parameter identification for state estimation—application to bioprocess software sensors

Chemical Engineering Science 59 (2004) 2465–2476www.elsevier.com/locate/ces

Parameter identi!cation for state estimation—application tobioprocess software sensors

Ph. Bogaertsa, A. Vande Wouwerb;∗

aService de Chimie g�en�erale et Biosyst�emes, Universit�e Libre de Bruxelles, Av. F.-D. Roosevelt, 50 C.P. 165/61, Brussels B-1050, BelgiumbService d’Automatique, Facult�e Polytechnique de Mons, 31 Boulevard Dolez, Mons B-7000, Belgium

Received 5 June 2003; received in revised form 23 October 2003; accepted 15 January 2004

Abstract

Together with some on-line measurements, a reliable process model is the key ingredient of a successful state observer design. Incommon practice, the model parameters are inferred from experimental data so as to minimize a model prediction error, e.g. so as tominimize an output least-squares criterion. In this procedure, no care is actually exercised to ensure that the unmeasured model states aresensitive to the measured states. In turn, if sensitivity is too low, the resulting state observer will probably generate poor estimates of theunmeasured states. To alleviate these problems, a new parameter identi!cation procedure is proposed in this study, which is based on a costfunction combining a conventional prediction error criterion with a state estimation sensitivity measure. Minimization of this combinedcost function produces a model dedicated to state estimation purposes. A thorough analysis of the procedure is presented in the context ofbioreactor modeling, including parameter identi!cation, model validation and design of extended Kalman !lters and full horizon observers.? 2004 Elsevier Ltd. All rights reserved.

Keywords: Parameter identi!cation; State estimation; Kalman !lters; Observers; Biotechnology

1. Introduction

First-principles modeling is usually driven by the concernof process simulation, i.e. the model structure and parame-ters are adjusted following some performance criterion so asto mimic faithfully the behavior of the real plant. Based onthe resulting process model and some on-line measurements,state observers (or software sensors) can be designed to re-construct unmeasured state variables. At this stage, varioustechniques can be applied, e.g. Kalman !ltering, recedinghorizon observers, high-gain observers, etc. State estimationtechniques play a key role in many areas of science and engi-neering, and particularly in biotechnology, where hardwaresensors are extremely costly and have stringent operatingconditions (sterilization, long processing times, etc.).However, experience reveals that an accurate process

model can be a poor basis for the development of a softwaresensor. The reason for this observation lies in the identi!-cation procedure itself, which is usually aimed at reducing

∗ Corresponding author. Tel.: +32-6537-41-41; fax: +32-6537-41-36.E-mail address: [email protected] (A.V. Wouwer).

the prediction error, but does not take care of the “internalmodel connections”. Indeed, the unmeasured states—as re-produced by the process model—should be sensitive to themeasured ones. Otherwise, even if the system is observable,the software sensor might produce poor estimates of the un-measured states.The objective of this study is to suggest a new parameter

identi!cation procedure yielding a model dedicated to stateestimation purposes. Based on the concept of canonical ob-servability forms of nonlinear systems (Zeitz, 1984, 1989;Gauthier and Kupka, 1994), a “measure of observability” isderived, which quanti!es the ability to detect in the outputtrajectories any diHerences in the initial states. State esti-mation sensitivity is enforced by minimizing a cost func-tion combining a conventional maximum-likelihood crite-rion with this observability measure.The proposed identi!cation procedure is quite general and

can be applied to a wide range of engineering processes de-scribed by nonlinear !rst-principles models (i.e. grey-boxmodels). Here, based on the authors’ experience, the eHec-tiveness of this parameter identi!cation procedure is demon-strated in the context of bioprocess modeling and softwaresensor design. Two applications are considered. The !rst

0009-2509/$ - see front matter ? 2004 Elsevier Ltd. All rights reserved.doi:10.1016/j.ces.2004.01.066

