Nonlinear Dynamic Lettuce Growth Model: Parameter Selection andEstimation for NLimited Experiments
Ilya Ioslovich, Marco Ivan RamirezSosa Moran, and PerOlof Gutman
Abstract
—Regulations of European Commission have imposed upper limits of nitrate in lettuce due to possible healthhazards for consumers. These nitrate limits create problems forgrowers in Northern countries. Therefore an effective modelingtool for lettuce growth is necessary. A parameter estimation fora nonlinear threestate model for Nlimited lettuce growth ingreenhouses is presented in this paper. The variable structuralnitrogen concentration and water content are included in themodel. The experimental data collected for identiﬁcation doesnot allow to readily estimate all parameters of the model, and arecently proposed Dominant Parameter Selection (DPS) method[8], [9] is used to select the set of parameters to be effectivelyestimated. The calibrated model was perfectly ﬁtted to existingexperimental data.
I. I
NTRODUCTION
High concentrations of nitrate in leafy vegetables such aslettuce constitute a health hazard. Concerned for consumersafety, the EC issued a directive (194/97), deﬁning maximumlevels for nitrate in fresh vegetables, including lettuce andspinach. At present, low light levels in winter often make itdifﬁcult for growers to meet the required standard. The aimof decreasing winter nitrate levels in crop plants has been thesubject of research already for decades. However, a generalmethod, incorporating both accurate control of the internalnitrate concentration and an optimal production managementstrategy, have not been available.An useful approach was suggested over a decade agoby [1], showed that the nitrate concentration in lettuce wasnegatively correlated with the level of soluble carbohydrates.Thus, increased carbohydrate concentrations, associated withstimulated photosynthesis at higher light levels, led to lowernitrate concentrations in lettuce.This work was generalized in [12] assumed that thepool size of nonstructural carbohydrates (including solublesugars) is determined by the balance between source activity(supply by photosynthesis) and sink activity (demand bymaintenance and growth). In view of the strong negativecorrelation between nitrate and carbohydrate contents, theplant is assumed to posses a regulation mechanism whichadjusts the nitrate concentration in its cells to the ﬂuctuatingcarbohydrate level, in order to meet the turgor requirement.
This work was supported by EC Project FAIR6CT984362 (NICOLET).The authors acknowledge the work of M. Warmenhoven, who conducted theexperiments, and useful discussions with I. SeginerI. Ioslovich and P.O. Gutman are with Faculty of Civil and EnvironmentalEngineering, TechnionIsrael Institute of Technology, Haifa 32000, Israel
agrilya,peo@tx.technion.ac.il
M.I. RamirezSosa M. is with Departamento de Ingenieria Mecanica yMecatronica Tecnologico de Monterrey Campus Estado de Mexico, Mexico
miramire@itesm.mx
According to this view, any change in growing conditionswhich leads to an increase of source activity relative tothe sink activity, should result in accumulation of solublecarbohydrates, thus diminishing the need for nitrate as acellular
osmoticum
. Hence, the scope for controlling nitrateaccumulation in vegetables could be considerably largerthan previously thought. Experiments conﬁrm that not onlyincreasing the light level, but also a high atmospheric CO2concentration, low temperature and growth reduction througha limitation of Nnutrition result in lower levels of nitratein lettuce. As expected, decreased concentrations of nitratecoincided with increased soluble carbohydrate levels in allcases. A dynamic model, formulated on the basis of thisnew concept [12], was able to simulate published results of seasonal effects on the nitrate concentration in lettuce [4]with reasonable accuracy. Optimal control policy includingreduction of Nsupply was presented in [10], [5]. Modelingof crop quality is a relatively new ﬁeld, in which there isconsiderable potential for progress. Development of a genericmethod to accurately control an important quality aspectof greenhouse vegetables, i.e. internal nitrate concentration,constitute a signiﬁcant contribution to scientiﬁc advancementin this ﬁeld. The EC NICOLET project aimed at furtherdevelopment of the dynamic model of nitrate accumulationin lettuce. Within the scope of the project, the model wasstudied for different climatic zones of Europe. Independentset of experiments was carried out in UK and describedin [2]. It was demonstrated that the model can provide thenutritional decision support for fertilizer. The validation of the model for practical greenhouse