extended kalman filter for identification of parameters in geotechnical models_1st review

8/12/2019 Extended Kalman Filter for Identification of Parameters in Geotechnical Models_1st Review

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Extended Kalman Filter for Identification of

Parameters in Geotechnical Models

Tamara NestorovicRuhr-Universitat BochumMechanik adaptiver Systeme

Universitatsstr. 150

D-44801 Bochum, Germany

Email: [email protected]

Luan T. NguyenRuhr-Universitat BochumMechanik adaptiver Systeme



Miroslav TrajkovRuhr-Universitat BochumMechanik adaptiver Systeme



AbstractDirect measurement of relevant system parametersoften represents a problem due to different limitations. Ingeomechanics, measurement of geotechnical material constantswhich constitute a material model is usually a very difficult taskeven with modern test equipment. Back-analysis has proved tobe a more efficient and more economic method for identifying

material constants because it needs measurement data such assettlements, pore pressures, etc., which are directly measurable,as inputs. Among many model parameter identification methods,the Kalman filter method has been applied very effectivelyin recent years. In this paper, the extended Kalman filter local iteration procedure incorporated with finite elementanalysis (FEA) software has been implemented. In order toprove the efficiency of the method in geotechnical applications,model parameter identification has been performed for variousgeotechnical problems. In this paper, the examples of identifiedparameters include elastic constants of linear elastic materialmodels and model geometry parameters.

KeywordsParameter identification; extended Kalman filter;parameters of geotechnical models.

I. INTRODUCTION

The back analysis procedures require reliable and suffi-cient measurement data, a robust numerical model and anefficient method for the solution of the inverse (optimization)problem [1]. However in situ measurements are susceptibleto uncertainties due to instrument inaccuracies, environmentand/or measurement noise or human handling. Furthermore,for geotechnical problems the back analysis employs directapproach based on iterative solution of a forward problemby means of a numerical approximation (e.g. finite elementanalysis) and therefore the exact model response is basicallyunknown [1][3].

The Kalman filter (KF) method was developed by RudolfE. Kalman in early 1960s [4]. Since that time it has beenwidely employed in signal processing, mechanical systems,etc. as a very successful method for state estimation andparameter identification. In recent years, some applications ofthe KF method have also arisen in the field of geotechnicalengineering for estimation of material parameters [5][8]. TheKF method is based on the least-squares estimation methodadapted to time updates and observation updates of the meanstate and its covariance, taking into account uncertainties ofsystem modeling and measurements [4], [9]. It starts froma priori estimates and utilizes a set of observation data to

calculate a posterioriestimates. The sequence is repeated untilconvergence has been ascertained. One of the main features ofthe KF method is that the estimation of parameters at anytime instant is accompanied by relevant covariance matrices,which represent the uncertainty of the estimated parameters,as well as of the process noise and observation noise. Dueto this mechanism of the KF, it is classified as a statisticalidentification approach. Murakami [6] has pointed out that theKalman filter and the Bayesian estimator are equivalent.

In this paper, the KF method for model parameter iden-tification has been successfully implemented for applicationsin geotechnical engineering. The method is based on extendedKF technique combined with FEA. The underlying idea relieson the parameter estimation using the extended Kalman filter(EKF) algorithm, a nonlinear version of the Kalman filter,which is described in detail in the subsequent section. Themethod has been successfully applied for different linearand nonlinear geotechnical problems. Observations necessaryfor the EKF estimation are provided through finite element

analysis and they include simulated measurements requiredin geotechnical applications. A demanding task of incorpo-rating the EKF algorithm within the FEA is solved throughappropriate software coupling in order to provide a smoothbi-directional exchange of the required data both for the EKFexecution as well as for the FEA. The novelty represents alsothe verification of the proposed method for model parameteridentification by means of its performance with respect todifferent geotechnical problems, e.g. strip footing on multi-layer soil structure and retaining wall structure - the problemswhich have according to the authors best knowledge not beentreated from this point of view sufficiently detailed before.Through the examples the feasibility and the efficiency of theproposed technique have been successfully proven and highrobustness with respect to the choice of the initial parametervalues have been observed.

A formulation of the Kalman filter along with finite elementanalysis requires two equations of a state-space model: thestate equation and the observation equation.

