Research Article The PVC Stripping Process Predictive Control Based on the Implicit Algorithm

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1 Mathematical Problems i Egieerig, Article ID , 8 pages Research Article The PVC Strippig Process Predictive Cotrol Based o the Implicit Algorithm Shuzhi Gao ad Liagliag Lua College of Iformatio ad Egieerig, Sheyag Uiversity of Chemical Techology, Sheyag , Chia Correspodece should be addressed to Shuzhi Gao; szg686868@126.com Received 12 December 2013; Revised 24 Jauary 2014; Accepted 27 Jauary 2014; Published 4 March 2014 Academic Editor: Youqig Wag Copyright 2014 S. Gao ad L. Lua. This is a ope access article distributed uder the Creative Commos Attributio Licese, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly cited. Accordig to the oliear ad parameters time-varyig characteristics of stripper temperature cotrol system, the PVC strippig process Geeralized Predictive Cotrol based o implicit algorithm is proposed. Firstly, supportig vector machie is adopted to dyamically modelize for the stripper temperature; Secodly, combiig with real-time model liearized of oliear model, a predictive model is liearized for real-time olie correctio. The, the implicit algorithm is used for optimal cotrol law. Fially, the simulatio results show that the algorithm has excellet validity ad robustess of temperature cotrol of the stripper. 1. Itroductio Polyviyl chloride (PVC) resi is a kid of bulk basis chemical raw material, oe of the five commo plastics. It is geerated by the polymerizatio of viyl chloride moomer PVC. Sice viyl chloride moomer has some toxicity, so the residual chloride i the PVC resi must be cotrolled withi a certai rage, which requires high precisio of stripper temperature cotrol. The removal of viyl chloride moomer i PVC commoly uses strippig process, which is a typical complex idustrial process with characteristics of highly oliear, time-varyig ad couplig. Domestic PVC stripper temperature cotrol system usually uses cascade cotrol scheme as commo [1, 2], which is difficult to achieve highly precise cotrol result. Therefore, it ca improve the PVC product quality, reduce productio costs ad protect the eviromet to use advaced itelliget cotrol techology i the strippig process. Geeralized Predictive Cotrol (GPC), a computer cotrol method developed with the Adaptive Cotrol, has bee successfully used i idustrial process cotrol. I recet years, for oliear predictive cotrol system, may foreig scholars have proposed model predictive cotrol method based o piecewise liear [3 6]. But domestic scholars use liear method or hierarchical optimizatio method, or directly use Hammersteia model, Wieer model, Volterra model, fuzzy reasoig ad eural etwork as a predictive model for oliear model predictive cotrol. Ad based o structural risk miimizatio support vector machie (SVM) regressio, solved the small sample, oliearity, high dimesio ad local miima problems, ad it has strog geeralizatio ability. The researches of support vector machie used i Geeralized Predictive Cotrol are gradually icreased, ad there are some o-liear support vector machie based predictive cotrol method [7, 8]. Literature [9] for oliear predictive cotrol process at coloy algorithm is proposed rollig optimizatio least squares support vector machie (LS-SVM) predictive cotroller. Literature [10], i the process of treatmet of sewage biochemical reactio, geeralized the fuzzy adaptive predictive cotrol method to achieve water quality idicators ammoia cocetratio ad itrate cocetratio of effective cotrol. The article puts forward the implicit algorithm PVC strippig process geeralized predictive cotrol based o high accuracy of PVC stripper temperature cotrol. It builds models adoptig support vector machie model as a predictive model after liearizatio, real-time olie correctio. Ad it adopts the implicit algorithm to solve for optimal cotrol law. Simulatio results show that the algorithm based o the implicit geeralized predictive cotrol for stripper temperature cotrol with a good validity ad robustess.

