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Application of An Adaptive Neuro Fuzzy Inference System (ANFIS) for a Class of Fed-Batch Fermentation Processes
APPLICATION OF AN ADAPTIVE NEURO FUZZY INFERENCE SYSTEM (ANFIS) FOR A CLASS OF FED-BACTH FERMENTATION PROCESSES
l
Endra Joelianto', Basuki Rahmae, Kusmayanto Kadiman , and Tjandra Setiadi
J
I Department of Engineering Physics, Ins/ilu! Teknologi Bandung, Jf. Ganesha la, Bandung 40132 Indonesia
2 Department of Computer Systems Sekolah Tinggi Manajemen Informatika & Teknik Komputer Surabaya. Jf. Raya Kedung Baruk 98, Surabaya 60298 Indonesia J Department of Chemical Engineering institut Teknologi Bandung, Jf. Ganesha 10, Bandung 40132 Indonesia
This paper considers an implementation of a modified adaptive neuro fuzzy inference system (ANFIS) controller to find substrate feedrate for the growth of Saccharomyces cerevisiae in a fed-batch fermentation process from a given profile of output responses. The adaptive capability of the modified ANFIS is an advantage to handle nonlinear and un certain dynamics of the fed-batch fermentation process. The hybrid learning algorithm of the modified ANF[S comprises two steps, i.e. forward pass by using fuzzy inference mechanism based on the neural architecture and Least-squares Estimator (LSE), and backward pass to update the premise parameters by the error a modified backpropagation algorithm. Keywords: fuzzy systems, simu lation, validation.
l. INTRODUCTION
While a perfect model is impossible in rea lity, a relative accurate model is also difficult to obtain. Hence, the modeling result is a model which involves uncertainty and unmodeled dynamics. Since the feed rate profile must be found based on this incomplete information, the input must be carefully designed in order to induce as much information as possible in the yielding the output trajecto!),. This points out the importance of non-model based control method.
The fed-batch fermentation processes have attracted attention in recent years because of the effectiveness in overcoming substratc inhibition, catabolite repression and glucose effects by applying of the appropriate substrate feed rate (Shuler and Kargi, 1992) (Bastin and van Impe, 1995) (Chattway, et. al., 1992). Depending on the objective of the fermentation, it is necessary to determine the substrate feed rate profile, which minimizes the specified performance cost. [n industrial fermentation, for example the production of Saccharomyces cerevisiae known as baker's yeast, the objective is aimed to achieve maximum yie ld or productivity.
Adaptive Neuro Fuzzy Inference System (ANFIS) is a class of adaptive networks that are functionally equivalent to fuzzy inference systems, where the parameters of fuzzy inference systems are updated by neural networks from a set of training data (Jang.
271
era!., 1996). The adaptive capability of ANFIS makes it almost directly applicable to adaptive control and learning control. In fact, ANFIS can replace almost any neural network in a control system and perform the same function. The non linearity and structural knowledge representation of ANFIS are its primary advantages over classical Iinear approaches in adaptive fi Itering and signal process ing, such as identification, inverse modeling, predictive coding, and so on. The growth and metabolism of S. cerevisiae shows a complex interaction between the specific growth rate, the availability of glucose and oxygen, cell yield, cell morphology, and intracellular enzyme levels. In this paper, based on the growth model considered in (Boskovic and Narendra, 1995), we propose a control strategy for the fed-batch fermentation process using a modified ANFIS algorithm (Rahmat, el.al., 2001). Simulation results show the effectiveness of the modified ANFIS algorithm compare to the standard ANFIS CJang, et. aI., 1996) when adequate prior information concerning the dynamics of the process is not available and accuracy is critical issue.
2, I. Process Model
Fermentation is a slow process in which microorganisms utilize available nutrients (substrate) for growth, biomass maintenance and product formation , One of typical fermentation processes is S.cerevisiae known as baker's yeast. The process is assumed to be described by the non linear state space equation :=
x(/)E9~j
lI(1)E91
2
XJ
I
O,0023xj + 0,007 Pm k" +X2 I +
(4)
Xl
(5) (6)
In equations (2)-{5), the unknown process parameter ks, ky and km were assumed to vary with time as described by the following equation )lm,
~Lm(t) == 004\ + 0.01 cos (2/5;r/),
k.(t)
:=
0.Q3 - 0.005 cos (21l5;r t),
ky(t)
==
0.4 + 0.1 cos (2110;rt),
(7)
km(t) == 0,04 - 0.0 1 cos (2120;rt).
!
Jre (X2 ) (X2 < 0,28 danx4 >0, atauJr.(xJ >()),
~I(X)
0
(x 2$0,28danx 4 ==(),
o
(X2 ~ 0,28 atau Jre(x2)~0).