conditions has been a veryimportant task. In particular, low Nsupply experiments werecarried out in the Division of Plant Growth and Developmentof the Research Station for Floriculture and GlasshouseVegetables, Aalsmeer, The Netherlands. These experimentsrevealed several phenomena that were not been successivelypredicted by the basic model [12]. Among them are essentialincrease in dry matter content and decrease in Nitrogen instructure (reduced Nitrogen) for severe limitations in Nsupply. To cope with this problem an extension of the model[15] was designed, where the water uptake was assumed tobe proportional to N ﬂow into the structure. This model hasto be calibrated with existing experimental data, which is thecontents of the present paper. We note that the alternativeextension of the model was presented in [14], and [11].
A. Model for lettuce growth
The model of the lettuce growth in a greenhouse with Nlimitation, described in detail in [15], is a multiinput multi
Proceedings of the44th IEEE Conference on Decision and Control, andthe European Control Conference 2005Seville, Spain, December 1215, 2005
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output nonlinear ODE system whose states are the molarmass per
m
2
[
ground
]
of C in the vacuoles,
M
Cv
, and inthe structure of the plant,
M
Cs
, and N in the structure,
M
Ns
.The inputs are the greenhouse climate variables: lightintensity (PAP),
I
, carbondioxide concentration in the air,
C
Ca
, and the air temperature,
T.
The measured outputs are fresh matter,
F
M
,
(
g/head
)
,
dry matter,
D
M
,
(
g/head
)
,
Nitrate concentration,
N
it
,
(
mg
[
NO
3
/kg
[
FW
]])
,
and reduced Nitrogenconcentration,
N
red
(
g/g
[
DW
])
.
The exchange ﬂuxes of C and N between each twocompartments, the atmosphere to the vacuoles and fromvacuoles to the structure, (subindexes
a,v
and
s
respectively)are represented by the following nonlinear functions:
F
Cav
=
p
{
I,C
Ca
}
f
{
M
Ns
}
h
p
{
C
Cv
}
,F
Cvs
= min
F
DCvs
,F
SCvs
,F
Nvs
=
h
N
{
C
Cv
}
r
N
F
Cvs
,
(1)where the concentration
C
Cv
is deﬁned below. The demandﬂux,
F
DCvs
,
and the real supply ﬂux,
F
SCvs
,
of C from thevacuole compartment to the structure, are deﬁned as follows:
F
DCvs
=
g
{
T
}
f
{
M
Ns
}
h
g
{
C
Cv
}
,F
SCvs
=
β
N
F
SNnv
+
β
C
(
F
Cav
−
F
Cm
)
λ
Π
V
+
B
C
(1 +
θ
) +
h
N
{
C
Cv
}
r
N
(
β
N
+
µ
Π
V
)
The ﬂux
F
SNnv
for the nonlimited N supply is adjusted tothe demand for carbon, and for the Nlimited case the rate of growth is adjusted to this ﬂux, as deﬁned in [13]. The ﬂuxfor maintenance respiration is given by
F
Cm
=
f
{
M
Ns
}
e
{
T
}
,
where the basic internal functions of the plant are modeledas:
p
{
I,C
Ca
}
=
εIσC
Ca
εIσ
+
C
Ca
,e
{
T
}
=
k
exp(
c
(
T
−
T
∗
))
,g
{
T
}
=
me
{
T
}
,f
{
M
Ns
}
= 1
−
exp
−
aM
Ns
r
N
.
(2)For more details on these functions see [13]. The states of the model must hold the algebraic restrictions,
Γ
Nv
+ Γ
Cv
= 1;Γ
Nv
=
β
N
C
Nv
Π
V
,
Γ
Cv
=
β
C
C
Cv
Π
V
;
C
Nv
=
M
Nv
V
V
, C
Cv
=
M
Cv
V
V
;
V
V
=
µM
Ns
+
λM
Cs
.
(3)The
V
V
represents the volume of the plant, and the parameter
µ
is responsible for the variable water content. It is animportant element of the extended model [15]. These algebraic relationships involve some of the parameters, thus thoseparameters will also have restrictions. Initially all parametersshould be strictly positive and for some cases, such as
b
p
,b
g
and
b
N
must be less than 1 as will be seen later. Thus wehave to use a constrained optimization procedure to estimatethe parameters.Using these notations the differential equations of themodel are:
dM
Cv
dt
=
F
Cav
−
F
Cm
−
(1
−
θ
)
F
Cvs
,dM
Cs
dt
=
F
Cvs
,dM
Ns
dt
=
F
Nvs
(4)There are three different inhibition factors used in themodel, and related to photosynthesis, growth and the Nuptake. They are deﬁned respectively as:
h
p
(Γ
Cv
) =11 +
1
−
b
p
1
−
Γ
Cv
s
p
,h
g
(Γ
Cv
) =11 +
b
g
Γ
Cv
s
g
,h
N
(Γ
Cv
) =11 +
1
−
b
N
1
−
Γ
Cv
s
N
,
(5)Due to the deﬁnition of
b
g
, instead of having a nominalparameter between
0
.
5
and
1
, as in the case of
b
p
and
b
N
,
we have a positive nominal value less than
0
.
5
, so in orderto analyze its sensitivity we use the transformed parameter
b
g
= (1
−
b
g
)
which is also in the range from
0
.
5
to
1
.Expressions for the outputs can be found in [13] or [15].
B. Experimental data features
This model has to estimate the measurements of FreshMatter,
F
M
(
g/head
)
,
Dry Matter,
D
M
(
g/head
)
,
Nitrate,
N
it
(
mg
[
NO
3
/kg
[
FW
]])
,
Nitrogen,
N
red
(
g/g
[
DW
])
,
taken from the experiments carried out in Aalsmeer in 1999,with four treatments: Control (abundant
N
supply),
N
supply
12%
of the growth demand
N
,
6%
and
0%
, respectively, during two periods, JulyAugust (
J

