Application of a Fuzzy Neural Network for Modeling of the MassTransfer Coefficient in a Stirred Tank Bioreactor
ABSTRACT A type of a fuzzy neural network for mathematical modeling of the volumetric masstransfer coefficient is presented in the paper. Performed investigations show that the presented fuzzy neural network can be successfully used for modeling of such a complex process, like masstransfer.
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ABSTRACT: Polyβhydroxybutyrate (PHB) is synthesized by some microorganisms under stressful conditions. Despite its properties being comparable to those of synthetic polymers, and its biocompatibility and biodegradability, low productivities have dampened commercial interest in microbial PHB production. To increase production efficiency, a fedbatch fermentation with Ralstonia eutropha was optimized recently through a neuralcumdispersion model (Dmodel) incorporating incomplete dispersion and noise in the feed streams. The approach described in the work has been improved in two ways: first by a model comprising neural networks only (Nmodel) and then by a hybrid neural model (Hmodel) with a mathematical component. At optimum dispersion, PHB production through the Nmodel optimization was 35% more than by the Dmodel, and this was enhanced by a further 58% using hybrid optimization. Recognizing that the Dmodel itself more than doubled the PHB production compared to a noisefree fully dispersed bioreactor, the present results establish hybrid neural optimization as a viable method for PHB production improvement under realistic conditions.Bioremediation Journal 01/2008; 12(3):117130. · 0.40 Impact Factor
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1
Application of a Fuzzy Neural Network for Modeling
of the MassTransfer Coefficient in a Stirred Tank
Bioreactor
Mitko Petrov*1, Tatiana Ilkova1, Stoyan Tzonkov1, Uldis Viesturs2
1Centre of Biomedical Engineering “Prof. Ivan Daskalov”  Bulgarian Academy of
Sciences, 105, Acad. George Bonchev Str., 1113 Sofia, Bulgaria
Email: mpetrov@clbme.bas.bg
2Latvian State Institute of Wood Chemistry, 27 Dzerbenes Str., LV1006 Riga, Latvia
Email: koks@edi.lv
∗ ∗Corresponding author
Received: December 14, 2004 Accepted: April 19, 2005
Published: April 28, 2005
Abstract: A type of a fuzzy neural network for mathematical modeling of the volumetric
masstransfer coefficient is presented in the paper. Performed investigations show that the
presented fuzzy neural network can be successfully used for modeling of such a complex
process, like masstransfer.
Keywords: Artificial neural networks, Modeling, Fuzzy neural network, Volumetric mass
transfer coefficient
Introduction
The volumetric oxygen masstransfer coefficient (KLa) defines the bioreactor effectiveness
for aerobic biotechnological processes. The KLa magnitude depends on a considerable
number of constructive and regime parameters, as well on physicschemical parameters of
culture medium. The increase of KLa value is a basic problem for the bioreactors design.
Recently, an increasing number of publications concerned with artificial neural networks
(ANN) and fuzziness have appeared [6, 9, 10]. Interesting and promising algorithms for
training ANN were proposed by using paradigms of fuzzy sets theory [7, 11].
The main advantage of ANN, known as a “flexible” model, is that they allow modeling of
complex and illdefined objects. However,
(backpropagation, reinforcement learning etc.) are a lot of time consuming [4, 8, 12].
A simplified type of ANN, that consists of two layers, is considered in the paper. Transfer
functions (somatic mapping) of every neuron from the second layer are considered to be
piecewise linear.
The weights of the neurons the first layer are random chosen by between 1 and +1. The
consideration of the transfer function as a crisp value is an idealization. A powerful tool for
more "flexible" description, which can be considered as more appropriate and closely to the
biological nature of the neurons action, are fuzzy relations [6].