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2466 P. Bogaerts, A.V. Wouwer / Chemical Engineering Science 59 (2004) 2465–2476

one is a simulation study, which allows the usefulness of the“identi!cation for state estimation” concept to be demon-strated, even in the situation where the exact model struc-ture is known and large sets of good-quality measurementdata are available. The second application is a real-casestudy, e.g. experimental measurements in batch CHO ani-mal cell cultures. In both applications, several types of soft-ware sensors are considered, including Kalman !lters andreceding-horizon observers, which demonstrates the wideapplicability of the proposed method.To cope with reduced observability properties in nonlin-

ear systems, another approach could have been consideredas well, which consists in designing high-gain observers(Gauthier and Kupka, 1994; Vargas et al., 2002) based on aconventional model (i.e. derived using a conventional identi-!cation procedure) in canonical observability form. Thoughquite attractive, this alternative approach has severe limita-tions in the case of discrete-time measurements with rela-tively long sampling periods, as it is the case in most bio-process state estimation problems.This paper is organized as follows. The next section dis-

cusses some observability notions in the case of nonlinearsystems, presents a useful canonical form, and introducesan observability measure, which, combined with a conven-tional prediction error criterion, provides a new criterionfor parameter identi!cation. In Section 3, the alternativeapproach of designing a high-gain observer is discussed.Section 4 deals with the simulation study of a fed-batch bio-process, whereas Section 5 focuses attention on an experi-mental case-study, e.g. batch animal cell cultures. Finally,Section 6 is devoted to concluding remarks.

2. Identi�cation for state estimation

Consider nonlinear systems in the form

x = f(x; u); x(0) = x0 (1)

y = h(x); (2)

where x∈Rn, u∈Rm, y=Rp. The vector-functions f and hare assumed analytic, i.e. each component and its derivativesof arbitrary order exist and are continuous.In the following, observability of nonlinear systems is

analyzed and a practical observability measure is intro-duced, which, combined with a conventional predictionerror criterion, provides a new criterion for parameteridenti!cation.

2.1. Observability analysis of nonlinear systems

The observability analysis of nonlinear systems is basedon the observability map (Isidori, 1989; Nijmeijer andVan der Schaft, 1990; Zeitz, 1984,1989; RothfuN and

Zeitz, 1995)

q(x) =

q1(x)

q2(x)

· · ·qp(x)

with

qi(x) =

L0fhi(x)

L1fhi(x)

· · ·Lki−1

f hi(x)

=

yi

y i

· · ·ki−1yi

;

k =p∑

i=1

ki; k¿ n; (3)

where Lkfhi is the kth Lie derivative along the vector !eld

f , which is de!ned as

Lkfhi(x) =

@(Lk−1f hi(x))

@xf(x; u); k¿ 1;

L0fhi(x) = hi(x): (4)

The observability map q(x) establishes a link between thestate x(t) and the time derivatives of the outputs yi(t), i =1; : : : ; p. The fact that observability depends on the states xand the inputs u, and that the number of derivatives ki − 1,i=1; : : : ; p, is not !xed a priori, is completely diHerent fromthe situation for linear systems.The analysis of the system observability involves the

solution of a nonlinear system of equations in form (3), withk¿ n. A system is said globally observable if the inverseqI(x) of the observability map exists everywhere (in thestate and input space). Global observability can however bediOcult to assess in practice, as the solution of (3) is not atrivial problem. In those cases, a local observability analy-sis, which amounts to study the local invertibility of q(x)in the neighborhood of a point x, provides useful insight. Asystem is said locally observable when

Rank Q(x) = n with Q(x) =@q(x)@x

(5)

which leads to the conventional Kalman’s rank test, i.e.,rank[C CA : : : CAn−1], when k = n.The analysis of global observability for nonlinear systems

can be considerably simpli!ed through the introduction ofcanonical forms, for which the invertibility of the observ-ability map is guaranteed (and hence the global observabil-ity of the system). This is the case when the system can beput in the following form (Zeitz, 1984,1989; Gauthier and

P. Bogaerts, A.V. Wouwer / Chemical Engineering Science 59 (2004) 2465–2476 2467

Kupka, 1994).

x =

x1

· · ·xi

· · ·xq−1

xq

=

f1(x1; x2; u)

· · ·f i(x1; : : : ; xi+1; u)

· · ·fq−1(x1; : : : ; xq; u)

fq(x1; : : : ; xq; u)

;

y =

h1(x11)

h2(x11 ; x12)

· · ·hn1 (x

11 ; : : : ; x

1n1 )

; (6)

where

∀i ∈ {1; : : : ; q}; xi ∈Rni ; n1¿ n2¿ · · ·¿ nq;∑16i6q

ni = n (7)

and

∀j ∈ {1; : : : ; n1}: @hj

@x1j�= 0; (8)