A
) and SeptemberOctober(
S

O
), denoting (
2
nd
line) and enumerating (
3
rd
line) themrespectively in (6) and (7) as:
J

A
:
Cntrl 12% 6% 0%
C

1 12

1 6

1 0

11 2 3 4
(6)during the ﬁrst period of July to August and
S

O
:
Cntrl 12% 6% 0%
C

2 12

2 6

2 0

25 6 7 8
(7)for the second one from September to October. In the ﬁrstexperiment the four treatments taken together and consideredas one measured experiment will be denoted as
J
−
A
and enumerated as
9
. In the second experiment the fourtreatments taken together and considered as one measuredexperiment will be denoted as
S
−
0
and enumerated as
10
.The eight treatments of two experiments will be denoted as
J
−
O
and enumerated as
11
. Each variable was measured
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ﬁve times during each one of the experiments
1
−
8
: theinitial day, the 4th (in the case of
S
−
O
on the 3rd), the7th, the 14th and the 21st days. Some of these treatmentswere carried out using different addition rates of nutrientsfrom those intended. More details on the experiments can befound in [3].II. M
ETHODOLOGY OF SENSITIVITY AND CORRELATIONANALYSIS
The objective is to ﬁnd the optimal set of parameterswhich minimizes a performance index of the model that isa function of the errors between the measurements and theoutputs of the model. For each treatment in the experimentsof Aalsmeer we have ﬁve sampling instants where the fouroutputs are measured with different sampling periods, three,four and seven days.In this case, due to the existence of a limited set of experimental data it is not possible to estimate all theparameters of the model, and it is required to ﬁnd subsetof parameters that can be readily estimated. This will bedone using the approach introduced by [9] using the Fisherinformation matrix to study the sensitivity of the outputs andstates of the model on each parameter and the correlationbetween each pair of them. The condition number of theFisher matrix related to the complete set of parameters hasa very large value, so the matrix is nearly singular, dueto the almost zero sensitivity to some parameters in sometreatments of periods of the experiments. Another reasonis the high correlation between different parameters. Withthis methodology the set of most sensitive and uncorrelatedparameters is selected to be estimated and the computationalburden is avoided.
A. Sensitivity and correlation analysis
In order to ﬁnd the optimal set of parameters (the costfunction will be deﬁned later) for the model (4), it isnecessary to ﬁnd the maximal set of parameters to whosevariations the outputs of the model are most sensitive, andeffects are uncorrelated one to the other. The methodologyto be used is the one introduced by [9] which will be shortlydescribed in this section.Usually in the studies on plant growth models the relativesensitivities are used (see [6], [7]) in order to equalize thedata related to different plant ages, and also to eliminatethe inﬂuence of units. For the outputs of the model, whichare nonlinear functions of the three states, which in turn arenonlinear functions of the parameters of the model,
Y
=
Y
{
M
Cv
,M
Cs
,M
Ns
}
=
Y
1
{
P,t
}
Y
2
{
P,t
}
Y
3
{
P,t
}
Y
4
{
P,t
}
(8)where
P
=
p
1
p
2
···
p
,
(9)we can ﬁnd an approximation of the relative sensitivity oneach parameter of the model
p
j
∀
j
= 1
,...,.
For each output
Y
i
{
P,t
}
;
i
= 1
,...,
4
, we deﬁne itsrelative sensitivity to the vector of parameters in the instant