We use fuzzy relation in order to achieve more adequate somatic mapping. Therefore, the
training task is a fuzzy optimization problem. Applying new results in this field [1, 2] and
usually used learning algorithms
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their possibilities for neural networks learning [3], a noniterative algorithm for training of
mentioned above type of ANN is proposed in the paper. Using a fuzzy neural network (FNN),
which input signals are the constructive and regime bioreactor parameters, is an alternative
approach for modeling and investigation their influence under the KLa.
The aim of this paper is to be synthesized of a model of the volumetric masstransfer
coefficient using a fuzzy neural network.
Structure of proposed FNN
The structure of proposed FNN is shown on Fig. 1.
2
u1
uI
xJ
x2
x1
y
.
.
.
.
.
.
Fig. 1 Type of the FNN
The transfer function at the first layer is sigmoidal:
⎡
⎟
⎠⎝
where: xjsignal in jth neuron in the hidden layer; "≅"fuzzy relation represented by its
membership function; uiinput signal; Inumber of input signals; ai,jweights of the connections
ith neuron in the first layer to jth neuron in hidden layer, ai,j∈[1,1].
The somatic mapping at the second layer is represented as a piecewise linear function on the
basis of the fuzzy equation:
∑
=
1j
where: yjoutput signals; Jnumber of hidden signals; wjweights of the neurons.
The membership function of (1) is as follows:
1
I
1i
i j , i
a
j
uexp1x
−
=
⎥⎦
⎤
⎢⎣
⎞
⎜
⎛−
+≅
∑
, (1)
≅
J
jjj
xwy
, (2)
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3
2
J
1j
jjxwy1
−
=
⎥
⎦
⎥
⎤
⎢
⎣
⎢
⎡
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−−=
∑
µ
(3)
Training of the proposed type of FNN
The training task includes a determination of the weights which minimizes the total error:
(
∑
=
1r
where: Rsize of the training set, yvector of the experimental outputs (from the training set),
nm
denotes fuzzy minimization.
In general, the training task is presented as follows:
nmE →
∑
∑
=
⎠
⎝
1i
This problem belongs to class of fuzzy mathematical programming problems and a theorem
[1, 2] introduced recently can be here applied. As a result, the best possible weights could be
found as a solution of the following linear equations [1, 2]:
1JxJxJ
R,R,
∈∈=
BMBwM
,
The elements of the matrixes M and B are determined by the following relations:
b
m...mm
==
BM
,
)
→−=
R
2
rr
ni~mYyE
, (4)
i~
i~
(5)
=
⎟
⎞
⎜
⎛
−+
≅
J
1j
I
r
ij , i
j
r
uaexp1
w
y
(6)
(7)
J
2
1
J ,R2 ,R1 ,R
J , 3
...
2 , 3
...
1 , 3
...
J , 22 , 21 , 2
b
...
b
;
m...mm
...
m... mm
(8)
where:
R,...,2r ; J,...,1j ,yyb
The best possible weights of the neurons in the second layer are obtained, received after the
solving equation (7). As far as the number of neurons in the first layer is known (it is equal to
uaexp1
1
uauaexp1
1
m
r)1r(
r
I
1i
r
ij , i
I
1i
1r
ij , i
r
ij , i
j , r
==−=
⎟
⎠
⎞
⎜
⎝
⎛−
+
+
⎟
⎠
⎞
⎜
⎝
⎛−
+
=
−
==
−
∑∑
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the number of input signals), then the number of neurons has to be determined. It will
determined simultaneously with the training of the neural network.
A simple noniterative algorithm for training this FNN is developed by authors.
Training algorithm
1. The initial data (uri, yr, I, J, R) is inputted. The coefficients ai,j∈[1, 1], R=J+1 are
determined by randomization;
2. The elements of the matrixes M and B are calculated;
3. The matrix M is inverted (M1);
4. The weights vector is calculated from w=M1B;
5. The simulation of FNN is realized (yr are printed);
6. Stop.
On the basis of the propose algorithm a program on FORTRAN 77 v. 5.0 is developed.
