Research Article Modeling, Prediction, and Control of ...downloads.hindawi.com/journals/mpe/2015/576813.pdfResearch Article Modeling, Prediction, and Control of Heating Temperature

Research ArticleModeling Prediction and Control of HeatingTemperature for Tube Billet

Yachun Mao1 Dong Xiao2 and Dapeng Niu2

1College of Resources and Civil Engineering Northeastern University Shenyang 110004 China2Information Science and Engineering School Northeastern University Shenyang 110004 China

Correspondence should be addressed to Dong Xiao xiaodongiseneueducn

Received 26 May 2014 Revised 4 September 2014 Accepted 7 October 2014

Academic Editor Yi Jin

Copyright copy 2015 Yachun Mao et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Annular furnaces have multivariate nonlinear large time lag and cross coupling characteristics The prediction and control of theexit temperature of a tube billet are important but difficult We establish a prediction model for the final temperature of a tube billetthrough OS-ELM-DRPLS method We address the complex production characteristics integrate the advantages of PLS and ELMalgorithms in establishing linear and nonlinear models and consider model update and data lag Based on the proposed model wedesign a prediction control algorithm for tube billet temperature The algorithm is validated using the practical production data ofBaosteel Co Ltd Results show that the model achieves the precision required in industrial applications The temperature of thetube billet can be controlled within the required temperature range through compensation control method

1 Introduction

A seamless tube should be heated to a given temperature in anannular furnace prior to piercing The heating quality of thetube billet directly influences the quality of the seamless tubeIn seamless tube production the main evaluation criterionfor tube billet heating quality is the final exit temperature ofthe tube billetThe heating reactionmechanism of an annularfurnace is complicated It has nonlinear large time lag anduncertainty characteristics The temperature distribution onthe surface of a tube billet within the furnace cannot bedirectly measured and controlled [1] Thus controlling theoutlet temperature of the billet is difficult

Measuring the temperature of the billet in the furnace isalso difficult A commonmethod is to establish a temperatureprediction model for the billet Several scholars have pro-posed mechanism models for billet temperature in furnacesChen et al [2] established a temperature dropmodel of a tubebillet during the transfer of the tube billet from the heatingfurnace to the rolling mill based on the partial differentialequation of heating transfer Jaklic et al [3] established amathematical model of deformation and heat flow duringrough rolling Considering that mechanism modeling is

complex and modeling consumes much time a model isusually establishedwith a single furnace type and undermanyrestricting conditions Zhang et al [4 5] estimated the tubebillet exit temperature through dynamic modeling of furnacetemperature however the prediction error was too largeto meet the requirements of the heating process Wick [6]applied Kalman filter technique to estimate the temperaturedistribution of a tube billet inside a heating furnace Thedisadvantage of this method is that the surface temperatureof the tube billet inside the furnace must be measured whichis difficult to perform in practical production Xiao et al[7] employed production data and applied PCR method toestablish a softmeasurementmodel of tube billet temperatureinside a furnace However this model has poor predictionprecision because of the strong nonlinearity of productiondata Chen and Chai [8] designed a preprocessing systemfor production process data This system can predict severalvariables that are difficult tomeasure through the use of a self-adapting fuzzy-neural network Cui and Ding [9] establisheda soft measurement model of tube billet temperature basedon RBF neural network however model update was notconsidered Iwamoto et al [10] designed an automatic controlsystem for a tube billet reheating rotary hearth furnace

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 576813 10 pageshttpdxdoiorg1011552015576813

2 Mathematical Problems in Engineering

The system consists of a component that calculates thetube billet temperature and a component that calculatesthe optimal furnace temperature set point In large Chinesesteel companies such as Baosteel Co Panzhihua Iron andSteel Co Ltd Anshan Iron and Steel Co Ltd and CapitalIron and Steel Co Ltd temperature prediction models areutilized in several heating furnacesHowever their predictionmodels are almost entirely engineering models importedfrom abroadTherefore thesemodels are difficult tomaintainand transplant and the costs of doing so are high

With the development of configuration software anddatabase technique increasing amounts of production dataare being collected and stored Therefore data-driven mod-eling and control methods are eliciting more and moreattention He et al [11] and Lv et al [12] established datamodels for the Ladle furnace through data-driven methodsIn the present work the production data of an annularfurnace were obtained from the seamless tube subcompanyof Baosteel Industrial process data contain noise whichreduces the modeling accuracy of the extreme learningmachine (ELM) algorithm By contrast the capability ofthe partial least square (PLS) algorithm to process linearrelevant data is suitable Moreover some cyclical changesoccur in production Thus the model should be updatedin real time Online sequential (OS) ELM and recursivePLS algorithm can realize model update Online sequentialextreme learning machine dynamic recursive partial leastsquare (OS-ELM-DRPLS) algorithm was proposed in thisstudy A tube billet temperature prediction model basedon this algorithm was established and a strategy for theoptimization and control of tube billet temperature wasproposed based on this model OS-ELM-DRPLS not onlyhas the advantages of the OS-ELM algorithm (eg rapidnonlinear modeling and update) but also has the capability ofthe RPLS algorithm to process linear relevant data The largetime lag and reduced model precision were solved throughdynamic processingThe algorithm is easy to implement withadvanced computer language Configuration software suchas WINCC61 can be utilized to compile the modeling andcontrol algorithms into special modules for use in industrialsites Simulation and actual experiments prove that tubebillet temperature can be predicted and controlled withinthe scope of the production process requirements with theestablished temperature prediction model and the proposedstrategy of optimization and control based on OS-ELM-DRPLS algorithm

2 Annular Furnace

An annular furnace is a type of rotary hearth furnaceutilized for tube billet heating It consists of the furnace shaftand auxiliary equipment for charging and discharging Thefurnace shaft consists of a fixed furnace roof a ring-typetunnel surrounded by a fixed furnace roof wall and a circularring-like rotary hearth as shown in Figure 1

External and inner ring seams are placed between thefixed furnace wall and rotary hearth in an annular fur-nace Internal and external water seal tanks are arranged

6

45

1

2

3

(1) Roof(2) Wall(3) Rotary hearth

(4) Charger(5) Discharger(6) Pipeline system

Figure 1 Schematic of annular furnace

beneath the external and inner ring seams to maintainnormal temperature and pressure in the furnace cavity andprevent external cold air from entering the furnace cavityFuel gas and combustion-supporting air are blown into thefurnace through burning nozzles mounted on the externaland internal walls or furnace roof tomake the gas burnwithinthe furnace and heat the tube billet The fumes producedby gas burning within the furnace move conversely throughthe rotary hearth to the tail end of the soaking zone enterthe flue and chimney outside the furnace and exit to theatmosphereThe external wall of the furnace has charging anddischarging furnace doors in which a charger and dischargerare placed respectively Charging and discharging proceedsimultaneously When a tube billet is placed in the furnacethe bottom rotates at a certain angle Tube billets follow aradial layout inside the furnace and are arranged either in asingle row or in multirows

