Model Reference Adaptive Estimation Applied to a Bioprocess

Copyright © IFAC Integrated Systems Engineering. Baden-Baden. Gennany. \994

MODEL REFERENCE ADAPTIVE ESTIMATION APPLIED TO A BIOPROCESS M. MAHER, I. QUEINNEC*, F.Y. ZENG+ and B. DAHHOU· Laboratoire cl' Analyse et d' Architecture des Systemes du CNRS 7, Avenue du Colonel Roche, 31077 Toulouse Cedex - France. +Centre de BioengMnierie Gilbert Durand INSA. URAICNRS 544, 54. Avenue de Rangueil. 31077 Toulouse Cedex - France .

Abstract. We are concerned in this paper by some experimental studies of an adaptive estimation algorithm based on the reference model approach, using a non linear process model. The adaptive estimation theory is then exposed before to be applied on a real-life batch fennentation process. Particular emphasis is placed on the experimental evaluation of the method. Key words. adaptive estimation; bioprocess; Nonlinear systems; Narendra's method

1. INTRODUCTION

Narendra theorem (Duarate et al. 1989). from estimation error equations and Strictly Positive Real condition (SPR) . It allows to simultaneously estimate a part of the process state and some parameters which are involved in the state space modelling. Our objective is to recover all the process state evolution from only the on-line measured variables.

The critical issue in controlling bioprocesses is that efficient and cheap sensors for on-line measurements of the main biological variables are most often not available. Observers must then be implemented to bypass the lack of sensors. Another bottleneck to perfonnant control of bioprocesses is that physical models are non linear with time-varying process parameters. Mathematical modelling of these process is then far from being an easy task, and leads to complex non linear state space representations based on mass balances considerations. Adaptive estimators can then constitute a valuable alternative with the aim of controlling the bioprocess.

The paper is organized as follows. After a brief description of the process, some comments are given about the process model. The state and parameters estimation method is then presented. Stability and convergence are not proven here as this has been done in a previous paper (Zeng et al. 1993a). Algorithms are then designed and two experimentations of a batch fennentation on the real pilot plant are presented and discussed. Our conclusion emphasizes the good perfonnances of the estimator which can then be viewed as the estimation part of perfonnant adaptive control of the real-life process.

In recent years considerable work has been done on the problem of combined state and parameter estimation for adaptive controllers. Numerous algorithms have been implemented and for some of them successfully applied to engineering systems. The extended Kalman filter was applied by San and Stephanopoulous (1984). Nihtila et al. (1984); Chamilotoris and Sevely (1988) used a gradient type estimation; Bastin and Dochain (1990) proposed an extended Luenberger adaptive observer; Flaus et al. (1991) suggested a recursive prediction error method.

2. PROCESS ANALYSIS AND MODELLING Cultures of microorganisms are used in the fennentation industry, e.g., for producing enzymes. in biological degradation of wastes. and in many bacteriological experiments. Cultivations are characterized by the biological degradation of substrate by a population of microorganisms (biomass) into metabolites, primary or secondary, consisting. for example. of alcohol in the case of ethylic fermentation. The experimental alcoholic

Our contribution in this paper is to present an experimental evaluation of the reference model adaptive estimation (Zeng tt al. 1992. 1993a. b) The adaptive estimation law is obtained. using • Members of the GR-Automatique-SARTA-CNRS.

429

fennentation plant we are concerned with is a typical batch fennentation process.

The equations of a batch fennentation process are:

The strain used for experiments was Saccharomyces cerevisiae UOS. The carbon source was cerelose. The reactor consisted of a 2 I SOl 2M fermentor equipped with a magnetic agitator and temperature and pH control. Typical operating variables and parameters for the experimental plant are given in Table 1.

dC(t) / dt = p(t)C(t) dS(t)/ dt=-J....p(t)C(t)

(1)

Yx 1

dP(t) / dt = --p(t)C(t)

YP

where eft), S(t) and P(t) are the biomass, substrate and product concentrations (g/l) respectively; pet)

Table 1 Typical operating Parameters Active volume

1.5 I

is the specific biomass growth rate (h -1); Y x and Y p are the substrate-biomass and the substrate-

Temperature

30°C

product conversions yields respectively.

Stirrer speed

200 rpm

pH

Many analytical laws have been suggested in the literature (Bastin and Dochain, 1990) for specific growth rate modelling, which take into consideration the limitation and/or the inhibition of the growth by certain variables of the process.

