ADAPTIVE ALGORITHMS FOR ESTIMATION OF MULTIPLE BIOMASS GROWTH RATES AND BIOMASS CONCENTRATION IN A CLASS OF BIOPROCESSES

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1 ADAPTIVE ALGORITHMS FOR ESTIMATION OF MULTIPLE BIOMASS GROWTH RATES AND BIOMASS ONENTRATION IN A LASS OF BIOPROESSES V. Lubenova, E.. Ferreira Bulgarian Academy of Sciences, Institute of ontrol and System Research, Acad. G. Bonchev str, P.O. Box 79, Sofia Bulgaria, tel fax:+59 76, lubenova@iusi.bas.bg entro de Engenharia Biologica - IBQF, Universidade do Minho, Braga, Portugal ecferreira@deb.uminho.pt Abstract. An approach for multiple biomass growth rates and biomass concentration estimation is proposed for a class of bioprocesses characterizing by on-line measurements of dissolved oxygen concentration and off-line measurements of biomass concentration. The approach is based on adaptive observer theory and includes two steps. In the first one, an adaptive estimator of two biomass growth rates is designed using on-line measurements of dissolved oxygen concentration. In the second step, the third biomass growth rate and the biomass concentration are estimated on the basis of other estimator using additionally off-line measurements of biomass concentrations. The simulation results demonstrated the good performance of the estimators under fed-batch conditions. Keywords: adaptive systems, biotechnology, estimators. INTRODUTION The kinetic rates are ones of the most important parameters of the bioprocesses. Their exact estimation allows to be realized on-line by adaptive control algorithms. Moreover, the information about their changes in the time is important and useful with regard to the investigations and study of every bioprocess. In many practical cases, it is impossible to measure the kinetic rates directly. Their estimation becomes a necessary step. Some approaches, proposed in the literature, are based on extended Kalman filter (Dochain and Pauss, 988; Lubenova, 99), adaptive system theory (Lubenova, 99; Bastin and Dochain, 989; azzador and Lubenova, 995; Ferreira, 995; Oliveira et al., 996; Pomerleau and Perrier, 99), high gain approach (Farza et al., 997), hybrid neural network based approach (hen et al., 995) etc. A considerable part of the authors direct their efforts to estimation of the specific growth rate in simple culture. Multiple rates estimation is realized in small number of works (Ferreira, 995; Oliveira et al., 996; Pomerleau and Perrier, 99). Pomerleau and Perrier (99) used the observer-based estimator (Bastina and Dochain, 989) and a pole placement based tuning in the estimation of three specific growth rates of baker s yeast fed-batch fermentation. Ferreira (995) and Oliveira et al. (996) proposed a second order dynamic based tuning for the design parameters of the same general estimator to estimate these rates in the same fermentation process. In both cases, two partial models are used to describe the process. The estimation is carried out using a switch between two partial observer based estimators using on-line measurements of the dissolved oxygen and carbon dioxide concentrations. This paper is dedicated to the estimation of the multiple biomass growth rates and the biomass concentration of a class of bioprocesses, which is characterized by the following reaction network: S + O S µ S (P) +O µ X + () X + + S (P) () µ X + ()