Different from the conventional use of the Kalman filteralgorithm which has applications e.g. in estimation of dynamicstate variables for active control purpose in systems and struc-tures (like in [10], [11]), the estimation of parameters presentedin this paper is aimed at identification of geotechnical model


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parameters which do not change in time. Thus, a stationarytransition of state estimates is suitable for our purpose [5].

Stationary system is expressed by the following stateequation contaminated with noise at the k -th observation data(imaginary time tk):

xk+1= xk+wk, (1)

where xk is the state variable vector containing model pa-

rameters to be identified, dimension n 1; wk is a systemnoise vector contaminating the state equation, dimension n1.System noise is treated as white noise having zero meanE(wk) = 0 and a covariance matrix E(wkw

Ts) = Qkks.

ks is the Kronecker delta: ks = 1 ifk = s, and ks = 0 ifk =s.

Most materials involved in geomechanical applicationsbehave nonlinearly [12]. Even for a finite element formu-lation, resulting in a discretized system of linear constitutiveequations, the nonlinearities will arise when solving inverseproblems. In general, the constitutive relations depend non-linearly on the material and model parameters. The demand ofour estimation task, the inverse formulation, is to find material

parameters incorporated within the constitutive laws from e.g.the known displacements. The finite element formulation for ageneral static problem is given by a relation between appliedloads and displacements:

K(x) u= f, (2)where:

K is the stiffness matrix, dimension l l;x is the state vector associated with the material

model used, as in equation (1) x has dimensionn 1;

u is the displacement vector, dimensionl 1;f is the load vector, dimensionl 1.

The model response is observed at a finite number ofnodal points within the finite element model. Measurementdata in our case can involve displacements, stresses, pore waterpressures, etc. The choice of the type of measurements dependson the particular geotechnical problem and the available mea-surement tools. In this paper, displacements at a finite numberof nodal positions are chosen to be compared with measure-ment data. The reason for this choice is that displacement(settlement) is the quantity of interest in many geotechnicalproblems and hence due to the well developed equipment formeasurement of settlements it would be possible to obtain realdata as input for the model parameter identification procedure.

In agreement with the measurement data, we take thesimilar quantities from the FEA as observation data. The

observation vector at observed positions at time tk is relatedwith the displacement vector in the following manner

yk = T uk+vk, (3)

where:

yk is the observation vector (measurement), dimen-sion m 1;

T is the transfer matrix, dimensionm l;vk is measurement noise vector caused by uncertain-

ties in measurement equipment, dimensionm1.Observation noise is also white noise having

zero mean E(vk) = 0 and a covariance matrixE(vkvTs) =Rkks.

From equation (2) we have uk = K1(xk)f. Thus,

equation (3) becomes

yk = T K1(xk)f+ vk. (4)

The stiffness matrix, K(xk), is generally a non-linearfunction of the material parameters xk and for a compactpresentation here we express the equation (4) in a more generalform:

yk = h(xk) +vk. (5)

For simplicity, the system noise wk and the observationnoisevk are assumed uncorrelated. Furthermore, it is assumedthat the parameter uncertainty is independent. The same ap-plies to observation uncertainties. Thus the covariance matricesQkand Rk are diagonal.

The state equation (1) and the observation equation (5)are the basic equations representing the state-space system.

Our goal is to estimate the state vector of the nonlinearsystem based on the knowledge about the modelled systemand observation availability which are both contaminated byuncertainties. If the system is linear, the standard linear Kalmanfilter is the best estimator to be used. For our case ofhighly nonlinear system (geomechanical structure), a nonlinearKalman filter (extended Kalman filter) is required. Unlike thestandard Kalman filter, the extended Kalman filter linearizesthe system at the current state. The extended Kalman filter ispresented in the next section based on the state-space systemformulation given in this section.

II. EXTENDED K ALMAN FILTER (EKF) - LOCALITERATION PROCEDURE INCORPORATED WITH FINITE

ELEMENT ANALYSIS

First we introduce definition of a priori estimate and aposteriori estimate. A priori estimate is the estimate of thestate before observation data are available. Each time the dataobservation occurs, a priori estimate will be corrected to thenew state, which is called a posteriori estimate, that betterrepresents the system. In mathematical expressions, a prioristate estimate (estimate) and its covariance matrix at time tkare estimated as follows given the observation data up to timetk1:

xk =E[xk|y1, y2, , yk1] ;

Pk =E(xk x

k)(xk x

k)T

.