3.8

Off-line glucose concentration was measured, every 15 minutes, by using a semi-automatic YSI 27A enzymatic analyser which consists of an immobilized glucose oxydase membrane with an oxygen sensor. Ethanol concentration was measured by gas chromatography using isopropanol as internal standard with a period of 30 minutes. Biomass concentration was evaluated by dry weight (every 3 hours) and turbidimetry at 620 nm (every 15 minutes).

The model we are concerned with is then the following: p(t)= Pm

S

Ks+S

(1-~)

(2)

Pm

where Pm is the maximum growth rate (h-1) and Ks is the "Michaelis-Menten" constant (g/l) and Pm is the ethanol inhibition factor (g/1). The retained model parameters are given in Table 2. They have been determined such that simulation roughly correspond to real-life experiments. The structure of this model will be used to elaborate the following adaptive estimation method.

The on-line glucose analysis was carried out by the same enzymatic analyser (YSI 27 A) by association of the glucose analyser with an automated injection syringe (Queinnec et al. 1992). A set of pneumatic jacks around the syringe ensured sampling of the culture medium from a loop disposed on the culture vessel and injection into the glucose analyser. By first selecting the distance of syringe travel, i.e., the volume to be injected, the working scale of glucose concentrations to be measured could be set up to 200 (g/l). All procedure for sampling, injection and control of the analyser were managed by a Progrnmmable Logic Computer ( PLC ).

Table 2 Modelling parameters

The plant was linked to a PC/AT compatible microcomputer. Process monitoring and control were achieved using dedicated software written in Turbo Pascal composed of a group of tasks including data acquisition and storage, graphic display, printing. PLC management, parameter estimation and evaluation of the control signal if necessary. Interfacing between the microcomputer and the process was carried out by a RTI 815 board from Analog Devices' family.

Pm

0.38 h-1

Ks

5g/l

Pm

100g/l

Yx

0.07

Yp

0.16

3. ADAPTIVE ESTIMATION 3.1. Problem Statement The reproducibility of experiments is often uncertain as it may be difficult to obtain the same environmental conditions and prevent changes in the internal state of the organism. Morever the state variables, i.e. the substrate, biomass and product concentrations are mainly determined by laboratory analysis. The expense and duration of laboratory detenninations limit measurement frequency. That is

2.1. Process Model

In many bioprocesses, the original process model is usually described by a set of non1inear differential equations derived from material balances considerations, and involving modelling of the growth rate.

430

a serious obstacle for process monitoring and control. Due to those facts. it is important to investigate practical estimation.

where

Eee = Oe - O. Eec = Ce - C. 11 Hp = [dJ.l/ dO]e,s' 11 E R+ is the positive common factor of the components of [dJ.l/de]e,s'

In our case. only substrate concentration S(t) was measured on-line, and therefore the specific growth rate was expressed as the following:

J.l S J.l(t) =J.l(O,S) = _m_ Ks+S

(3)

Substituting equation (6) in equation (5) the estimation error equation can be obtained by subtraction of equations (5) - (4) that is:

It is clear that in this expression J.lm takes into consideration the influences of all process components. and in particular the inhibition of the growth by the product formation. It is then a timevarying parameter.

(7)

dEe/ dt= AeEe+BE;ew Ees =hT Ee where

The adaptive estimation method proposed in this paper is elaborated from the nonlinear structure of the process model and the reference model approach. The adaptive estimation law is obtained. using Narendra theorem (Duarate et al. 1989), from estimation error equations and SPR condition. It allows to estimate on-line both non-measured process state (C(t). P(t) and one component of parameter vector 0 T =[J.l m K s], the other one being a priori known.

E;=[Eec Ees]. Eec=Ce-C, Ees=Se- S • Ee8=8 e -O, Ae =

BT=11[l W

=Ce Hp

;

-:J.

J.

hT = [0

1],

is a bounded time function.

In equation (7) matrix Ae is a time-varying matrix composed of both physical components and estimation gains. Consequently, gains a and f3 can be chosen to obtain a stable estimator system matrix Ae.

3.2.~

The process dynamics is described by:

dC(t) / dt =J.l(t)C(t)

[-J.l~ Y x

(4)

1

The error transfer function We (p) of system (7) is:

dS(t) / dt = --J.l(t)C(t) Yx We assume that the structure of the specific biomass growth rate J.l(t) is a known function of substrate concentration S( t) and biological parameters

OT = [J.l m Ks].