2 where S is the first substrate; O dissolved oxygen; X biomass; S (P) second substrate or product depending on the metabolic pathway; carbon dioxide; µ, µ, and µ are the specific growth rates for the three metabolic pathways. In the sequel O, X, S, S, will represent the concentrations of these variables. Pathways (), () and () refer respectively to the respiratory growth on substrate S, fermentative growth on substrate S and the respiratory growth on S. It is considered the case when the processes, belong to the class above, are described using only one model. Two adaptive algorithms are proposed. The first one estimates two of the biomass growth rates R and R using on-line measurements of dissolved oxygen concentration. The second algorithm estimates the biomass growth rate R and the biomass concentration using additionally off-line measurements of biomass concentration. The second estimator need the estimates R and R, obtained by the first algorithm. The performance of proposed estimators is investigated by simulation.. PROBLEM STATEMENT A general dynamical model of a stirred tank reactor (Bastina and Dochain, 989) has the form: dξ = K r( ξ ) Dξ + F - Q (4) where ξ is the n vector of state variables; K - the n m yield coefficient matrix; D - the dilution rate; F - n feed rate vector, Q n gaseous outflow rate vector, r (ξ)- a m r reaction rate vector. The dynamics of the class of bioprocesses, which is characterized by the scheme (), () and () is given in the matrix form (4), (5) by the following vectors and matrix: X S k k ξ = S ( P) ; K = k k4 ; O k5 k6 k 7 k8 k9 DS in µ X R F = ; Q = r ( ξ ) = µ X = R (5) OTR µ X R TR where k k 9 are yield coefficients, OTR is the oxygen transfer rate, TR is the carbon dioxide transfer rate, and S in is the first substrate concentration in the feed. For the system (4), (5), it is assumed that: A. Only dissolved oxygen concentration O, the oxygen transfer rate OTR, and the carbon dioxide transfer rate TR, and carbon dioxide concentration are measured on-line. A. The elements of the yield coefficient matrix K are known and constants. A. The off-line measurements of biomass concentration X are available. For the class of bioprocesses, given by the system (4), (5) and under the assumptions A A, the following problem is considered: Estimation of the biomass growth rates R, R and R, as well as the concentration of biomass on the basis of adaptive algorithms, derived by the theory of adaptive estimation using on-line measurements of the dissolved oxygen concentration and the off-line X measurements.. ADAPTIVE ALGORITHMS DESIGN. Estimator of R and R (Estimator I) According to Eq. 5, the dynamics of the dissolved oxygen concentration is presented by: do = k5 R k R DO OTR (6) 6 + From Eqs. (6), it follows that the oxygen uptake rate (OUR) is given by: OUR = k + (7) 5R k6 R Then the time derivative of the reaction rate R can be presented in the following way: dr dour k6 dr = (8) k5 k5 Define the auxiliary parameter: dour k dr 6 I R II k5 k5 ρ = + (9) with I and II - positive constants or time-varying parameters. ombining Eqs. (6), (8), and (9), it is possible to derive the following adaptive estimator of R and R : doˆ = k Rˆ k Rˆ 5 6 DO + OTR + O Oˆ (a) drˆ II I = ρˆ Rˆ + O Oˆ V O Oˆ 4 (b)

3 drˆ dρ = O Oˆ (c) ( O O) ˆ = ˆ 4 (d) V I II = V + (e) where O ˆ, R ˆ, R ˆ and ρˆ are estimated values for O, R, R and ρ, while I, II,,, and 4 are estimator parameters, which must be chosen according to stability conditions, V is an auxiliary variable.. Stability analysis Defining the estimation errors: O = O Oˆ, R = R Rˆ, R = R Rˆ, ρ = ρ ˆ ρ consider the following error system: dx = A x + u where R x = ; R ρ A - = V 4 It is assumed that: dr u =. d ; ρ - k 5 - k 6 - A5. >; = h + h - I ; A6. >; =[(h - h ) - ( - I ) ]/4.k 5 A7. c >; c =- /k 5 A8. = c k 6 ; 4 = c k 5 V I II () where h = -h ', h = -h ' with h, h positive constants, h ', h ' - eigenvalues of the matrix: B - k 5 = I A9. The vector [k 6, k 5 V ] T is a persistently exciting signal. dr A. dρ M, and A. where M and M are upper bounds. M Statement : Under assumptions A to A, there exist positive finite constants l, l and l such that the error x is bounded as follows, for all t: x (t) l x () + l M + l Proof:. Defining R * = R V ( ρ ) the following system is equivalent to the error system (): - - k - k - V k 5 6 * 5 I * d R = - - R R + R ρ 4 ρ dr. d () ρ. The homogeneous part of () can be written: * d R T * R = B - B R () B P R ρ ρ with P = - k5 B = I ; c k6 k 5 V B = ; The matrix T c k5 c P B + B P = I k5 c is negative definite by the assumptions A5 A8. The matrix B is stable by the same assumptions. Then the exponential stability of () follows from Theorem. (Bastin and Dochain, 989).. The forcing term of () is bounded by assumptions A and A. 4. Then it is a result of adaptive system theory that the state of () is bounded (Theorem A.6, Bastin and Dochain, 989).. Estimator of R and X (Estimator II) Define the auxiliary parameters: ρ = OUR X (4) + III IV ( R X ) ρ = (5)