Whenever observation data at timetk are available, a posterioriestimate and its covariance matrix can be calculated as follows:

x+k =E[xk|y1, y2, , yk] ; P+k =E

(xk x

+k)(xk x

+k)

T

The estimation procedure should be initiated by best initialknowledge of the considered geomechanical model. As initialcondition, the initial state estimate and the covariance of theestimation error should be available:

x+0 =E(x0); P+

0 =E

(x0 x+0)(x0 x

+0)

T

.


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The initial state, which consists of model parameters, is chosenbased on engineering experience or preliminary examinationof the structure. The closer the initial parameter values to thetrue ones are, the better it is to begin the EKF. But this isnot necessary. The initial covariance matrix is assigned ratherarbitrarily large due to the lack of confidence in the initialchoice of the state vector.

Before observation data of the model are available, the EKFpropagates the mean and error covariance of the state throughtime. The time update equations of the mean and covarianceof the state are calculated in the following way:

xk = x+k1; P

k =P+k1+Q, (6)

where xk is the mean of state, xk = E(xk), andPk is the estimation error covariance matrix, Pk =E

(xk xk)(xk xk)

T

at time tk. The superscript denotes a priori estimate, and the superscript + denotes aposteriori estimate.

As soon as the observation data calculated by FEA areavailable, observation update procedure is applied. We apply

here the local iteration scheme for the observation updates inorder to continuously get better a posteriori estimate withoutcalculating time update. This local iteration observation updatecan be repeated as many times as desired, but for most systemssignificant improvement of the state estimate is obtained byonly one iteration [14]. At time tk in the local iterative loopi of the EKF - local iteration procedure, the Kalman gainKk,i, observation update of the state x

+k,i+1 and estimation

error covariance matrix P+k,i+1 are calculated according to thefollowing equations:

Hk,i = h

x

x+k,i

Kgk,i = Pk HTk,i(Hk,iPk HTk,i+R)1 (7)

P+k,i+1 = (I Kgk,i)P

k

x+k,i+1 = x

k +Kgk,i

yexp h(x

+k,i, 0) Hk,i(x

k x+k,i)

whereH, which is termed sensitivity matrix, is composedof the derivatives of observation data with respect to the statevariables. Hcannot be calculated analytically in our problemsbecause in the observation equationh(x)is not known. Instead,a numerical approximation of H will be performed usingforward results from FEA. Kg is the Kalman gain matrix.Derivation of Kalman gain formula based on the theory ofleast-squares estimation may be found in [9], [13], [14]. yexpis the expected measurement. Expected data are measured fielddata for the geotechnical structure being studied. In this paper,measurement data are produced synthetically by finite elementsimulations assuming that the model parameters are alreadyknown.

In the method used here, the error covariance matrix ofestimation Pk is enlarged with a modification weight W inevery global iteration to obtain fast convergence as proposed byHoshiya et al. [5]. The aim is to make the fictitious differencebetween the true state and the estimated state more significantas it can be seen from the function J in equation (8). J is

named cost function, objective function or return function.

Jk = E[(x1 x1)2] + +E[(xn xn)

2]

= E(2x1,k+ +2xn,k)

= E(Tx,kx,k)

= E[T r(x,kx,k)T]

= T r(Pk) (8)

In a stable system, the EKF always attempts to minimizethe cost function J. The state error covariance matrix P isenlarged at observation step k, which means that an amountof pseudo-error is added to the filter. This means we make EKFthink that its current state estimate xk is too far away fromthe true statexk. The new Kalman gainKk, equation (7), willalso be enlarged with enlarging Pk. The updated observationequation in turn uses Kk to scale up the innovation (yk h(x+k,i, 0)Hk,i(x

kx+k,i))and adds it to thea prioriestimate

xk. Thus, state variable estimate x changes more quickly, andthis makes convergence rate of the filter faster.

An independent treatment of the FEA program provides

a significant advantage to the proposed method, in the sensethat any FEA program can be incorporated into the processafter the communicative interface between the main algorithmflow and the FEA program is created. In this paper, the mainalgorithm flow and the communicative interface are realisedvia MATLAB computing language and Python scripts, whereasthe ABAQUS FEA software is used for performing the forwardsimulations.