(8)
~2(X):= {V,.(X2'X ]) (x 2 > 0.28
I(x, u, p) + I;(x),
o
(\)
x(O):= xo
where
(3)
The non linearities C;j (.) and c; 2 (.) correspond to ethanol consumption and format ion rates. These non linearities are genera ll y not known precisely, and represent the unmodeled dynamics of the system. Following (Boskovic and Narendra, 1995), these non Iinearities are of the form
(I. PROBLEM FORMULATION
x
(2)
ataUVe (X2 ,x)> 0).
(x 2 ~ 0 ,28 atau ve (X 2• Xl ) S; O. atau X4 == 0)
(9)
where denotes
the
state at time t,
00028 - 0, 28
ve(x2 'x)) == 0138-0,062-\) + - - -
denotes the control at time t, the parameter
X2
vector is denoted by pE 91' . The process unmodeled dynamics, whose exact form is not known, is denoted by c; . The elements of the state vector (i ,e. the state variab les) xl) (j = 1,2,.",5) are defined as follow: XI is the concentrat ion of microorganisms, X2 is the concentration of substrate, x) is the concentration of inhib itory substance, X4 is the concentration of ethanol and Xs is the volume of the fermentation broth, The elements of the control vector 1I,(.) and ul,) are the feed of substrate and the influent sllbstrate concentration respectively. The process starts at time t 0 and terminates at time T, where T is generally of the order of a few hours to a few days. In this prob lem, T is assumed to be 15 hours.
(10)
The model (2)-(6) has attracted a lot of atten tion because it shows hig h yield on biomass. In (Boskovic and Narendra, 1995), it has been shown that the high yield can be obtained by the so ca lled "nom inal regime" which is characterized by the low concentration of the ethano l. In this paper, this resu lt wi ll be used as the optimal feed rate profile, 2.2, Control Scheme
The objective of the control is to determine u I(t) and that maximize the production of microorganisms in the interval [O,T] whi le maintaining that the concentration of ethanol X4(t) is low as it has a deleterious effect on the final product. This is a difficult control prob lem since the both the nonlinear nature of the plant and the uncertainty U2(t)
The model for a baker's yeast fed-batch fermentation process used in this paper is taken from equations in (l3oskovic and Narendra, 1995) and described by the following differential:
272
the appropriate controller output by fixing the output and then finding the input. [n this paper, the fixed output is the output that is produced by the nominal model operated in the nominal regime. The pair of the arranged training data is obtained from the simulation results in Figure I. For the training purposes, the state variables are arranged as follow:
caused by unmodeled dynamics of the system (';1 (.) and ';2 (.») are encountered in the process. Specifically, the control design is aimed to produce controller output that results in the satisfaction of the design requirements by the plant output. In this paper, we follow the method developed by (Okada, el. a/. , 1981) and (Takamatsu, et.al., 1985) which has also been used in (Boskovic and Narendra, 1995). In this approach, the feed rate does not need to be solved via the solution of the optimal control problem. Instead, it is found by minimizing the effects of the unmodeled dynamics.
U,(k)
Controller
From (2)-(6), as is shown in (Boskovic and Narendra, 1995), the problem comes from the presence of uncertainty which arises when X2(t) has value more than its critical value x2'=0.28. When X2(0) is selected to be 0.28 and the control input UI(t)""U'(t) is determined such that dxidt=O over the time interval [O,T], we obtain .; (x(t))=O. Also it has been found that choosing U2' =200, the parameter vector p=p'=[0.42 0.025 0.5 0.03] and the initial vector x(O)=x' (0)=[ I 0 0.28 0 0 10]T yield the trajectory of the state x(t) satisfies the optimal performance of the process denoted by x'(t). The process described by (2)-(6) x(t)=x (t), u2=200 and UI=UI' given by
with
Ill. MODIFIED ANFIS
3.1. Architecture of Modified ANFlS
The standard ANFIS structure represents Sugeno fuzzy model. For simplicity, it is assumed that the considered fuzzy inference system has two inputs X dan y and one output f. A common rule set for a first-order Sugeno fuzzy with two fuzzy if-then rules is as follows:
p=p',
Rule I: If X is AI and y is BI ' then
J; = PI +qly+rl Rule 2: If X is A2 and y is B2 , then
1; = P2 + q2Y + r2 Figure 3.a shows the reasoning mechanism for this Sugeno model and the corresponding standard ANFIS architecture which is shown in Figure 3.b where nodes of the same layer have similar functions. The detail explanation can be found in (Jang, et. a/. , 1996).
·~El ..
"LZ--
I"
I....
.
''-----
LA", . . . . . . . 4
1·01--------1
,_,~__ ..... ~
i
·
'.
......
-
x(k+I)
Figure 2. Control system
is referred as "nominal model". Figure I shows the simulation result of the system (2)-(5) is operated in the nominal regime. Notice that the biomass xl(T) at the final time T is 8 times the initial biomass XI(O).
..
Plant~-~
Mod. ANFIS
,
..
................... .
•
. -
f,
C'
p.,X
:- '\ ,=
·B.
1
.