∀i ∈ {1; : : : ; q − 1}; ∀(x; u)∈Rn × Rm:

rank@f i(x; u)@xi+1 = ni+1: (9)

Conditions (8) imply that the !rst n1 state variables can beinferred from the measurements, whereas conditions (9) en-sure that any diHerences in the state trajectory can be de-tected in the measurements. In other words, if we considertwo diHerent initial states x(0) and x′(0), which only diHerby the components x2(0) and x2′

(0), the !rst state equationsx1 = f1(x1; x2; u) of system (6) show that the trajectoriesx1(t) and x1′

(t) will be diHerent, provided that property (9)is satis!ed for i= 1. As the components of x1(t) can be re-constructed through inversion of the measurement operatorh(x) thanks to conditions (8), the deviation between the tra-jectories x1(t) and x1′

(t) will be observable in the output.Of course, the same reasoning holds for any couple xi(t)and xi′(t), going from one set of equations to the precedingone in system (6).Conditions (9) can also be conveniently expressed in

terms of the square matrices Mi(x; u)

∀i ∈ {l; : : : ; q − 1}; ∀(x; u)∈Rn × Rm:

rankMi(x; u) = ni+1 with (10)

Mi(x; u) =(

@f i(x; u)@xi+1

)T (@f i(x; u)@xi+1

)∈Rni+1×ni+1 : (11)

2.2. Parameter estimation and observability of theresulting model

A process model is often identi!ed from discrete-timemeasurements, so that the model andmeasurement equationscan be rewritten as

x = f(x; u; #); (12)

y(tk) = h(x(tk)) + �(tk); (13)

where x(t) is the state, u(t) is the input, and # is the unknownparameter vector, y(tk) and �(tk) are the measurement andnoise vector, respectively.If �(tk) is a white noise sequence, normally dis-

tributed, with zero mean, and covariance Q(tk), themaximum-likelihood estimation # of # is given by

#=Argmin#

Fml(#)

= Argmin#

12

N∑k=1

(y(tk) − h(x(tk)))T

×Q(tk)−1(y(tk) − h(x(tk))); (14)

where h(x(t)) is the output estimate obtained by integrationof the model equations (12–13) with the parameters #= #.If the experimental data are suOciently informative, the

minimization of Eq. (14) using an appropriate numericalprocedure yields a satisfactory process model, which canbe used for simulation purposes. To develop a state ob-server (e.g. an extended Luenberger observer or an extendedKalman !lter) the system observability must be investigatedusing the identi!ed model and the methods reviewed in theprevious section. If the system can be put in the observabilitycanonical form (6) and the rank test (10–11) is successful,then the system is globally observable. However, some ma-trices Mi(x; u) can be ill-conditioned, and even though thesystem is theoretically observable, a diHerence in the initialstates will then be extremely diOcult to detect in the outputtrajectories. This observation leads to the de!nition of an“observability measure” Fobs quantifying this ability to de-tect, in the output trajectories, any diHerences in the initialstates. Assuming that the state trajectory x(t) is measured atdiscrete times tk , (k =1; : : : ; N ), a candidate scalar measureis given by

Fobs =N∑

k=1

q−1∑i=1

√cond(Mi(x(tk); u(tk))); (15)

where “cond” represents the condition number of the matrix,i.e., the ratio of its largest to its smallest eigenvalue.Based on these results, a new form of the cost function

combining the conventional maximum likelihood criterionFml (14) with the observability measure Fobs (15) can be


de!ned as

#=Argmin#

F(#) = Argmin#

{Fml(#) + �Fobs(#)}; (16)

where � is a weighting factor.From Eq. (15) and remembering that the lower bound of

a condition number is 1, a lower bound of the cost functionF is given by

Fmin = Fml(#) + �N (q − 1): (17)

A problem that arises when using combined cost functions,is the choice of the weighting factor �. One way to proceedis to make some a priori choice of � and to compute #according to (16). Repeating this procedure and plottingthe separate terms Fml and Fobs as functions of � (the !rstterm increasing and the second one decreasing with �), it ispossible to choose a trade-oH value for � which allows Fobs

to be reduced while keeping an admissible value for Fml.