4 where III and IV are parameters, which can be positive constants or time-varying parameters. Write the dynamical model of dissolved oxygen concentration, given by Eq. (6) and using Eq. (4) as: do = ρ + X DO + OTR (6) According to Eq. (5), the dynamical model of X can be written as: dx IV III = R + R ρ X DX (7) Using Eqs. (6), (7) and on-line measurements of the dissolved oxygen concentration and the off-line measurements of the biomass concentration, the following adaptive estimator can be derived: doˆ = ˆ ρ + Xˆ DO + OTR + ( O Oˆ) (8a) 5 dxˆ = ˆ + ˆ IV III R R ˆ ρ Xˆ DXˆ + ( O Oˆ ) + I I 6 + V ( O Oˆ) + ( X Xˆ ) (8b) 8 d ˆ ρ = ( O Oˆ) 7 d ˆρ x ( O Oˆ ) X X = ˆ 8 x L L (8c) (8d) dv III IV = ( + D) V + (8e) where Oˆ, Xˆ, ρˆ and ρˆ are estimated values for O, X, ρ and ρ, while III, IV, 5, 6, 7, 8, x and x are estimator parameters, V - an auxiliary variable, X L and Xˆ L - the off-line measurements and the estimates of X, Rˆ, Rˆ - the estimated values of I I the R, I R I obtained by the first algorithm. An estimation of R is obtained using ρ, X estimates and the relationship (5) as follows: Rˆ IV III = ˆ ρ Xˆ (8f) The stability of the estimator (8) can be proved like estimator I under the following assumptions: a) in the case where the off-line X measurements are not available: A. 5 >; 5 = h - ( III + D), h positive constant. A. 6 >; 6 =[ 5 -( III +D)] /4 A4. c >; c = 6 A5. 7 = c, 8 = c V A6 The vector [, V ] T is a persistently exciting signal A7. A8. dρ M ; dρ M4 where M and M 4 are upper bounds. b) in the case where the off-line X measurements are available: A9. 5 >; 5 = h - ( III + x + D), with h positive constant. A. 6 >; 6 =[ 5 -( III + x +D)] /4 A. c >; c = 6, A. 7 = γ A. 8 = A4. x = γ IV with γ, γ positive constants. 4. SIMULATION INVESTIGATIONS The behaviour of proposed adaptive estimators is investigated by simulations of a process model, which belongs to the class defined by Eqs. (4), (5) where specific growth rates are given by Monod models with constant parameters µ max, µ max, µ max maximum specific growth rates, K s, K s, K s saturation constants. The values of the model parameters are given in Table. It is considered fed-batch process. The initial values of variables and parameters X, S, R, R, R are given in Table. In Fig., the simulation results from estimator I under different values of design parameter I are shown. The simulations results of estimator II are plotted in Fig.. On the basis of the obtained results, the following conclusions can be drawn:. Although the proposed estimators have not small number parameters, the proposed tuning procedures reduce their number to: three for estimator I: I, h I and V (); four for estimator II: III, h II, x and x. The estimates under different values of the sampling period T x coincide.. More accurate estimates are obtained for R, R and X in comparison with R. 4. In the considered simulation investigations, the values of the tuning parameters are chosen using a trial-and-error approach. It is possible to be proposed and applied other tuning methods under the