In order to ensure that the above formula is strictly positive real. we can fix the two poles of the transfer function Weep) such that they are both equal to ( -gJ.l ). This condition is obtained for a and f3 expressed as follows:

An estimation reference model is described by the following equations:

(5)

a=Y x (1+g)2J.l

(9)

f3 =-(1 + 2g)J.l where the same process model structure. equation (4), is used. Parameters a and f3 are estimator gains.

where J.l > 0 and g> O. Coefficient g is chosen in order to assure convergence of the algorithm.

The first order Taylor expansion of J.l(t) allows to obtain the approximate linear relationship among the specific growth rate estimation error Eep. and the parameters estimation error Ee8 as follows:

The parameter vector adaptive adjustment algorithm can then be obtained: (lO)

(6)

431

with r =rT is a positive definite matrix of design parameters .

r

=

Since. in equations (9) and (10). J.l.(t) and Hp are inaccessible. an approximate method using their estimate val ues J.l. e and He is adopted.

[Ylo O' 0]

and the explicit parameter adjustement law becomes: (14)

3.3. Algorithm

and

In summary. the adaptive estimation algorithm. which recovers all the process state and parameter evolutions. consist of the following equations: Values of the design parameters and initial conditions are listed in Table 3.

d Ce I dt =J.l.eC e + a(Se- 5) 1

dSel dt=--J.l.eC e+ P(Se- S)

Table 3 Initial conditions and design parameters

Yx

=Y x(1+g)2 J.l. e P =-(1+ 2g)J.l. e a

(11)

d Oel dt= rCeHeEes J.l.e=J.l.(Oe'S) where the formulas of J.l. e and He are identical to J.l.(t) and Hp. but the variables are replaced by Ce• S and Oe.

Two applications on data from real-life bioreactor are presented. The adaptive estimation algorithms have been implemented numerically by simply using Euler discretization . The substrate concentration S(t) is measured with a sampling period of 15 minutes. In order to use the above described adaptive estimation algorithm (11) we deduce. from formula (3). the following expressions:

OT =[J.l. m Ks] J.l.",.

8.,5

2

89 g/l

Pe (0)

0.38 g/l

J.l. me (0)

0.38 1/h 0.005 12/ g2 h 2 1.25

The results of filtered substrate concentration. estimated biomass concentration. product concentration and specific growth rate obtained by the adaptive estimation algorithm are shown in Fig.l and Fig.2. respectively for experiment 1 and experiment 2. The substrate concentrations evolutions were superimposed. This imply that. with the same initial conditions. the estimated based on the substrate measurement will have exactly same prom in the two experiments. One can observe the good agreement between the on-line estimations and a few measurements obtained by off-line analysis of biomass concentration. The validation. of this estimation method. was based on a comparaison between the estimate of pet) obtained via the estimated specific growth rate J.l. e and the estimated biomass concentration Ce • and the off-line measurement values of pet). The validation results are shown in Fig.l.c and Fig.2.c. The results clearly showed a realistic behaviour of estimate product concentrations. especially during the first experiment. The choice of the design parameters Y1 and g were made empirically after a set of simulations of the process model. equations (1) and (2). which was assumed to behave approximately as the "true"

4. APPLICATION TO REAL PROCESS DATA

J.l. «).,S)]T J.l.""S

Se (0)

g

(12)

=[J.l.(O.,S)

0.1 g/l

Yl

The product concentration pet) is determined by the following equation:

[dJ.l.I dO]

Ce(O)

(13)

HJ(Oe,S)=[1 _J.l.(O;,S)]

process. The same design parameters were used in the two applications. Fig.l and Fig.2. clearly leading to satisfactory results in both cases. The choice of

In the following. we consider the case where the parameter Ks ( 5 g/l) is known a priori and we estimate J.l. m. Then. we can choose a diagonal matrix for

=

rl

r.