In order to begin the filter process, an initial value x0 forthe state vector along with the corresponding estimation errorcovariance matrix P0 have to be assigned. Usually, the initialstate values are taken to be the most likely parameter valuesthat can be inferred from prior knowledge or preliminarygeomechanical tests. Values of the corresponding estimation

error covariance matrix explains how much confidence thefilter designer may have on the geotechnical model underconsideration.

The effects of the process noise wk and the observationnoise vk are omitted from system equation and observationequation for the sake of simplicity. However, their correspond-ing covariance matrices Q and R remain in the equations oftime update and observation update of the EKF process. Thesecovariance matrices are written without subscript k, whichindicates that they remain unchanged over iterations.

The calculation of observation vector derivatives with re-spect to the state values at the current state estimate, Hk,i,is crucial as the observation derivative matrix determines thedirection in which the state estimate xk,i+1 progresses. Cal-culation of the observation derivative matrix is approximatedby forward 2-points finite difference employing the forwardsolution of FEA. Sensitivity matrix H is calculated for eachglobal iteration k and local iteration i.

The iterative procedure explained consists of a local it-erative loop for calculating observation update and a globaliterative loop for the filtering process. The local iterative loopiterates a certain amount of times according to user setting andthe global iterative process will continue until a stop criterionis met. Possible stop criteria are: 1) cost function achieves


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Layer 1

Layer 2

Layer 3

Symmetryline

Pressure

Fig. 1. Finite element mesh and observation positions for the plane strainfooting structure

TABLE I. TRUE ELASTIC CHARACTERISTIC VALUES FOR THE THREESOIL LAYERS

Pa ra meter La ye r 1 L ay er 2 La ye r 3

E[N/m2] 2.0e7 1.0e7 0.5e7 0.3 0.2 0.4

a predefined minimum value; 2) absolute difference of twoconsecutive estimated state vectors is less than a predefinedtolerance; 3) the specified number of iterations is exceeded.In this paper, the third stop criterion is selected since thenumber of iterations is usually small and it also serves thepurpose of examining the filter process at some later time afterconvergence has been reached.

III. RESULTS AND DISCUSSIONS

A. Strip footing load on elastic soil - Identification of elastic

constants

The first geotechnical problem to be used as an example isthe strip footing on an isotropic linear elastic soil. The planestrain model with three layers is considered. The model takesits horizontal limit of 2 m and the vertical limit is 0.5 m depth.A strip load of 2 MPa is applied on the soil surface overthe length of 10 cm. Due to the symmetry of the problemonly a half of the structure is considered for the finite elementanalysis and a symmetry boundary condition is imposed alongthe symmetry line as shown in Figure 1.

The soil is assumed to obey an isotropic linear elasticmaterial law. A set of predefined model parameters servesthe purpose of synthetic generation of measurement data bynumerical simulation. This set is considered as the exact andtrue soil model parameter set and it is used as a basis forverification of the final estimated parameter values. The trueelastic soil parameters are given in Table I.

The observation positions are selected at the dotted points,as shown in Figure 1, and they are grouped into observationgroup 1 and observation group 2. Observation group 1 consistsof 4 (round-dotted) points located along the model symmetryline. Observation group 2 consists of 4 (square-dotted) pointslocated at the same depths as the observation points of group 1and 10 cm apart from them in horizontal direction. At points

from observation group 1, because of the symmetry boundarycondition, only vertical displacements are used. At points fromgroup 2, both vertical and horizontal displacement componentsare observed.

The observation positions are intentionally chosen at depthlevels that are distributed to all the three soil layers. Thus wemay evaluate the sensitivity of observation data with respectto model parameters involved in modeling the behavior ofthe different soil materials composing the examined structure.Different combinations of observation data will be selected totest the influence of the number of measurement or observationpoints and their location within the model on the convergencerate of the EKF method and its effectiveness.

Successful identification of the total six material modelparameters should prove the capability of the EKF to identifya large number of parameters. The assessment of the capacityof the EKF to model identification for this example is doneby means of three different tests:test 1: all observation data from all groups are taken intoaccount, i.e. four vertical displacements of observation group 1and four vertical and four horizontal displacements of obser-

vation group 2 (total number of observation data is twelve);test 2: only observation data from group 1 and the verticaldisplacement components from observation group 2 are takeninto account, (total number of observation data is eight);test 3: only observation data from group 1 is taken intoaccount, (total number of observation data is four).

