PWM-Digital-Control-of-Aeration-in-Wastewater-Treatment-Process

oxygen,moreenergyconsumptionandmorecostonehastopayaswell.Soitisnotdifficulttomaketheconclusionthataerationprocessshouldbecontrolledaccuratelyinordertosavemoreenergy.Inthissection,weutilizethelinearquadraticregulator(LQR)methodtodesigntheanalogcontrollawinPAMform,andthenthePAMcontrollawistransformedtoaPWMdigitalcontrollawusingthedigitalredesigntechnique[8].TheLQRperformanceindexis0{()()()()}TTJXtQXtutRutdt∞=+�(8)where0Q≥and0R>areweightingmatricesandthecorrespondingdesignedanalogcontrollawincontinuous-timeformis()()()CCututKXt==−(9)where1TCttKRCP−=.tPisthesolutionofthefollowingRiccatiequation10TTttttttttAPPAPCRCPQ−+++=(10)Asthecontrollawin(9)iscontinuous-time,soitisunpracticaltoputitintousesincethemotorspeedwillbetime-varyingimposedbytheanalogcontrollawin(9).ThePWMcontroller,whichproducesaseriesofdiscontinuouspulseswithafixedamplitudeandvariablewidth,hasbecomepopularinindustryforon-offcontrolofDCpowerconvertersandsteppermotors.Hence,wefocusonthedigitalredesignapproachforthedevelopmentofconvertingthePAMcontrollawtothePWMcontrollaw,whichtakesintoaccounttheinter-samplebehavioroftheclosed-loopsampled-datasystem[8].3.2PWMDigitalControlLawDesignThePWMdigitalcontrollaw,whichisformulatedfromthePAMdigitalcontrolusingtheChebyshevquadraturemethod[3],isdescribedasfollows.()()dddutKXkT=−�1()()dCCCnKKATGI−=−(11)1TCttKRCP−=whereTisthesamplingperiod.ThegraphicalillustrationofthePAMandPWMcontrollerinputsareshowninFig.2.ThecorrespondingPWMcontrolledsampled-datasystemcanbedepictedas()()()wwwXtAXtBut=+�(12)wherethePWMcontrollawis_0,(),0,kMwkkkkkkTtkTutukTtkTkTtkTTτττδτδ≤<+���=+≤≤++��++<<+��(13)_()/MkdTukTuτ=,()/2kkTδτ=−andTissamplingperiod.Mu()dukTkTkkTτ+kkkTδτ++kTT+Fig.2.IllustrationofPAMandtheequivalentPWMcontrolinputs4SIMULATIONS4.1ControllerDesignTodemonstratetheeffectivenessoftheproposedmethod,weconsideranillustrativeexamplewiththeprocessmodeldepictedin(3)-(5)andthesystemparametersarelistedinTable1.Thesamplingtimeinoptimallinearmodel(7)ischosenas12min.TheinflowratecurveisgiveninFig.