Soft sensor modelling by time difference, recursive partial least squares and adaptive model updating

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1 Soft sensor modelling by time difference, recursive partial least squares adaptive model updating Y Fu 1, 2, W Yang 2, O Xu 1, L Zou 3, J Wang 4 1 Zijiang College, Zejiang University of ecnology, Hangzou 3124, Cina 2 Scool of Electrical Electronic Engineering, University of Mancester, Mancester M13 9PL, UK 3 Scool of Automation, Nanjing University of Science ecnology, Nanjing 2194, Cina 4 Scool of Electronic Information Electrical Engineering, Sangai Jiaotong University, Sangai 224, Cina fuyongfeng@zjut.edu.cn Abstract o investigate time-variant nonlinear caracteristics in industrial processes, a soft sensor modelling metod based on time difference moving-window recursive partial least square (PLS) adaptive model updating is proposed. In tis metod, time difference values of input output variables are used as training samples to construct te model, wic can reduce te effects of te nonlinear caracteristic on modelling accuracy retain te advantages of recursive PLS algoritm. o solve te ig updating frequency of te model, a confidence value is introduced, wic can be updated adaptively according to te results of te model performance assessment. Once te confidence value is updated, te model can be updated. e proposed metod as been used to predict te 4-CBA (i.e. carboxy-benz-aldeyde) content in te PA (i.e. purified tereptalic acid) oxidation reaction process. e results sow tat te proposed soft sensor modelling metod can reduce computation effectively, improve prediction accuracy by making use of process information reflects te process caracteristics accurately. Key words:soft sensor, ime difference, Recursive PLS, Model updating List of symbols a B Number of latent variables Regression coefficient matrix x i, N i, N 1 c Stardised feature vector x t x Mean of te it sample in te data window of lengt N Online updated mean Auxiliary variable values of new data e Noise vector xt ' ime difference values of x t ' E Residual matrix X Input data matrix F Residual matrix X t Present values of auxiliary variables l Number of output variables X t i Values in some time i before X t m Number of input variables Xt ime difference values of X t n Number of available samples y t Present values of objective variables y Values in some time i before y t N Window lengt t i p Loading vector of te t latent variable yt ime difference values of y t Predicted objective variable of new data P Loading matrix y t ' q Loading vector of te t latent variable y t ' ime difference value of y t ' Q Loading matrix Y Output data matrix R Error matrix β Regression coefficient matrix t Score vector of t latent variable Estimate of β β 1

2 Score matrix Confidence limit u Score vector of t latent variable e Adaptive confidence limit U Score matrix 2 Variance of it sample in data window of lengt N i, N w Stardised feature vector 2 Online updated variance i, N1 1. Introduction In industry, some process variables are closely related to te product quality. However, in many cases, it is difficult to measure tose variables online for tecnical or economic reasons. erefore it is difficult to monitor control te processes in real time, te product quality cannot be ensured. o address tis issue, soft sensors ave been developed in recent years used to estimate te process variables. Wit a soft sensor, an inferential model can be constructed between some variables tat can be measured online (also called auxiliary variables) oter variables tat cannot be measured online (also called objective variables). en an objective variable can be estimated using te model. If te objective variables can be estimated accurately by te use of soft sensors, optimal control of tose industrial processes can be implemented [1-3]. Among various soft sensor modelling metods, te partial least squares (PLS) metod is most popular because it is simple in calculation, not very sensitive to noise can provide good correlation between te measured variables (i.e. te input variables) te estimated variables (i.e. te output variables) [4-6]. However, te PLS metod can only reflect a linear relationsip between te input output variables. Considering tat industrial processes in general ave time-varying caracteristics, te prediction accuracy of te soft sensors based on PLS would gradually decrease due to various reasons, e.g. te canges in te state of a process, sensor drift, loss of catalysing performance. o overcome te problem tat te PLS metod does not reflect te non-linear relationsip between te input output variables, Rosipal et al. proposed te Kernel PLS metod [7-8]. Wile tis metod can overcome to some extend te sortcomings of PLS, its non-linear approximation depends largely on te complexity of te model, its calculation is complicated. o reduce te degradation caused by te cange in a process, Hell et al. proposed a recursive PLS algoritm, wic can update te parameters of te PLS model online [9]. Qin modified supplemented te recursive PLS algoritm, used tis metod to model estimate some cemical processes [1]. A recursive algoritm can make use of information from old new data, can effectively track te dynamic caracteristics of a process. e problem wit a recursive algoritm is tat a large number of repetitive calculations are needed wit te continuous increase in te number of modelling samples, tere is a data saturation problem. In addition, too many old samples ide te information of new samples. Moving window [1] or forgetting factor [5, 11] is an effective way to solve tese problems. However, a moving window metod treats new data old data equally in model updating. erefore, it cannot keep a good track of te dynamic caracteristics of an industrial process. As for te forgetting factor, it is difficult to be set or optimised. Mu et al. proposed a moving-window recursive PLS algoritm tat combines a moving-window a recursive algoritm [12]. is algoritm can modify te mean variance online, wic 2