The furnace cavity is divided into preheating heating andsoaking zones according to the heating process of the tube bil-let in the annular furnace Burning nozzles are not mountedin the preheating zone A flue opening is arranged on theside wall near the charging furnace door in this zone High-temperature exhaust gas flows toward the opposite directionof hearth rotation and exits into the atmosphere throughthe flue opening in the heating and soaking zones Duringthe flow process of high-temperature exhaust gas the tubebillets in the preheating zone are mainly convection heatedThe length of the preheating zone accounts for approximatelyone-fourth of the peripheral length of the annular furnaceTemperature differences between the surface and center andbetween both ends exist in the tube billet rapidly heatedin the heating zone To reduce the temperature differenceof the tube billets and eliminate their male-female facesthe tube billets must be heated in the soaking zone Thelength of the soaking zone is approximately three-twentiethsof the peripheral length of the annular furnace In additionno tube billet and burning nozzle are present between thecharging and discharging furnace doors A partition wall isplaced in the middle The distance between the charging and

Mathematical Problems in Engineering 3

discharging furnace doors is approximately one-tenth of theperipheral length of the annular furnace

3 Dynamic Nonlinear PLS Method

Given that a linear PLS model cannot correctly describe thenonlinear relationship between independent variable X anddependent variableY (X is the variable matrix that affects theheating temperature of the tube billet and Y is the variablematrix of the heating temperature) nonlinear PLS (NLPLS)method is required Wold et al extended the PLS methodto the nonlinear field [13 14] Two feasible methods existin nonlinear PLS methods One method is to perform arrayextension for the input matrix introduce several nonlinearterms of the original variable (eg the square term) and thenregress the extended input and output matrix through PLSmethod If prior knowledge on the relationship of originalinput variable does not exist this method cannot be utilizedas a reference in the selection of the combined mode andmay lead to an oversized dimension of the input matrix anddifficulties in processing The other method is to reservethe linear external model of the PLS method The internalmodel is nonlinearThe effect of various input variables on thefinal tube billet temperature has a different time lag becauseof the large time lag characteristics of tube billet heatingAccurately predicting the tube billet temperature throughtraditional PLSmethod is difficult In this study dynamic PLSmethod was utilized to calculate the lag time of various inputchannels and significantly improve themodelrsquos precisionThealgorithm is as follows

X = [119909

1198701

1 119909

1198702

2 119909

119870119901

119901] (1)

where 11987011198702 119870119901 are the ratio of lag time to samplingperiod for sampling variables 119909

1 1199092 119909

119901

(1) The external relation model is

119883 = 119879119875

119879+ 119864 =

119860

sum

119886=1

119905119886119901

119879

119886+ 119864

119884 = 119880119876

119879+ 119865 =

119860

sum

119886=1

119906119886119902

119879

119886+ 119865

(2)

where 119860 is the number of reserved eigenvector 119905119886(119899 times 1) and

119906119886(119899times1) are the score vectors ofX andY respectively 119901

119886(119898times

1) and 119902119886(119901 times 1) are the load vectors of X and Y respectively

119879(119899 times 119860) and 119880(119899 times 119860) are the score matrixes of X and Yrespectively 119875(119898 times119860) and119876(119901 times 119860) are the load matrixes ofX andY respectively and119864 and119865 are the fit residualmatrixesof X and Y respectively

(2) The internal relation model is

119886= 119891 (119905

119886) + 120576 (3)

where 119891() is the nonlinear function and 120576 is the residualGiven that a neural network is capable of nonlinearity

fitting during the modeling of the batch process nonlinearMPLS method where the internal model adopts a neuralnetwork has gained extensive applications Considering that

a traditional feed-forward neural network adopts a gradientlearning algorithm during training the parameters in thenetwork need iteration and updating Training not onlyconsumes much time but also easily results in issues of localminimum and excessive training [15]

4 OS-ELM-DRPLS Algorithm

41 ELM Algorithm In supervised batch learning the learn-ing algorithms employ a finite number of input-outputsamples for training [16ndash22] For119873 arbitrary distinct samples(119909119894 119905119894) isin 119877

119899times 119877

119898 where 119909119894is a 119899 times 1 input vector and 119905

119894is

a 119898 times 1 target vector if a single hidden layer feed-forwardneural network (SLFN) [23ndash26] with

119873 hidden nodes canapproximate these119873 samples with zero error then 120573

119894 119886119894 and

119887119894exist such that

119891(119909119895) =

sum

119894=1

120573119894119866(119886119894 119887119894 119909119895) + 120576119895= 119905119895 (4)

In the expression above 119895 = 1 119873 119886119894 and 119887

119894are the

learning parameters of the hidden nodes (the weight vectorconnecting the input node to the hidden node and thethreshold of the hiddennode) randomly selected according tothe proof provided byHuang et al 120573

119894is the weight connecting

the 119894th hidden node to the output node Error term 120576119895is added

to avoid overfitting the noise in the data 119866(119886119894 119887119894 119909) is the

output of the 119894th hidden node with respect to input 119909 and119873 is the number of hidden nodes that can be determined bytrial and error or prior experience Equation (4) can then bewritten compactly as

119867120573 = 119879 (5)

where

119867(1198861 119886

1198871 119887

1199091 119909

119873)

=

[

[

[

119866 (1198861 1198871 1199091) sdot sdot sdot 119866 (119886

119887 1199091)

sdot sdot sdot

119866 (1198861 1198871 119909119873) sdot sdot sdot 119866 (119886

119887 119909119873)

]

]

]119873times

120573 =

[

[

[

120573

119879

1

120573

119879

]

]

]times119898

119879 =

[

[

[

119905

119879

1

119905

119879

119873

]

]

]119873times119898

(6)

In the expressions above119867 is the hidden layer output matrixof the network the 119894th column of 119867 is the 119894th hidden nodersquosoutput vector with respect to inputs 119909

1 1199092 119909

119873 and the

119895th row of 119867 is the output vector of the hidden layer withrespect to input 119909

119895 The hidden node parameters 119886

119894and

119887119894need not be tuned during training and may simply be

assigned with random values Equation (5) then becomes alinear system and the output weights 120573 are estimated as

120573 = 119867

+119879 (7)

where119867+ is theMoore-Penrose generalized inverse of hiddenlayer output matrix119867 [27 28]


