Engineering Innovations, ISSN: 2813-1002, Vol. 5, pp 39-48

doi: 10.4028/p-si0156

© 2023 The Author(s). Published by Trans Tech Publications Ltd, Switzerland.

Submitted:
2022-10-25

Revised:
2023-03-14

Accepted:
2023-03-17

Online:
2023-04-19

Forward-Backward Propagation to Identify the Maximum Specific Growth Rates of a Bioreactor

The original version of this paper is available on https://www.scientific.net/EI.5.39.pdf

BORSALI Salima ,

¹Automatic Laboratory, Department of Electrical Engineering University of Tlemcen, Algeria

borsali.salima@gmail.com

Keywords: Estimation, Bioreactor, Interval arithmetic method, Forward backward technique.

Abstract

In this article, we are interested in identifying the parameters of an aerobic bioprocess model used for wastewater treatment. In the field of biotechnology, various computer bugs caused by round- ing errors can induce an error interval that is too wide during data acquisition. For this reason, we are testing a new identification method using a set method based on interval arithmetic. The process studied is the chemical transformation of ammoniacal nitrogen which takes place in two stages: Re- action of nitrification-denitrification. The parameters chosen for the identification are the yields and the maximum growth rates. Initially, the study of observability by a differential algebraic method will simplify the study of the mathematical model. This nonlinear model is described by six differential equations. Subsequently, we apply a set method, in particular the propagation of constraints also called forwardbackward propagation, this technique allowed us to determine intervals containing the variable returns as well as the maximum specific growth rates defined from the Monod model which describes the operation of the bioreactor. This method also guarantees the result by rejecting all inconsistent values.

Introduction

The original version of this paper is available on https://www.scientific.net/EI.5.39.pdf

In order to estimate the parameters of a bioreactor, we are interested in this study at a new identification method. This choice is motivated for several reasons. The first is due to an important problem faced by the scientific community and which concerns the various computer bugs caused by rounding errors [1]. The other reasons for our motivation is to find solutions to the problems of convergence and of estimates caused by the errors of modeling and noises of measurements [2], therefore, one cannot rely only on conventional methods of identification. The set approach based on interval arithmetic is well suited in the biotechnology field because in these cases the error interval introduced during data acquisition is too wide as is also the case in biology, chemistry or medical sciences. In this context, a variable is represented by a set called the likelihood set or domain, supposed to contain the real value x . The objective is to identify the parameters of a bioreactor as part of an ammoniacal nitrogen nitrification process, chemical reaction observed during depollution of wastewater. The first models to describe such processes are those reported by the international Association on Water Pollution Re- search and Control(IAWPRC) [3]. One of them is the activated sludge model [4] using Monod kinetics. It is a complete model describe by many parameters making it very complex and therefore identifi- cation by statistical methods is difficult or even impossible. These models are also not suitable for online control. Other models can be considered for the online control of an activated sludge process, using models like ARMAX [5], fuzzy logic [6] and neural networks [7]. The last two methods allow easier handling of nonlinearities. The identification of the parameters of a bioreactor using genetic al- gorithm are cited in [8]. The set techniques used in this study are an approach known for many years. The bases of these methods were established by R. Moore [9], then Neumaier [10] and Hansen [11]. There are applications in many fields for estimating uncertain parameters. This is the case in robotics for the identification of dynamic parameters [12, 13]. Due to the difficulties of acquiring measurements, the use of interval arithmetic is not new in the field of biotechnology [14, 15]. First, we apply a differential algebraic method [16,17],[18], in order to study the observability and identifiability of the model. Indeed, models of biotechnological processes are often described in terms of differential
algebraic equations which therefore lend themselves well to this approach. The article is organized as follows, section 2 describes the essentials of the algebraic approach, and it will be followed in section 3 by the basic definitions of interval arithmetic as well as the principle of constraint propagation. In section 4, we present the bioreactor model whose dynamic parameters we want to identify. The results of the simulation and the discussion are given in section 5 followed by a conclusion.