2 2 Mathematical Problems i Egieerig 2. Modelig Method of Supportig Vector Machie Regressio 2.1. Priciple of Support Vector Machie Regressio. PVC strippig process is a complex idustrial process with strog oliear characteristics. For o-liear support vector regressio, the basic idea is to map the data ito a high dimesioal feature space through a oliear mappig, ad the liear regressio i this space. Thus, liear regressio of the high dimesioal feature space correspods to the low dimesioal iput space oliear regressio. The specific method is implemeted by the kerel fuctio K(x i,x j ) = Φ(x i )Φ(x j ). Fially it gets global optimal solutio by solvig the followig quadratic programmig problem: W(α,α )=max { { { s.t. 1 2 i,j=1 + (α i α i )(α j α j )k(x i,x j ) (α i α i )y i (α i α i )=0, 0 α i, α i C,,...,. (α i +α i )ε} } } The substitute the obtaied parameters α i, α i ito the followig formula: ω= Regressio fuctio ca be derived: f (x) = = (1) (α i α i )Φ(x i). (2) (α i α i )(Φ(x i) Φ(x))+b (α i α i )K(x i,x j )+b. Whe α i α i is ot equal to zero, the correspodig sample data is the support vector. Whe b is take o the boudary poit, the mea value, that is,: b=average k [σ k +y k (3) (α i α i )K(x i,x k )] (4) σ k is the predictio error, which varies differet loss fuctios for differet values Costructio of Stripper Model. PVC strippig process has the characteristic of oliear, ad the object model is difficulttobeaccuratelyestablished.firstofall,toestablish the procedure for support vector regressio model, the followig oliear system model is itroduced: Y (k) =f(y(k 1),...,y(k ),u(k 1),..., u (k m),w(k 1)) y R,u R,m. (5) Amog them, u(k), y(k) ad w(k) represet the iput, output, ad disturbace of the cotrolled object, take iput U(k) = (u k m,u k m+1,...,u k ) ad output Y(k) = (y k,y k +1,...,y k 1 ) to form the iput vector of SVM form: X(K) = [y(k 1),...,y(k ),u(k),...,u(k m)], the form of the traiig samples: (X(k), Y(k)), igorigthe disturbace w(k), by support vector regressio we ca get the followig model: y M (k) = l (α i α i )K(X i,x k )+b. (6) I the formula, X i is the support vector, l is the umber of support vectors. Accordig to the process of PVC strippig aalysis, it is kow that stripper top temperature accuracy must be strictly cotrolled to remove viyl chloride moomer cotet i PVC. Therefore, this paper selects the stripper top temperature as the output vector of the model y(k). Ithe slurry flow ad steam flow uiformity coditios, the slurry flow ad steam flow iput vectors that comprise the model u(k), that is to say, SVM iput vectors X(k) is costituted by the stripper top temperature y(k),composedvectorofslurry flow ad steam flow u(k),thevalueof ad m is 2 ad 4. Accordig to the site of the 106 group field data collectio, to be ormalized, of which 53 groups are used SVM traiig sample, with the other 53 group as the test samples. I the simulatio, selectig the polyomial kerel fuctio q=2, the isesitive coefficiet ε=0.2,capacitycotrolc=40, 53 traiig sample get obtaied SVM 39, the the established model ca be expressed as: y (k) = 39 (α i α i )[(X i X k +1)] 2 +b. (7) I the formula, X i is the support vector b is the threshold value, i this experimet b = , [(X i X k )+1] 2 is the polyomial kerel fuctio. The parameters of support vector machie model that supports vector coefficiets is show i Table 1,Ascabe see from Table 1, whe parameter α i α i is zero, it meas that the vector is o-support vector, otherwise α i α i is the support vector. The formula (7) with a o-liear model is trasformed ito a form with characteristic of quadratic programmig, quadratic programmig algorithm i Matlab support vector machie regressio. Figures 1 ad 2 are output model ad the real value of the cotrast of the output curve obtaied i the traiig ad testig samples uder the iput. Meawhile, itroducig the variace as the evaluatio idexes: (1/( 1)) (Y m Y r ) 2. Y m is the output of the support vector model, Y r is the actual output value. Traiig

3 Mathematical Problems i Egieerig 3 Table 1: The parameter of SVM model. Number of group α i α i error is calculated as ,testerroris Ituses support vector regressio model to meet the requiremets of the strippig process modelig. Output y Sample System model output The actual output Figure 1: The simulatio of modelig o SVR (Traiig sample). Output y Sample System model output The actual output Figure 2: The simulatio of modelig o SVR (Testig sample). 3. Geeralized Predictive Cotrol Based o SVM 3.1. Predictio Model. Usig a support vector machie method to create predictive models, it turs the oliear systems ito liear time-varyig systems, ad thus adopts the geeralized predictive algorithm based o liear model, ad realizes the geeralized predictive cotrol of oliear systems [11]. The strippig process SVM model of PVC is show as the followig formula: y (k) = 39 (α i α i )[(X i X k )+1] 2 +b. (8)