42 OS-ELM Algorithm In actual applications training datamay arrive chunk by chunk or one by one Hence thebatch ELM algorithm has to be modified and made onlinesequential for this case [29 30]

Output weight matrix 120573 (

120573 = 119867

+119879) provided in (7) is

a least-squares solution of (5) We consider the case whererank(119867) =

119873 is the number of hidden nodes Under this

condition119867+ of (7) is provided by

119867

+= (119867

119879119867)

minus1

119867

119879 (8)

If 119867

119879119867 is singular one can make it nonsingular by

selecting a small network size 119873 or increasing data number

119873 in the initialization phase of OS-ELM Substituting (8) to(7) 120573 becomes

120573 = (119867

119879119867)

minus1

119867

119879119879 (9)

Equation (9) is the least-squares solution to 119867120573 = 119879Sequential implementation of (9) results in OS-ELM [31]

Given a chunk of initial training set alefsym0= (119909119894 119905119894)

1198730

119894=1and

1198730ge119873 if the batch ELMalgorithm is employed the solution

of minimizing 1198670120573 minus 119879 which is given by 120573

0= 119870

minus1

0119867

119879

01198790

where1198700= 119867

119879

01198670 must be considered

We consider another chunk of data alefsym1= (119909119894 119905119894)

1198730+1198731

119894=1198730+1

where 1198731is the number of samples in this chunk The

problem involves minimizing

10038171003817100381710038171003817100381710038171003817

[

1198670

1198671

] 120573 minus [

1198790

1198791

]

10038171003817100381710038171003817100381710038171003817

(10)

Considering both alefsym0and alefsym

1 output weight 120573 becomes

1205731= 119870

minus1

1[

1198670

1198671

]

119879

[

1198790

1198791

] where 1198701= [

1198670

1198671

]

119879

[

1198670

1198671

] (11)

For sequential learning 1205731should be expressed as a

function of 1205730 1198701 1198671 and 119879

1and not as a function of dataset

alefsym0 1198701can be written as

1198701= [119867

119879

0119867

119879

1] [

1198670

1198671

] = 1198700+ 119867

119879

11198671 (12)

[

1198670

1198671

]

119879

[

1198790

1198791

] = 119867

119879

01198790+ 119867

119879

11198671= 1198700119870

minus1

0119867

119879

01198790+ 119867

119879

11198791

= 11987001205730+ 119867

119879

11198791= (1198701minus 119867

119879

11198671) 1205730+ 119867

119879

11198791

= 11987011205730minus 119867

119879

111986711205730+ 119867

119879

11198791

(13)

Combining (11) and (13) 1205731is obtained with

1205731= 119870

minus1

1[

1198670

1198671

]

119879

[

1198790

1198791

] = 119870

minus1

1(11987011205730minus 119867

119879

111986711205730+ 119867

119879

11198791)

= 1205730+ 119870

minus1

1119867

119879

1(1198791minus 11986711205730)

(14)

where1198701= 1198700+ 119867

119879

11198671

When the (119896 + 1)th chunk of dataset

alefsym119896+1

= (119909119894 119905119894)

sum119896+1

119895=0119873119895

119894=(sum119896

119895=0119873119895)+1

(15)

is received where 119896 ge 0 and 119873119896+1

denotes the number ofsamples in the (k+1)th chunk we have

119870119896+1

= 119870119896+ 119867

119879

119896+1119867119896+1

120573119896+1

= 120573119896+ 119870

minus1

119896+1119867

119879

119896+1(119879119896+1

minus 119867119896+1

120573119896)

(16)

119870

minus1

119896+1rather than 119870

119896+1is utilized to compute 120573

119896+1from

120573119896in (16) The update formula for 119870minus1

119896+1is derived with the

Woodbury formula

119870

minus1

119896+1= (119870119896+ 119867

119879

119896+1119867119896+1

)

minus1

= 119870

minus1

119896minus 119870

minus1

119896119867

119879

119896+1(119868 + 119867

119896+1119870

minus1

119896119867

119879

119896+1)

minus1

times 119867119896+1

119870

minus1

119896

(17)

We let 119875119896+1

= 119870

minus1

119896+1 The equation for updating 120573

119896+1can

be written as

119875119896+1

= 119875119896minus 119875119896119867

119879

119896+1(119868 + 119867

119896+1119875119896119867

119879

119896+1)

minus1

119867119896+1

119875119896

120573119896+1

= 120573119896+ 119875119896+1

119867

119879

119896+1(119879119896+1

minus 119867119896+1

120573119896)

(18)

Equation (18) provides the recursive formula for 120573119896+1

43 OS-ELM-DRPLS Modeling Steps The difference of non-linear DRPLS modeling method based on OS-ELM from lin-ear PLS method is that the former employs ELM to establishthe internal nonlinear model and updates the internal andexternal models This method reserves the linear externalmodel extracts the attributive information of the processthrough PLS eliminates the colinearity of data reduces thedimension of the input variable and then adopts ELM toestablish a nonlinear internal model between the input scorevector matrix and the output score vector the nonlinearprocessing capability of the internalmodel is enhancedThusOS-ELM-DRPLSmethod has the advantages of PLS andELM(ie the robustness and feature extraction capability of PLSmethod and quick nonlinear processing capability of ELM aswell as precision accuracy through real-time model update)

The modeling and testing steps of nonlinear DRPLSmethod based on OS-ELM are as follows


(1) Two standardized data matrices X isin 119877

119899times119898 and Y isin

119877

119899times119901 are assigned The dynamic nonlinear PLS regressionmodel can be expressed as follows

X = [119909

1198701

1 119909

1198702

2 119909

119870119901

119901] (19)

where11987011198702 119870119901 are the ratio of lag time to the samplingperiod for sampling variables 119909

1 1199092 119909

119901

(2) The batch data of the batch process are deployedcross-validation method is implemented to determine thenumber of latent variables and linear PLS method is appliedto calculate score vector matrices 119879 and 119880 and load vectormatrices 119875 and 119876 for modeling samples X and Y

119883 = 119879119875

119879+ 119864 =

119860

sum

119886=1

119905119886119901

119879

119886+ 119864

119884 = 119880119876

119879+ 119865 =

119860

sum

119886=1

119906119886119902

119879

119886+ 119865

(20)

(3) A node number is assigned to the ELM hidden layerand activation function (eg sigmoid function) ELM isemployed to establish a nonlinear model between internalmodels 119879 and 119880 and 119880 = 119891ELM(119879) is obtained where119891ELM(sdot) is the nonlinear function indicated by ELM Thehidden nodes in SLFN transform the feature space intoanother feature space The original ELM regards the numberof nodes as a parameter to be defined We increase thenumber of hidden nodes until stop criteria (eg residualerror reduction) are reached Meanwhile the number ofhidden nodes is less than119873

