisoffractionalordersystems

ConsiderthesystemdescribedbyEq.(4)withrandomindepen- dentcoefficients,a₀,...,a_l,b₀...,bm,withthegivendistributions.

ThespectralcharacteristicsofitsinputandoutputarelinkedbyEq.

(17).Assumethattherandomforcinginputwithagivenmeanand covariancefunctionisasfollows:

MU(t)=E[U(t)]=(CmU)^T␺(t)

UU=E{[U(t1)−MU(t1)][U(t2)−MU(t2)]}=





i(t1) j(t2)cij=␺(t1)^TCK_UU␺(t2). (18)

Aftersomemanipulationoftheoperationalmatrix,thespectral characteristicsofthemeanandcovarianceofthesystem’soutput canbeexpressedas[14]

CmY =E[AG]CmU

C_YY=E[A_G{C_UU+(Cm_U)(Cm_U)^T}AG]^T−Cm_Y(Cm_Y)^T

, (19)

or,

mY(t)=␺(t)^TE[AG]CmU

YY(t1,t2)=␺(t1)^TE[AG{CUU+(CmU)(CmU)^T}AGT␺(t2)]−␺(t1)^TCmY(CmY)^T␺(t2) . (20)

Randomparameters, a_i,b_j, resultin therandomoperational matrix,A_G,inEqs.(19)and(20),anditsmomentcanbeestimated usingastochasticcollocation[14].

Whenparametersa_i,b_jaredeterministic,Eq.(20)becomes

mY(t)=␺(t)^TAGCmU

YY(t1,t2)=␺(t1)^TAG{C_UU+(Cm_U)(Cm_U)^T}AGT␺(t2)−␺(t1)^TCm_Y(Cm_Y)^T␺(t2) .

(21)

3. Optimaldesignoffractionalordersystemwith stochasticinput/parametricuncertainties (a)Shapingfilterdesignbyparametricoptimization

A shaping filteris normallyusedto shape thecovariance functionofagiveninputnoiseintotoadesirednoisewitha referencecovariance.

AssumethatthesystemisdescribedbyEq.(4),wherethe coefficientsa_i,b_j,˛_k,ˇ_larethedeterministicdesignparame- ters.Theinputisarandomprocesswithazeromeanandgiven covariancefunction.Themeanandcovariancefunctionofthe inputcanbeexpandedtoaseriesofBFF,asexpressedinEq.

(18).Therefore,themeanandcovariancefunctionfortheout- putaregivenbyEq.(21).Thecovarianceofdesiredisgivenand expandedinspectralformasfollows:

Y_DY_D(t1,t2)=E{[YD(t1)−MY_D(t1)][YD(t2)−MY_D(t2)]}

=





i(t1) _j(t2)c_ij^YDYD=␺(t1)^TCK_YDYD␺(t2).

(22)

FromEq.(21)itcanbeseenthattheoutputhasazeromean.

The design parametercanbeobtainedbyminimizing the objectivefunctionintermsofthespectralcharacteristicsofthe covariancefunctionoftheinputandoutput,

min

a_i,b_j,˛_k,ˇ_l





2= min

a_i,b_j,˛_k,ˇ_l

 

YDYD

 

2

, (23)

where2denotetheEuclideannormofamatrix.

(b)Optimalcontrollerdesignbyparametricoptimization Assume that the system is described by Eq. (4), where coefficients a_i,b_j are independentrandom variables with a givendistribution.Thesetpointinputisarandomprocesswith agivenmeanandgivencovariancefunctionasfollows:

MR(t)=E[R(t)]=(CmR)^T␺(t)

RR=E{[R(t1)−MR(t1)][R(t2)−MR(t2)]}=





i(t1) j(t2)c_ij^R=␺(t1)^TCKRR␺(t2).

(24)

Thesystemisintheclosed-loop,asshowninFig.1,witha fractionalorderPI^D^controller[18]:

C(s)=Kp+K_i

s^+K_ds^. (25)

TheparametersofthePI^D^controllerareobtainedbyopti- mizingthecostfunctionasfollows:

min

Kp,Ki,Kd,,J= min

Kp,Ki,Kd,,E



(y_d(t)−y(t))²dt

= min

Kp,K_i,K_d,,



{E[y_D²(t)]+E[y(t)²]−2E[y_d(t)]E[y(t)]}dt.