3. The high-gain observer: an alternative approach?

If the nonlinear systems (1) and (2) are globally observ-able, then there exists a transformation into the observabil-ity canonical form (6), which is based on the observabilitymap (3), i.e.

z =

z1

z2

· · ·zp

=

q1(x)

q2(x)

· · ·qp(x)

with

zi =

L0fhi(x)

L1fhi(x)...

Lki−1f hi(x)

=

yi

y i

...ki−1yi

;

k =p∑

i=1

ki; k¿ n: (18)

The transformed system in zi can be written as follows:

zi =

zi2

· · ·ziki

’i(z; u)

=

0 1 0...

. . .. . .

. . . 1

0 : : : : : : 0

zi

+

0...

0

’i(z; u)

= Aizi + �i(z; u); (19)

yi = zi1; i = 1; : : : ; p; (20)

where the system nonlinearity is contained in the function’i(z; u).

The complete system in z, which is called the kthorder-extension of the system if k ¿n (Vargas et al., 2002),takes therefore the form

z = Az + �(z; u);

y = Cz; (21)

where A is block-diagonal matrix (with the blocks Ai) andC is a diagonal matrix (with 0’s and 1’s).The high-gain observer is based on

˙zi =

0 1 0

.... . .

. . .

. . . 1

0 : : : : : : 0

zi +

g1=�

...

gki−1=�ki−1

gki =�ki

(yi − zi

1);

i = 1; : : : ; p (22)

in which the gains g1; : : : ; gki are positive and � 1.The system nonlinearity ’i(z; u) does not appear in the

observer formulation (22) and it is easily shown that theestimation error dynamics (i.e., the dynamics of ei= zi −zi)does not depend on ’i(z; u) if � is suOciently small. Onthe other hand, the gains g1; : : : ; gki determine the speed ofconvergence of the observer.The relative independence of the estimation error dynam-

ics with respect to the system nonlinearity confers robust-ness to the high-gain observer, i.e. insensitivity to mod-eling errors, and allows reduced observability properties(which often originates in the system nonlinearity) to behandled.However, as � → 0, the observer gain becomes larger and,

as a consequence, the sensitivity to measurement noise sig-ni!cantly increases. SchaHner and Zeitz (1995) have shownthat the observer practical applicability is restricted to cases,where n=p6 3, in order to limit the order of diHerentiationof the measurement outputs.The preceding analysis has been developed for

continuous-time systems with continuous-time measure-ments (1–2). It is possible to apply high-gain observersin the case of continuous-time systems with discrete-timemeasurements (12–13). However, high-gain observers usu-ally perform very poorly when low sampling periods areused and/or high measurement noise occurs. Indeed, thecorrection at the sampling instant is very brutal (as mea-surement extrapolation has to be used before the next mea-surement sample becomes available, and possibly large pre-diction errors and measurement noise are ampli!ed by highgains), which leads to strong oscillations in the estimatetrajectory.


In bio-chemical applications, where measurements sam-ples are often collected with relatively long sampling pe-riods (several hours to 1–2 days) and are corrupted bynoise, the design of high-gain observers is delicate andoften leads to unsatisfactory results. For this reason, theproposed approach of “identi!cation for state estimation”is considered as an appealing alternative. The resultingmodel can indeed be used to design any kind of observers,and particularly continuous-discrete extended Kalman !l-ters or receding-horizon observers which provide rigorousstochastic treatments of discrete-time, noisy measurements.As these latter observers are comparatively less robust thanhigh-gain observers (i.e. more strongly dependent on thenonlinear model, and in turn, more demanding in termsof observability), the “identi!cation for state estimation”allows an intrinsic lack of sensitivity with respect to someof the state variables to be compensated.

4. Demonstration through model falsi�cation

The idea of this section is to demonstrate the usefulnessof the proposed identi!cation procedure, even in the situa-tion where the exact model structure is known and large setsof good-quality measurement data are available for param-eter identi!cation. This ideal situation, which can only beexplored in simulation, leads to the concept of “model fal-si!cation”, i.e. the fact that the best simulation model mightnot be the best for state estimation, and that, in this case, theprocedure devised in the Section 2 results in an adjustment(or falsi!cation) of the model parameters in order to en-hance the state estimation sensitivity. This simulation studyconsiders a fed-batch bioprocess, which is detailed in thesequel.Consider a simple macroscopic reaction scheme

Growth : Vs2 S2’g→∧ X: (23)