The Algebraic Method

The original version of this paper is available on https://www.scientific.net/EI.5.39.pdf

The study of observability and identifiability problems using differential algebraic methods dates back to the early 1990s [19, 20],[21]. The principle of the method used in this article consists in verifying the observability of a latent variable compared, for example, to two variables and , if each com- ponent of is solution of an algebraic equation not differential with coefficients dependent on and and a finite number of their time derivatives . In fact, to verify the observability of a variable with respect to and of a system:

Then, for we compute a characteristic set of all differential polynomials we compute a characteristic set of all differential polynomials with respect to a classification whose order must be respected. This classification means that all the derivatives of v and w are less than x and all the derivatives of x are less than . This characteristic set will be represented by a set E of differential polynomials, each directed by one and only one of the variables. We conclude that x is observable with respect to v and w if and only if, each component of it leads to a differential polynomial in E. To do this, a REDUCE package called astb (Diop S, 2002) is used.

Constraint Propagation

The original version of this paper is available on https://www.scientific.net/EI.5.39.pdf

Interval arithmetic

The original version of this paper is available on https://www.scientific.net/EI.5.39.pdf

An interval, denoted by , is a bounded and connected set of , which is defined by:

The real numbers and are the lower and upper bounds of , respectively all intervals within R are denoted by , basic athematical operations are extended to intervals.
Let , then, we define.
Its lower bound:
Its upper bound:

Its Middle: mid
Its radius:
A box (or interval vector) is a compact of , defined by the Cartesian product of n intervals.Therefore,

Elementary mathematical operations are extended to intervals. The result of an operation between two intervals of finite terminals is an interval obtained by working only on their bonds.
Let and , then

In practice, we use less abstract equations:

The division is defined by:

In order to reduce an interval or block, bisection techniques are used. However, the result of operations between intervals is not minimal because of pessimism (Tarek Raissi, 2004) .Contraction techniques have been developed (Luc Jaulin, Michel Kieffer, Olivier Didrit, and Eric Walter, 2001), (Tarek Raissi, Nacim Ramdani, and Yves Candau ,2003) in order to reduce pavement without sometimes having to use bisections. So, a contractor is an operator who makes it possible to contract or reduce an initial search pavement without bisection and keeping all the solutions, i.e. eliminating inconsistent values. Constraint propagation on intervals allows on a predefined domain of variables, a considerable reduction in the size of a box without resorting to bisections. The contractor that we use in our study is based on the principle of projection constraints (Isabelle Braems, 2002).
Consider a set , let us note s containing the smallest pavement . s is called the contraction operator. This operator is able to replace a box by a box s with lower size while preserving the entire of solution set.
A contractor must therefore verify the following two properties (Fig1):

Fig. 1: Contraction of a set

Fig. 1. Contraction of a set

Application to the bioreactor

The original version of this paper is available on https://www.scientific.net/EI.5.39.pdf

In practice, modeling bacteria is difficult due to its living nature. Consequently, they are described by complex functions involving poorly known parameters. We focus on the interval approach, in order to represent the uncertainties that affect these parameters. For this application, a mathematical model is used to describe a biological wastewater treatment with activated-sludge process. Ammonia nitrogen is treated by nitrification-denitrification reaction, involving two populations of
autotrophic bacteria. Nitrification is the biological oxidation of ammoniacal nitrogen, it occurs through two steps: First, nitratation by ammonium-oxidizing bacteria, Nitrosomonas (AOB) and second, nitritation by nitride- oxidizing bacteria,Nitrobacter (NOB).

Fig. 2: The two stages of nitrification

Fig. 2. The two stages of nitrification

The state of this bioreactor is described by six variables: the bacterial concentrations, called biomass and denoted and , as well as the concentrations of substrate and is the concen- tration of the incoming substrate. The above model represents the dynamic evolution of these state variables. The model represents the dynamic evolution of these state variables.

with:

and are the biomass growth rates. They are modeled by Monod kinetics[26]. and are the maximum specific growth rates, and are the stoichiometric coefficients which represent the respective yields of biomasses and and are the half-saturation con- stants for the growth corresponding to the cell affinity for the substrate of each bacterial population. D is the dilution rate, such that , where is the input flow of the bioreactor and V its volume.

Results and Discussions

The original version of this paper is available on https://www.scientific.net/EI.5.39.pdf

The observability of the yields and of the system described by Equations (8) cited above is tested by calculating the characteristic set of the following set of differential polynomials. First, we simu- lated the model with parameters and measurements taken from the literature [27] over a period of 220 days. The yields and are obtained by the algebraic method [28]:

In order to illustrate the propagation of constraints, we construct the directed acyclic for and which are represented by Figure 3

Fig. 3: The directed acyclic graph (DAG) for and

Fig. 3. The directed acyclic graph (DAG) for and

We apply the forward-backward propagation after having developed the expressions (10) in primitive functions (see Appendix). For the simulations, we have chosen ;
and for the additive error due to disturbances and the modeling error, we chose . The principle of forward-backward propagation is to select the primitive constraints to be used for the contractions according to an optimal order, in the sense of the size of the domains obtained at the end.
All the intervals of are chosen and are initialized with
The intervals are obtained by simulation on Matlab, using the Intlab toolbox.