(26) whereE[y(t)²]andE [y(t)] areobtainedfromEq.(20).

Becausethevarianceisameasureofthevariabilityofaran- domprocess,alargevarianceofthesystemoutputimpliesa largedeviationfrom thenominalresponse understochastic input/parametricuncertainties.Anadditionalconstraintonthe variancecanbeusedincombinationwiththeobjectivefunction (26)

0≤t≤TmaxDy(t)=YY(t,t)≤Dmax. (27)

Fig.3.AbsoluteerrorsurfacefortheapproximatedcovariancefunctionwithEq.

(20)andtheexactdesiredcovariancefunctionforexample1.

Theupperbound,D_max,ofthevarianceofthesystemoutput canbeconsideredthemaximumallowablesensitivityofthe systemresponsetothestochasticuncertainties.

4. Examples

4.1. Example1:shapingfilterdesign

Assumetheshapingfiltercanbedescribedas

D^˛₀y(t)+ay(t)=bu(t), (28)

whereu(t)is a zeromeanrandomprocess withthecovariance function,UU(t1,t2)=ı(t1−t2),andıistheDiracdeltafunction.

Thedesiredcovariancefunctionofoutputcanbeexpressedas anOrnstein–Ulenbeckprocess:

Y_DY_D(t1,t2)=1

2(e⁻|^t1−t2| −e^−(t1+t2)); t1,t2≥0. (29) Thedesignparameters˛,a,andbareobtainedbyminimizing Eq.(23).Theresultsare˛=1.001;a=0.996;b=0.998,whichare sufficientlyclosetotheexact(analytical)result,˛=1;a=1;b=1 [19].

Theoperationalmatrixforsystem(28)is

A_G=(B˛+aI)⁻¹bI, (30)

whereIisanidentitymatrix,andB˛isanoperationalmatrixofthe derivative.

1024blockpulsefunctionswereusedforthisexample.

Fig.3showstheerrorsurfaceofthecovariancefunctiongiven byEq.(20)fortheoptimalparameterswithrespecttotheexact desiredcovariancefunctionofEq.(29).Asshowninthefigure,the levelofrelativeerrorissmallenough,<0.5%.Thissimpleexample highlightstheeffectivenessoftheproposedmethodindesigning theshapingfilter.

4.2. Example2 4.2.1. Example2a

Considersystemdescribedbythedouble-fractionalorderdif- ferentialequation,

D^3/4₀ y(t)+D¹₀y(t)=u(t), (31)

whereu(t)israndomwhite noisewithazeromeancovariance function,UU(t1,t2)=0.1ı(t1−t2).

Fig.4.Varianceofthesystemoutputforexample2aandtheabsoluteerrorofthe proposedmethod.

Fig.5.Meanandvarianceofsystemoutputforexample2bwiththeproposed controller.

Theexactvarianceofthesystemoutputcanbeexpressedas[20]

D_y(t)=²(t)= 0.1 a22



u^2(˛2−1)



ε˛2−˛1,˛2



−a₁u^(˛2−˛1) a₂



du,

(32) whereε˛2−˛1,˛2()isaMittag–Lefflerfunctionthatcanbecalculated usingMATLABfunctionmlf.m[21],a₁=a₂=1;˛₁=3/4,and˛₂=1.

1024BPFswereusedfortheapproximationinthisexample.The operationalmatrixforthissystemcanbeexpressedas

A_G=(B_3/4+B₁)⁻¹. (33)

Fig.4showsthevarianceofthesystemoutputbytheproposed methodandanalyticalvarianceinEq.(32).Thefigurealsoshows theabsoluteerrorofthevariancebytheproposedmethodwith respecttotheanalyticalresults.Theproposedmethodcanprovide accurateresultsforpredictingthevarianceofthesystemoutput.

4.2.2. Example2b

Considertheabovesysteminaclosedloopconfigurationwith aPI^D^controller.Thesetpointr(t)isarandomprocesswithunit meanandthesamecovariancefunctionasinthepreviousexam- ple.API^D^controllerisdesignforthissystembyminimizingthe objectivefunction(26)withthedesiredoutput,y_d(t)=1(determi- nisticunitstepresponse).Theobjectivefunctionisminimizedusing thepatternsearchfunctionoftheoptimizationtoolboxofMATLAB.