3 enables te metod to keep te old sample information effectively track te dynamic caracteristics of a process. However, every time a new sample is collected, te model needs to be updated. e updating rate is terefore ig, resulting in low computational efficiency [13]. A data block-based recursive PLS metod combines te moving-window forgetting factor [5]. In tis metod a model is updated only wen new data are accumulated to a fixed lengt. Altoug tis metod can reduce te model updating rate, te data saturation problem is still remaining it is difficult to set te forgetting factor. In addition, te lengt of te data blocks can only be set by experience because tere is no establised rule to follow. o address te above issues, we propose a soft sensor modelling metod based on time difference, moving-window, recursive PLS adaptive model updating. is metod firstly uses te time difference values between te current sample data te previous sample data to build a moving-window recursive PLS model. is time difference moving-window recursive PLS model not only can fully retain te good caracteristics of linear PLS model, improve te model s approximation accuracy to te nonlinear caracteristics of te industrial process. en, tis metod can be automatically generated model confidence limit based on te initial caracteristics of te process. is confidence limit can update adaptively wit te time-varying nature of te process te results of te model performance assessment. If te model prediction error is greater tan te confidence limit, te model will update. Once te model updated, te confidence limit value will update too. e proposed metod is applied to build a soft sensor model based on actual industrial process data to predict te 4-CBA content in PA oxidation process. e results indicate tat te metod is effective. It can reduce computation improve te prediction accuracy effectively. 2. Basic algoritms 2.1 PLS algoritm Let s consider a pair of input data matrix, nm X R, an output data matrix nl Y R, wic ave been centralised normalised, wit n being te number of available samples, m te number of input variables, l te number of output variables. e relationsip between X Y can be represented by were β is te coefficient matrix e is te noise vector. Y Xβ e (1) A PLS algoritm decomposes matrixes X Y into two external relations, called external model: X P E a t 1 p E (2) Y UQ F a 1 u q F (3) were t, t 2,, U u, u2,, are called score matrix, P p, p2,, 1, q2, q a 1 t a 1 u a 1 Q q, are called loading matrix, E F are corresponding residual matrix, a is te number p a 3

4 of latent variables, t corresponding loading vectors. were Score vectors E t u are te score vectors of t latent variable, u can be solved by E 1 t p 1,2,,a, E X t E1w, u F 1c ; p q are te (4) q F F 1 u 1,2,,a, F Y ; w c are te stardised feature vectors corresponding to te main caracteristic values of E 1F 1F 1E1 1E1E 1F 1 F respectively. e relationsip between U is called te internal model, wic can be expressed as were b, b,, matrix R. U B R a 1 b t B diag 1 2 b a is te regression coefficient matrix, wic is determined by te minimised error R (5) In a PLS algoritm, te estimate β of te regression coefficient matrix β is β W BQ k 1 were W w, w2,, w, k I w p 1 w a, 1 w1 1 w, k 2,, a. (6) e detailed description of te PLS algoritm can be found in references [9-1, 14]. 2.2 ime difference algoritm In a traditional soft sensor modelling metod, wen te auxiliary variables set variables set metod. y t are given, te relationsip between X t t In a time difference modelling metod [15, 16], te time difference values y t are firstly calculated by were X t t X t i t i t Xt Xt i X t te objective y can be found using a regression Xt yt of X t X (7) t yt yt i y (8) y are present values of auxiliary variables objective variable respectively, y are values in some time i before te present time. e relationsip between process, wen new data calculated by Xt yt can be modelled by regression metods. In te prediction x t ' are collected, te time difference value xt ' of te new data is firstly 4