Fig. 4: Yield (right), (left)

Fig. 4. Yield (right), (left)

In Figure (4) are represented on the right the yield and on the left the yield , in blue framing obtained by propagation of constraints, in red star model a priori We obtain the intervals for the yields:

Such as:

We apply the same principle to determine interval for the maximum specific growth rates and . The directed acyclic graph (DAG) are represented by figure 4 In the same way, all the intervals of the primitives constraints are chosen and are initialized with [0, + [

Fig. 5: DAG for and

Fig. 5. DAG for and

In Figure (5) are represented on the right the the maximum specific growth rates and on the left the maximum specific growth rates , in blue framing obtained by propagation of constraints, in red star model a priori We obtain the intervals for the maximum specific growth rates

Fig. 6: The maximum specific growth rates and

Fig. 6. The maximum specific growth rates and

As a result, we have been able to associate with the yield parameters and , as well as with maximum specific growth rates and the smallest intervals contain the a priori values obtained by the propagation of constraints

Conclusion

The original version of this paper is available on https://www.scientific.net/EI.5.39.pdf

One of the difficulties in the study of biological or biotechnological systems is due to the experimental samples which can be counted in days. Therefore, the error interval may be too wide. The set methods based on interval arithmetic that we have already used are well suited in these areas. However, it has a drawback which concerns the problem of pessimism due to two phenomena
which can lead to an overestimation of the desired interval. There is a phenomenon of wrapping (wrapping effect) and of dependence or multi-occurrence because each variable is considered to be different. In this study, we tested this technique to identify the parameters of an aerobic bioprocess model for wastewater treatment. In order to avoid pessimism by studying a model with five equations, an algebraic method allowed us to determine simpler expressions for the yield and then to apply the principle of propagation constraints. The Forward-backward propagation technique allowed us to find a minimum interval for the return and as well as for the maximum unobservable specific growth rates and The advantage of this method is that it allows to obtain an interval containing the real value by rejecting the inconsistent values. We can extend our method to the identification of half-saturation constants for the growth and unobservable parameters

Appendix

The original version of this paper is available on https://www.scientific.net/EI.5.39.pdf

Fig. 7: Notations

Fig. 7. Notations

Forward propagation

Backward propagation

Fig. 8: Simulink

Fig. 8. Simulink

References

The original version of this paper is available on https://www.scientific.net/EI.5.39.pdf

S.M.Rump. "Algorithms for verified inclusions theory and practice". In Reliability in In R.E. Moore, editor, editor, volume 19 of Perspectives in Computing, page 109-126. Academic Press, 1988, https: //doi.org/10.1016/B978-0-12-505630-4.50012-2

E.Walter and M.Kieffer "Sur quelques idées fausses ayant des conséquences en identification". J. Européen des Systèmes Automatisés, 42(2) :181-210, 2008

N.J Carstensen and H. Madsen. Identification of waste water processes. PhD thesis, Technical University of Denmark Danmarks Tekniske Universitet, Administration Office for Study Programmes and Student Affairs Afdelingen for Uddannelse og Studerende, 1993

M.Henze, CPL.Grady Jr, W.Gujer, G.Marais, and T.Matsuo. Activated sludge model no. 1 : Iaw- prc scientific and technical report no. 1. IAWPRC, London, 1987

G.Olsson, B.Andersson, B.Hellstrom, H.Holmstrom, L.Einius, P.Vopatek. "Measurements data analysis and control methods in wastewater treatment plants state of the art and future trends". Water Science and Technology. 21(10/11), 1989, pp.1333-1345. https:

D.Couillard and S.Zhu. "Control strategy for the activated sludge process under shock loading". Water Research, 26(5) :649-655, 1992. https : (92) 90241 —

N.Bhat, T.McAvoy, "Use of neural nets for dynamic modeling and control of chemical process systems" Computer and Chemical Engineering 14(5), 1990, pp.573-583. https :