(4) When one new batch of data 1198831 1198841is obtained

PLS decomposition is performed and score and load vectors1198791 1198801 1198751 1198761are obtained

1198831= 1198791119875

119879

1+ 119864

1198841= 1198801119876

119879

1+ 119865

(21)

According to (18) the OS-ELM algorithm is adoptedto update the output layer weight value and the internalmodel Weighted mean is conducted on the load matrix ofthe external model and external RPLS update where119908 is theweight value factor is achievedThe above steps are repeatedand model update is conducted for every batch

119875

119879= 119908119875

119879+ (1 minus 119908) 119875

119879

1

119876

119879= 119908119876

119879+ (1 minus 119908)119876

119879

1

(22)

(5) Testing data are utilized to verify themodelrsquos precisionPLS decomposition is conducted on testing data 119883

2 and

score vector 1198792is obtained

1198832= 1198792119875

119879+ 119864 (23)

1198792

is introduced into the OS-ELM model 1198802

=

119891OS-ELM(1198792) is obtained and the model prediction value isdetermined through

119884 = 119880119876

119879

(6) A system error is obtained by comparing 119884 with the

practical output 11987011198702 119870119901 can vary within 1 minus 119899 Aftereach variation 1199091198701

1 119909

1198702

2 119909

119870119901

119901 are substituted back to (19)for calculation and to obtain an estimation error Finally oneis obtained by exhausting a group of optimal 119870

1 1198702 119870

119901

values to minimize the model estimation error

119882 =

1198992

sum

119894=1

10038161003816100381610038161003816

Y (119894) minus Y (119894)

10038161003816100381610038161003816

(24)

Model parameters 1198701 1198702 119870

119901and 120573 of the OS-ELM-

DRPLSmodel are then obtained through the aforementionedcalculation

5 Prediction and Control ofTube Billet Heating Quality Based on OS-ELM-DRPLS Model

51 Introduction of the Site and Selection of Measuring PointsIn the seamless tube subcompany of Baosteel the designedoutput of an annular furnace was 160 th Its intermediatediameter was 35m and the effective width of hearth was45m The hearth was divided into six burning controlsections The diameter of the heated tube billet was 178mmThe temperature upon entering the furnace was 20∘C and themaximum temperature upon leaving the furnace was 1280∘CIn the annular furnace mixed gas that consists of 52 blastfurnace gas 13 converter gas and 348 coke oven gaswas utilized The composition of the blast furnace gas was235 CO 2 H

2 195 CO

2 and 35 N the composition

of coke oven gas was 53 H2 292 CH

4 28 weight carbon

hydride 75 CO 20 CO2 06 O

2 and 44 N

2 The

composition of converter gas was 56 CO 24 N2 and

197 CO2 The specific technical parameters are shown in

Table 1The final exit temperature of the tube billet was predicted

through OS-ELM-DRPLS method First the variation inthe tube billet temperature was reflected and the measuredvariables were easily obtained On one hand gas could notbe obtained through the peep holes because the peep holesin the furnace were closed On the other hand opening ofthe furnace door to obtain gas affects the testing precisionbecause of the absorption of cold air Therefore the measur-ing points in the site were set at the lighting holes of burningnozzles in the external surrounding furnace walls Six flowrate detecting points were set for the burning nozzles Ninethermocouples were mounted in the six working sectionsto measure the temperature inside the furnace cavity Thespecific positions of flow rate and furnace temperature areshown in Figure 2 Fifteen measuring variables were selectedto predict the final tube billet exit temperature 119909

1ndash1199096were

measuring points for numbers 1ndash6 burning nozzle flow rateand 119909

7ndash11990915

were measuring points for numbers 1ndash9 furnacecavity temperature The variable table is shown in Table 2After selecting the measuring variables and gathering site


Table 1 Main technical parameters of annular furnace

Furnace output 160 thSpecification of tube billet Φ175mm 860ndash4500mm in length maximum weight per piece 850 kg

Furnace sizeIntermediate diameter 35m effective width of furnace cavity 5m height of furnacecavity 3m (one section in the preheating zone) 25m (2 sections) 2m (3ndash6sections) total number of hearth batch bins 391 number of tube billets 381 pieces

Calorific value of combustion Heavy oil 37620Kjkg mixed gas 9196Kjm3

Demand of combustion Heavy oil 6755 kgh mixed gas 30545m3h

Arrangement of burning nozzle Total of 96 side burning nozzles for either oil or gas used in sections 1ndash6 heat is notprovided in the preheating zone

Maximum furnace cavity temperature Approximately 1400∘C

Temperature of tube billet Enter furnace at 20∘C leave furnace at 1280∘C cross-section temperature differenceof leaving furnace plusmn10∘C

Charging and discharging rhythm Maximum 270 pieceh equivalent to discharging interval of 133 spiece

Direction ofbottom rotated

F1

F2

F3

F4

F5F6

Number 1 Number 2

Num

ber3

Number4

Number 5

Number 6

Charging

Discharging

T1T2

T3

T4

T5

T6

T8T7T9

Measuring point for temperatureMeasuring point for flow rate

Preh

eat z

one

Figure 2 Measuring point distribution diagram for the annularfurnace

production data OS-ELM-DRPLSmethodwas applied to theprediction model of tube billet temperature

52 Establishment and Checking of the Tube Billet FinalTemperature Prediction Model The production data for 70pieces of tube billets produced by Baosteel in March 2013were utilized The first forty samples were utilized as trainingdata to establish the prediction model of tube billet finaltemperature The last thirty samples were used for modelupdate Lump update was employed Every group of five wasconsidered a lump The model was updated The last thirtysamples acted as the testing samples to check the precisionof model prediction Prior to modeling data were expandedthey were standardized-processed and they underwent cross

Table 2 Variables in the modeling of tube billet final temperature

Ser number Variablename Variable meaning Unit

1 1199091

Number 1 burning nozzle flowrate m3h

2 1199092


3 1199093


4 1199094


5 1199095


6 1199096


7 1199097

Number 1 furnace cavitytemperature

∘C

8 1199098


∘C

9 1199099


∘C

10 11990910


∘C

11 11990911


∘C

12 11990912


∘C

13 11990913


∘C

14 11990914


∘C

15 11990915


∘C

checking The number of PLS potential variables was deter-mined to be 4 The number of ELM hidden layer nodes was10 The excitation function was a sigmoid function The ratio