5 Using te time difference yt '. en t ' because yt i ' y can be calculated by is given previously. ' ' ' t xt xt i xt ', te constructed model can predict te time difference value of t ' x (9) y ' ' ' t yt yt i y, (1) 2.3 ime difference PLS algoritm e PLS algoritm as many advantages. It is simple as clear pysical meaning. It can effectively overcome te collinearity problems can make te model contain te minimum number of independent variables. A problem wit te PLS algoritm is tat it is essentially a linear regression metod, but industrial processes are often non-linear, tus te prediction accuracy of te model built directly by a PLS algoritm is not ig. In tis paper, te above-mentioned time difference algoritm is used to build a soft sensor model, i.e. instead of directly using te auxiliary variables set difference values Xt yt Suppose a nonlinear process model as follows. At any point x x x, x 1, 2, m equation can be obtained, X t te objective variables set y t are used to build a soft sensor model. e principle is as follows. x, teir time y f (11), using aylor s expansion omitting te iger-order terms, te following f f f x f x x1 x1 xm xm (12) x1 xm x x f f f y f x f x x1 x2 xm (13) x1 x2 xm x x According to te above derivation, te relationsip between y x is linear. us te PLS model built by te time difference values y x as good nonlinear regression performance, can improve te prediction accuracy [17]. Furtermore, because te time-difference PLS algoritm uses time difference values of te input output data to build a soft sensor model, slow canges in an industrial process will not ave significant impact on te prediction accuracy. 3. Recursive PLS algoritm adaptive online updating As described above, te time difference PLS algoritm as good nonlinear regression performance. Wen te production process as strong time-varying caracteristics, or te operating point canges frequently, owever, te prediction accuracy will gradually decrease. o reduce te degradation of time difference PLS 5

6 model, it is necessary to update te model online [18]. e moving-window recursive algoritm as been widely used because it can effectively track te dynamic caracteristics of a process solve te data saturation problem. However, te updating rate of te ordinary moving-window model is ig, leading to a eavy computational load reduced real-time performance. Considering te above issues, an adaptive updating strategy is used in tis paper. First, a confidence limit is set according to te initial caracteristics of te process. en te confidence limit is updated adaptively wit te dynamic cange in te process te prediction results. Wen te prediction error is bigger tan te confidence limit, te model parameters will be updated ence te confidence limit is updated. Oterwise, te model will not be updated. 3.1 Moving-window recursive PLS algoritm based on variance mean online updating Because te PLS algoritm can be used for modelling ig-dimensional data wit few samples, te data lengt generally used for modelling is not large, wit te ordinary moving-window recursive metod used. Wen a new sample is collected, te oldest sample will be aboned, wic may lose information. In te moving-window recursive algoritm based on online updating of variance mean, te variance mean will be updated online, wen a new sample is collected. us, part of information of old data can be retained in te model troug variance mean. en, te oldest sample will be discarded, keeping constant modelling sample lengt N. Online recursive equations of te mean te variance are given by N 1 x i, N 1 xi, N xi, N 1 (14) N 1 N N 1 2 N 1 i, N 1 i, N x 2 i, N 1 xi, N 1 (15) N N were x i, N 2 i,n are te mean te variance of te it sample in te data window of lengt N, x i, N 1 2 i, N 1 are te online updated mean variance of te it sample after te arrival of a new sample [19]. e data lengt N of te moving window is very important for te computational efficiency, but till now, tere is not an effective metod to determine it. Currently, te setting of N is totally depend on experience experiment. In tis work, we did many experiments cose a proper N. en, te regression coefficient matrix of te PLS model can be updated online by using te updated mean variance. 3.2 Adaptive confidence limit setting e moving-window recursive algoritm can effectively solve te data saturation problem, it can retain part of information of old samples troug online updating te mean variance of samples wile tracking te dynamic cange in te process, tus improving te prediction performance. e problem wit te moving-window algoritm is te ig model updating rate. Wenever a new sample is collected, te model will 6