We begin with test 1, in which the largest number ofobservation data is considered. We want to give the easiestcondition for the EKF to work by feeding it with as muchinformation as the measurement set-up can offer. Following,in test 2 and test 3, we constrain the EKF by providing lessand less number of observation data to it. As a result, weexpect to assess the capability of the EKF to identify elastic

soil parameters when the amount of measurement data vary.When varying the number of observation data we still keepthe strategy of spreading the observation positions to all thethree elastic layers.

The state value vector in this example reads:

x= [ E1 1 E2 2 E3 3 ] ,

whereE1 and 1 are Youngs modulus and Poissons ratio ofelastic soil layer 1, E2 and2 are those of elastic soil layer 2and E3 and3 are those of elastic soil layer 3.

The EKF in test 1 converges to a solution after about 15iterations and the rate of convergence is examined for three

different sets of initial values for the state values. In orderto cover wide range of initial state values for the EKF, thesets of an extreme initial state values are considered. In thefirst case the EKF begins with very small Youngs moduli(E1 = E2 = E3 = 0.5 107N/m2) and very large Poissonsratios (1 =2 = 3 = 0.45), whereas in the second case verylarge Youngs moduli (E1 = E2 = E3 = 3.0 107N/m2)and very small Poissons ratios (1 = 2 = 3 = 0.1) areconsidered. In the third case, a random predicted initial setof parameters in between the two extreme cases is chosento initialize the KF process, namely (E1 = 1.0 107, 1 =0.1, E2 = 2.0 107, 2 = 0.3, E3 = 1.5 107, 3 = 0.2).


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Fig. 2. KFiteration result - Test 1 (number of observations 12), initialstate value case 1

Fig. 3. KFiteration result - Test 1 (number of observations 12), initialstate value case 2

Figures 2 and 3 show EKF converging behavior in test 1for two different cases of initial state values. In both cases,the elastic soil parameters are identified after 16 iterations.Parameters of the upper soil layer converge faster to theirtrue values compared to those of the lower one. It means,more information may be captured observing the upper surfaceof the structure and this is reasonable because settlementmeasurements at places close to the surface of the strip footing

are most sensitive to the soil material characteristics.In test 2, we use only the initial state value case 2 and run

the KF with eight observations. The progressing of the EKFparameter identification process depicted in Figure 4 showsthat convergence rate is much slower than it is in test 1. For theworst converging parameter, Poissons ratio of the first layer1, over 40 iterations are required for the EKFprocess toconverge. The same initial state parameters are taken for test 3,where only four vertical displacements are taken as observationdata. The state value estimates converge to wrong values as canbe seen in Figure 5 and Table II where the deviation of theidentified parameter values from the true ones is also given.

The results from the three tests of EKF application to

linearelastic strip footing problem have proved that the moreobservations are used, the better the convergence is and moreaccurate the estimated state is. However, to collect more mea-surement data more equipment and human effort are requiredfor a real construction site. The authors recommend that oneshould take the number of measurement a few more thanthe number of identified parameters for easy convergence.Choosing observation locations where displacement responsesare sensitive also helps to reduce the number of observationlocations. Murakami et al. [6], [15] proposed a method forcomputing sensitivity distribution of the observed displacementfor a linear analysis of an elastic soil medium.

TABLE II. EKFCONVERGENCE PERFORMANCE AND QUALITY OF

THE SOLUTION TEST3Parameter Initial state Estimated state Deviation from true state

E1 3.0 107 2.0891 107 4.45%1 0.1 0.0246 91.8%E2 3.0 107 0.9842 107 1.58%2 0.1 0.2283 14.15%E3 3.0 107 0.6457 107 29.14%3 0.1 0.2751 31.22%

B. Retaining wall - Identification of Mohr-Coulomb elasto-plastic constitutive parameters

An identification of Mohr-Coulomb constitutive modelparameters is carried out employing an example of a 4 mhigh retaining wall structure as depicted in Figure 6. A 30 kPa

pressure and a 500 kPa pressure are applied on the surface ofthe backfill over the lengths of 6 m and 3 m and the problem isconsidered as 2D plane strain problem. The backfill structuredefined in this way is similar to the one discussed in [16], butwith modified geometry, materials and loads.

