Numerical Models for the Simulation of the Fractio

arXiv:math/0204108v1[math.OC]10Apr2002c1994,Ing.ˇLubom´ırDorˇc´ak,CSc.ThispublicationwastypesetbyLATEX.Contents1Abstract22Introduction23Definitionofthefractional-ordercontrolsystem34Numericalandanalyticalcomputationoftheunitstepresponseofafractional-ordersystem35Approximationoffractional-ordersystemwithinteger-ordersystem66Designoftheregulator67Conclusion11References1NUMERICALMODELSFORTHESIMULATIONOFTHEFRACTIONAL-ORDERCONTROLSYSTEMSˇLubom´ırDorˇc´akDepartmentofManagementandControlEngineering,BERGFaculty,TechnicalUniversityofKosiceBozenyNemcovej3,04200Kosice,Slovakiae-mail:[e-mail],phone:[phone]AbstractThiscontributiondealswiththecreationofnumericalmodelsforthesimulationofthedynamiccharacteristicsoffractional-ordercontrolsystemsandtheircomparisonwithanalyticalmodels.Wegivetheresultsofthecomparisonofdynamicpropertiesinfractional-andinteger-ordersystemswithacontroller,designedforaninteger-ordersystemasthebestapproximationtogivenfractional-ordersystem.Otheropenquestionsarepointedout,whichshouldbeansweredinthisareaofresearch.2IntroductionThestandardcontrolsystemsusedsofarwereallconsideredasinteger-ordersys-tems,regardlessofthereality.Intheiranalysisanddesign,theLaplacetransformwasusedheavilyforsimplicity.Becauseofthehighercomplexityandtheabsenceofadequatemathematicaltools,fractional-orderdynamicalsystemswereonlytreatedmarginallyinthetheoryandpracticeofcontrolsystems,e.g.[1,2,3].Theiranalysisrequiresfamiliarityofworkwithfractional-orderderivativesandintegrals[4,5,6,9,10,11,12].Byremovingtherestrictionstointeger-ordersystemsitispossibletoobtainsystemswhosepropertiesareacombinationofsystemsoftheclosestinteger-order,butalsointermediatetypesofsystems,whichbroadenstheclassofthesystemsconsiderably[1].Withdifferentfractional-ordersystemsthenotionsarisesuchasweakorstrongintegratorordifferentiator,weakorstrongfractional-typepole,orzero,withinterestingcontributiontothedynamicsofthesystem(stability,phaseshiftetc.),assomepropertiesareretained,othersareeliminated.Afractional-ordersystemcombinessomecharacteristicsofsystemsoftheorderNand(N+1).Hencebychangingtheorderasarealandnotonlyintegervaluewehavemorepossibilitiesforanadjustmentoftherootsofthecharacteristicequationaccordingtospecialrequirements.Inthiscontributionwewillanalyzedynamicpropertiesofsystemsinthetimedomainwithanemphasisonthenumericalmethodsofsimulationoffractional-ordersystems.Wewillpointoutproblemsofinadequateapproximationoffractional-order2systemswithinteger-ordersystemsandthedifferencesindynamicpropertiesofsuchsystemsinclosedcontrolsystemswithcontroller.3Definitionofthefractional-ordercontrolsystemForthedefinitionofthecontrolsystemweconsiderasimpleunityfeedbackcontrolsystemshowninFig.1.Gs(s)denotesthetransferfunctionofthecontrolsystemwhichiseitherinteger-type(Gis(s))ormoregenerallyfractional-type(Gfs(s))andGr(s)isthetransferfunctionofthecontroller,alsoeitherinteger-type(Gir(s))orfractional-type(Gfr(s)).Figure1:SimpleunityfeedbackcontrolsystemW(s)+−E(s)U(s)Y(s)-6@@-Gr(s)-Gs(s)-Forlaterpurposesconsiderafractional-ordercontrolledsystem,whichrepresentsourrealsystem,withthetransferfunctionGfs(s)=1a2sα+a1sβ+a0(1)whereαandβareingeneralreal(αβ).Inthesimulation,thecoefficientvaluesa2=0.8,a1=0.5,a0=1,α=2.2,β=0.9werechosen[10,12].Tothefractional-typetransferfunction(1)therecorresponds,intimedomain,thefractional-orderdifferentialequationa2y(α)(t)+a1y(β)(t)+a0y(t)=u(t)(2)withinitialconditionsy(β)(0)=0andy(0)=0.4Numericalandanalyticalcomputationoftheunitstepresponseofafractional-ordersystemForthenumericalcalculationoftheunitstepresponseofthefractional-ordersystem(2)weemploy,fortheapproximationofthefractionalderivativesinequation(2),therelation(3)with”shortmemory”principle,formulatedin[11]y(α)(t)≈(t−L)Dαty(t)=h−αN(t)Xj=0bjy(t−jh),(3)whereLis”memorylength”,histimestep,N(t)=minth,Lh,[z]istheintegerpartz,3bj=(−1)jαj!(4)whereαj
isbinomialcoefficient.Tocalculatebjitisconvenienttousethefollowingrecurrentrelationb0=1,bj=(1−1+αj)bj−1(5)Itfollowsfromtheestimatesderivedin[11]thatinourcasethenormederrorofsuchapproximationisδ0=y(α)(t)−(t−L)Dαty(t)M=1√LΓ(α),M=max[0,∞]|y(t)|(6)whencewehavethefollowingconstraintforthechoiceof”memorylength”L:L≥1δ20Γ2(α)(7)whereδ0isthemaximumadmissiblenormalizederrorandΓ(α)istheGammafunction.Byusingtherelation(3)wecanapproximatethedifferentialequation(2)adifferentmode.Ourapproximation[10,12]ofequation(2)indiscretetimestepstm(m=2,3,...)hasthefollowingforma2h−αmXj=0bjym−j+a1h−βmXj=0cjym−j+a0ym=um(8)ora2h−α(b0ym+mXj=1bjym−j)+a1h−β(c0ym+mXj=1cjym−j)+a0ym=um(9)Fromtheapproximation(9)wecanderive[10,12],thefollowingexplicitrecurrentrelationforthecalculationofthevaluesym(m=2,3,...)ym=um−a2h−αmPj=1bjym−j−a1h−βmPj=1cjym−ja2h−αb0+a1h−βc0+a0(10)withy0=0,y1=0,u0=0andum=1form=1,2,....Thisalgorithmdoesnotrequireiterationalcalculations,incontrasttothepro-ceduregivenin[4].Fortheanalyticalcalculationoftheunitstepresponseoffractional-ordersystems(2)weapplytheanalyticalformoftheimpulseresponseofsuchsystem[5].Byintegratingtheimpulseresponseofsuchsystem