7 be updated, leading to a eavy computational load reduced computational efficiency. o determine weter it is necessary to update te model, a confidence limit of te prediction error is used. A commonly used model updating metod based on te confidence limit of prediction error is to set a small positive number as te confidence limit. Wen te prediction error is larger tan, te model will be updated. Oterwise, te model remains uncanged. Altoug tis metod can effectively reduce te model updating rate, it is difficult to set te confidence limit, wic can balance between te prediction accuracy te updating rate. Currently, te setting is totally dependent on experience or experiment. After te confidence limit is set, it remains constant trougout te wole process. erefore, it cannot reflect te dynamic cange caracteristics of te process. In tis paper, an adaptive confidence limit, wic is te stard deviation of prediction errors of te objective variables, is used as te criteria to determine weter te model sould be updated [4, 2], as defined by l yij yij j1 i1 e (16) N a A l N 2 were a is te number of latent variables, l is te number of output variables, yij is te measured value for te i t measurement of te j t output variable, yij is te corresponding predicted value given by te model, A is set to be 1 if te model is stardised, oterwise ; N is te number of modelling samples, tat is te lengt of te moving window. e value of N can not only affect te prediction accuracy of te model, but also can affect te value of te confidence limit, so as to affect te updating rate of te model. Because tere is not an effective metod to determine te value of N, we compreensively consider te model's prediction accuracy update rate, determine a suitable value of N troug experiments. e difference between tis confidence limit te conventional confidence limit is tat te confidence limit is a statistical variable associated wit te process caracteristics. It can update adaptively wit te dynamic cange in te process caracteristics. erefore, te difficulties wit te conventional metod in setting te confidence limit no reflection of te dynamic canges, are resolved. 4. Soft sensor modelling based on time-difference, moving-window, recursive PLS adaptive model updating 4.1 e proposed metod e soft sensor modelling metod based on time-difference, moving-window, recursive PLS adaptive model updating makes use of te time difference values of te input output variables to build a moving-window, recursive PLS model, makes use of a statistical variable, wic can cange adaptively wit te dynamic caracteristics of a process te canges of te model s prediction accuracy, as te confidence limit to determine weter te model sould be updated. e advantages of tis metod include (1) improving te model s approximation accuracy to te nonlinear caracteristics of industrial processes, (2) enancing te model s tracking capabilities to dynamic canges in industrial processes, 7

