# Soft Sensor Based on Relevance Vector Machines for Microbiological Fermentation.

код для вставкиСкачатьDev. 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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