Table 3 RMSE and modeling time of different models

Method RMSE (test) TimesRPLS 102 02132RBF-PLS 42 30692OS-ELM-DRPLS 31 06239

of lag time of1198701 1198702 119870

119901in (19)was calculated in formulas

equation (25)The same data were tested with RPLS RBF-PLS and

OS-ELM-DRPLS methods The predicted mean square errorand modeling time are shown in Table 3 Although the threemethods meet the requirements of industrial applicationOS-ELM-RPLS method exhibits better expansion capabilityprediction precision and nonlinear fitting capability forindustrial application than RPLS method Compared withnonlinear RBF-PLS method the training time in OS-ELM-RPLS method is shorter OS-ELM-RPLS method can achieverapid modeling and model update and is significant to theintermittent production processes such as tube billet heating

[1198701 1198702 119870

15]

= [58 55 51 46 42 36 61 56 51 45 39 35 52 58 60]

(25)

The unit of [1198701 1198702 119870

15] is the sample time Figure 3

shows a comparison between regression data and practicalmodeling data using RPLS and OS-ELM-DRPLS modelsThe maximum error was 69∘C and the mean error was23∘C which meet the requirements of the production siteTo further verify the accuracy of the model new data wereintroduced into the model and substituted into the following

equation to obtain estimation value 119884new of the new dataThecomparison with 119884new is shown in Figure 4 The maximumerror was 98∘C and the mean error was 31∘C which meetthe requirements of the production site

119884new = 119891OS-ELM-RDPLS (119883new) (26)

53 Predicted Control of Tube Billet Final Temperature Theaforementioned data indicate that tube billet exit temperatureoften fluctuates in the temperature range of 1200∘C to 1300∘Cand often deviates from the ideal piercing temperature(1270∘C) Such condition degrades the quality of the tubeThe tube billet exit temperature should be controlled withinthe temperature range of 1255∘C to 1295∘C The gas flowrate can be adjusted according to the prediction practicalmeasuring and target temperatures Its control periodwas 1 sAn ELM model predicted controller (EPC) was designed forthe annular furnace system with the OS-ELM-DRPLS modelpredictor (EMP) as shown in Figure 5

The basic operating principle of predictive control isto generate a sequence of control signals at each sampleinterval that optimize the control effort to follow the referencetrajectory exactly [32 33]The ELMmodel predictive controllaw was obtained by minimizing the following predictiveperformance criterion

119869 (119896) =

1

2

119873119901

sum

119901=0

(119903 (119896 + 119901) minus 119910 (119896 + 119901))

2

=

1

2

(119877 (119896) minus 119884 (119896))

119879(119877 (119896) minus 119884 (119896)) =

1

2

119864

119879(119896) 119864 (119896)

(27)

where

119877 (119896) = [119903 (119896) 119903 (119896 + 1) sdot sdot sdot 119903 (119896 + 119873119901)]

119879

119884 (119896) = [119910 (119896) 119910 (119896 + 1) sdot sdot sdot 119910 (119896 + 119873119901)]

119879

119864 (119896) = [119903 (119896) minus 119910 (119896) 119903 (119896 + 1) minus 119910 (119896 + 1) sdot sdot sdot 119903 (119896 + 119873119901) minus 119910 (119896 + 119873

119901)]

119879

(28)

119873119901is the predictive output horizon 119903(119896 + 119901) is the input

reference signal at discrete time 119896 + 119901 and 119910(119896 + 119901) is the119901 step-ahead prediction of 119910(119896) In general 119873

119901is selected

to include all responses that are significantly affected by thepresent control In this study119873

119901is min(11987011198702 11987015) =

35The control 119906(119896) = [119906(119896) 119906(119896 + 1) sdot sdot sdot 119906(119896 + 119873

119901)]

119879was obtained from the optimization of the cost function (29)based on gradient descent method that is

119906 (119896) = 119906 (119896 minus 1) + Δ119906 (119896) = 119906 (119896 minus 1) + 120578

120597119884

119879(119896)

120597119906 (119896)

119864 (119896)

= 119906 (119896 minus 1) + 120578119862

119879(119896) 119864 (119896)

(29)

where

119862 (119896) =

120597119884

119879(119896)

120597119906 (119896)

=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

120597119910 (119896)

120597119906 (119896)

120597119910 (119896)

120597119906 (119896 + 1)

sdot sdot sdot

120597119910 (119896)

120597119906 (119896 + 119873119901)

120597119910 (119896 + 1)

120597119906 (119896)

120597119910 (119896 + 1)

120597119906 (119896 + 1)

sdot sdot sdot

120597119910 (119896 + 1)

120597119906 (119896 + 119873119901)

d

120597119910 (119896 + 119873119901)

120597119906 (119896)

120597119910 (119896 + 119873119901)

120597119906 (119896 + 1)

sdot sdot sdot

120597119910 (119896 + 119873119901)

120597119906 (119896 + 119873119901)

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(30)


5 10 15 20 25 30 35 401200

1220

1240

1260

1280

1300

Batch

Measured valueOS-ELM-RDPLS model valueRPLS model value

Tem

pera

ture

of t

ube(

∘ C)

Figure 3 Comparison diagram of modeling data

5 10 15 20 25 301200

1220

1240

1260

1280

1300

Batch


Tem

pera

ture

of t

ube(

∘ C)

Figure 4 Comparison diagram of checking data

To reduce the computational load of EPC we let 119906(119896 +119873119901) =

sdot sdot sdot = 119906(119896 + 1) = 119906(119896) The EPC controller is expressed in theform

119906 (119896) = 119906 (119896 minus 1) + 120578119862

119879(119896) 119864 (119896) (31)

where

119862 (119896) = [

120597119910 (119896)

120597119906 (119896)

120597119910 (119896 + 1)

120597119906 (119896)

sdot sdot sdot

120597119910 (119896 + 119873119901)

120597119906 (119896)

]

119879

(32)

A schematic of the proposed PLC-based temperaturecontrol system is shown in Figure 6 The actual tempera-ture control system of the annular furnace is depicted inFigure 7 SIMATIC S7-400 was selected as the PLC of thecontrol system The entire system is mainly composed ofa PLC master station a remote IO station an operatorstation a programmer and communication bus and othercomponents The main modules of PLC include nine slotbases (UR2) a 4 A power supply module (PS407) a central

EPC Annularfurnace

EMP

r(k) u(k)

+ +minus

minusy(k + p)

y(k)

e(k)

y(k)

Figure 5 Architecture of the annular furnace employing OS-ELM-DRPLS-based predictive control