8 (3) reducing te model s update rate, (4) improving te model s computational efficiency. is modeling metod can be described in detail as: Select a modelling sample set, calculate te time difference values of te input output variables build a time-difference moving-window PLS model; According to te statistical caracteristics of prediction errors of te modelling sample set, calculate te stard deviation of te prediction errors by equation (16), tis calculation result will be te initial confidence limit; Incorporate te new sample into te modelling sample set wenever a new sample is collected update te mean variance of te new modelling sample set, ten discard te oldest sample; Calculate te prediction error of te new sample after te prediction value of te new sample is obtained. If tis prediction error is larger tan te confidence limit, te model will be updated, te confidence limit will be updated too. Oterwise te model will remain uncanged. 4.2 Modelling steps of te soft sensor model based on time-difference, moving-window, recursive PLS adaptive model updating e modelling steps of te soft sensor model based on time-difference, moving-window, recursive PLS adaptive model updating are as follows. Step 1: Determine te window lengt N of sample data (note tat te N samples constitute te modelling sample set X t ), calculate te time difference values Xt yt in t t 1 1, y X 1 t, y1 according to equations (7) (8); Step 2: Stardise variance of tese N -1 samples; Xt yt to obtain te training sample set t yt of te samples X, calculate te mean Step 3: Build time-difference, moving-window, recursive PLS model using te sample set t yt X,, to obtain te regression coefficients β, (note tat te prediction value y t of yt can be obtained by te equation yt X t β ); Step 4: Calculate te stard deviation te model (note tat according to y t yt i yt obtained); e by equation (16), use e as te initial confidence limit of, te predicted value of te objective variable can be Step 5: Incorporate te new sample into te sample set X t wen a new sample, collected; calculate te time difference value (8); t 1 1, y t t x new y new x is new y new of te new sample by equations (7) Step 6: Recursively update te means variances of te modelling sample set t te oldest sample; Step 7: According to y new t of te new sample can be obtained; x new t X 1 t, y1 discard β, y t yt i y t, te prediction value y t new new new 8

9 Step 8: Calculate te root means squared (RMS) error; Step 9: If te condition of RMS> e is satisfied, ten go to Step 3. Oterwise, go to Step Industrial application 5.1 Process In a purified tereptalic acid (PA) production process, 4-carboxy-benz-aldeyde (CBA) content reflects te process of oxidation. 4-CBA is one of te main by-products in te oxidation process, is one of te important quality indicators. According to te studies on te reaction mecanism, if te 4-CBA content is too ig, te oxidation reaction is insufficient te p-xylene (PX) conversion rate is too low, wic must be controlled in te production process. If te 4-CBA content is too low, per oxidation te side reaction is increased, leading to te increase in consumption of energy, acetic acid PX. o save energy ensure te purity of PA, te 4-CBA content must be controlled witin a certain range. erefore, it is important to implement online control of 4-CBA content. However, currently te 4-CBA content cannot be measured on-line. An alternative metod is to use a soft sensor. e detail of te PA oxidation reaction process can be found in [21]. According to analysis on te process oxidation reaction mecanism, 1 parameters are selected as te auxiliary variables (i.e. input variables). e objective variable (i.e. output variable) is te 4-CBA content in tereptalic acid (A). e input output variables of te soft sensor model are listed in able 1. able 1 Input output variables of te model Attribution No. Description input variables output variable 1 Mixing tank feed flow 2 Oxidation reactor feed flow 3 Catalyst concentration 4 Heigt of reactor liquid 5 Reactor temperature 6 Last oxygen content of reactor 7 Discarge water of condenser 1 8 Discarge water of condenser 2 9 Meld temperature 1 Last oxygen content of mould 11 4-CBA content 5.2 Results discussion Because te 4-CBA content cannot be measured online, it is usually sampled tree times eac day measured manually, so, only tree samples per day can be used to build te soft sensor model. A large amount of process data were collected from a PA oxidation reactor of a cemical fibre factory. After pre-processing, a data set wit 231 samples was obtained. e data set is divided into two sets. One wit 2 samples is used as a 9

10 modelling sample set, te oter wit 211 samples as a testing sample set to evaluate te performance of te model. o verify te effectiveness of te proposed metod, different models were used, including (1) a PLS model (2) a moving-window, recursive PLS model (3) a time-difference, moving-window, recursive PLS model (4) an adaptive model updating, moving-window, time-difference, recursive PLS model. eir prediction results were compared te models prediction accuracy was evaluated by tree indicators: (1) te maximum relative error, (2) te minimum relative error (3) te RMS error. Eac model uses 2 modelling samples (or te moving-window lengt), 4 latent variables. 4 CBA concentration (ppmw) real value predict value CBA concentration (ppmw) real value predict value relative error (a) PLS model relative error (b) Moving-window recursive PLS model 4 CBA concentration (ppmw) real value predict value relative error (c) ime-difference moving-window recursive PLS model Figure 1 Measured values predicted values wit different models o compare te prediction accuracy of te models, te comparison of te prediction results of te models te measured values are sown in Figure 1. It can be seen tat te coincidence of te prediction values of te PLS model te measured values is poor. In particular, wen te data cange intensely, te error is large. Because te PLS algoritm is a linear algoritm, its regression accuracy to a nonlinear process is poor. e 1