The Mohr-Coulomb constitutive model of the soil backfillis assumed to have a friction angle = 37, a dilation angle = 5 and a cohesion c = 10 kPa. Before the soil undergoesplastic deformation it behaves linearelastically with a Youngsmodulus of 50 MPa and a Poissons ratio of 0.3. The concreteretaining wall is assumed to be linear elastic with a Youngs


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Fig. 4. EKFiteration result - Test 2 (number of observations 8), initialstate value case 2

modulus of 21.3 GPa and a Poissons ratio of 0.2. Density of

the soil is 1900 kg/m3 and of the concrete is 3000 kg/m3.

The finite element simulation is performed by means ofABAQUS dynamic analysis module and it is carried out in twosteps, each step lasts for 5 seconds. Pressure loads are appliedinstantaneously at the beginning of step 1. Gravity load and

prescribed displacementu

are ramped over step 1 and step 2as it can be seen in Figure 7.

Four-node linear planestrain finite elements are used inthis finite element model. Observation data are taken fromvertical displacements at four positions that are marked withdark color dots in the soil region as shown in Figure 6.

Observation data is composed of the observation pointvertical displacements at the end of step 2 when the responseof the model has attained its steady state. Figure 8 presentsthe displacements obtained with the set of true constitutiveparameters of the observed points over the simulation time.

It has to be pointed out that for the purpose of this examplenot all parameters involved in the Mohr-Coulomb constitutive

model are subject to identification. The elastic parameters,Youngs modulus and Poissons ratio, and soil cohesion areassumed to be known, whereas the friction angle and thedilation angle, are parameters that are selected to be identified

and therefore they form the state vectorx = [ ]

,wherefriction angle and dilation angle are assumed as knownin order to generate the synthetic measurement data. Frictionangle and dilation angle are 37 and5 respectively.

The EKF estimation process is run with only one localiteration. One single setting for noise covariance matrices Qand R and weight factor W for both initial state cases has

Fig. 5. EKFiteration result - Test 3 (number of observations 4), initialstate value case 2

Fig. 6. Soil backfill supported by a concrete wall

shown difficulty in obtaining fast convergence. Thus, thesesettings are adjusted according to each initial state case.

The EKFconvergence performance was tested using dif-ferent initial conditions sets. Converged parameter estimatesfor initial case 1 are obtained after 15 iterations and as itcan be seen from Table III they show quite large deviationfrom the true parameter values. The identified parameter valuesfor initial case 2 indicate faster EKFconvergence and lessdiscrepancy from the true parameter values. In table IV theconverged results in case 2 are obtained after 6 iterations.

TABLE III. EKFCONVERGED PARAMETER ESTIMATES AFTER 15ITERATIONS INITIAL CASE1

Parameter Initial state Estimated state Deviation from true state

40 37.32 0.32 10 4.66 0.34


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Fig. 7. Loading history for the retaining wall example

Fig. 8. Displacements at observed points over simulation time

IV. CONCLUSIONS

Good convergence of the parameter estimation processdemonstrated through various geotechnical problems presentedin this work has proved that the EKF incorporated withmodern FEA program has prospects to be a reliable methodfor parameter identification in geotechnical engineering. Theimportant advantageous feature of the EKF method, namelythe utilisation of the measurement uncertainties, which can-not be avoided in large scale geotechnical structures, makesthis method very attractive and powerful. In addition, EKF-procedure performs well even in the environment corrupted by

noise and due to the underlying characteristic of the algorithm,the state-value vector estimate (i.e. identified parameters)represents the values that best minimize the estimation errorcovariance.

The implementation of the EKF incorporated with FEA

TABLE IV. EKFCONVERGED PARAMETER ESTIMATES AFTER 6ITERATIONS INITIAL CASE2

Parameter Initial state Estimated state Deviation from true state

40 36.98 0.02 10 5.02 0.02

with regard to implementation efforts against the resultingoutcome is relatively simple but still presents a very powerfultool since using modern programming languages the EKFalgorithm can be realized within several code lines includingvector and matrix manipulations. However, the tuning and ad-justing the EKFconfiguration plays a decisive role regardingits successful implementation and therefore requires a specialattention. Good understanding of the model and the influence

of each parameter on the model is crucial for the configurationof the EKF in order to achieve its successful convergence.

ACKNOWLEDGMENT

The authors gratefully acknowledge the funding by theGerman Research Foundation (DFG) within Collaborative Re-search Center under grant SFB-837.

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extended kalman filter for identification of parameters in geotechnical models_1st review

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