IT,
'y"
~
... _ /
f,_p,x..q,Y+'1
W,',+W,', W, .. W,
= W, ',. ",',
Y
(al
t.y.,
I-
2
Layer
1
!
Layer 2
!
Laye, 3
I
! ,y
Loyo,S
x<'GJ~~w, .o-w, I, ! ";< ..:::. ">.( ----@-. Y ~·-···gl· )\DJ;; -~-C1-W;" ""';::::, /' ' ' .. xy
".
Figure I. Nominal model
1
~/
Ibl
Figure 2 shows the diagram block of the applied control system by using inverse dynamic method. The modified ANFIS controller is aimed to obtain
Figure 3. (a) A two-inputs first-order Sugeno fuzzy model; (b) equivalent ANFIS architecture.
273
The architecture of ANFIS controller with modification in the learning algorithm that is implemented in this fermentation control problem is described in the following figure.
Layer 2: Fuzzy logic AND is used as the node function in this layer. The outputs of this layer are:
Laye, .. .,(t)I,(k.I) ... 1
Every output from this node is labeled by a, so the output is given by n I a to 020a. The symbol a is used to differentiate with new symbol b (after the correction).
~(tt:II.p.11
~~
n21a = min(nla, n3a)
( 12) n30a = min(nISa, n20a)
Layer 3: The input signals of this layer are normalized. Hence n31a = n21a /(n2Ia + ... + n30a)
(13)
u,(k)
"" t---+
n40a =n30a /(n2Ia + ... + n30a)
Layer4: From the incoming signals, we obtain matrix A obtained as follows:
A I = [(n31 a in I )(n3 Ia in2) (n32a in I) (n32a in2) (n33a in3) (n33a in4)]; A2 = [(n34a in3) (n34a in4) (n35a inS) (n35a in6) (n36a inS) (n36a in6)]; A3 = [(n37a in7) (n37a inS) (n3Sa in7) (n3Sa inS) (n39a in9) (n39a nlO)]; A4 = [(n40a in9) (n40a in 10)] Then
t ... t Figure 4. Modified ANFIS structure
A =[AI A2A3A4] 32. Learning Algorithm
The important part of Modified ANFIS is the modification of the error correction rules of error back propagation (EBP) by using a mapping function to replace membership function in the standard ANFIS. The learning system of the Modified ANFIS is similar to the standard ANFIS. It uses hybrid learning algorithm which consists of two steps, i.e. forward pass by using fuzzy inference mechanism based on the neural architecture and Least-squares Estimator (LSE), and backward pass to update the premise parameters by the error backpropagation algorithm which has been modified. I.
Forward Pass
The mechanism in the forward pass can be explained as follows: Layer 1: The membership function (usually the bell function) is used in this layer is given by I J1A(x)=----
(11 )
x - C ?h a, The parameters of the membership function {a, ,b" c,}, i = 1,2,3,4 are predetermined.
1+1 - - ' 1-'
274
( 14)
By using Least Squares Estimator (LSE) method, we then obtain the consequent parameters of P. We use ( 15) where U I is the desired output of the controller. We obtain the parameter P = [pe I) ... p(20)] and fl
=
p(I) inI + p(2) in2;
f2 = p(3) in I + p(4) in2; = p(5) in3 + p(6) in4; = p(7) in3 + peS) in4; = p(9) inS + p( 10) in6; = p(II) inS + p(I2) in6; = p(13) in7 + p(14) in8; = p( IS) in7 + p( 16) in8; ~ = p(J7) in9 + p(IS) in 10; fl 0 = p( 19) in9 + p(20) in 10; G f4 f5 f6 f7 f8
After that, the output are n41a = n3Ia fl; n43a = n33a G: n45a = n35a f5; n47a = n37a f7; n49a = n39a f9;
( 16)
of the node n41 a ... n50a n42a = n32a n44a = n34a n46a = n36a n4Sa = n3Sa n50a = n40a
f2; f4; f6; f8; fI O.
( 17)
Layer 5: The output of this layer is the summation of all incoming signals: nSI a = n41 a+n42a+n43a+n44a+n45a+n46a +n47a+n48a+n49a+n50a (18) 2.
Backward Pass
output that yields the output response following the response of the nominal model in Fig. I. Figure 5 shows the initial membership function and final mapping function of the modified ANFIS learning process. The response of the state variables and the control signal are shown in Figure 6. Simulation results show that the output responses follow the given output profile, but these responses are achieved by an oscillatory control input. The average percentage error is given in Table I.
Define the error between the output of the network and the desired output dk . The sum of the squared
~W~OO i "lYJ"X! ..~, . :' O:[ j\'
error is given by
""'o.t NU)
Erl
2)d( - xL)2
=
(19)
lOO
where we have defined Ef' =
5 51 , XI
in this layer is
~
=
U I • Hence
c51 =
-2(U I -175Ia)
(20)
lOO
o.a
'[f
.f
0.'
;
028 Q.l5
~
\
0..