11 prediction results of te moving-window, recursive PLS model are muc better tan te PLS model, because te moving-window, recursive PLS model used can update its means variances online, it can track te dynamic caracteristics of te process, tus enancing te modelling accuracy. e prediction results of te time-difference, moving-window, recursive PLS model are te best, because tis model uses te time difference values between te input output variables to build te soft sensor model online, it can improve te model s approximation accuracy to te nonlinear process, enancing te prediction accuracy. Comparing Figure 1 (a), (b) (c), it can be seen tat te time-difference, moving-window, recursive PLS algoritm is effective. It can greatly reduce te prediction errors, te curve of prediction values is consistent wit te curve of te measured values. But because te online updating metod used in time-difference, moving-window, recursive PLS model updates te model s parameters wenever a new sample is collected, altoug tis metod can improve te prediction accuracy, it is at te expense of computational efficiency. e prediction results of adaptive updating, time-difference, moving-window, recursive PLS model are sown in Figure 2. It can be seen tat te prediction accuracy of adaptive updating, time-difference, moving-window, recursive PLS model is sligtly lower tan te time-difference, moving-window, recursive PLS model. But overall, te improvement is obvious compared wit te ordinary moving-window, recursive PLS model. ink of te computational efficiency, we can use model updating rate to compare it. e ordinary moving-window, recursive PLS model te time-difference, moving-window, recursive PLS model updated wenever a new sample is collected. From able 2, we can see tat te number of updates is 211. e adaptive model updating, time-difference, moving-window, recursive PLS model is only updated 157 times, te computational cost saving is about 25.6%, so te computational efficiency of te adaptive updating, time-difference, moving-window, recursive PLS model as been greatly improved. It can be concluded tat te adaptive model updating, time-difference, moving-window, recursive PLS model proposed in tis paper is effective as it can not only improve te prediction accuracy of te model, but also improve te computational efficiency. 4 CBA concentration (ppmw) real value predict value relative error Figure 2 Real values te predict values of adaptive updating time-difference moving-window recursive PLS model 11

12 Model able 2 Performance comparison of different models for modelling 4-CBA content Maximum relative error Minimum relative error Root mean square error Model updating rate PLS Moving-window, recursive PLS ime-difference, moving-window, recursive PLS Adaptive model updating, time-difference, moving-window, recursive PLS able 3 Overall qualitative comparison of eac model Model Prediction accuracy Algoritmic complexity Calculating time PLS Low Low Sort Moving-window, recursive PLS Middle Middle Long ime-difference, moving-window, recursive PLS Hig Hig Long Adaptive model updating, time-difference, moving-window, recursive PLS Hig Hig Middle e performance of different models for 4-CBA content is compared in able 2, including te maximum relative error, te minimum relative error, te root mean square error of prediction, te model updating rate. It can be seen tat te prediction accuracy of te time-difference, moving-window, recursive PLS model as been greatly improved, but its computational efficiency is also low. e computational efficiency of te adaptive model updating, time-difference, moving-window, recursive PLS model as been greatly improved wit a little loss of prediction accuracy. e overall qualitative comparison of eac model is given in able 3, including prediction accuracy, algoritmic complexity calculating time. From Fig.1, Fig.2 able 2, able3, we can see tat te proposed metod as ig prediction accuracy ig computational efficiency. Because tis metod uses time difference moving-window recursive PLS model, it can not only fully retain te good caracteristics of linear PLS model, but also improve te model s prediction accuracy to nonlinear process. Furtermore, tis metod uses adaptive confidence limit tat can update adaptively wit te time-varying nature of te process to reduce te model updating rate, so to improve te computational efficiency. 6. Conclusions e ordinary PLS algoritm is popular in soft sensor modelling, but cannot reflect te nonlinear relationsip between te input output variables. Altoug te recursive PLS algoritm can reflect te nonlinear relationsip to some extent, te model updating rate is ig, ence te computational efficiency is low. o deal wit tese problems, a soft sensor modelling metod based on time-difference, moving-window, recursive PLS adaptive model updating is proposed in tis paper. is metod firstly uses te time difference values between te input output variables to build a time-difference moving-window recursive PLS model. en a confidence limit is generated automatically based on te initial caracteristics of te process, wic can be updated adaptively wit te time-varying nature of te process te prediction accuracy of te model. 12