The result parameters of controller are K_p=0.8286; K_i=0.8137;=0.2660;K_d=0.0056;=0.9996. Fig. 5 shows themeanandvarianceofthesystemwiththedesignedcontroller.

Fig.6shows100MonteCarlosimulationsofthissystemwiththe proposedcontroller. By comparingthe variancesof the system outputs in Figs. 4 and 5, the designed controller can stabilize this stochastic fractionalorder system.The performance ofthe proposedcontrollerissatisfactory.

Fig.6. 100MCsimulationofexample2b.

4.3. Example3

TheproposedmethodwasusedtodesignaPI^controllerfor afractionalordersystemthathasbeenstudiedinthenumerical referencesonafractionalordersystem[14,22–24]:

P(s)= k₁

k₂s^0.5+1. (34)

where k₁ and k₂ are uniform random variables in the interval [0.5,1.5].ThisfractionalordermodelinEq.(34)isknownasthe Cole–Colemodel, which provides a parameterof thefractional orderthatinterpretstheunderlyingmechanismofdielectricrelax- ation[22–24].

Thesetpointinput,R(t),isaband-limitedGaussianwhitenoise processwithaunitmeanandacovariancefunctionof

RR(t1,t2)=W_B sinc



1−t₂)W_B



, (35)

withW_B=0.02 ,andthesinc()functionisdefinedas

sinc(x)=



1 for x=0 . (36)

ItcanbeseenfromFig.1thatthetransferfunctionfrommea- surement noise n(t) to output y(t) is the same as the transfer functionfromr(t)toy(t),i.e.,Y(s)=T(s)R(s)−T(s)N(s).

Letusdefineanewsignalz(t)=r(t)−n(t).Itcanthenbeshown that:

Theexpectation ¯z(t)=mz(t)ofz(t)isequaltor(t).Thecovari- anceofz(t)is

E{(z− ¯z)²}=E{z²−2z ¯z+ ¯z²}=E(z²)− ¯z²=E{(r−n)²}−r²

=E{r²+n²−2rn}−r²=E{n²} . In the above equation, the dependentof signals ontime is dropped;r(t)andn(t)areindependent.Thecovarianceofnewsig- nalisthesameasthecorrelationfunctionofmeasurementnoise.

Notethatthecovarianceandcorrelationofthemeasurementnoise isthesamebecausethemeasurementnoiseiszeromean.Thus, incombination,thedeterministicsetpointandrandommeasure- mentnoisecanbeviewedasarandomsetpointwiththemeanas adeterministicinputandthecovarianceasameasurementnoise.

A PI␭ controller was obtained by minimizing the objective function (26) with the desired output is y_d(t)=1. The search space for optimal parameter of the controller was limited to 0≤Kp≤1;0≤K_i≤40;0.9≤≤1.5 for the sake of simplicity, as inotherstudiesontheprobabilisticapproach[25,26].Theinitial controller,C0(s)=0.1817+(36.3528/s^1.216),wasobtainedfrom[24].

Theresultingcontrollerparametersare

C₁(s)=1+(40/s^0.9). (37)

Fig.7. Meanandvarianceofthesystemoutputforexample3withtheproposed andinitialcontrollers.

Fig.7showsthemeanandvarianceofthesystemoutputwith theinitialandproposedcontrollers.Fig.8showsaboundedregion for 2000 possible responses of the uncertain system with the proposedandinitialcontrollers.Thesefiguresshowthatthepro- posedcontrolleroutperformstheinitialoneundertheinfluenceof stochasticuncertainties.

4.4. Example4:systemswithinfiniteH²norm

4.4.1. Example4a

Integratorisanelementaryblockonmanycontrolsystems.Itis adequateforpresentingabasicinventorysysteminprocesscontrol suchasaliquidlevelcontrolsystem[27,28].Consideranintegral system:

G(s)= 1

s, (38)

subjecttotheinputwithazeromeanfirstorderGaussMarkov processwithacovariancefunctionof:

K_UU(t₁,t₂)=²e^{−ˇ|t1−t2|}. (39) Thevarianceofthesystemoutputcanbecalculatedanalyticallyas [29]:

DYY(t)=2²

ˇ² (ˇt−(1−e^−ˇt)). (40)

TheanalyticalresultisshowninFig.9.Theresultsandtheabso- luteerrorswithrespecttotheexactvariancebythehybridspectral (proposed)andQuasiMonteCarlo(QMC)methodsarealsoshown inFig.9.TheerrorsbyboththeQMCandproposedmethodsgrow withtimesincetheoutputvarianceincreaseswithoutboundast goestoinfinity.TheQMCmethodused5000samplesofrandom processandthecomputationaltimeneededwas117sonanIntel corei3laptopwith2GBRAM.Ontheotherhand,thecomputational

Fig.8. Boundedregionsfor2000MCsimulationsofthestochasticsystemwiththe proposedandinitialcontrollers.