\
0; .'
ou . . o.JII
'f!(!rol)
o.lS
0
Q.JIS
,
2
~
411'01'
i]/"[],'
"::'/ :'["~' // :: y : .' : ./ i
0.1
n51 and d k
0."
..
"/
0
rlN
0..
;
0.4
0: / ::c
Q
I
(lA
o~ /
k~1
(UI
Q.I
0.4'
0.2
...
o
o.a
i
M
0
'"
III
/
I
0.1
J
QJ
/
:
/
0,2
0..1
.
~
~')
~~.
~'.lIO·
G.1.: 0
~O
O'l0Q510iU'O
Next, dSI is defined as
•
~
~
~II
(a)
d 51
= -E: 51
/2 = U I
-
(21 )
n5 I a
The output of the node n5 I becomes n51 b = n51 a + dSI
(22)
In fact, n51 b = n41 b + ... + n50b since n5 I a = n4 I a + ... + n50a. A II of mechanisms in forward pass in layer 1 until layer 5 are used in the same manner for backward pass to produces a corrected value (labelled b after added by corrected factors dj, i = 21 .. 51). Further, by extend ing the correction rule of backward pass modified ANFIS learning algorithm in (Jang, er af, 1996) for ten inputs, it will be obtained the corrected value of each node. Finally, if the corrected value in layer I obtained, then all inputs are mapped to this value by interpolation technique. The mapping function then becomes the mapping function of the learning mechanism of the modified ANFIS.
"IXJlilIXl11JIJ o
Sl
1(005)
~I(kI
.1XlQ28
~I(Io.')
0.211 0,280Je dI;IO
G.2J
oleo
1
)
~
'I2(t;")
(b)
Figure 5. a. Initial Membership Function b. Final Mapping Function of Modified ANFIS learning process
~-
lE]:
3. 3. Simula/iol1s
~
To show the effectiveness of the modified ANFIS controller, simulation is carried out by using experimental data in (Joelianto, et. aI., 200 I). The data was obtained from experiment held in a fermentor with physical states: temperature 30 e, pH 4.5 , air sreed flow I vvm, initial volume 1.5 L, feed concentration (U2) 10 glL, feed speed (UI) 0.104 glL, sal11rling time 2 hours, and sampling volume 20 mL. From I" hour until 14,h, baker's yeast was grown in batch condition, and the fed-batch condition started at 14'11 hour until 36'11 hour. D
In this simulation, the modified ANFIS controller is trained by using state variables from the experimental data. Then, the modified ANFIS finds the controller
275
.
~
U.
I
..
..
"171 '~ .
.
.
.
.,,<----;-----.----i,
Figure 6. Simulation results
Chattaway, T., G.A. Montague and A.J. Morris (1992). Fermentation Monitoring and Control. In: Biotechnology (H.J. Rehm and G. Reeds. (Ed»,Vol. 3,319-354. VCH Weinheim. Fukuda, H., T. Shiotani, W. Okada and H. Morikawa (1978). A Mathematical Model of The Growth of Baker's Yeast Subject to Product Inhibition, J. Fermentation Technology, 56, 361-368. Jang, J.-S.R., C.-T. Sun and E. Mizutani (1996). Neuro-Fuzzy and Soft Computing. Prentice Hall, Int. Edition. Joelianto, E., R. Dwihandayani and L. Ling (200 I). Parameter Estimation using Genetic Algorithm for Fed-Batch Growth of Saccharomyces cerevisiae. Proc. 2 nd IFAC-C1GR Workshop on Intell. Contr. for Agricultural Application. Okada, W., H. Fukuda and H. Morikawa (1981). Kinetic Expression of Ethanol Production Rate and Ethanol Consumption Rate in Baker's Yeast Cultivation. 1. Fermentation Technof. 59, 103-109. Rahmat, B., E. Joelianto and K. Kadiman (2001). Mapping Function for Error Backpropagation Algorithm of Adaptive Neuro Fuzzy Inference System (ANFIS). Proc. 2"" Int. Seminar on Numerical Analysis in Engineering (NA E), Batam-Indonesia. Shuler, M. L. and F. Kargi ([ 992). Bioprocess Engineering: Basic Concepts. Prentice-Hall [nc., New Jersey. Takamatsu, T., S. Shioya, S. Okada and M. Kanda ([ 985). Profi le Control Scheme in a Baker's Yeast Fed-batch Culture. Biotechnology and Bioengineering, 27, [675-[686.
Table I. The average percentage error (APE)
Output plant XI (t) X2 (t) X3 (t) X4 (t)
APEJo/~
1.5670 7.0361 x 10. 7
3.3392 0
IV. CONCLUSIONS In this paper, a recently developed modified ANFIS controller has been implemented to the control problem in a class of fed-batch fermentation processes which is characterized by non Iinear and uncertain dynamics. Simulation results showes that the modified ANFIS leads to a good output responses with small error with respect to the given output response. REFERENCES
Bastin, G. and J.F. van Impe (1995). Nonlinear and Adaptive Control in Biotechnology. Eur. J. Control, I, 37-53. Boskovic, 1.D. and K.S. Narendra (1995). Comparison of Linear, Nonlinear and Neural Network-based Adaptive Controllers for A Class of Fed-batch Fermentation Processes. Automatica,31,817-840.