PLC1 PLC2

Printer

PROFIBUS-DP1

PROFIBUS-DP2

PROFIBUS-DP3

Industrial Ethernet

LII serverOperator station 2Operator station 1

middot middot middot

Figure 6 Schematic of the PLC-based temperature control system

processor (CPU416-2DP) 1M memory card and a networkcommunication module (CP443-1) The main modules ofIO expansion include a power supply module (PS307) aninterface module (IM153-1) a digital input module (SM321DC24V times DI16) a digital output module (SM322 DC24Vtimes DO16) a counter function module (8CH FM350-2) aneight-thermocouple input module (SM331) an eight-RTDinput module (SM331) and a four-output module (SM332)The main modules of the workstation include a CPU (IntelCore i7-930 28 GHz times 4) hard disk (WD 2TB) memory(Kingston 8GB) color LED (2410158401015840 1280 times 1024 resolution)and a net card (Siemens 10100MB) The main module ofcommunication includes Ethernet SINEC H1 and field busPROFIBUS-DP The main Software programs are Windows2003 Prof STEP7 V54 and WINCC61

The tube billet exit temperature should be controlled asbest as possible within the temperature range of 1255∘C to1295∘C Thirty tube billets were controlled by ELM modelpredicted control A thermocouple was ldquoburiedrdquo in a tubebillet The temperature course of the tube billet with theburied thermocouple is shown in Figure 8 In position 30the predicted temperature of the tube billet is 1211 OS-ELM-DRPLS-based predictive control algorithm was employed tomake the tube billet reach the lowest required temperature(1255∘C) By adjusting the input the temperature of the tubebillet reached 1265∘CThe variation in tube billet temperatureafter introducing temperature compensation control is shownin Figure 9 The variation in tube billet temperature after


Figure 7 Actual temperature control system of the annular furnace

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 380

100200300400500600700800900

1000110012001300

Position

Tem

pera

ture

of t

ube(

∘ C)

Figure 8 Temperature course of the tube billet with a thermocou-ple

introducing PID temperature control is shown in Figure 10The effect of predicted control is better than that of the PIDmethod

Figure 9 shows that the tube billet exit temperaturebasically fluctuates in the range of [1255∘C 1295∘C] the tubebillet heating quality is better than that before predictioncontrol and meets the requirements of piercing productionfor tubes

6 Conclusion

Measuring and controlling tube billet heating temperature aredifficult because of the complex reaction mechanism duringthe heating process in an annular furnace A tube billet finaltemperature prediction model was established in this studythrough OS-ELM-DRPLS modeling method An OS-ELM-DRPLS-based predictive controller for the control of tubebillet temperature was also systematically developed Thetube billet heating quality increased to a certain extent Thisfinding lays the foundation for the improvement of seamless

5 10 15 20 25 301255126012651270127512801285129012951300

Batch

Predicted control

Tem

pera

ture

of t

ube(

∘ C)

Figure 9 Temperature of the tube billet after introducing tempera-ture compensation control

5 10 15 20 25 301200

1220

1240

1260

1280

1300

Batch

PID control

Tem

pera

ture

of t

ube(

∘ C)

Figure 10 Temperature of the tube billet of PID method

tube quality After the developed model was compiled into auniversal module through the advanced computer languageof the configuration software the modules not only assistedin production by guiding front line workers to operatemanually but also formed a perfect close loop control circuittogether with the heating furnace model and controllerHence tube billet heating quality was improved effectivelyExperimentation proved that this method is feasible Thismodeling method is also versatile and can be extended toother processes with a large time lag

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research is supported by the National Natural ScienceFoundation of China (Grant nos 61203214 41371437 and61304121) and Provincial Science and Technology Depart-ment of Education Projects the General Project (L2013101)


References

[1] A D Acharya and S Chattopadhyay ldquoReheat furnace temper-ature control and performance at Essar Steelrdquo Iron and SteelEngineer vol 75 no 11 pp 31ndash36 1998

[2] W C Chen I V Samarasekera A Kumar and E B HawboltldquoMathematical modelling of heat flow and deformation duringrough rollingrdquo Ironmaking and Steelmaking vol 20 no 2 pp113ndash125 1993

[3] A Jaklic B Glogovac T Kolenko B Zupancic and B TezakldquoA simulation of heat transfer during billet transportrdquo AppliedThermal Engineering vol 22 no 7 pp 873ndash883 2002

[4] B Zhang Z G Chen and L Y Xu ldquoThe modeling and controlof a reheating furnacerdquo in Proceedings of the American ControlConference 2002

[5] B Zhang J C Wang and J M Zhang ldquoDynamic model ofreheating furnace based on fuzzy systemand genetic algorithmrdquoControl Theory amp Application vol 20 no 2 pp 293ndash296 1998(Chinese)

[6] H J Wick ldquoEstimation of ingot temperature in a soakingpit using an extended Kalman filterrdquo in Proceedings of the8th Triennial World Congress of the International Federation ofAutomatic Control 1981

[7] D Xiao Y H Yang and Z Z Mao ldquoA model for billet temper-ature of prediction of heating-furnace based on improved PCRmethodrdquo Information and Control vol 34 no 3 pp 340ndash3432005 (Chinese)

[8] Y-W Chen and T-Y Chai ldquoPreprocessing of operation data inheating furnacerdquo Control Theory and Applications vol 29 no 1pp 114ndash118 2012 (Chinese)

[9] G M Cui and G B Ding ldquoResearch on the optimal controlof tube billet temperature for rotary reheating furnacerdquo inAdvanced Electrical and Electronics Engineering vol 87 ofLecture Notes in Electrical Engineering pp 471ndash477 SpringerBerlin Germany 2011

[10] H Iwamoto O Sugiyama R Nakanishi and T OkuyamaldquoAutomatic control system of billet reheating rotary hearthfurnacerdquo in Proceedings of the International Conference onIndustrial Electronics Control Instrumentation 1992

[11] F He A Xu H Wang D He and N Tian ldquoEnd temperatureprediction of molten steel in LF based on CBRrdquo Steel ResearchInternational vol 83 no 11 pp 1079ndash1086 2012

[12] W Lv Z Mao and P Yuan ldquoLadle furnace steel temperatureprediction model based on partial linear regularization net-works with sparse representationrdquo Steel Research Internationalvol 83 no 3 pp 288ndash296 2012

[13] S Wold N Kettaneh-Wold and B Skagerberg ldquoNonlinear PLSmodelingrdquo Chemometrics and Intelligent Laboratory Systemsvol 7 no 1-2 pp 53ndash65 1989

[14] S J Qin ldquoRecursive PLS algorithms for adaptive data model-ingrdquo Computers amp Chemical Engineering vol 22 no 4-5 pp503ndash514 1998