The proposed FNN will be used for modeling of the KLa in dependence on some constructive
and regime parameters of the bioreactor. On the basis of a preliminary analysis of the factors
and assessment of the conditions for realization of the experiment, (in this paper) the
following constructive and regime parameters of the bioreactor are considered:
Name of the parameter Symbol
Eccentricity of impeller toward
its rotation axis u1
Width of baffles u2
Slope angle of stirrer’s blades u3
Number of impeller u4
Impeller speed u5
Gas flow rate u6
They are coded in the interval [1, +1]. The coding is performed based on the equation:
(
0 , imax, i0 , i
i
i
uu/uuu
−−=
,
where:
()
max, imin, i0 , i
uu 5 . 0u
+=
,
maximum, minimum and mean real values of the examined parameters.
The chosen constructive and regime parameters of the bioreactor ui=u[u1,…,uI], I=6 are the
input signals of the FNN. The output signal from the FNN is y=KLa (Fig. 1).
Experimental investigations
The experiments are carried out in a laboratory bioreactor 2LM with a magnetic coupling
with maximal volume 2 L. The bioreactor is included in an automatic control system (ACS).
The ACS has been developed by а scientific team from Centre of Biomedical Engineering. It
gives a possibility for control of two bioreactors. The scheme of the experimental is presented
in [5].
4
Min. value Max. value
0.0 mm
10.0 mm
1.5 mm;
14.0 mm;
450 900;
1 3
2 s1
50 l/h
20 s1;
300 l/h.
) ()
iu , ui,min, ui,max, ui,0 and ui are respectively the current,
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In order to compare the results, all experiments are performed at constant conditions (Table
1). The measurement of the KLa is realized based on the method of degasation in pattern
medium – distillated water [5].
Table 1. Basic measurement and conditions for experiment investigations
Basic measurement and conditions for experiment investigations
Vessels diameter, D
Impeller diameter, d
Width of paddle impeller, b
High of paddle impeller, h
Distance between vessel bottom and impeller, h1
Distance between two impellers, h2
Diameter of aerator, Da
Number of baffle assembly
Number of perforation of aeration
Height of liquid in the bioreactor, L
Volume of liquid in the bioreactor, V
Temperature, T
The obtained experimental results are shown in Table 2.
Table 2. Experimental investigations
N0 u1 u2 u3 u4 u5 u6 KLa, h1
1 1 1 1 1 1 1 103.0
2 1 1 1 1 1 1 121.6
3 1 1 1 1 1 1 113.1
4 1 1 1 1 1 1 94.8
5 1 1 1 1 1 1 83.7
6 1 1 1 1 1 1 91.0
7 1 1 1 1 1 1 103.4
8 1 1 1 1 1 1 107.1
9 1 1 1 1 1 1 110.1
10 1 1 1 1 1 1 120.1
11 1 1 1 1 1 1 123.7
12 1 1 1 1 1 1 110.0
13 1 1 1 1 1 1 100.1
14 1 1 1 1 1 1 112.0
15 1 1 1 1 1 1 117.6
16 1 1 1 1 1 1 92.5
17 1 1 1 1 1 1 69.6
18 1 1 1 1 1 1 81.1
19 1 1 1 1 1 1 100.0
20 1 1 1 1 1 1 96.0
21 1 1 1 1 1 1 87.6
22 1 1 1 1 1 1 96.4
23 1 1 1 1 1 1 107.4
5
Value
120.0 mm
58.0 mm
14.0 mm
12.0 mm
58.0 mm
58.0 mm
50.0 mm
3
120
120.0 mm
1.2 l
250C
N0
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
u1 u2 u3 u4
1
u5 u6
1
1
KLa, h1
111 1
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
1
1
0
0
1

104.0
101.0
114.1
131.5
139.1
140.9
142.5
137.2
121.5
117.7
139.3
139.5
98.6
105.0
107.0
147.5
148.2
171.9
44.8
172.4
109.6
229.6
1
1
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
0
0
1
1
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
0
0
0
0
1
1
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
0
0
0
0
0
0
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
0
0
1
1
1

1
1
0
0
0
0
1

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Training of FNN and analysis of results
The training of FNN is realized on the basis of the developed algorithm and program. After
the training of the network, the following values for the number of the neurons in the second
layer and their weights are obtained:
J=8;w1=11.97, w2=2.94, w3=3.53, w4=10.77, w5=14.76, w6=12.84, w7=0.49, w8=1.06,
i.o. for only the first eight experiments, listed in Table 2, are used for training of FNN.