A.Sorsa, R.Peltokangas, K.Leiviska." Real coded Genetic Algorithms and Nonlinear Parameter Identification", International IEEE Conference, Intelligent system.(2008), DOI: 10.1109/IS.2008.4670495

R.E.Moore, "Interval Analysis", Prentice Hall, Englewood Cliffs, New Jersey, 1966 DOI: 10.1126/science.158.3799.365

E.Hansen and R.Greenberg. "An interval newton method". Applied Mathematics and Computation, 12(2): 89-98, 1983. https:

A.Neumaier, "Interval Methods for Systems of Equations", Encyclopedia of Mathematics and its Application, Cambridge 1990,https : . 19920721114

X.Baguenard, M.Dao, L.Jaulin, and W.Khalil. "Méthodes ensemblistes pour l'étalonnage géométrique". Journal Europeen Des Systemes Automatises., 37(9): 1059-1074, 2003.

M.Milanese, and A.Vicino."Estimation theory forNonlinear models and set Membership Uncertainty". Automatica, 27(2),1991, pp.403-408 90090-O

N.Noykova, T.Müller, M.Gyllenberg and J.Timmer " Quantitative Analysis of anaerobic waste water treatment, Identifiability and Parameter Estimation" Biotechnology and Bioengineering, Vol.78, .

J.L.Gouzé, A.Rapaport, and M.Z.Hadj-Sadok. "Interval observers for uncertain biological systems". Ecological modelling, 133(1): 45-56, 2000. (00)00279-9

J.F. Pommaret, "Géométrie différentielle algébrique et théorie du contrôle".C.R. Acad. Sci. Paris Ser. I 302,1986, pp. 547-550.

M.Fliess, "Quelques remarques sur les observateurs non lineaires". In: Proceedings Colloque GRETSI Traitement du Signal et des Images, GRETSI, 1987, pp. 169-172

S.Diop, M.Fliess: "On nonlinear observability" In: Commault, C., Normand-Cyrot, D., Dion, J.M., Dugard, L., Fliess, M., Titli, A., Cohen, G., Benveniste, A., Landau, I.D. (eds.). Proceedings of the European Control Conference, Hermes, Paris, 1991, pp. 152-157 DOI: 10.1109/CDC. 1991. 261405

S.Diop "From the geometry to the algebra of nonlinear observability" Contemporary Trends in Nonlinear Geometric Control Theory and its Applications, A. Anzaldo-Meneses, B. Bonnard, J. P. Gauthier, and F. Monroy-Perez, Eds. Singapore:

Diop S, Simeonov I "On the biomass specific growth rates estimation for anaerobic digestion using differential algebraic techniques". Int. J. BIO automation, 13(3), 2009, pp.47-56, https :

E.Chorukova, S.Diop, I.Simeonov, "On differential algebraic decision methods for the estimation of anaerobic digestion models". Lecture Notes in Computer Science, Springer, Verlag, 2007, 4545, pp.202-216, DOI : 10.1007/978-3-540-73433-815

T.Raissi."Méthodes ensemblistes pour l'estimation d'état et de paramètres". PhD thesis, Univer- sité Paris XII Val de Marne, 2004

L.Jaulin, M.Kieffer, O.Didrit, and E.Walter."Applied Interval Analysis." Springer, 2001.

T.Raissi, Nacim Ramdani, and Yves Candau."Parameter estimation for non-linear continuoustime systems in a bounded error context." In Decisionand Control, 2003. Proceedings. 42nd IEEE Conference on, volume 3, pages 2240-2245. IEEE, 2003.

I.Braems." Méthodes ensemblistes garanties pour l'estimation de grandeurs physiques". Phd of, 2002.

J.Monod. "La technique de culture continue". Annales de l'Institut Pasteur 79: 390-410, 1950

M.Dumont. " Apports de la modélisation des interactions pour compréhension fonctionnelle d'un écosystème: application à des bactéries nitrifiantes en chémostat" (Doctoral dissertation, Mont- pellier 2). 2008

S.Borsali "Algebraic method and LSCR technique for estimating the parameters of a bioreactor". J. Fundam. Appl. Sci., 2020, 12(2), 683-699.DOI : 10.4314/jfas.v12i2.11

This ePub file was created using machine-generated methods and automated software to convert the source text into an ePub format and may contain formatting inconsistencies from the original version. Formatting may vary depending on the device used.