Research of Methods for Extracting Principal Components

Sensors & Transducers, Vol. 162 , Issue 1, January 2014, pp. 221-226

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Research of Methods for Extracting Principal Components Responding to Sucrose Supersaturation Based Soft Sensors in Cane Sugar Process Yanmei MENG, Guancheng LU, Kangyuan ZHENG, Zhihong TANG, Xiaochun WANG College of Mechanical Engineering, Guangxi University, University East Road 100, Nanning City, Guangxi Province, 530004, China Tel: 07713273600 E-mail: [email protected] Received: 18 September 2013 /Accepted: 9 January 2014 /Published: 31 January 2014 Abstract: Based on the soft sensor techniques, five external factors that affect the cane sugar crystallization process are used as auxiliary variables, i.e., the syrup Brix and temperature, vacuum, steam pressure and feed flow, while the sucrose supersaturation is considered as the key variable. Then a soft sensor method for the sucrose supersaturation is developed based on kernel partial least squares. However, the cane sugar crystallization process is very complex, with features of multiple nonlinearity among the auxiliary variables. Besides, the auxiliary variables obtaining from the on-line sensors involves lots of uncertain components because of the limits of the sensors themselves and the environmental influence in cane sugar process, which weakens the generalization capacity of the soft sensor. Therefore, a kernel partial least squares method is adopted to extract the principle components of the auxiliary variables, which improves the accuracy and generalization capacity of the soft sensor model for the sucrose supersaturation. The experiment results proved that the soft sensor values were close to the actual values where the maximal relative error was nearly 2.91 %. This method is of high performance. Copyright © 2014 IFSA Publishing, S. L. Keywords: Soft sensor, Sugar supersaturation, Kernel partial least squares, Principal component analysis.

1. Introduction Boiling sugar cane is the last and the key link in the sugar industry field, sucrose supersaturation is the main process parameter in the sugar boiling process. However, it's difficult to stably measure sucrose supersaturation through an online instrument directly [1-3], nowadays. Soft sensor technology combines computer technology and knowledge of industrial processes organically, replacing hardware with software, using instrumental variables to estimate dominated variables, which makes the

Article number P_1774

measurement of unpredictable variable possible [4]. Nowadays, domestic and foreign research of soft sensor technology is mainly focused on the process model, system state space model [4], statistical analysis and artificial intelligence, studying the construction method and application of soft sensor model. Some domestic scholars also used neural networks and least squares method to study soft sensor of sugar cane crystallization supersaturation [5, 6], but there are no monitoring method of actual sugar cane crystalline supersaturation nowadays. Based on kernel partial

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Sensors & Transducers, Vol. 162 , Issue 1, January 2014, pp. 221-226 least squares, this paper analyzed the principal components of instrumental variables in sucrose supersaturation soft sensor, and improved the accuracy of soft sensor by eliminating the multiple nonlinear correlations among the instrumental variables and noise interference.

2. Method of Extracting the Principal Components of Instrumental Variables 2.1. A Scheme for Soft Sensor The factors that affect sugar cane crystalline process state are: syrup brix, syrup temperature, vacuum, steam pressure and feed syrup flow. It’s important to control all this parameters to balance sucrose deposition rate and moisture evaporation rate, so as to ensure that the sugar crystal can grow fast without spurious grain and sticky crystals. We used five instrumental variables: syrup brix, syrup temperature, vacuum, steam pressure and feed syrup flow as input, cane sugar supersaturation as the output. The scheme of cane sugar supersaturation soft sensor is shown in Fig.1.

2.2. Algorithm Considering there are multiple nonlinearity features between the complicated instrumental variables in the sugar crystalline process, meanwhile

sensors are affected by inherent characteristics and environmental noise, resulting that the instrumental variable data obtained by sensors has more uncertain factors. It compromises the ability of generalization. In order to obtain useful information from instrumental variables, this paper adopted kernel partial least squares to analyze the principle components of instrumental variable in order to improve the accuracy of soft sensor. We set the sugar syrup supersaturation soft measurement sample as

 xk , yk k 1 , where M

M is the number of samples; xk

represents the soft sensor instrumental variables vector, xk  [ xk1 , xk 2 , xk 3 , xk 4 , xk 5 ] , xk 1 , xk 2 , xk 3 ,

xk 4 , xk 5 respectively, represents the vacuum, syrup temperature, steam pressure, syrup brix, and feed syrup flow of sample i ; and yk represents the sugar supersaturation sample value that corresponds to the instrumental variable k . This paper maps the sugar supersaturation and nonlinear input space of the soft sensors into high-dimensional space [7]. Therefore, the nonlinear input space is transformed into a linear high-dimensional space. Using kernel function, this paper replaces inner product in the high dimensional feature with kernel function computing [8-9], so the inner product of the two instrumental variables from two original input space mapped into high dimensional feature is:

 



 x i  T x j  K x i ,x j



(1)

Fig. 1. A scheme of the cane sucrose supersaturation soft sensor in the sugar boiling process.