[15] B Hu Z Zhao and J Liang ldquoMulti-loop nonlinear internalmodel controller design under nonlinear dynamic PLS frame-work using ARX-neural network modelrdquo Journal of ProcessControl vol 22 no 1 pp 207ndash217 2012

[16] G-B Huang Q-Y Zhu and C-K Siew ldquoExtreme learningmachine theory and applicationsrdquoNeurocomputing vol 70 no1ndash3 pp 489ndash501 2006

[17] Y Yu T-M Choi and C-L Hui ldquoAn intelligent quick pre-diction algorithm with applications in industrial control and

loading problemsrdquo IEEE Transactions on Automation Scienceand Engineering vol 9 no 2 pp 276ndash287 2012

[18] J Zhai H Xu and Y Li ldquoFusion of extreme learning machinewith fuzzy integralrdquo International Journal of Uncertainty Fuzzi-ness and Knowlege-Based Systems vol 21 supplement 2 pp 23ndash34 2013

[19] J-H Zhai H-Y Xu and X-Z Wang ldquoDynamic ensembleextreme learning machine based on sample entropyrdquo SoftComputing vol 16 no 9 pp 1493ndash1502 2012

[20] J W Cao T Chen and J Fan ldquoFast online learning algorithmfor landmark recognition based on BoW frameworkrdquo in Pro-ceedings of the 9th IEEE Conference on Industrial Electronics andApplications June 2014

[21] Y Jin J W Cao Q Q Ruan and X Q Wang ldquoCross-modality2D-3D face recognition via multiview smooth discriminantanalysis based on ELMrdquo Journal of Electrical and ComputerEngineering vol 2014 Article ID 584241 9 pages 2014

[22] J Cao and L Xiong ldquoProtein sequence classification withimproved extreme learning machine algorithmsrdquo BioMedResearch International vol 2014 Article ID 103054 12 pages2014

[23] Y Yang Y Wang and X Yuan ldquoBidirectional extreme learningmachine for regression problem and its learning effectivenessrdquoIEEE Transactions on Neural Networks and Learning Systemsvol 23 no 9 pp 1498ndash1505 2012

[24] G-B Huang H Zhou X Ding and R Zhang ldquoExtremelearning machine for regression and multiclass classificationrdquoIEEE Transactions on Systems Man and Cybernetics Part BCybernetics vol 42 no 2 pp 513ndash529 2012

[25] G-B Huang ldquoAn insight into extreme learning machinesrandom neurons random features and kernelsrdquo CognitiveComputation vol 6 no 3 pp 376ndash390 2014

[26] G Huang S Song J N D Gupta and C Wu ldquoSemi-supervised and unsupervised extreme learningmachinesrdquo IEEETransactions on Cybernetics 2014

[27] H-X Tian and Z-Z Mao ldquoAn ensemble ELM based on mod-ified AdaBoostRT algorithm for predicting the temperature ofmolten steel in ladle furnacerdquo IEEE Transactions on AutomationScience and Engineering vol 7 no 1 pp 73ndash80 2010

[28] G Feng Z Qian and N Dai ldquoReversible watermarking viaextreme learningmachine predictionrdquoNeurocomputing vol 82no 4 pp 62ndash68 2012

[29] N-Y Liang G-B Huang P Saratchandran and N Sundarara-jan ldquoA fast and accurate online sequential learning algorithmfor feed forward networksrdquo IEEE Transactions on Neural Net-works vol 17 no 6 pp 1411ndash1423 2006

[30] J Zhao Z Wang and D S Park ldquoOnline sequential extremelearning machine with forgetting mechanismrdquo Neurocomput-ing vol 87 pp 79ndash89 2012

[31] S J Xie J Yang H Gong S Yoon and D S Park ldquoIntelligentfingerprint quality analysis using online sequential extremelearning machinerdquo Soft Computing vol 16 no 9 pp 1555ndash15682012

[32] M Khalid S Omatu and R Yusof ldquoMIMO furnace controlwith neural networksrdquo IEEE Transactions on Control SystemsTechnology vol 1 no 4 pp 238ndash245 1993

[33] C-H Lu C-C Tsai C-M Liu and Y-H Charng ldquoNeural-network-based predictive controller design an application totemperature control of a plastic injection molding processrdquoAsian Journal of Control vol 12 no 6 pp 680ndash691 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


The system consists of a component that calculates thetube billet temperature and a component that calculatesthe optimal furnace temperature set point In large Chinesesteel companies such as Baosteel Co Panzhihua Iron andSteel Co Ltd Anshan Iron and Steel Co Ltd and CapitalIron and Steel Co Ltd temperature prediction models areutilized in several heating furnacesHowever their predictionmodels are almost entirely engineering models importedfrom abroadTherefore thesemodels are difficult tomaintainand transplant and the costs of doing so are high

With the development of configuration software anddatabase technique increasing amounts of production dataare being collected and stored Therefore data-driven mod-eling and control methods are eliciting more and moreattention He et al [11] and Lv et al [12] established datamodels for the Ladle furnace through data-driven methodsIn the present work the production data of an annularfurnace were obtained from the seamless tube subcompanyof Baosteel Industrial process data contain noise whichreduces the modeling accuracy of the extreme learningmachine (ELM) algorithm By contrast the capability ofthe partial least square (PLS) algorithm to process linearrelevant data is suitable Moreover some cyclical changesoccur in production Thus the model should be updatedin real time Online sequential (OS) ELM and recursivePLS algorithm can realize model update Online sequentialextreme learning machine dynamic recursive partial leastsquare (OS-ELM-DRPLS) algorithm was proposed in thisstudy A tube billet temperature prediction model basedon this algorithm was established and a strategy for theoptimization and control of tube billet temperature wasproposed based on this model OS-ELM-DRPLS not onlyhas the advantages of the OS-ELM algorithm (eg rapidnonlinear modeling and update) but also has the capability ofthe RPLS algorithm to process linear relevant data The largetime lag and reduced model precision were solved throughdynamic processingThe algorithm is easy to implement withadvanced computer language Configuration software suchas WINCC61 can be utilized to compile the modeling andcontrol algorithms into special modules for use in industrialsites Simulation and actual experiments prove that tubebillet temperature can be predicted and controlled withinthe scope of the production process requirements with theestablished temperature prediction model and the proposedstrategy of optimization and control based on OS-ELM-DRPLS algorithm

2 Annular Furnace

An annular furnace is a type of rotary hearth furnaceutilized for tube billet heating It consists of the furnace shaftand auxiliary equipment for charging and discharging Thefurnace shaft consists of a fixed furnace roof a ring-typetunnel surrounded by a fixed furnace roof wall and a circularring-like rotary hearth as shown in Figure 1