276
l
Endra Joelianto', Basuki Rahmae, Kusmayanto Kadiman , and Tjandra Setiadi
J
I Department of Engineering Physics, Ins/ilu! Teknologi Bandung, Jf. Ganesha la, Bandung 40132 Indonesia
2 Department of Computer Systems Sekolah Tinggi Manajemen Informatika & Teknik Komputer Surabaya. Jf. Raya Kedung Baruk 98, Surabaya 60298 Indonesia J Department of Chemical Engineering institut Teknologi Bandung, Jf. Ganesha 10, Bandung 40132 Indonesia
This paper considers an implementation of a modified adaptive neuro fuzzy inference system (ANFIS) controller to find substrate feedrate for the growth of Saccharomyces cerevisiae in a fed-batch fermentation process from a given profile of output responses. The adaptive capability of the modified ANFIS is an advantage to handle nonlinear and un certain dynamics of the fed-batch fermentation process. The hybrid learning algorithm of the modified ANF[S comprises two steps, i.e. forward pass by using fuzzy inference mechanism based on the neural architecture and Least-squares Estimator (LSE), and backward pass to update the premise parameters by the error a modified backpropagation algorithm. Keywords: fuzzy systems, simu lation, validation.
l. INTRODUCTION
While a perfect model is impossible in rea lity, a relative accurate model is also difficult to obtain. Hence, the modeling result is a model which involves uncertainty and unmodeled dynamics. Since the feed rate profile must be found based on this incomplete information, the input must be carefully designed in order to induce as much information as possible in the yielding the output trajecto!),. This points out the importance of non-model based control method.
The fed-batch fermentation processes have attracted attention in recent years because of the effectiveness in overcoming substratc inhibition, catabolite repression and glucose effects by applying of the appropriate substrate feed rate (Shuler and Kargi, 1992) (Bastin and van Impe, 1995) (Chattway, et. al., 1992). Depending on the objective of the fermentation, it is necessary to determine the substrate feed rate profile, which minimizes the specified performance cost. [n industrial fermentation, for example the production of Saccharomyces cerevisiae known as baker's yeast, the objective is aimed to achieve maximum yie ld or productivity.
Adaptive Neuro Fuzzy Inference System (ANFIS) is a class of adaptive networks that are functionally equivalent to fuzzy inference systems, where the parameters of fuzzy inference systems are updated by neural networks from a set of training data (Jang.
271
era!., 1996). The adaptive capability of ANFIS makes it almost directly applicable to adaptive control and learning control. In fact, ANFIS can replace almost any neural network in a control system and perform the same function. The non linearity and structural knowledge representation of ANFIS are its primary advantages over classical Iinear approaches in adaptive fi Itering and signal process ing, such as identification, inverse modeling, predictive coding, and so on. The growth and metabolism of S. cerevisiae shows a complex interaction between the specific growth rate, the availability of glucose and oxygen, cell yield, cell morphology, and intracellular enzyme levels. In this paper, based on the growth model considered in (Boskovic and Narendra, 1995), we propose a control strategy for the fed-batch fermentation process using a modified ANFIS algorithm (Rahmat, el.al., 2001). Simulation results show the effectiveness of the modified ANFIS algorithm compare to the standard ANFIS CJang, et. aI., 1996) when adequate prior information concerning the dynamics of the process is not available and accuracy is critical issue.
2, I. Process Model
Fermentation is a slow process in which microorganisms utilize available nutrients (substrate) for growth, biomass maintenance and product formation , One of typical fermentation processes is S.cerevisiae known as baker's yeast. The process is assumed to be described by the non linear state space equation :=
x(/)E9~j
lI(1)E91
2
XJ
I
O,0023xj + 0,007 Pm k" +X2 I +
(4)
Xl
(5) (6)
In equations (2)-{5), the unknown process parameter ks, ky and km were assumed to vary with time as described by the following equation )lm,
~Lm(t) == 004\ + 0.01 cos (2/5;r/),
k.(t)
:=
0.Q3 - 0.005 cos (21l5;r t),
ky(t)
==
0.4 + 0.1 cos (2110;rt),
(7)
km(t) == 0,04 - 0.0 1 cos (2120;rt).
!
Jre (X2 ) (X2 < 0,28 danx4 >0, atauJr.(xJ >()),
~I(X)
0
(x 2$0,28danx 4 ==(),
o
(X2 ~ 0,28 atau Jre(x2)~0).
(8)
~2(X):= {V,.(X2'X ]) (x 2 > 0.28
I(x, u, p) + I;(x),
o
(\)
x(O):= xo
where
(3)
The non linearities C;j (.) and c; 2 (.) correspond to ethanol consumption and format ion rates. These non linearities are genera ll y not known precisely, and represent the unmodeled dynamics of the system. Following (Boskovic and Narendra, 1995), these non Iinearities are of the form
(I. PROBLEM FORMULATION
x
(2)
ataUVe (X2 ,x)> 0).