Fig.9.Varianceofsystemoutputforexample4a.

Fig.10.Varianceofsystemoutputforexample4bforK=1,=1.7,=1.

timeoftheproposedmethodwith2048BPFswasonly0.73s.Since theconvergencerateoftheQMCmethodisproportional tothe magnitudeofvariance,theQMCmethodcangivebetteraccuracy atthebeginningofthetransienttimeduetothesmallvariance atthebeginning.Itcanbeseenthattheproposedmethodgives betterperformanceincomputationalload.Hence,itismoresuit- ableforthedirectdesignviaoptimizationthantheQMCmethod.

Ingeneral,theanalyticaloutputvariancecanbecalculatedonly fora few particularcases, especiallywhen a fractionalorder is involved.

4.4.2. Example4b

Considerafractionalordersystem:

G(s)= K

s+, (41)

subject toa zero mean ideal white noise witha covariance of ı(t₁−t₂).Fromthefrequency method,thesteadystatevariance oftheoutputisthesameassystemH²norm[30]

Dyss=





2= 1 2



−∞





⎧ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎨

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎩

∞ if 0<≤1/2

−K²





cot



2



sin





2 if v₌1

. (42)

Fig.10showstheexactsteadystatevarianceofsystemoutputby thefrequencymethodforK=1,=1.7,=1.Fig.10alsoshowsthe varianceofsystemoutputbytheproposedmethod.Itcanbeseen thatwhentheH²normofsystemisfinite,theproposedmethod givesaconsistentresultwiththefrequencymethodforasteady statevariance.

Fig.11showsthevarianceoftheoutputcalculatedbythepro- posemethodwithdifferentnumberofblockpulsefunctionsfor K=1,=0.4,=1.Thefrequency methodgives thesteadystate variance





2=∞byEq.(42).Itseemsthattheproposedmethod stillhasalimitationforobtainingtheaccurateresultforthiscase.

Totheauthors’knowledge,thereisnogeneralmethodavailablefor thecasewherethesystemhasafastresponseandinfiniteH²norm.

Fig.11. Varianceofsystemoutputforexample4bforK=1,=0.4,=1.

5. Conclusions

Fractionalcalculus hasalready been acceptedas a powerful toolformodelingandcontrolofcomplicatedprocess[24,30–33].

Inthiswork,anapproachfortheoptimaldesignforlinearfrac- tionalorderprocesseswithstochasticuncertaintieswasproposed.

Theuseofthehybridspectralmethodleadstoalgebraicrelation- shipsbetweenthefirstandsecondorderstochasticmomentsof asystem’sinputandoutput.Therefore,thespectralmethodcan obtainahighlyaccuratesolutionwithlesscomputationaldemand thanthetraditionalMonteCarlomethod.Hence,itismoresuit- ablefortheoptimaldesign.Theproposedmethodcanbeusedfor thesynthesiscolorednoisefromthearbitrarygivennoise.More- over, thestability of uncertainsystems can beinferred from a statisticalevolutionofthesystemoutput,androbustPI^D^ con- trollerscanbedesignedefficientlywithrespecttothestochastic uncertainties.Dependingonwhethertherobustnessornominal performanceisofprimaryconcern,differentparameterdistribu- tionsand stochastic forcing can be assumed for the stochastic uncertainties.Basedonthedifferentassumptionsofthestochastic uncertainties,theoptimalparameterswerecalculatedusingasim- plenonlinearoptimization.Integerordersystemscanbetreated asa particular case of fractionalorder systems.The simulation examplesconfirmedthattheproposeddesignmethodgivesrobust closed-loopresponsesforarangeofstochasticsystems.Extension oftheproposedtothenonlinearcaseisaninterestingissuefor futureworks.

Acknowledgement

ThisstudywassupportedbytheBasicScienceResearchPro- gramthroughtheNationalResearchFoundationof Korea(NRF) funded by the Ministry of Education, Science and Technology (2012012532).

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