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[8].C02incubatoranalyzersaredesignedtomonitorCO2 fortheveri白cationofincubators.However，measurableCO2 uptakeisnotforeachcel，butasumofalleafs.Fordesigna fe edback control system，it is desirable to obtain a C O2 uptake foreachcel.
Thebiologicalc10ckisaspatiotemporalproductofmany weaklycoupledindividualoscilators，definedbythemetabolic constraints of C A M [12]. Takahashi et al. showed that the sho ot apex is composed of an ensemble of coupled c10cks that influence rhythms in rots [13]. We make folowing asumptionsaboutCAMdynamics:
• CO2 uptakeforeachcelisproportionaltocelvol- ume.
• Celsareconectedbyvascularbundletoexportpho- tosyntheticproducts.Delaysinceldynamicsresults fromthesetransportdelays.
WegivethewholeCO2 uptakemodelas
Jg;&(t) = 乞 α kJ~02(t - Lk) (4)
k
whereJco2 istheCO2 uptakeofasinglecelgivenby(3)， Jg~& isthewholeCO2 uptakeofaCAMplant，kisacel number，and αゎ L k are the volume and the transfer delay of k-thcel.
Wedefineanidentificationproblemofthedelaysequence oftheCO2 uptakesasfolows:
Estimate the parameters L k's and αk'S so as to minimizetheobjectivefunctionP(t):
P(t) (Jg;&(t)-Jg;&(t)2 (5) J込(t) L&kJ~02(tーら (6)
II. MOMENTUMMETHODINOPTIMIZATIONUSING DIFFERENTIALFILTER
Real-time optimization (RTO) is a category of c1osed- lopcontrolthataimsatoptimizingprocesperformancein realtimeforsystems.WholeCO2 uptakesarecolectedon- lineusingCO2 analyzer.Weestimatetheparametersofthe wholeCO2 uptakesusingareal-timeoptimizationmethod. In particular，we propose a gradient descent method with momentumaplicabletoRTObyreplacingagradientofan objectivefunctionwithrespecttoadecisionvariablewitha ratio between the time derivatives of the objective function andthedecisionvariable.Thetimederivativesareestimated byusingtheadaptivediferentialfilter[1η.
B.AdaptiveD~万セrentiαl Filter
W e have proposed an adaptive observer to estimate the
time-deri翌日vewhoseupperboundisknown[16，] [17η].The 叫 m削叫at叫
A.GradientCαlculation Ifthedecisionparameterissingleandafunctionoftime，
d'(t) =
This optimizer can be implemented on Simulink shown in
Fig.3.
Fig.3. MomentumMethodinRea¥-TimeOptimizationon臼S1mπnu凶 n
D.NumericalExample
Asanumericalsimulation，wesearchtheglobalminimum of the multimodal objective function P(υ) created by "l-D
thegradientofthe'objectivefun-;tionwithrespecttothe decisionvariablecanbeもwritenby
df(z(t)- dtipi
dx(t) 一等辻 (7)
Thisequationalowsustoestimatethegradientinreal-time using the time derivatives of data sequences.
the削f010、川W吋Vi昭n屯1gadaptiveobserverandthe叩up白dat白el制a、明W附刊，vJ川sぽぶ: x -k(x-x)+D(t)-E(t)sgn(x-x)
。 where1>0，k>O.Theestimateof土(t)isgivenby
1
l(t) -r1(X(t)-x(t)， E(t)=Ix(t)-x(t)1
ーか
x=ム(t)= C.MomentumMethodinReal-TimeOpti・mization
d"榊
(X(T) 一 片 ) )dT
(8) Wecalthisestimatortheadaptivevelocityestimator.
De自ningP E R asaobjectivefunctionanddεR as adecisionvariable，thecontinuous-timemomentumgradient descentforsearchingaminimumoftheobjectivefunctionis givenby
d勺 )+α d'(t) = c¥1P(d(t) ) = cPd(d(t) ) (9)
where，d'(t)andd勺)arethefirstorderandsecondorder de巾 ativesofd(t)withrespecttotime，respectively.Pd(d(t) isthefirstorderderivativeofP withrespecttod.Replacing Pd(d(t) to the ratio between the time derivatives of the objectivefunctionandthedecisionvariable，wehave
d勺)=-αd'(t)-cP'(d(t)jd'(t). (10)
In the above equation，we can use the estimate P'(d(t) obtainedbytheadaptivediferentialfilterinsteadofthetime derivative，P'(d(t).Thetimederivatived'(t)canbeobtained by a dif ferential equation solver. Moreover，to improve the convergenceprope町 ，d'(t)ismodi白edas:
J