(x 2 ~ 0 ,28 atau ve (X 2• Xl ) S; O. atau X4 == 0)
(9)
where denotes
the
state at time t,
00028 - 0, 28
ve(x2 'x)) == 0138-0,062-\) + - - -
denotes the control at time t, the parameter
X2
vector is denoted by pE 91' . The process unmodeled dynamics, whose exact form is not known, is denoted by c; . The elements of the state vector (i ,e. the state variab les) xl) (j = 1,2,.",5) are defined as follow: XI is the concentrat ion of microorganisms, X2 is the concentration of substrate, x) is the concentration of inhib itory substance, X4 is the concentration of ethanol and Xs is the volume of the fermentation broth, The elements of the control vector 1I,(.) and ul,) are the feed of substrate and the influent sllbstrate concentration respectively. The process starts at time t 0 and terminates at time T, where T is generally of the order of a few hours to a few days. In this prob lem, T is assumed to be 15 hours.
(10)
The model (2)-(6) has attracted a lot of atten tion because it shows hig h yield on biomass. In (Boskovic and Narendra, 1995), it has been shown that the high yield can be obtained by the so ca lled "nom inal regime" which is characterized by the low concentration of the ethano l. In this paper, this resu lt wi ll be used as the optimal feed rate profile, 2.2, Control Scheme
The objective of the control is to determine u I(t) and that maximize the production of microorganisms in the interval [O,T] whi le maintaining that the concentration of ethanol X4(t) is low as it has a deleterious effect on the final product. This is a difficult control prob lem since the both the nonlinear nature of the plant and the uncertainty U2(t)
The model for a baker's yeast fed-batch fermentation process used in this paper is taken from equations in (l3oskovic and Narendra, 1995) and described by the following differential:
272
the appropriate controller output by fixing the output and then finding the input. [n this paper, the fixed output is the output that is produced by the nominal model operated in the nominal regime. The pair of the arranged training data is obtained from the simulation results in Figure I. For the training purposes, the state variables are arranged as follow:
caused by unmodeled dynamics of the system (';1 (.) and ';2 (.») are encountered in the process. Specifically, the control design is aimed to produce controller output that results in the satisfaction of the design requirements by the plant output. In this paper, we follow the method developed by (Okada, el. a/. , 1981) and (Takamatsu, et.al., 1985) which has also been used in (Boskovic and Narendra, 1995). In this approach, the feed rate does not need to be solved via the solution of the optimal control problem. Instead, it is found by minimizing the effects of the unmodeled dynamics.
U,(k)
Controller
From (2)-(6), as is shown in (Boskovic and Narendra, 1995), the problem comes from the presence of uncertainty which arises when X2(t) has value more than its critical value x2'=0.28. When X2(0) is selected to be 0.28 and the control input UI(t)""U'(t) is determined such that dxidt=O over the time interval [O,T], we obtain .; (x(t))=O. Also it has been found that choosing U2' =200, the parameter vector p=p'=[0.42 0.025 0.5 0.03] and the initial vector x(O)=x' (0)=[ I 0 0.28 0 0 10]T yield the trajectory of the state x(t) satisfies the optimal performance of the process denoted by x'(t). The process described by (2)-(6) x(t)=x (t), u2=200 and UI=UI' given by
with
Ill. MODIFIED ANFIS
3.1. Architecture of Modified ANFlS
The standard ANFIS structure represents Sugeno fuzzy model. For simplicity, it is assumed that the considered fuzzy inference system has two inputs X dan y and one output f. A common rule set for a first-order Sugeno fuzzy with two fuzzy if-then rules is as follows:
p=p',
Rule I: If X is AI and y is BI ' then
J; = PI +qly+rl Rule 2: If X is A2 and y is B2 , then
1; = P2 + q2Y + r2 Figure 3.a shows the reasoning mechanism for this Sugeno model and the corresponding standard ANFIS architecture which is shown in Figure 3.b where nodes of the same layer have similar functions. The detail explanation can be found in (Jang, et. a/. , 1996).
·~El ..
"LZ--
I"
I....
.
''-----
LA", . . . . . . . 4
1·01--------1
,_,~__ ..... ~
i
·
'.
......
-
x(k+I)
Figure 2. Control system
is referred as "nominal model". Figure I shows the simulation result of the system (2)-(5) is operated in the nominal regime. Notice that the biomass xl(T) at the final time T is 8 times the initial biomass XI(O).
..
Plant~-~
Mod. ANFIS
,
..
................... .
•
. -
f,
C'
p.,X
:- '\ ,=
·B.
1
.
IT,
'y"
~
... _ /
f,_p,x..q,Y+'1
W,',+W,', W, .. W,
= W, ',. ",',
Y
(al
t.y.,
I-
2
Layer
1
!