4 4 Mathematical Problems i Egieerig The liearize the followig formula (8) atthesamplig time usig Taylor equatio, the parameters of geeralized predictive cotrol model is obtaied: 39 a j (k) = (α i α i )2[(X i X k )+1]X i (j), j = 1, 2, 3, b j (k) = 39 The liearized model is: (α i α i )2[(X i X k )+1] X i (j + 4), j = 0, 1, 2, 3. y (k) +a 1 (k) y (k 1) +a 2 (k) y (k 2) +a 3 (k) y (k 3) =b 0 (k) u (k 1) +b 1 (k) u (k 2) +b 2 (k) u (k 3) +b 3 (k) u (k 4) +Y c (k 1). (9) (10) I the formula, parameters a j ad b j are related to support vector machies. Number of support vectors is also decided by the umber of data iput ad output, after the liearizatio, CARIMA geeralized predictive cotrol model based o liear cotrol algorithms for predictive cotrol of oliear systems ca be used. 3.2.DesigofthePredictiveCotrollers.The article begis with the strippig process of PVC for support vector machie modelig, with model olie correctio, ad the liearized as Geeralized Predictive Cotrol predictio model ad solves the optimal cotrol law whe usig implicit algorithm, avoidig olie solvig Diophatie equatios, thereby reducig the amout of computatio. PVC strippig process is a typical oliear system. Whe iput is u(k), outputisy(k), thesystemoutputy m (k) cabeobtaiedbythesvmpredictmodelthroughthepast iput ad output data of systems ad the amout of curret cotrol iput u(k). The the deviatio betwee the actual system output ad predicted output is: Predict model is: e (k) =y(k) y m (k). (11) y p (k+i) =y m (k+i) +e(k). (12) I the feedback correctio, the use of SVM modelig ca be corrected olie, but i order to reduce the amout of computatio repeated correctio model, the followig correctio strategies ca be used: (1) Whe the error e betwee the actual output value ad predicted output is bigger tha the allowable error (take SVM isesitive loss fuctio ε), reestablishmet of the model. (2) Whe the error e betwee the actual output value ad predicted output is smaller tha the allowable error, i order to reduce the computatioal model recostructio, the geeral error model for feedback correctio should be used. Referece trajectory chooses oe-order filter equatios yields: y r (k+i) =αy(k) + (1 α) y r,,2,...,. (13) I the formula, y r, y(k) ad y r (k + i) are Set value, the system output ad the referece trajectory respectively; α is the softe modulus, 0<α<1. Performace idex fuctio select: J= [y (k+i) y r (k+i)] 2 + m Writte i vector form as follows: λ i [Δu (k+i 1)] 2. (14) J=(Y Y r ) T (Y Y r ) + λδu T ΔU. (15) Derivatio of future cotrol icremet, that is The optimal cotrol law is: J =0. (16) ΔU ΔU = (G T G + λi) 1 (G T (Y r f)). (17) Expad the above equatio, the cotrol icremet sequece of the ope-loop cotrol from time k to time k+ m 1, Δu(k), Δu(k + 1),..., Δu(k + m 1) cabeget: Δu (k+i 1) =d T i (Y r f). (18) I the formula, d T i is the i row vector of (G T G + λi) 1 G T, ad d T i =[d i1 d i2 d i ]. Whileiactualpractice,eachtimeolythefirstcompoet added to the system, while the cotrol icremet momets later recalculated each step, closed-loop cotrol measureisachieved,theweolyeedtocalculatethefirst row d T 1 of (GT G + λi) 1 G T. Now, the actual implemetatio is: u (k) =u(k 1) +d T 1 (Y r f). (19) Accordig to the actual iput ad output data of the Stripper, directly idetify the matrix G ad vector predictive of ope loop f, ad the get the cotrol icremet ΔU. The best predictive value ca be draw from projectios theory: Y =GΔU+f. (20)