Maintenance : S1 + #X X’m→ #X X + #PP (24)

where X; S1; S2 and P represent the biomass, substrate1–2 and product, respectively, and #S2 ; #X and #P arepseudo-stoichiometric coeOcients. The symbol “→∧”means that the growth reaction is auto-catalyzed by X andthe presence of “#X X ” in both sides of the maintenancereaction means that X catalyzes this latter reaction.The growth rate ’g and the maintenance rate ’m are de-

scribed by classical Monod laws and inhibition factors

’g(X; S1; S2) = %gmax

S2Kg

M + S2

Kgi

Kgi + S1

X; (25)

’m(X; S1) = %mmax

S1Km

M + S1

Kmi

Kmi + X

X: (26)

These growth and maintenance reactions take place in abioreactor operated in fed-batch mode with a time-varyinginlet Sow rate, Fin(t) and constant inlet substrate concentra-

Table 1Model parameters

#S2 = 0:2 mM=(105 cell=ml) Kgi = 70 M

#P = 1:7 %mmax = 0:1 h−1

%gmax = 0:05 h−1 Km

M = 0:2 mMKg

M = 0:1 mM Kmi = 3 × 105 cell=ml

tions S1; in and S2; in. Simple mass balances allow the follow-ing dynamic model to be derived:dXdt

= ’g(X; S1; S2) − DX; (27)

dS1dt

= −’m(X; S1) + D(S1; in − S1); (28)

dS2dt

= −#S2’g(X; S1; S2) + D(S2; in − S2); (29)

dPdt

= #P’m(X; S1) − DP; (30)

dVdt

= Fin ; (31)

where X; S1; S2; P now denote the respective component con-centrations, V is the culture volume and D(t) = Fin(t)=V (t)is the dilution rate.The model equations (27)–(31) together with the numer-

ical values of the several parameters listed in Table 1 de!nethe reference system, which is investigated in the continua-tion of this simulation study.

4.1. Parameter identi<cation

In a classical bioprocess application, the !rst task is todevelop a process model and to estimate the model pa-rameters from experimental data. Here, it is assumed thatthe exact model structure (27–31), the numerical values ofthe pseudo-stoichiometric coeOcients as well as the modelinitial conditions (initial concentrations and volume) areknown exactly and that only the values of the kinetic pa-rameters must be inferred from measurement data.To this end, component concentrations can be measured

oH-line at regular time intervals, e.g. every 5 h. Measure-ments are corrupted by normally distributed, white noiseswith zero mean and variance matrix Q. Constant relative er-rors are considered, e.g. �relX = 0:1, �relS1 = 0:05, �relS2 = 0:05,�relP = 0:05. Three experiments are performed, which diHerin their initial conditions, inlet substrate concentrations andSow rate (taken, for instance, in the form Fin(t)=0, for t ¡ tband Fin(t)=�(t−tb), for tb6 t6 tfb, with �=5×10−4 l=h2)according to Table 2. The data from the !rst two experi-ments are used for parameter estimation, the third experi-ment provides a cross-validation test.Based on these measurement data, the maximum-

likelihood estimation # of # given by Eq. (14) allows theparameters of Table 1 to be retrieved very satisfactorily (theonly small diHerences are due to a speci!c realization of


Table 2Experimental conditions

Exp 1 Exp 2 Exp 3

X (0) (105 cell=ml) 1 1 1S1(0) (mM) 10 10 8S2(0) (mM) 8 1 5P(0) (mM) 1 1 1V (0) (l) 0.5 0.5 0.5S1; in (mM) 5 5 10S2; in (mM) 1 1 3tb (h) 30 35 50tbf (h) 80 80 90

the measurement noise). Several optimization runs are per-formed starting from diHerent initial parameter guesses, inorder to check global identi!ability. In essence, the model isperfectly identi!able with the data at hand, and Fig. 1 showsthe cross-validation test performed with experiment 3. Inthis graph, the circled points are the measured data and thebars represent the 99% con!dence intervals. The solid linesare the concentration trajectories predicted by the identi!edmodel.