t
as follows:
y
i
{
t
}
=
P Y
i
{
P,t
}
∂Y
i
{
P,t
}
∂P .
(10)To ﬁnd a numerical approximation of matrix
y
i
, we ﬁrstapproximate the partial derivatives of the output
Y
i
withrespect to each parameter
∂Y
i
{
P
}
∂p
j
≈
12
Y
i
{
P
+∆
p
j
}−
Y
i
{
P
}
∆
p
j
+
Y
i
{
P
}−
Y
i
{
P
−
∆
p
j
}
∆
p
j
,
where
P
+ ∆
p
j
=
p
1
p
2
···
(1 + ∆)
p
j
···
p
.
For several treatments (
i.e.
J
−
A
,
S
−
O
, and
J
−
O
)the matrices
y
i
have to be vertically concatenated. In sucha way, we repeat the process for every parameter in orderto construct the matrix
y
i
of relative sensitivities of the
i
th output to all parameters. In the matrix
y
i
the column
j
corresponds to the parameter
j,
and the row
k
correspondsto the measurement instant
k
.With the relative sensitivity matrix,
y
i
,
we can run thefollowing procedure for the
i
th output:1) Construct a sensitivity matrix
y
i
as shown above.2) Find the diagonal values of the Fisher informationmatrix
F
i
= (
y
i
)
T
y
i
, sort the parameters according tothe corresponding diagonal value of the Fisher matrix.3) Find the eigenvalues of the Fisher information matrix
F
i
.4) Select an appropriate maximal threshold conditionnumber
α
1
for the Fisher matrix.5) Calculate the normalized sensitivity matrix
y
iN
=
y
i
/
(
y
i
)
2
and construct the modiﬁed Fisher matrix
F
iN
= (
y
iN
)
T
y
iN
6) The offdiagonal elements of the modiﬁed Fisher matrix indicate the pairs of parameters that are correlated.Select the maximum threshold
α
2
for the correlationcoefﬁcient.7) Select the
R
parameters which correspond to thehighest diagonal values of the Fisher matrix. Take off each parameter that has correlation coefﬁcient with thepreviously selected parameters larger than
α
2
. Thentake the next candidate parameter from the sorted list.Enlarge
R
until upper threshold
α
1
can be kept.8) Choose these
R
parameters and go to step 1 to proceedwith the next output.The algorithms for calculating matrices
y
i
, and the selection of parameters were implemented in
MATLAB.
To analyze the four outputs together their sensitivity matrices
y
i
must be summed with appropriate weighting coefﬁcients.
B. Selection of the parameters to be estimated
In the experiments of Aalsmeer the following variableswere measured:
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TABLE IO
UTPUT MEASUREMENTS
Nitrate
N
it
ppm (mg [NO
3
] / kg[FW])Dry Matter
D
M
g / headFresh Matter
F
M
g / headReduced N
N
red
g / g[DW]In this section we present the results obtained whenapplying the sensitivity analysis to each of the correspondingoutputs of the model. The parameters to be used as thevector of parameters,
P,
are:
ε,σ
(photosynthesis),
m,k,θ,c
(respiration and growth),
a
(growth and photosynthesis),
r
N
,λ,µ,β
N
,β
C
(composition),
s
p
,s
g
,s
N
,b
p
,b
g
,b
N
(inhibition functions),
M
Cv
0
,M
Cs
0
,M
Ns
0
(initial conditionsin two experiments). The nominal values are taken as:
TABLE IIN
OMINAL
P
ARAMETERS
ε
= 0
.
035
σ
= 120
.
96
m
= 13
k
= 0
.
0216
c
= 0
.
0693
a
= 1
.
7
r
N
= 0
.
12
λ
= 9
.
7
e
−
5
µ
= 5
e
−
3
θ
= 0
.
3
β
N
= 6
β
C
= 0
.
61
s
p
= 10
s
g
= 10
s
N
= 40
b
p
= 0
.
95
b
g
= 0
.
2
b
g
= 0
.
8
b
N
= 0
.
88
M
1
Cv
0
= 0
.
061
M
1
Cs
0
= 0
.
251
M
1
Ns
0
= 0
.
034
M
2
Cv
0
= 0
.
031
M
2
Cs
0
= 0
.
255
M
2
Ns
0
= 0
.
033
Now we present the sensitivity analysis for each outputin the different experiments, which was carried out using arecursive function implemented in