5 Mathematical Problems i Egieerig T ( C) Iput t (mi) Time Normal GPC Implicit GPC SP (a) (b) Figure 3: The cotrol result of PVC strippig process i proper operatio Geeralized Predictive Cotrol Implicit Algorithm Steps. The geeralized predictive algorithm strikes the optimal cotrol law algorithms with idetificatio of the cotroller parameters directly from the iput ad output data. It avoids the olie solvig Diophatie equatios ad iverse matrix to improve the speed of operatio ad save computig time. Step 1. Algorithm Iitializatio: The legth of time domai =7, Forecast legth =6, Cotrollig legth m=2, the weightig coefficiets of cotrollig parameter t 0 = 1.8, softe coefficiet α = 0.85 ad forgettig factor K 1 =1. 4. PVC Strippig Process Cotrol Simulatio I the cotrol system of PVC strippig process, the support vector machie modelig ad Geeralized Predictive Cotrol Implicit algorithm are combied with olie correctio of the model ad error models feedback correctio, ad PVC strippig process is studied accordig to the actual situatio to simulatio. PVC strippig process accordig to the actual process, the stripper top temperature optimum temperature of C, so the simulatio sigal is a give value C. Usig support vector machie model obtaied by the liearized expressio: Step 2. Set square P to a diagoal matrix; Set the iitial value of the iput output sequece X(K) = [y(k 1),...,y(k ),u(k),...,u(k m)], Produce a give value the sigal Y r. y (k) = 2.5y (k 1) 2.2y (k 2) +0.7y(k 3) u (k 1) u (k 2). (21) Step 3. Accordig to formula (7), calculate the output values y m (k) of time k,coserve output values before time k to modelize calculatio; whe error e > ε, rebuild the model; whe error e<ε, use the error to correct the model. Step 4. Accordig to the recursive least squares equatio, y(k) calculate the elemet of Gg 0,g 1,...,g 1, get the matrix G. Step 5. Accordig to the vector Y 0 latter time, calculate the forecast vector f;theoutput y(k) of time k ad set value Y r, get the referece trajectory after time k. Step 6. Calculate ad reserve m cotrol icremets after time k;drawthegivevalue,theoutputvalue. Compare the Simulatio curve of the ormal algorithm GPC ad implicit algorithm GPC. Simulatio predicted legth = 6, cotrollable legth m = 2, the weightig coefficiets of the cotrol parameter t 0 = 1.8, softe coefficiet α = 0.85 ad forgettig factor K 1 =1. Accordig to the actual situatio may arise durig the strippig process, four mai cases are cosidered i the simulatio. (1) Good coditio durig operatio, show i Figure 3. Curve ca be see from Figure 3.Thesystemcatracka smooth chage of the referece sigal output. Figure 3 also shows the geeral algorithm for GPC, Implicit algorithm for GPC effect curves, it ca be see that the implicit algorithm by reducig the amout of computatio, cotrol effect ca be improved. (2) Set value temperature chages because of the differet grades of polyviyl chloride resi, or process requiremets,

6 6 Mathematical Problems i Egieerig T ( C) u t (mi) Normal GPC Implicit GPC SP (a) t (mi) (b) Figure 4: The cotrol result of PVC strippig process whe set value chaged. T ( C) Iput t (mi) Normal GPC Implicit GPC SP (a) Time (b) Figure 5: The cotrol result of PVC strippig process by a sudde oise. the optimum temperature of the strippig colum top is differet. The simulatio is give the first 15 miutes C, i 15 miutes chagig the resi grades; the optimum temperature was chaged to 105 C, as the obtaied cotrol effect curve shows i Figure 4. AscabeseefromFigure 4, whe the give value is chaged, the system ca quickly respod to ad track ew set value. I additio, the use of implicit algorithm geeralized predictive has faster respose speed. (3) Whe the cotrolled object was disturbed, such as the ueve heatig of the slurry ad other factors led to sudde abrupt chages i temperature. The simulatio time of 15 miutes, addig the amplitude of a sudde disturbace 5 C, as cotrol effect curve show i Figure 5, aditcabeseethatthesystemisdisturbed suddely, the system ca quickly overcome the disturbace, the cotrol to a give values, cotrol of implicit algorithm has better cotrol effect.