4.2. State estimator design

Based on the model identi!ed in the previous sec-tion, two diHerent state estimators are designed, e.g., acontinuous-discrete extended Kalman !lter (Gelb, 1974) anda full horizon observer (AllgUower et al., 1999; Bogaerts andHanus, 2001). The objective is to obtain a continuous-timeestimation of biomass and substrate 1 concentrations fromdiscrete-time measurements of substrate 2 and product con-centrations, and to show that the main observations areindependent of a speci!c observer design.As only two component concentrations are measured

on-line, the measurement equation (13) takes the form

y(tk) = Cx(tk) + �(tk); (32)

where C is a measurement matrix C and �(tk) is awhite measurement noise vector with variance matrixQ= diag(*2

S2 ; *2P).

4.2.1. System observabilityThe model equations (27–30,32) are already in the canon-

ical observability form (6), i.e.

x =

[x1

x2

]=

[f1(x1; x2; u)

f2(x1; x2; u)

]; y = �x1� (33)

with

x1 = [S2 P]T; x2 = [X S1]T; u = [S1; in S2; in D]T; (34)

f1(x1; x2; u) = [ − #S2’g(X; S1; S2) + D(S2; in − S2) #P’m(X; S1) − DP]T; (35)

f2(x1; x2; u) = [’g(X; S1; S2) − DX − ’m(X; S1) + D(S1; in − S1)]T; (36)

Note that the volume described by Eq. (31) is not explicitlyconsidered as a state variable here, as it is directly relatedto the dilution rate D, which a known system input.The global observability condition is

rank@f1(x; u)

@x2 = n2 = 2: (37)

As

@f1(x; u)@x2

=

−#S2@’g(X; S1; S2)

@X−#S2

@’g(X; S1; S2)@S1

#P@’m(X; S1)

@X#P

@’m(X; S1)@S1

; (38)

the global observability condition is satis!ed if X; S1 and S2do not vanish.

4.2.2. Continuous-discrete extended Kalman <lterThe continuous-discrete extended Kalman !lter is the

generalization of the Kalman !lter to nonlinear systemsdescribed by continuous-time state equations (12) anddiscrete-time measurement equations (13) or (32).Prediction step (between samples):

dxdt

= f(x; u); tk ¡ t ¡ tk+1; (39)

dPdt

= A(x)P+ PA(x)T: (40)

Correction step (at sampling times):

K(tk) = P(t−k )CT[CP(t−k )CT +Q(tk)]−1; (41)

x(t+k ) = x(t−k ) + K(tk)(y(tk) − Cx(t−k )); (42)

P(t+k ) = P(t−k ) − K(tk)CP(t−k ): (43)

The extended Kalman !lter requires the on-line numericalintegration of the state equation (39) and the Ricatti equa-tion (40). The latter involves the matrix A(x) = (@f=@x)xresulting from the model linearization along the predictedstate trajectory.These equations are solved starting with the initial con-

ditions x(0) = x0 and P(0) = P0. For substrate 2 and prod-uct, these values are best taken from the measured concen-trations and the measurement error variances at the initialtime, respectively. For the unmeasured component concen-trations, these initial values can only be guessed based oncommon sense and process knowledge. They represent thetuning parameters of the Kalman !lter, which are taken hereas, e.g. X (0) = 3× 105 cell=ml; S1(0) = 5 mM; P0(1; 1) =*2

X = 106 (cell=ml)2 and P0(2; 2) = *2S1 = 106 mM2.


Fig. 1. Maximum-likelihood identi!cation of models (22–26)—cross-validation with experiment 3.

Fig. 2. State estimation with a continuous-discrete extended Kalman !lter using a model identi!ed with the maximum-likelihood criterion (14)(experiment 3).

Fig. 2 shows that the Kalman !lter accurately estimatesthe measured substrate 2 and product concentrations as wellas the unmeasured biomass concentration. However, it pro-duces poor estimates of the unmeasured substrate 1 concen-tration, with the exception of the !nal times where substrate1 disappears and growth limitation occurs. This latter obser-vation will become clear in the next section.

