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Identification of a Nonlinear Dynamic Biological Model Using the Dominant Parameter Selection Method

Identification of a Nonlinear Dynamic Biological Model Using the Dominant Parameter Selection Method
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  Nonlinear Dynamic Lettuce Growth Model: Parameter Selection andEstimation for N-Limited Experiments Ilya Ioslovich, Marco Ivan Ramirez-Sosa Moran, and Per-Olof Gutman  Abstract —Regulations of European Commission have im-posed 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 three-state model for N-limited 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 non-structural 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 FAIR6-CT-98-4362 (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, Technion-Israel Institute of Technology, Haifa 32000, Israel agrilya, M.I. Ramirez-Sosa M. is with Departamento de Ingenieria Mecanica yMecatronica Tecnologico de Monterrey Campus Estado de Mexico, Mexico 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 N-nutrition 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 N-supply 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 N-supply 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 N-supply. 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 multi-input multi- Proceedings of the44th IEEE Conference on Decision and Control, andthe European Control Conference 2005Seville, Spain, December 12-15, 2005 WeIB20.1 0-7803-9568-9/05/$20.00 ©2005 IEEE 5534 Authorized licensed use limited to: Tec de Monterrey. Downloaded on February 4, 2009 at 12:38 from IEEE Xplore. Restrictions apply.  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  , carbon-dioxide 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 non-limited N supply is adjusted tothe demand for carbon, and for the N-limited 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 alge-braic 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 N-uptake. 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 ) , Ni-trate, 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, dur-ing two periods, July-August ( J  - A ) and September-October( 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 5535 Authorized licensed use limited to: Tec de Monterrey. Downloaded on February 4, 2009 at 12:38 from IEEE Xplore. Restrictions apply.  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 off-diagonal elements of the modified Fisher ma-trix 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 selec-tion 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: 5536 Authorized licensed use limited to: Tec de Monterrey. Downloaded on February 4, 2009 at 12:38 from IEEE Xplore. Restrictions apply.  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 sub-matrixand (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 fourJuly-August, ( J  − A ) treatments, and in the Table V forthe second four September-October, ( 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 5537 Authorized licensed use limited to: Tec de Monterrey. Downloaded on February 4, 2009 at 12:38 from IEEE Xplore. Restrictions apply.  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 con-ditions from a nonlinear model, in such a way that enablesthe selection of those outputs of the real system whose mea-surement 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 sub-matrix 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 pa-rameters remain “near” the nominal ones. The fitting of the 5538 Authorized licensed use limited to: Tec de Monterrey. Downloaded on February 4, 2009 at 12:38 from IEEE Xplore. Restrictions apply.
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