432

experimental results. Biotechnol. Bioeng .• 26, 1189-1197. Zeng. F.Y .• B. Dahhou. G. Goma. and M.T. Nihtila (1992). Adaptive estimation and control of the specific growth rate of a non linear fermentation process via MRAC method. Proc. of 5th ICCAFT & 2nd IFAC-BIO symp., USA. Zeng, F.Y .• B. Dahhou, and M.T. Nihtila (1993a). Adaptive control of nonlinear fermentation process via MRAC technique. App/i ed Mathematical Modelling, 17, 58-69. Zeng, F.Y., B. Dahhou. M.T. Nihtila, and G. Goma (1993b). Microbial specific growth rate control via MRAC method. Int. J. Systems Sci., 23 , l-

and g could also be validated from off-line additional measurements as in Fig.1.c and Fig.2.c. The results obtained from two real process data showed that the adaptive estimation algorithm was quite appropriate for the on-line estimation of the state and parameter in a batch fermentation process. from only the on-line substrate measurement. S. CONCLUSION Experimental reSUlts, concerning a method for estimating the biomass concentration. the product concentration and the specific growth rate in a nonlinear batch fermentation process have been presented in this paper. Based on the knowledge of process model structure. this adaptive estimation used the measured substrate concentration for the recovering all the process state evolution. The methodology was based on a model reference adaptive technique. The estimate errors gains were chosen without dynamics to obtain prescribed stable eigenvalues for the estimate error system. The adaptive adjustement law of the estimator was derived from Narendra Theorem. The experimental results pointed towards the performance of the adaptive estimation method proposed in this paper. It is worth noting that this algorithm is easy to tune and to implement and that the proposed estimation method could be coupled with adaptive control of fed-batch and continuous fermentation processes (Zeng et al. 1993a).

B.

6. REFERENCES Bastin, G., and D. Dochain (1990) . On-line estimation and adaptive control of bioreactors. Elsevier science publishers B.V., Amsterdam, The Netherlands. Chamilothoris, G., and Y. Sevely (1988). Adaptive control of biomass and substrate concentration in a continuous-flow fermentation process. APTIRA/RO, 22 , 159-175. Duarate, M.A.. and K.S. Narendra (1989) . A new approch to model reference adaptive control. 1nl. J . Adaptive Control and Signal Processing, 3, 53-73. Flaus, J.M., A. Cheruy, J.M. Engasser, M. Poch and C. Sola (1991). Estimation of the state and parameters in bioprocesses from indirect measurements. Proc. of the 1st ECC, Grenoble, France, vol. 2,1642-1647. Nihtila, M.T.• P. Harmo. and M. Perttula (1984). Real-time growth estimation in batch fermentation. Proc. of 9th IFAC World Congress, Budapest, Hungary, 225-230. Queinnec, I., C. Destruhaut. J.B. Pourciel, and G. Goma (1992). An effective automated glucose sensor for fermentation monitoring and control". World Journal of Microbiol. Biotechnol., 8, 713. San, K .• and G. Stephanopoulos (1984). Studies on on-line bioreactor identifi~tion-II; Numerical and 433

lOO

C(gII)

S(

80

eo 7ll

eo

+ measu~d

eo

4

• estimated

40

+ measu~d

30

...

• estimated

10

Time b

o~=r==r=~~__r_~~--~-r~r-~T~i~m~e~(h~~

,.

IZ

10

IZ

10

0

++ I.

Fig.l.b. Concentrations of substrate for experiment 1.

Fig.l.a Concentrations of biomass for experiment I.

~.~--------------------------------------.,

411

P(gII)

40

J.l(IIb)

» ~

211

... + measu~d • estimated

I. 10

• +

..

+ +

0 .1

Time h

0 0

0.2

10

I.

12

10

Fig.l.c. Concentrations of product for experiment I.

12

I.

Fig.l.d. The estimated specific growtb rate for experiment I.

100,---------------------------------------------,

.~--------------------------------------~ C(gII)

• eo 4

o

measu~d

10 40

o measu~d • estimated

I

10

O+--r__r-~~--r_~~--~~--r_~T~im~e~b)~++ 10 12 I.

IZ

10

Fig.2.a Concentrations ofbiomass for experiment 2.

Fig.2.b. Concentrations of substrate for experiment 2. ~.

45

P(gII)

40

J.l(llb) 0..

»

•

~

0.'

•

25

o.J 20

o measu~d • estimated

I. 10

cu

~1

•

Time b

0

0

10

12

Time(b

0

I.

D

10

Fig.2.d. The estimated specific growtb rate for experiment 2.

Fig.2.c. Concentrations of product for experiment 2.

434

12

I.