Supposed the kernel function of kernel partial least squares and soft sensor were the same, the followings will be named as soft sensor algorithm kernel function. They are all Gaussian radial basis







kernel function K xi , x j  exp  xi  x j



2



, and

are parameter of radial basis kernel function. The steps of extracting the principal component of instrumental variables in sucrose supersaturation soft measuring is described as followings: 1) Build the kernel matrix of instrumental variables of sample set

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The instrumental variables matrix of sample set is:  x1   x11        X   x k    xk 1         xM   xM 1

x12 x13 

x14





xk 2 xk 3

xk 4

   xM 2 x M 3 xM 4

x15     xk 5 ,1  k  M    xM 5 

(2)

2) The cane sugar supersaturation vector of sample set is:

Sensors & Transducers, Vol. 162 , Issue 1, January 2014, pp. 221-226

y [ y1,, yk ,, yM ]T ,1 k  M ,

(3)

3) Map the instrumental variable of original input space into high dimensional characteristic space through nonlinear mapping, and the instrumental variables matrix of the high dimensional characteristic space is:   x1        X    xk  ,1  k  M        x M 

(7)

where y is the mean value of supersaturaion while  is the standard deviation of it. 5) Extraction of Principal Components. Suppose Wi as a column vector to represent the correlation coefficient vector of sample i ; ti as a column vector to represent the scoring vector of sample i ; vi as a column vector to represent the

(4)

regression coefficient vector of ti to yi ; pi as a

The kernel matrix of instrumental variables of sample set is: K  X  X T T   x1     x1                x k      x k               x M    x M    x1  x1   x1  xi   x1  x M           xi  x1   xi  xi   xi  x M         x x x              x M i   x M  x M 1 M   K x1 , x1  K x1 , xi  K x1 , x M           K xi , x1  K xi , xi  K xi , x M          K x M , x1  K x M , xi  K x M , x M

T y  y y y y  y , , k , , M ycenter   1  ,1  k  M     

    ,1  i  M   

      

represent the residuals of instrumental variables kernel function and sugar supersaturation vector after the extraction of principal components of sample i . h indicates the number of principal components. K feature represents the characteristic transformation matrix of instrumental variable kernel function after the extraction has been operated by kernel partial least squares. K feature can be used to transform the instrumental variable kernel function matrix of sugar supersaturation so as to analyze the principal component of the instrumental variable [3].

2.3. Implementation 2.3.1. Characteristic Transformation Matrix of Instrumental Variable (5)

4) Centralize the kernel matrix Kr of instrumental variables of sample set and cane sugar supersaturation vector y. The application of kernel function makes it unnecessary to know the specific forms of transition from the original input space of instrumental variable to the high dimensional characteristic space. Since the high dimension characteristic space of instrumental variables can’t not be centralized directly, the centralization of kernel matrix Kr would be operated by correcting and indirect calculation [9-12], and the method is: 1 1    T  K center   I M  eM eT M  K  I M  M eM eM  M 

column vector to represent the load vector of principal components of sample i ; K i 1 and yi 1

,

(6)

where K center is the centralized kernel matrix; I M is the M-square matrix which all its elements is 1; eM is the column vector which all its M elements is 1. The supersaturation vector of sample set after centralization is:

Kernel partial least squares kernel function parameter  and parameters of soft sensor are the same, all of which will be referred to as soft sensor kernel function parameter. It comes from the optimization of genetic algorithm and the improved leave-one-out cross validation after mixing kernel partial least squares into soft sensor algorithm. The basic thought of kernel parameter optimization is to use soft sensor algorithm to study instrumental variable principal component based on kernel partial least squares [9], whose purpose is to minimize the error of cross validation. Therefore the best kernel function parameters will be obtained and the characteristic transformation matrix of instrumental variable kernel function matrix will be saved. When the kernel function parameters of sugar supersaturation soft sensor changes, the characteristic transformation matrix of instrumental variable kernel function matrix will be calculated again to adjust the sugar supersaturation soft sensor model. Consequently, it’s more precise to extract valid information of instrumental variable so as to improve accuracy of soft sensor. The flow process of characteristic transformation matrix of instrumental variable kernel function matrix is shown as Fig. 2.

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Sensors & Transducers, Vol. 162 , Issue 1, January 2014, pp. 221-226 square matrix which has all element of 1. eM is a Mcolumn vector which has all element of 1. Project vector tnew into characteristic transformation matrix K feature , then the scoring vector tnew is obtained after calculating the centralized kernel function vector knew . The calculation formula is:

t new  k new K

fe a tu re

(10)

2.4. Principle Component Analysis

Fig. 2. Flow process of the characteristic transformation matrix.

