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基于EKF的BLDCM状态观测器设计
Introduction:
BrushlessDCmotors(BLDCM)arewidelyusedinvariousindustrialandconsumerapplicationsduetotheiradvantagessuchashighefficiency,compactsize,andlowmaintenance.However,theprecisecontrolofBLDCMischallengingasitrequiresaccurateestimationofthemotorstatevariables,suchasrotorposition,speed,andvoltage,whichareusuallynotdirectlymeasurable.Therefore,designingarobuststateestimatorforBLDCMiscrucialforachievingaccuratecontrolperformance.
ExtendedKalmanfilter(EKF)isapopularmethodforstateestimationofnonlineardynamicsystems,suchasBLDCM.Inthispaper,wewilldescribethedesignandimplementationofanEKF-basedobserverforestimatingthestatevariablesofBLDCM.
Systemmodeling:
ThefirststepindesigningtheobserveristomodelthedynamicsofBLDCM.Themostcommonlyusedmodelisthestate-spacerepresentation,whichcanbedescribedasfollows:
ẋ(t)=f(x(t),u(t))
y(t)=h(x(t),u(t))
Wherex(t)=[θ(t),ω(t),i(t)]Tisthestatevariablevector,u(t)=[Va(t),Vb(t),Vc(t)]Tistheinputvoltagevector,andy(t)=[θ(t),ω(t)]Tistheoutputmeasurementvector.Thestateequationscanbedefinedas:
θ̇(t)=ω(t)
ω̇(t)=-(R/L)i(t)-(K/L)ω(t)sin(θ(t))+(K/(LJ))τ(t)
i̇(t)=-(1/C)(Va(t)+Vb(t)+Vc(t))-(R/L)i(t)
WhereR,L,K,J,andCarethemotorparameters,andτ(t)istheloadtorque.
EKFdesign:
TheEKFalgorithmisusedtoestimatethestatevariablesbasedontheinputandoutputmeasurements.Thealgorithmconsistsoftwostages:predictionandupdate.Inthepredictionstage,thestateestimateispropagatedbasedonthesystemdynamics,andthecovariancematrixisupdatedbasedontheprocessnoise.Intheupdatestage,thestateestimateiscorrectedbasedonthemeasurementresidual,andthecovariancematrixisupdatedbasedonthemeasurementnoise.
TheEKFalgorithmrequiresthenonlinearityofthemodeltobelinearizedaroundthecurrentstateestimate.Therefore,theJacobianmatricesofthesystemdynamicsandobservationfunctionsneedtobecomputedateachtimestep.TheJacobianmatrixofthesystemdynamicsisdefinedas:
F(x(t),u(t))=∂f(x(t),u(t))/∂x(t)
TheJacobianmatrixoftheobservationfunctionisdefinedas:
H(x(t))=∂h(x(t))/∂x(t)
TheEKFalgorithmcanbesummari