5 experimental validation of the proposed software sensors. One possibility to be searched is a reasonable trade-off between noise sensitivity and convergence using criteria proposed by laes and Van Impe (997) on the basis of the experimental data. 5. ONLUSION An approach for estimation of multiple biomass growth rates and biomass concentration for a class of aerobic bioprocesses is proposed. It is based on adaptive observer theory and requires on-line measurements of dissolved oxygen concentration and off-line measurements of biomass concentration. The practical applicability of the proposed approach is a direct consequence of several important factors. The estimators (i) are not depend on any particular models of the biomass growth rates, which are assumed to be unknown time-varying parameters; (ii) require only on-line measurements of dissolved oxygen concentration, which can be performed easily using cheap and reliable sensors. The results by simulation demonstrated the good performance of the proposed estimators under fedbatch conditions. The values of tuning parameters were chosen using a trial and error approach. The experimental validation of the proposed strategy gives the possibilities to be applied other tuning procedures, connecting with the use of experimental data. Table : Model parameters µ max =.5 h - µ max =.8 h - µ max =. h - K s = g l - K s = g l - K s = g l - k =.5 k =. k 5 =. k 6 =.8 k 7 =. k 8 =.5 k 9 =. S in = g l - OTR=. g l - min - TR=.5 g l - min - 6. REFERENES Bastin, G. and Dochain, D. (989). On-line Estimation and Adaptive ontrol of Bioreactors. Elsevier Science. azzador, L. and Lubenova, V. (995). Nonlinear Estimation of Specific Growth Rate for Aerobic Fermentation Processes. Biotechnology and Bioengineering, 47, hen et al., (995). ombining Neural Networks with Physical Knowledge in Modelling and State Estimation of Bioprocesses. Proc. of rd European ontrol onference, Rome, Italy, September, 46. laes, J. E. and J. F. Van Impe, (997). Experimental Validation of Software Sensors for Pure and Mixed ulture Microbial onversion Processes. Technical Report, BioTe. Dochain, D. and Pauss, A. (988). On-Line Estimation of Microbial Specific Growth Rates: an illustrative case study. anadian Journal of hemical Engineering, 47, 7-6. Farza, M., S. Othman, H. Hammouri, J. Biston, (997). A Nonlinear Approach for the On-Line Estimation of the Kinetic Rates in Bioprocesses. Bioprocess Engineering, 7, 4-5. Ferreira, E.. (995). Identification and Adaptive ontrol of Biotechnological Processes, Ph. D. Thesis, University of Porto, Portugal. Lubenova, V. (99). State and Parameter Estimation of Biotechnological Processes. Ph. D. Thesis, Technical University of Sofia. Oliveira, R., E.. Ferreira, F. Oliveira and S. Feyo de Azevedo, (996). A Study of the onvergence of Observer-Based- Kinetics Estimators in Stirred Tank Bioreactors. Journal of Process ontrol, 6, Pomerleau, Y. and M. Perrier, (99). Estimation of Multiple Specific Growth Rates in Bioprocesses, AIhE Journal, 6,, 7-5. Table : Initial conditions X()=4.56 g l - S ()=.86 g l - R ()=.694 g l - min - R ()=.86 g l - min - R ()=.445 g l - min - D=. h -

6 Biomass growth rate R (g/lmin) (a) Biomass growth rate R (g/lmin) (b) Figure. Estimates of R, R, and ρ by estimator I and the true value of the same parameters (lines) under different values of the design parameter I ( II =): I =.5 (o points), I =. (+ points), h, I = (a).5 (b) Biomass concentration (g/l) Biomass growth rate R (g/lmin) Figure. Estimates of X, R, ρ, and ρ by estimator II and the true value of the same parameters (lines) under different sampling times T x of biomass measurements. Values of design parameters: estimator I: I =.5, II =, h, I = -5, estimator II: III = -D+., IV =, h, II =- and x =, x = 7 =5 (o points for T x =5 min and with + points for T x = 5 min).

ON-LINE METABOLIC FLUX ANALYSIS IN A PHB PRODUCTION PROCESS. João Dias, Mário Eusébio, Luísa Serafim, Adrian Oehmen, Maria A.M. Reis, Rui Oliveira

ON-LINE METABOLIC FLUX ANALYSIS IN A PHB PRODUCTION PROCESS. João Dias, Mário Eusébio, Luísa Serafim, Adrian Oehmen, Maria A.M. Reis, Rui Oliveira th nternational FA ymposium on omputer Applications in Biotechnology reprints Vol., June 6, 7, ancún, Mexico LE METABL FLU AAL A HB RDUT RE João Dias, Mário Eusébio, Luísa erafim, Adrian ehmen, Maria A.M.