2.3.2. Extracting Principal Component of Soft Sensor Instrumental Variable To extract principal components of instrumental variable is an important process for sugar supersaturation soft sensor, which can eliminate uncertain components of instrumental variable and enhance the accuracy of soft sensor. Firstly, this paper uses the input instrumental variable vector xnew to build the corresponding kernel function vector

knew ; secondly, centralize the kernel function

vector; next, use kernel function characteristic transformation matrix to transform the centered kernel function vector so as to gain the scoring vector of kernel function vector. Then, it’s available to replace the original input vector with scoring vector and use sugar supersaturation soft sensor algorithm to study, model and soft measure. The construction method of instrumental variable kernel function vector is as follows when supersaturation soft sensor is operated: knew   xnew  x1 ,,  xnew  xi , ,  xnew  xM 

 K xnew , x1 , , K xnew , xi ,, K xnew , xM ,1  i  M

(8) The centralization method variable kernel function vector:

of

instrumental

1 1   T  k new   k new  eM eT M K  I M  M eM eM  M 

(9)

The knew in the right side of formula (9) indicates the instrumental variable kernel function vector which is not centralized. The knew in the left side indicates the instrumental variable kernel function vector which has been centralized. I M is the M-

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The soft sensor of sugar supersaturation should analyze its components based on the offline soft sensor model. The system uses online module algorithm to optimize the parameters and feedbacks online dynamic information to the model parameters, through which the real-time monitoring of main parameters is implemented. The modeling of soft sensor includes principal component analysis, construction of offline and online model and parameter optimization method. Using VC+ +6.0 to develop a sugar supersaturation soft sensor system, which is capable to achieve offline analyzing and online soft measuring the parameter of sugar boiling process based on kernel partial least squares method. The system provides two external interfaces, one is COM component interface, and the other is the interface of general dynamic link library, which makes it convenient to exchange data with external automatic control algorithm. It’s adaptive for any sugar boiling process automatic control system to use this system via corresponding interface and integrate this system into sugar boiling process automatic control system as long as the COM component statute or the general dynamic link library statute is called. For the sugar boiling process automatic control system which needs operating independently, the data exchange can be obtained by the data sharing cell without integrating the system into the process of automatic control system entirely. On the whole, the system can be divided into principal component analysis module, offline module, online module, interface management module, data sharing unit module and external interface module. The overall architecture of the system is shown in Fig. 3 [12].

3. Simulation and Experimental Verification Sugar supersaturation soft sensor must use Principal Component Analysis, take the offline soft sensor model as initial model, go through procedures of online algorithm fast parameter optimization and feed the online sample information back to the model parameter dynamically, which is how the real-time monitoring of the main parameters process.

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Fig. 3. Overall system architecture.

Soft sensor modeling includes principal component extraction, offline model and online model establishment and parameter optimization methods. Using VC+ +6.0 to develop sugar supersaturation soft sensor system, it’s capable to achieve offline analyzing and online soft measuring the parameter of sugar boiling process based on kernel partial least squares method. During the experiment, we chose 180 random samples. Select the previous 140 samples as a soft sensor model training set, followed by 40 samples as a testing set, we tested them in the self-developed sugar crystallization process comprehensive experimental platform. And the experimental results are shown in Fig. 4. The results showed that error of predicted value and actual value is small, the soft sensor values were close to the actual values where the maximal relative error was nearly 2.91 %. Thus, the system based on least squares support vector machine (SYM) measuring is feasible. It can give accurate and reliable measurement results. In order to verify the stability and accuracy of soft sensor online-monitoring, this paper made some experiment in the independently developed comprehensive experimental platform during the process of boiling sugar and estimated online sugar supersaturation by measuring the conductivity, making comparison between prediction value of soft sensor system and online measured value of sugar supersaturation. In the experiment, the online prediction value of sugar supersaturation and

prediction value of soft sensor was consistent during a period after beginning boiling sugar. But after the sensor was inserted into syrup for a long time, the difference between the measured value and the prediction value increased. We found some dirt in the surface of sensor, and the error between prediction value and measuring value became various, indicating that the dirt in the surface of sensor had great influence in the measuring result. The experimental result of soft sensor online prediction and supersaturation sensor test without dirt in the surface is shown in Fig. 4. The test result showed that the soft sensor values were close to the actual values where the maximal relative error was nearly 2.91 %. Thus, the prediction of soft sensor is stable. The soft sensor system is feasible based on least squares support vector machine, and it can give accurate and reliable measurement result.

4. Conclusion We used five instrumental variables syrup brix, temperature, vacuum, heating steam pressure and feed speed as input, cane sugar supersaturation as the output to build a model on purpose of analyzing the principal component of instrumental variables in sucrose supersaturation soft measuring, eliminating the multiple nonlinear correlation among the instrumental variables and noise interference, and improving the accuracy of soft sensor based on kernel partial least squares.

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Fig. 4. Result of sugar supersaturation online soft sensor.

Using VC+ +6.0 to develop sugar supersaturation soft sensor system, it’s capable to achieve offline analyzing and online soft measuring the parameter of sugar boiling process based on kernel partial least squares method. We tested them in the selfdeveloped sugar crystallization process comprehensive experimental platform. The soft sensor values were close to the actual values where the maximal relative error was nearly 2.91 %. The testing result is good.

Acknowledgements This research is supported by National Natural Science Foundation of China (51065004), Guangxi Natural Science Foundation (2011GXNSFA018168), and Nanning Scientific and Technological Project (Project No. 20131079).

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