MATLAB
which followsthe algorithm presented above.Using the methodology for each matrix
y
i
related to theeach treatment and each output respectively, with chosenmaximum condition number of the resulting Fisher matrixof
8
, and a maximum value of correlation between each pairof parameters of
0
.
95
,
the sorted uncorrelated parametersare presented in the following tables where the order of the parameters related to the diagonal values of the Fishermatrix (representing the most sensitive to the less sensitiveparameters) are given by the numbers in the boxes as:
1
,
2
,....
Here the number
1
represents the highest valueof the diagonal. In each table the last column contains thecondition number of the ﬁnally selected Fisher submatrixand (in the brackets) the maximum value of correlationbetween all pairs of parameters in the ﬁnal obtained set.
C. Sensitivity of the outputs on the parameters
First we consider the matrices
y
i
separately for each output
i
= 1
,...,
4
. The most sensitive selected parameters for eachoutput are shown in Table III:
TABLE IIIM
OST SENSITIVE PARAMETERS FOR EACH OUTPUT
b
p
b
N
b
g
µ
Cond.
N
it
1 1
D
M
1 2 4.3(0.26)
F
M
2 1 2.9(0.09)
N
red
1 1Here one can notice that only one or two parameters canbe estimated if only one of the outputs is available. To seethe differences between the data obtained from both periods,
J
−
A
and
S
−
O,
we separate the matrices
y
i
in two parts,and the results are shown in the Table IV for the ﬁrst fourJulyAugust, (
J
−
A
) treatments, and in the Table V forthe second four SeptemberOctober, (
S
−
O
) treatments. Theresults are very similar to the previous analysis.
TABLE IVM
OST SENSITIVE PARAMETERS
,
J
−
A
µ b
p
b
N
Cond.
N
it
1 1
D
M
1 1
F
M
2 1 2.6(0.42)
N
red
1 1
TABLE VM
OST SENSITIVE PARAMETERS
,
S
−
O
µ b
p
b
N
b
g
Cond.
N
it
2 1 2.0(0.28)
D
M
1 2 3.1(0.37)
F
M
1 1
N
red
1 1Thus we can conclude that it is important to include theinformation given by the all measurements.When we construct a sensitivity matrix by combiningmore than one output with a given number of treatments,each single matrix corresponding to each output shouldbe normalized, that is to say, the inverse of the absolutevalue of the maximum element of each matrix must bemultiplied every element in matrix. In this way the ﬁnalsensitivity matrix will have elements maximum value
1
andthe information given by the relative sensitivity matrices istaken into account in the same way for the four outputs.Such multipliers indicate how sensitive an output is to theparameters compared to other outputs, thus these values willbe referred as
weights
related to each output and will be usedin the performance index for the optimization procedure inthe parameter estimation.Let us consider the hypothetical case that one of theoutputs was not measured. Utilizing the eight treatments and
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the ﬁve sampling values, we construct the augmented matrixof sensitivities. Results for the different triplets of outputsare shown in Table VI.
TABLE VIM
OST SENSITIVE PARAMETERS FOR EACH TRIPLET OF OUTPUTS
b
p
b
N
b
g
µ
Cond.
N
it
D
M
N
red
2 1 5.5(0.69)
N
it
D
M
F
M
1 2 3 3 5.2(0.67)
N
it
F
M
N
red
2 1 4.3(0.62)
D
M
F
M
N
red
1 3 2 2.64(0.17)Here the augmented matrix is formed by all sensitivitymatrices of the outputs selected. For each treatment, usingthe enumeration of (6)(7), we have the following results:In each box, with an integer in it, the integer representsthe order number of the parameter from the diagonal of the Fisher matrix, and the condition number of the Fishermatrix, last column, was set to be less than
