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Soft Sensor Based on Relevance Vector Machines for Microbiological Fermentation.

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Dev. Chem. Eng. Mineral Process. 13(3/4),pp. 243-248, 2005.
Soft Sensor Based on Relevance Vector
Machines for Microbiological Fermentation
Zonghai Sun* and Youxian Sun
National Laboratory of Industrial Control Technology, Zhejiang
University, Hangzhou 310027, P.R. China
Microbiologicalfermentation is u fype of complicated butch process that is severely
nonlinear and time variant system. Many process parameters are required to be
monitored and controlled in order to make the microbiological fermentation
process successful. Therefore, it is particularly important to measure the process
variables, however the instrumentation and sensors available for fermentation
control do not cover all the desirable or necessary measurements. In fermentation
processes, many important internal variables cannot be measured. Soft sensor has
become an indispensable method that measures internal variables in fermentation
processes. In this paper, the relevance vector machine has been shown to provide
an effective method for constructing a soft sensor for microbiologicalfermentation.
The results obtained have demonstrated that the relevance vector machine soft
sensor can estimate the internal variables infermentation process.
Introduction
It is well known that instrumentation and sensors available for fermentation control
do not cover all the desirable or necessary measurements. In fermentation
processes, the important internal variables that present measurement difficulties, but
which characterize the state and progress of the fermentation, include biomass,
substrate and secondary product concentrations, heat of fermentation, the ratio of
growth rate to consumption rate of substrate. Despite recent encouraging
developments in ion selective and enzyme sensors, optical, and high frequencybased methods, most of the concentrations variables in the fermentation liquidphase cannot be measured accurately or reliably on-line. Therefore, laboratory
analyses are usually required to support fermentation supervision and control [ 1-41.
Soft senor may estimate the above variables on-line. Usually, there are five
types of soft sensor techniques in fermentation:
1, Modelling of mechanism estimates the variables by the elements’ equilibrium.
2. Adaptive observer is also a mechanism-based modelling approach, but the
lunetics parameters are updated by measurement errors on-line. The
fundamental equations of the observer [5] are:
* Authorfor correspondence (zhsun@iipc.zju.edu.cn).
243
Zonghai Sun and Youxian Sun
dxe
-=
dt
dke
-=
dt
g(Xe,u,ke)+ K,(M
K,(M
- f(Xe,ke)) ---- estimate X
. .(1)
---- estimate k
...(2)
- f(Xe,W)
*
where K,, K2are parameters of observer. Bastin and Dochain [6] proposed the
adaptive nonlinear observer for cell concentration and growth rate. The frame of
the adaptive observer is shown in Figure 1.
Filtering mainly includes Kalman filter and extended Kalman filter. When
estimating fermentation variables by filtering, the exact process model and prior
knowledge need to be known for the measurement noise and uncertainty of the
model. The extended Kalman filter suffers from the convergence problem
caused by model linearization.
Neural network mainly estimates the process variables by a construction blackbox model [7-111 and applying neural networks to soft sensors of
microbiological fermentation. Researchers have estimated the variables by
hybrid modelling in microbiological fermentation [12-14].
Method of multivariate statistics mainly adopts the PCA and PLS.
We will first discuss the soft sensor based on relevance vector machine (RVM)
for estimation of biomass concentration. "hen, experiments are shown to illustrate
the effect of soft sensor based on RVM.
Soft Sensor Based on RVM
In this paper we consider estimation of biomass concentration with the release rate
of carbon dioxide. We construct the soft sensor based on RVM that is used to
estimate biomass concentration via release rate of carbon dioxide. We assume the
model of biomass concentration is as follows:
Control
Measurement
noise
noise
-Sensor
+ model
244
-
Soft Sensor Based on Relevance Vector Machinesfor Microbiological Fermentation
Y&+,= f&,
&-I
,*** 9
. ..(3)
x&m, Y &Y*-l9***
,
9 Y&n)
where X , , X ~ - ~ , ~ ~ - , Xdenote
,-~
the release rate of carbon dioxide; y k , y k - , , - * - , y t - n
denote the biomass concentration; y,+,is the variable need to be estimated; m , n
denote the order of input and output respectively. We may view the
x, , x ~ - ~ , .x- ., ,- ~and y k ,y t - l , + .y,-"
- , as input variables.