The experimental results and the prediction after the training of the FNN are shown on Fig. 2.
6
8 12 1620 2428 32 364044
50
100
150
200
250
Experimental
Model
KLa, h
 1
Number of the experiments
Fig. 2 Experimental data and obtained results by ANN
A statistical analysis of the obtained results is performed. It gave the following results: an
experimental correlation coefficient R2=0.983, a theoretical correlation coefficient R2T=0.325
at degree of freedom ν=35 and a level of notability β=±5%. The experimental and the
theoretical Fisher quotients are: FE=1.01 and FT=2.11.
Fig. 2 and obtained results shows that FNN describes very well the experimental data. The
model is adequate and this network can be used for modeling of the volumetric oxygen mass
transfer coefficient.
Conclusions
1. The performed investigations show that the proposed type FNN can be successfully used
for modeling of the volumetric oxygen masstransfer coefficient in dependence on the
constructive and regime parameters of the bioreactor.
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2. Using of neural network is the advisable for description of such a complex process like
masstransfer. Therefore the necessity of solving of complex nonlinear differential
equations and further parameter identification, that needs more experimental data in
comparison with the training of the neural network, drops off.
Acknowledgments
The investigations are partially supported by National Fund for Scientific Investigations by
grant TH1314/2003.
References
1. Angelov P. (1993). An Analytical Approach for FMP Solving and its Application to
Neural Network Learning, 1st European Congress on Fuzzy and Intelligent Technologies 
EUFIT'93, Aachen, Sepember, 3, 12551264.
2. Angelov P. (1994). A generalized Approach to Fuzzy Optimization, International Journal
of Intelligent Systems, 9, 261268.
3. Angelov P., L. Kuncheva (1993). Analitycal Solving Fuzzy Optimization Problems,
International Conference MMSC'93, Sozopol, Bulgaria, 109112.
4. Berenji H. (1992). A Reinforcement LearningBased Architecture for Fuzzy Logic
Control, International Journal of Approximate Reasonings, 6, 267271.
5. Georgiev A., R. Nenov, M. Petrov (1992). Influence of Basic Constructive Parameters on
the Mass Transfer of Laboratory Fermenters, Bioautomation, 12, 5662.
6. Gupta M. (1992). Fuzzy Logic and Neural Networks, 2nd International Conference on
Fuzzy Logic & Neural Networks, Iizuka, Japan, June, 157160.
7. Keler J. C. Zhihong (1992). Learning in Fuzzy Networks Utilizing Additive Hybrid
Operators, 2nd International Conference on Fuzzy Logic & Neural Networks, Tizuka
Japan, May, 132140
8. Keler J., J. Thibault., V. Van Berusegen V., A. Cheruy (1990). Online Prediction of
Fermentation Variables Using Neural Networks, Biotechnology & Bioinstrumentation,
36, 10411050.
9. Kosko B. (1992). Neural Networks and Fuzzy Systems, PrenticeHall, Cl. Englewood
Cliffs.
10. McClelland J. D. Rumelhard (1988). Parallel Distributed Processing, Explorations in the
MIT Press, Cambridge, II Ed.
11. Nauck D., F. Klawoon, R. Kruse (1993). Combining Neural Networks and Fuzzy
Controllers, Lecture Notes in Computer Science, 3546.
12. Puyin Liu, Hongxing Li (2004). Fuzzy Neural Network Theory and Application, World
Scientific Publication, http://www.worldscientific.com.
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