7 Mathematical Problems i Egieerig T ( C) 60 Iput t (mi) 0.4 Time Normal GPC Implicit GPC SP (b) (a) Figure 6: The cotrol result of PVC strippig process by radom oise. (4) I simulatio, durig operatio subject to the radom disturbace of ucertaities factor is added, the value at 5 +5 C, for cotrollig the effect of the curve i Figure 6,cabe see i systems are kow i the case of radom disturbace. The system has better robustess. Cotrolled temperature value is ear the set value ad the deviatio is small, it ca be see from Figure 6 that the rate of chage of the implicit algorithm cotrol effect is small with better robustess. Abovefourcasescabeseefromthesimulatiocurve, usig the implicit algorithm for GPC was better tha geeral for GPC cotrol effect. 5. Coclusio The dyamic modelig is adopted based o the priciple of support vector machies ad the field data of PVC strippig process. Meawhile, it combies the model olie correctio ad error feedback correctio with the real-time liear ad oliear model. Ad the Geeralized Predictive Cotrol is adopted by usig implicit algorithm with strippig process of PVC. Simulatio results verify the validity of the model ad the feasibility ad robustess of the algorithm. Coflict of Iterests The authors declare that there is o coflict of iterests regardig the publicatio of this paper. Ackowledgmets This work was supported by the Key Program of Natioal Natural Sciece Foudatio of Chia ( ), the Postgraduate Scietific Research ad Iovatio Projects of Basic Scietific Research Operatig Expeses of Miistry of Educatio (N604001). Refereces [1] F. K. Zhu, Itroductio of stripper ad improvemet of viyl chloride moomer recovery system, Polyviyl Chloride, o. 5, pp , [2] X. Y. Li ad H. Y. Zhag, Applicatio of computer automatio cotrol i colophoy stripper, Chia Chlor-Alkali, o. 3, pp , [3] A. L. Cervates, O. E. Agameoi, ad J. L. Figueroa, A oliear model predictive cotrol system based o Wieer piecewise liear models, Joural of Process Cotrol,vol.13,o. 7,pp ,2003. [4] G. Shafiee, M. M. Arefi, M. R. Jahed-Motlagh, ad A. A. Jalali, Noliear predictive cotrol of a polymerizatio reactor based o piecewise liear Wieer model, Chemical Egieerig Joural,vol.143,o.1 3,pp ,2008. [5] G. Shafiee, M. Arefi, M. Jahed-Motlagh, ad A. Jalali, Model predictive cotrol of a highly oliear process based o piecewise liear wieer models, i Proceedigs of the 1st IEEE Iteratioal Coferece o E-Learig i Idustrial Electroics (ICELIE 06), pp , December [6] L. Özka, M. V. Kothare, ad C. Georgakis, Model predictive cotrol of oliear systems usig piecewise liear models, Computers ad Chemical Egieerig,vol.24,o.2 7,pp , [7] D.-C. Wag ad M.-H. Wag, O SVMR predictive cotrol based o GA, Cotrol ad Decisio,vol.19,o.9,pp , [8] J. Wag ad S. Y. Su, Predictive cotrol based o support vector machie model, i Proceedigs of the 6th World Cogress o Itelliget Cotrol ad Automatio (WCICA 06), pp , Dalia, Chia, Jue 2006.

8 8 Mathematical Problems i Egieerig [9] J. Wag ad M.-Z. Liu, Study of LS-SVM predictive cotrol usig at coloy algorithm rollig optimizatio, Cotrol ad Decisio,vol.24,o.7,pp ,2009. [10] S.-G. Xie ad L.-F. Zhou, Fuzzy adaptive geeralized predictive cotrol for wastewater treatmet process, Joural of East Chia Uiversity of Sciece ad Techology,vol.34,o.6,pp , [11] Q. Miao ad S.-F. Wag, Noliear model predictive cotrol based o support vector regressio, i Proceedigs of the 1st Iteratioal Coferece o Machie Learig ad Cyberetics, vol. 3, pp , Beijig, Chia, November 2002.

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