13 o verify te metod, te proposed metod was used to predict te 4-CBA content in a PA oxidation reaction process. e results sow tat te soft sensor modelling metod based on te time-difference, moving-window, recursive PLS adaptive model updating can not only improve te prediction accuracy significantly, but also greatly improve te computational efficiency. In industry, many process variables are difficult to measure online te processes ave time-variant nonlinear caracteristics. e soft sensor metod proposed in tis paper is an effective alternative coice to implement te real-time estimation of tese process variables, tus making online quality control, advanced control optimal control of tese processes possible. Acknowledgements e autors would like to tank te National Natural Science Foundation of Cina for supporting tis work (Grant No , ) te Open Researc Project of te State Key Laboratory of Industrial Control ecnology, Zejiang University, Cina (Grant No. IC16265). Yongfeng Fu would also like to tank te Cinese Scolarsip Council for supporting er to be an academic visitor at e University of Mancester in te UK. References [1] KANO M NAKAGAWA Y 28 Data-based process control, quality improvement: recent developments applications in steel industry Comput. Cem. Eng [2] KADLEC P, GABRYS B SRAND S 29 Data-driven soft sensors in process industry Comput. & Cem. Eng [3] FORUNA L, GRAZIANI S, RIZZO A XIBILIA M G 27 Soft sensors for monitoring control of industrial processes (Berlin: Springer-Verlag) [4] XU O G, CHEN X H, FU Y F LI L J 214 Recursive PLS modeling based on model performance assessment its application CIESC J., [5] WANG C P, YU Z J MENG F Q 213 Discount moving window recursive PLS algoritm its application to process of polypropylene production CIESC J [6] LIU J, CHEN D SHEN J 21 Development of self-validating soft sensors using fast moving window partial least squares Indus. & Eng. Cem. Res [7] ROSIPAL R 23 Kernel partial least squares for nonlinear regression discrimination Neural Network World [8] ROSIPAL R REJO L J 22 Kernel partial least squares regression in reproducing kernel Hilbert space J. of Macine Learning Res [9] HELLAND K, BERNSEN H E, BORGEN O S MARENS H 1992 Recursive algoritm for partial least squares regression Cemometrics & Intelligent Lab. System [1] QIN S J 1998 Recursive PLS algoritms for adaptive data modeling Compt. Cem. Eng [11] SONG K, WANG H Q LI P 24 Discounted-measurement RPLS algoritm its application to quality control of te rubber mixing process CIESC J [12] MU S J, ZENG Y Z, LIU R L, WU P, SU H Y CHU J 26 Online dual updating wit recursive PLS model its application in predicting crystal size of purified tereptalic acid (PA) process J. of Process Control [13] NI W, AN S K, NG W J BROWN S D 212 Moving-window GPR for nonlinear dynamic system modeling wit dual updating dual preprocessing Indus. & Eng. Cem. Res [14] GELADI P KOWALSKI B R 1986 Partial least squares regression: a tutorial Analytica Cimica Acta [15] KANEKO H FUNASU K 211 A soft sensor metod based on values predicted from multiple intervals of time difference for improvement estimation of prediction accuracy Cemometrics Intelligent Lab. Systems [16] KANEKO H FUNASU K 211 Maintenance-free soft sensor models wit time difference of process variables 13

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