Layer 2
!
Laye, 3
I
! ,y
Loyo,S
x<'GJ~~w, .o-w, I, ! ";< ..:::. ">.( ----@-. Y ~·-···gl· )\DJ;; -~-C1-W;" ""';::::, /' ' ' .. xy
".
Figure I. Nominal model
1
~/
Ibl
Figure 2 shows the diagram block of the applied control system by using inverse dynamic method. The modified ANFIS controller is aimed to obtain
Figure 3. (a) A two-inputs first-order Sugeno fuzzy model; (b) equivalent ANFIS architecture.
273
The architecture of ANFIS controller with modification in the learning algorithm that is implemented in this fermentation control problem is described in the following figure.
Layer 2: Fuzzy logic AND is used as the node function in this layer. The outputs of this layer are:
Laye, .. .,(t)I,(k.I) ... 1
Every output from this node is labeled by a, so the output is given by n I a to 020a. The symbol a is used to differentiate with new symbol b (after the correction).
~(tt:II.p.11
~~
n21a = min(nla, n3a)
( 12) n30a = min(nISa, n20a)
Layer 3: The input signals of this layer are normalized. Hence n31a = n21a /(n2Ia + ... + n30a)
(13)
u,(k)
"" t---+
n40a =n30a /(n2Ia + ... + n30a)
Layer4: From the incoming signals, we obtain matrix A obtained as follows:
A I = [(n31 a in I )(n3 Ia in2) (n32a in I) (n32a in2) (n33a in3) (n33a in4)]; A2 = [(n34a in3) (n34a in4) (n35a inS) (n35a in6) (n36a inS) (n36a in6)]; A3 = [(n37a in7) (n37a inS) (n3Sa in7) (n3Sa inS) (n39a in9) (n39a nlO)]; A4 = [(n40a in9) (n40a in 10)] Then
t ... t Figure 4. Modified ANFIS structure
A =[AI A2A3A4] 32. Learning Algorithm
The important part of Modified ANFIS is the modification of the error correction rules of error back propagation (EBP) by using a mapping function to replace membership function in the standard ANFIS. The learning system of the Modified ANFIS is similar to the standard ANFIS. It uses hybrid learning algorithm which consists of two steps, i.e. forward pass by using fuzzy inference mechanism based on the neural architecture and Least-squares Estimator (LSE), and backward pass to update the premise parameters by the error backpropagation algorithm which has been modified. I.
Forward Pass
The mechanism in the forward pass can be explained as follows: Layer 1: The membership function (usually the bell function) is used in this layer is given by I J1A(x)=----
(11 )
x - C ?h a, The parameters of the membership function {a, ,b" c,}, i = 1,2,3,4 are predetermined.
1+1 - - ' 1-'
274
( 14)
By using Least Squares Estimator (LSE) method, we then obtain the consequent parameters of P. We use ( 15) where U I is the desired output of the controller. We obtain the parameter P = [pe I) ... p(20)] and fl
=
p(I) inI + p(2) in2;
f2 = p(3) in I + p(4) in2; = p(5) in3 + p(6) in4; = p(7) in3 + peS) in4; = p(9) inS + p( 10) in6; = p(II) inS + p(I2) in6; = p(13) in7 + p(14) in8; = p( IS) in7 + p( 16) in8; ~ = p(J7) in9 + p(IS) in 10; fl 0 = p( 19) in9 + p(20) in 10; G f4 f5 f6 f7 f8
After that, the output are n41a = n3Ia fl; n43a = n33a G: n45a = n35a f5; n47a = n37a f7; n49a = n39a f9;
( 16)
of the node n41 a ... n50a n42a = n32a n44a = n34a n46a = n36a n4Sa = n3Sa n50a = n40a
f2; f4; f6; f8; fI O.
( 17)
Layer 5: The output of this layer is the summation of all incoming signals: nSI a = n41 a+n42a+n43a+n44a+n45a+n46a +n47a+n48a+n49a+n50a (18) 2.
Backward Pass
output that yields the output response following the response of the nominal model in Fig. I. Figure 5 shows the initial membership function and final mapping function of the modified ANFIS learning process. The response of the state variables and the control signal are shown in Figure 6. Simulation results show that the output responses follow the given output profile, but these responses are achieved by an oscillatory control input. The average percentage error is given in Table I.
Define the error between the output of the network and the desired output dk . The sum of the squared
~W~OO i "lYJ"X! ..~, . :' O:[ j\'
error is given by
""'o.t NU)
Erl
2)d( - xL)2
=
(19)
lOO
where we have defined Ef' =
5 51 , XI
in this layer is
~
=
U I • Hence
c51 =
-2(U I -175Ia)
(20)
lOO
o.a
'[f
.f
0.'
;
028 Q.l5
~
\
0..
\
0; .'
ou . . o.JII
'f!(!rol)
o.lS
0
Q.JIS
,
2
~
411'01'
i]/"[],'
"::'/ :'["~' // :: y : .' : ./ i
0.1
n51 and d k
0."