Let u(k) = [ul(k),~ 2 ( k ).,. .,un+m+t(k)l'
1I i 5 rn + 1
where u,(k)= x,-,+'
m +2 S i Sm +n +2
Figure 2 is the framework of RVM soft sensor for dynamic systems; TDL
denotes the Tapped Delay Line whose output vector is composed of time delay of
input. Therefore, we assume biomass concentration is defined as:
N
f(u,)=&K(u,,u,)+w,,
...(4)
k =1,2;-.N
,=I
Assume p(y,+,1 u , ) is Gaussian N(y,+, 1 f(u,),a'). The mean of this distribution
for a given u, is modelled by f(u,) in Equation (4) for support vector machine.
The likelihood of the data set can be written as [ 15, 161:
P(Y I ~
1
exp(-sl)y
0 '=)
--@q]')
. .(5)
*
where y = [ y l...yN]' , w = [w, wNIr, 0 E R N x N + ' , = K(u,,u,-,), a,'
= 1 . The
posterior over the weights is obtained from Bayes' rule [ 161:
(N+1)/2
p ( w 1 y , a , b * )= (21s)-
1
(Z(-"' exp{--(w - p)'C-'(w
2
- p ) ) ...(6)
with
...(7)
...(8)
1
TDL
JI
'
b
b
Figure 2. Identification model of dynamic system based on RVM.
245
Zonghai Sun and Youxian Sun
a =[n,,a,,..-,a,]'denotes a vector of N
+ 1 hyperparameters; A
=
diag{%, all
..., aN);B = 6-'1, .
Note that o1is also treated as a hyperparameter, which may
be estimated from the data.
By integrating out the weights, we obtain the marginal likelihood, or evidence
for the hyperparameters. Then maximizing the evidence, Tipping [15, 161 gave the
following results:
...(9)
..( 10)
...(11)
...(12)
Algorithm
(1) Acquire data samples, compute the @ ,randomly choose initial CT' ,a
(2) According to Equations (7) and (8), compute C and p respectively.
(3) The hyperparametersa, d a r e updated using Equations (9) and (10)
respectively.
(4) For a new datum, compute f ( x . ) according to Equation (12).
.
Experimental Details
For the ivermctin fermentation process, we develop the biomass concentration soft
sensor based on RVM.For given data set {x,,y,}f:, where x, denotes the carbon
dioxide release rate, and y , denotes the biomass concentration. Biomass
concentration is estimated by RVM soft sensor via carbon dioxide release rate.
Figure 3 shows the biomass concentration estimated by soft sensor based on RVM
and back propagation neural network (BP NN) respectively, and the sums of square
error of estimation are 0.0401, 0.0367 respectively. (In Figure 3, points shown as
stars denote the scatter plot of samples; solid line denotes output curve of soft
sensor based on RVM; and dashed line denotes the output curve of soft sensor
based on neural network.)
In the second example, we develop the ivermctin titer soft sensor based on RVM
for ivermctin fermentation. For given data set {x,, y ,
where xi denotes the sugar
246
Soft Sensor Based on Relevance Vector Machinesfor Microbiological Fermentation
Tim&
150
100
200
250
300
Figure 3. Results of biomass concentration estimated by sofr sensor based on RVM
and neural network via carbon dioxide release rate.
000
900.
700 800
600.
500.
400.
300.
Tirneh
SO
100
1 so
200
250
I
30
Figure 4. Results of ivermctin titer estimated by soft sensor based on RVM and
neural nenvork via sugar concentration.
concentration, y,denotes the ivermctin titer, the ivermctin titer is estimated by
R V M soft sensor via sugar concentration. Figure 4 presents the experimental results
for estimating ivermctin titer by R V M soft sensor and neural network soft sensor
respectively, and the sums of square error of estimation are 0.2666, 15.0218
respectively. (In Figure 4, points shown as stars denote the scatter plot of samples;
solid line denotes output curve of soft sensor based on RVM; and dashed line
denotes the output curve of soft sensor based on neural network.)
247
Zonghai Sun and Youxian Sun
Conclusions
For microbiological fermentation, in order to improve the yield, many process
parameters need to be monitored and controlled. But many important variables
cannot be measured on-line. Soft sensor has become an indispensable method that
measure internal variables in fermentation processes. In this paper we applied the
RVM to estimate internal variables in microbiological fermentation process, i.e.
construct soft sensor with RVM. In the experiments, this method of constructing
soft sensor was compared with that of BP NN. The experimental results indicate
that soft sensor based on RVM can measure the variables effectively in
microbiological fermentation.
Acknowledgments
China 973 Plan supported this work under grant No. 2002CB312200.
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1.
Received 10 November 2003; Accepted alter revision: 6 September 2004.
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