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ModelingandIdentificationofCO2 Uptakesof Multi-CelsinCrasulaceanAcidMetabolismUsing MomentumOptimizationMethod
KeisukeKawasaki，HaruoSuemitsu，ShoheiUeno，TakamiMatsuo DepartmentofMechatronics
OitaUniversity
Oita，JAPAN，870-1192 Email:{v18e6002，suemit叩 eno-shohei，r凶suo}@oita-u.ac.jp
TadashiKonishi InstituteofNationalColegesofTechnology OitaNationalColegeofTechnology Oita，JAPAN，870-0152
Abstract-The mechanism of endogenous circadian photo- synthesis oscilations ofplants performing crasulacean acid metabolism (CAM) is investigated in terms of a nonlinear theoretical modeI. B1asius et aI. used throughout continuous timediferentialequationswhichadequatelyreflecttheCAM dynamics.Wehaverecentlyproposedthereconstructionmethod oftheinternalC02concentrationbyusingtheCO2uptakefrom outsideasthemeasurementdata.However，theC02uptakefor eachcelcanotbemeasuredbutthesumoftheC02uptakes ofalcels.Itisnecesarytoestimatethestatesofsinglecelto controltheplantrhythm.Inthisreport，wesuposethatthesum oftheCO2uptakescanberepresentedastheIinearcombination ofdelayedoscilationwavesofcelsandperformthesimulation viaMATLAB/Simulink.
Keywords-Photosynthesis)timedelayestimation)real-timeop- timization.
1. INTRODUCTION
Inplant，thecircadianrhythmsplayimportantrolesingene expresions，photosynthesis，growth，andmanyotherphysi- ologicalproceses.Theplantcircadianrhythrniscomposed ofalargenumberofself-sustainedcelularoscilationsthat synchronizeeachother.Preciseandecologicalcontrolofthe circadian rhythm provides a key technology for enhancing theplantgrowthinaclosedcultivationsystemwherelight- darkcyclesdi百erfromthe24-hperiod[1].Climaticextremes threaten agricultural sustainability worldwide. Crasulacean acidmetabolism(CAM)plantsshowaremarkablemetabolic plasticity for modulating noctumal and diumal C O2 uptake andhavebenidentifiedascompetitivebiomasacumulators incomparisonwithmanyC3 andC4 crops.Oneaproachto increaseplantwater-useeficiencyistointroduceCAMinto C3 crops[2].ComputationalmodelingofCAMacelerates theimprovementofCAMcropsintermsofbiomasproduc- tivity and quality-related at tributes [3]. Blasius et α1.[4]，[5] investigatedthemechanismofendogenouscircadianphoto- synthesisoscilationsofplantsperformingCAMintermsofa non1ineartheoreticalmodel.Theyusedthroughoutcontinuous time diferential equations which modes adequately reftect the CAM dynamics. By incorporating results from both a complementaryandacontinuousmembranemodel，adetailed descriptionofthemolecularmalatetransportinandoutof the vacuole through the tonoplast membrane was achieved. Theiranalysisshowedthatthemembranee汀ectivelyactsasa
Email:konishi@oita-ct.acJ
hysteresis switch regulating the oscil lations. It thus provided
amolecularbasisforthecircadianclock.Themodelshowed regularendogenouslimitcycleoscilationsthatwerestable fora、，viderangeoftemperatures，inamannerthatcomplies welwithexperimentaldata.Thenon1ineardynamicalmodel ofCAMisdiscusedfromthecontroltheoreticalviewpoint. The state-variables of the nonlinear dynamic equations are anintemalCO2concentration，amalateconcentrationinthe cytoplasm，amalateconcentrationinthevacuole，andanorder ofthetonoplastmembrane.Theinputvariablesareanextemal CO2concentration，alightintensityandatemperature.
W e have recently presented a dynamic estimator of the tonoplastorderandafuzyidentifierofthenonlinearfunction in the dynamics of the tonoplast order [6]. Moreover，we proposedthefedbackcontrolerbycontrolingtheextemal CO2concentrationandthelightintensity[7].Thoughtthese estimatorandcontrolerrequirethestatevariablesexceptfor theorderofthetonoplastmembrane，theyarenotmeasurable directly.Then，weproposedthereconstructionmethodofthe intemalCO2concentrationusingtheCO2uptakefromoutside [8] ， because the C O2 uptake can be determined the C O2 exchange which is measured using a C O2 analyzer and an incubator[9].Wehaverecentlyproposedthereconstruction methodoftheinternalCO2concentrationbyusingtheCO2 uptakefromoutsideasthemeasurementdata.However，the CO2uptakeforeachcelcanotbemeasuredbutthesumof theCO2uptakesofalcels.Itisnecesarytoestimatethe statesofsinglecelintocontroltheplantrhythrn.
In this paper，we propose the model of the sum of the CO2uptakesasalinearcombinationofdelayedoscilation wavesofcels.Next，weformulatetheparameterestimation problemofdelayedoscilationwavesofcelsinviewofreaト timeoptimization.羽市oleCO2uptakesarecolectedon-line usingCO2analyzer.Itisnecesarytoupdatethemodelofthe CO2uptakesusingareal-timeoptimizationmethodbasedon theavailablemeasurements.Weproposeamomentummethod in the gradient descent algorithm by ading the adaptive di百erentialfilterandaplytheproposedmethodtoestimate delay sequences of the C O2 uptake. Final ly， we perform thesimulationviaMATLAB/Simulinktoverifytheproposed method.