4.2.3. Full horizon observerBetween two measurement times, numerical integration

of the state equations (12) from an estimate of the mostlikely initial conditions allows a prediction of the state vec-tor to be computed on-line. At the next sample time, a newestimate of the initial conditions can be obtained by min-imizing a maximum-likelihood criterion based on all the


Fig. 3. State estimation with a full-horizon observer using a model identi!ed with the maximum-likelihood criterion (14) (experiment 3).

measurements available up to this time. The procedure isrepeated from sample to sample and can be summarized asfollows:Prediction step (between samples tk ¡ t ¡ tk+1):

dxdt

= f(x; u); 06 t ¡ tk+1:

x(0) = x0=k : (44)

Correction step (at sampling times):

x0=k =Arg minx0

Jk(x0) with

Jk(x0) =12

k∑j=1

(y(tj) − Cx(tj))T

×Q(tj)−1(y(tj) − Cx(tj)): (45)

For the !rst time interval, the state equations (44) are solvedstarting with initial conditions determined in the same wayas for the Kalman !lter (i.e., S2(0) and P(0) are the mea-sured values while X (0) = 3 × 105 cell=ml and S1(0) =5 mM are a priori initial guesses). The minimization ofEq. (45) is performed repeatedly with “lsqnonlin” fromMATLAB. The tuning parameters of the observer are thelower and upper bounds on x0, which are taken here as, e.g.0¡X0 ¡ 5; 0¡S1;0 ¡ 20; 0¡S2;0 ¡ 10; 0¡P0 ¡ 5.

The observations are basically the same as with the ex-tended Kalman !lter, i.e., the unmeasured substrate 1 con-centration is poorly estimated (see Fig. 3).

4.3. Identi<cation for state estimation

It is now clear that, even with a high-quality model (forsimulation purposes), state observers might fail to producereliable estimates of unmeasured state variables. Attentionis now turned to the new parameter estimate proposed inEq. (16).Fig. 4 shows the evolution of the separate terms Fml and

Fobs as functions of �. For increasing �, Fobs decreases whileFml increases. A compromise solution must therefore beconsidered, e.g. �=0:1 (for this value, Fml doubles whereasFobs is reduced by a factor 2.6; for larger values of �; Fml

increases signi!cantly whereas Fobs only slightly decreases).With this trade-oH value for �, the combined cost function(16) is minimized using again “lsqnonlin” from MATLAB.The identi!ed kinetic parameters are listed in Table 3. Incomparison with Table 1, the values of some parameters,particularly Kg

i and KmM , have changed signi!cantly.

If these new model parameters are used in the extendedKalman !lter designed in Section 3.2, the results graphedin Fig. 5 are obtained. Signi!cant improvements in the esti-mation of the unmeasured substrate 1 concentration can beobserved. Indeed, the norm of the estimation error (de!nedas the sum of the squares of the estimation errors at eachsampling time weighted by the corresponding measurementvariances) has been reduced by a factor 30! Similar resultscan be obtained with the full-horizon observer as illustratedin Fig. 6.A closer look at the expressions of the reaction rates

(25)–(26) and a comparison of the numerical values of thekinetic parameters in Tables 1 and 3 allows a physical inter-pretation of these results. As Kg

i in Table 1 is much larger


Fig. 4. Evolution of Fml(�) (top) and Fobs(�) (bottom).

Table 3Identi!ed parameters (cost function (6))

%gmax = 0:062 h−1 %m

max = 0:11 h−1

KgM = 0:046 mM Km

M = 0:31 mMKg

i = 14 mM Kmi = 2:8 × 1015 cell=ml

than the concentration S1, the inhibition factor in Eq. (25)reduces to 1. On the contrary, S1 is much larger than Km

M inTable 1, at least at the beginning of the experiments 1–2,so that the Monod factor in Eq. (26) also tends to unity. Asa consequence, the system is intrinsically insensitive to thevariations of substrate 1! It is apparent from Table 3 that theminimization of the combined cost function (16) attempts tocompensate these eHects by signi!cantly reducing the valueof Kg

i and by increasing the value of KmM .

5. Real-case application

The previous observations are now con!rmed by anexperimental study, i.e. batch animal cell cultures.Rare and asynchronous measurements of biomass, glu-

tamine, glucose and lactate concentrations were performedin batch CHO (Chinese Hamster Ovaries) cell cultures (fordetails about the experimental conditions, see (Graefe et al.,1999)). In order to describe the macroscopic phenomenaoccurring in these cultures, a simple reaction scheme is pro-posed:

growth : #GlnGln’g→∧ X; (46)

maintenance : G + #X X’m→ #X X + #LL; (47)

where X; G; Gln and L represent the biomass (CHO cells),glucose, glutamine and lactate, respectively.The mass balances of these four components lead to the

system

dXdt

= ’g; (48)

dGdt

= −’m; (49)

dGln

dt= −#Gln’g; (50)

dLdt

= #L’m: (51)