8
. The maximalcorrelation coefﬁcient is shown in the brackets.The results for all four outputs and all treatments areshown in the Table VII. Enumeration of the experimentsaccording to the description in Subsection (1. B) above isshown in the ﬁrst column.
TABLE VIIM
OST SENSITIVE PARAMETERS FOR THE FOUR OUTPUTS ON DIFFERENTTREATMENTS AND PERIODS
.
ε β
N
b
p
b
N
b
g
µ M
0
Ns
r
N
Cond.
1
3 1 2 5.6
2
1 1
3
1 2 1.7
4
1 2 2.4
5
3 2 1 4.0
6
1 3 2 6.2
7
2 1 2.9
8
3 1 2 2.6
9
1 2 5.6
10
1 2 3 1.9
11
1 2 3 6.8In the last row it can be seen that three parameters canbe extracted to ﬁt the four outputs for the experimental dataavailable. The parameter
µ
could be included if the thresholdfor the condition number of the Fisher matrix is allowed tobe higher than
8
.III. P
ARAMETER ESTIMATION
The optimal set of parameters for the model depends onthe error function used as a criterion and on the nominalvalues of the parameters. This error function (performanceindex) was chosen as a quadratic logarithmic function of theerror between (weighted) output measurements and estimatedoutputs by the model, as it is usual for plant growth models.The methodology given in [9] makes it possible to analyzeeach measured output, realized treatment, and weather conditions from a nonlinear model, in such a way that enablesthe selection of those outputs of the real system whose measurement values would give the possibility to extract a highernumber of parameters, or even to design experiments forthe actual system and estimate the most sensitive parametersfor the purposes of optimal control design. In our modelwe found that all four output measurements should be usedto estimate a maximal set of parameters. Combining thesensitivity matrices of the four outputs as in [9], includingthe eight treatments and the ﬁve samplings, the following setof three parameters, using the methodology described before,was found to have the highest sensitivity values with an upperbound of the condition number of the Fisher matrix of
8
andan upper bound for the correlation coefﬁcient of
0
.
95
:
P
n
=
b
p
b
N
b
g
Here the associated Fisher submatrix is
F
=
⎡⎣
12
.
9
−
7
.
4 1
−
7
.
4 8
.
1
−
0
.
21
−
0
.
2 5
.
9
⎤⎦
,
the condition number is
6
.
83
, and eigenvalues are
2
.
69
,
5
.
87
,
18
.
35
.
The modiﬁed Fisher matrix for the correlation valuesbetween the parameters is
F
N
=
⎡⎣
1
−
0
.
7 0
.
1
−
0
.
7 1
−
0
.
030
.
1
−
0
.
03 1
⎤⎦
.
Scaling weights related to each output in each period
W
i
=1max
x,y
{
Y
i
(
x,y
)
}
with
x,y
coordinates of elements of the matrix
y
i
,
werefound to be:
W
D
M
F
M
N
red
N
it
J

A
1
/
3
.
1 1
/
2
.
0 1
/
3
.
5 1
/
21
.
2
S

O
1
/
8
.
2 1
/
3
.
9 1
/
10
.
1 1
/
13
.
5
These values were used to normalize the sensitivity matricesfor the correspondent outputs. The optimal set of estimatedparameters,
P
o
,
which minimizes the performance index wasfound to be:
b
p
= 0
.
99
b
N
= 0
.
965
b
g
= 0
.
215
.
If the dry matter content,
D
M
F
M
,
would have been used insteadof the dry matter,
D
M
,
the results are very similar:
b
p
= 0
.
986
b
N
= 0
.
957
b
g
= 0
.
22
b
g
= 0
.
78
.
For both cases the optimal estimated values of the parameters remain “near” the nominal ones. The ﬁtting of the
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