..
"/
0
rlN
0..
;
0.4
0: / ::c
Q
I
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k~1
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0.2
...
o
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i
M
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I
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J
QJ
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:
/
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.
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Next, dSI is defined as
•
~
~
~II
(a)
d 51
= -E: 51
/2 = U I
-
(21 )
n5 I a
The output of the node n5 I becomes n51 b = n51 a + dSI
(22)
In fact, n51 b = n41 b + ... + n50b since n5 I a = n4 I a + ... + n50a. A II of mechanisms in forward pass in layer 1 until layer 5 are used in the same manner for backward pass to produces a corrected value (labelled b after added by corrected factors dj, i = 21 .. 51). Further, by extend ing the correction rule of backward pass modified ANFIS learning algorithm in (Jang, er af, 1996) for ten inputs, it will be obtained the corrected value of each node. Finally, if the corrected value in layer I obtained, then all inputs are mapped to this value by interpolation technique. The mapping function then becomes the mapping function of the learning mechanism of the modified ANFIS.
"IXJlilIXl11JIJ o
Sl
1(005)
~I(kI
.1XlQ28
~I(Io.')
0.211 0,280Je dI;IO
G.2J
oleo
1
)
~
'I2(t;")
(b)
Figure 5. a. Initial Membership Function b. Final Mapping Function of Modified ANFIS learning process
~-
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3. 3. Simula/iol1s
~
To show the effectiveness of the modified ANFIS controller, simulation is carried out by using experimental data in (Joelianto, et. aI., 200 I). The data was obtained from experiment held in a fermentor with physical states: temperature 30 e, pH 4.5 , air sreed flow I vvm, initial volume 1.5 L, feed concentration (U2) 10 glL, feed speed (UI) 0.104 glL, sal11rling time 2 hours, and sampling volume 20 mL. From I" hour until 14,h, baker's yeast was grown in batch condition, and the fed-batch condition started at 14'11 hour until 36'11 hour. D
In this simulation, the modified ANFIS controller is trained by using state variables from the experimental data. Then, the modified ANFIS finds the controller
275
.
~
U.
I
..
..
"171 '~ .
.
.
.
.,,<----;-----.----i,
Figure 6. Simulation results
Chattaway, T., G.A. Montague and A.J. Morris (1992). Fermentation Monitoring and Control. In: Biotechnology (H.J. Rehm and G. Reeds. (Ed»,Vol. 3,319-354. VCH Weinheim. Fukuda, H., T. Shiotani, W. Okada and H. Morikawa (1978). A Mathematical Model of The Growth of Baker's Yeast Subject to Product Inhibition, J. Fermentation Technology, 56, 361-368. Jang, J.-S.R., C.-T. Sun and E. Mizutani (1996). Neuro-Fuzzy and Soft Computing. Prentice Hall, Int. Edition. Joelianto, E., R. Dwihandayani and L. Ling (200 I). Parameter Estimation using Genetic Algorithm for Fed-Batch Growth of Saccharomyces cerevisiae. Proc. 2 nd IFAC-C1GR Workshop on Intell. Contr. for Agricultural Application. Okada, W., H. Fukuda and H. Morikawa (1981). Kinetic Expression of Ethanol Production Rate and Ethanol Consumption Rate in Baker's Yeast Cultivation. 1. Fermentation Technof. 59, 103-109. Rahmat, B., E. Joelianto and K. Kadiman (2001). Mapping Function for Error Backpropagation Algorithm of Adaptive Neuro Fuzzy Inference System (ANFIS). Proc. 2"" Int. Seminar on Numerical Analysis in Engineering (NA E), Batam-Indonesia. Shuler, M. L. and F. Kargi ([ 992). Bioprocess Engineering: Basic Concepts. Prentice-Hall [nc., New Jersey. Takamatsu, T., S. Shioya, S. Okada and M. Kanda ([ 985). Profi le Control Scheme in a Baker's Yeast Fed-batch Culture. Biotechnology and Bioengineering, 27, [675-[686.
Table I. The average percentage error (APE)
Output plant XI (t) X2 (t) X3 (t) X4 (t)
APEJo/~
1.5670 7.0361 x 10. 7
3.3392 0
IV. CONCLUSIONS In this paper, a recently developed modified ANFIS controller has been implemented to the control problem in a class of fed-batch fermentation processes which is characterized by non Iinear and uncertain dynamics. Simulation results showes that the modified ANFIS leads to a good output responses with small error with respect to the given output response. REFERENCES
Bastin, G. and J.F. van Impe (1995). Nonlinear and Adaptive Control in Biotechnology. Eur. J. Control, I, 37-53. Boskovic, 1.D. and K.S. Narendra (1995). Comparison of Linear, Nonlinear and Neural Network-based Adaptive Controllers for A Class of Fed-batch Fermentation Processes. Automatica,31,817-840.
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