A general kinetic model describing the growth rate ’g andthe maintenance rate ’m is taken from (Bogaerts and Hanus,2000)

’g = �gX -g; X G-g;GG-g;Glnln e−.g; X X e−.g;GlnGe−.g;GlnGln e−.g; LL;

(52)

’g = �mX -m;X G-m;GG-m;Glnln e−.m;X X e−.m;GG

×e−.m;GlnGln e−.m; LL: (53)

On the basis of !ve experimental batches (correspond-ing to a total of 128 measurements), the model param-eters (i.e. pseudo-stoichiometric coeOcients and kineticparameters) are identi!ed in a two-step procedure. The!rst step yields the maximum likelihood estimate ofthe pseudo-stoichiometric coeOcients independently of thekinetic coeOcients, whereas the second step gives the


Fig. 5. State estimation with a continuous-discrete extended Kalman !lter using a model identi!ed with the combined criterion (16) (experiment 3).

Fig. 6. State estimation with a full-horizon observer using a model identi!ed with the combined criterion (16) (experiment 3).

maximum-likelihood estimation of the kinetic coeOcients(together with the culture initial concentrations); see(Bogaerts and Hanus, 2000) for more details.Based on the models (48–53), the maximum-likelihood

estimates of the parameters, and on measurements of glu-tamine and lactate concentrations, a full horizon observer(44–45) is implemented in order to estimate the biomass andglucose concentrations. The results corresponding to a batch

(whose measurements were not used for parameter estima-tion) are given in Fig. 7. The observer accurately estimatesthe measured part of the state, but produces bad estimatesof the unmeasured biomass and glucose concentrations.In order to improve state estimation, the second step of the

above-mentioned identi!cation procedure (i.e. identi!cationof the kinetic parameters) is performed with the combinedcriterion (16). The results obtained with the full horizon


Fig. 7. State estimation with a full horizon observer using a model identi!ed with the maximum-likelihood criterion (14).

Fig. 8. State estimation with a full horizon observer using a model identi!ed with the combined criterion (16).

observer using these new parameter values are given inFig. 8. Signi!cant improvements can be observed in the es-timation of the unmeasured biomass and glucose concentra-tions.Again, the proposed parameter identi!cation procedure

appears as a useful tool to overcome potential lack of sensi-tivity of the unmeasured states with respect to the measuredones. In these experimental applications, the degrees of free-dom available at the modeling stage, e.g. the model struc-ture and parameter values, can be used as “tuning knobs” toenhance the intrinsic model connections.

6. Conclusion

In this study, a new parameter identi!cation procedureyielding a model dedicated to state estimation purposes isproposed and thoroughly analyzed. Basically, a new costfunction is developed which combines a conventional maxi-mum likelihood criterion with a “measure of observability”.Based on the concept of canonical observability forms ofnonlinear systems (Zeitz, 1984, 1989; Gauthier and Kupka,1994), this measure quanti!es the ability to detect in theoutput trajectories any diHerences in the initial states.


The minimization of the proposed cost function enforceshigher model sensitivities and in turn, a better transfer ofinformation from measured to unmeasured variables. Thesimulation studies described in this paper show that thisgoal is achieved by modifying the numerical values of somemodel parameters in order to compensate an intrinsic lack ofsensitivity with respect to some of the state variables. Thismodel “falsi!cation” allows signi!cant improvements in thequality of state estimates provided by software sensors.These observations are also con!rmed by experimental

results obtained in the context of batch animal cell cultures.In these experimental applications, the degrees of freedomavailable at the modeling stage, e.g., the selection of themodel structure and parametrization, can be used to enhancethe sensitivity of the unmeasured states with respect to themeasured ones.Even though high-gain observers could be developed to

cope with bad observability properties of nonlinear systems,they usually perform poorly in the case of discrete-time(possibly rare), noisy measurements. In this latter situation,which is very important in bio-chemical applications, the“identi!cation for state estimation” appears as an appealingalternative approach, which allows continuous-discrete ex-tended Kalman !lters or receding-horizon observers to besubsequently designed. Both observers provide a rigorousstochastic treatment of discrete-time measurements and arebest suited to bio-chemical applications.

References

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parameter identification for state estimation—application to bioprocess software sensors

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