External and inner ring seams are placed between thefixed furnace wall and rotary hearth in an annular fur-nace Internal and external water seal tanks are arranged

6

45

1

2

3

(1) Roof(2) Wall(3) Rotary hearth

(4) Charger(5) Discharger(6) Pipeline system

Figure 1 Schematic of annular furnace

beneath the external and inner ring seams to maintainnormal temperature and pressure in the furnace cavity andprevent external cold air from entering the furnace cavityFuel gas and combustion-supporting air are blown into thefurnace through burning nozzles mounted on the externaland internal walls or furnace roof tomake the gas burnwithinthe furnace and heat the tube billet The fumes producedby gas burning within the furnace move conversely throughthe rotary hearth to the tail end of the soaking zone enterthe flue and chimney outside the furnace and exit to theatmosphereThe external wall of the furnace has charging anddischarging furnace doors in which a charger and dischargerare placed respectively Charging and discharging proceedsimultaneously When a tube billet is placed in the furnacethe bottom rotates at a certain angle Tube billets follow aradial layout inside the furnace and are arranged either in asingle row or in multirows

The furnace cavity is divided into preheating heating andsoaking zones according to the heating process of the tube bil-let in the annular furnace Burning nozzles are not mountedin the preheating zone A flue opening is arranged on theside wall near the charging furnace door in this zone High-temperature exhaust gas flows toward the opposite directionof hearth rotation and exits into the atmosphere throughthe flue opening in the heating and soaking zones Duringthe flow process of high-temperature exhaust gas the tubebillets in the preheating zone are mainly convection heatedThe length of the preheating zone accounts for approximatelyone-fourth of the peripheral length of the annular furnaceTemperature differences between the surface and center andbetween both ends exist in the tube billet rapidly heatedin the heating zone To reduce the temperature differenceof the tube billets and eliminate their male-female facesthe tube billets must be heated in the soaking zone Thelength of the soaking zone is approximately three-twentiethsof the peripheral length of the annular furnace In additionno tube billet and burning nozzle are present between thecharging and discharging furnace doors A partition wall isplaced in the middle The distance between the charging and





X = [119909

1198701

1 119909

1198702

2 119909

119870119901

119901] (1)


1 1199092 119909

119901


119883 = 119879119875

119879+ 119864 =

119860

sum

119886=1

119905119886119901

119879

119886+ 119864

119884 = 119880119876

119879+ 119865 =

119860

sum

119886=1

119906119886119902

119879

119886+ 119865

(2)



119886(119898times




119886= 119891 (119905

119886) + 120576 (3)






119899times 119877


119894is



119894 119886119894 and


119891(119909119895) =

sum

119894=1

120573119894119866(119886119894 119887119894 119909119895) + 120576119895= 119905119895 (4)


119894are the






119867120573 = 119879 (5)

where

119867(1198861 119886

1198871 119887

1199091 119909

119873)

=

[

[

[

119866 (1198861 1198871 1199091) sdot sdot sdot 119866 (119886

119887 1199091)

sdot sdot sdot

119866 (1198861 1198871 119909119873) sdot sdot sdot 119866 (119886

119887 119909119873)

]

]

]119873times

120573 =

[

[

[

120573

119879

1

120573

119879

]

]

]times119898

119879 =

[

[

[

119905

119879

1

119905

119879

119873

]

]

]119873times119898

(6)


1 1199092 119909

119873 and the



119894and



120573 = 119867

+119879 (7)





120573 = 119867





119867

+= (119867

119879119867)

minus1

119867

119879 (8)

If 119867




120573 = (119867

119879119867)

minus1

119867

119879119879 (9)



1198730

119894=1and



0= 119870

minus1

0119867

119879

01198790

where1198700= 119867

119879



1198730+1198731

119894=1198730+1



10038171003817100381710038171003817100381710038171003817

[

1198670

1198671

] 120573 minus [

1198790

1198791

]

10038171003817100381710038171003817100381710038171003817

(10)



1205731= 119870

minus1

1[

1198670

1198671

]

119879

[

1198790

1198791

] where 1198701= [

1198670

1198671

]

119879

[

1198670

1198671

] (11)


function of 1205730 1198701 1198671 and 119879



1198701= [119867

119879

0119867

119879

1] [

1198670

1198671

] = 1198700+ 119867

119879

11198671 (12)

[

1198670

1198671

]

119879

[

1198790

1198791

] = 119867

119879

01198790+ 119867

119879

11198671= 1198700119870

minus1

0119867

119879

01198790+ 119867

119879

11198791

= 11987001205730+ 119867

119879

11198791= (1198701minus 119867

119879

11198671) 1205730+ 119867

119879

11198791

= 11987011205730minus 119867

119879

111986711205730+ 119867

119879

11198791

(13)


1205731= 119870

minus1

1[

1198670

1198671

]

119879

[

1198790

1198791

] = 119870

minus1

1(11987011205730minus 119867

119879

111986711205730+ 119867

119879

11198791)

= 1205730+ 119870

minus1

1119867

119879

1(1198791minus 11986711205730)

(14)

where1198701= 1198700+ 119867

119879

11198671


alefsym119896+1

= (119909119894 119905119894)

sum119896+1

119895=0119873119895

119894=(sum119896

119895=0119873119895)+1

(15)



119870119896+1

= 119870119896+ 119867

119879

119896+1119867119896+1

120573119896+1

= 120573119896+ 119870

minus1

119896+1119867

119879

119896+1(119879119896+1

minus 119867119896+1

120573119896)

(16)

119870

minus1

119896+1rather than 119870


119896+1from



Woodbury formula

119870

minus1

119896+1= (119870119896+ 119867

119879

119896+1119867119896+1

)

minus1

= 119870

minus1

119896minus 119870

minus1

119896119867

119879

119896+1(119868 + 119867

119896+1119870

minus1

119896119867

119879

119896+1)

minus1

times 119867119896+1

119870

minus1

119896

(17)

We let 119875119896+1

= 119870

minus1


119896+1can

be written as

119875119896+1

= 119875119896minus 119875119896119867

119879

119896+1(119868 + 119867

119896+1119875119896119867

119879

119896+1)

minus1

119867119896+1

119875119896

120573119896+1

= 120573119896+ 119875119896+1

119867

119879

119896+1(119879119896+1

minus 119867119896+1

120573119896)

(18)







119877


X = [119909

1198701

1 119909

1198702

2 119909

119870119901

119901] (19)


1 1199092 119909

119901


119883 = 119879119875

119879+ 119864 =

119860

sum

119886=1

119905119886119901

119879

119886+ 119864

119884 = 119880119876

119879+ 119865 =

119860

sum

119886=1

119906119886119902

119879

119886+ 119865

(20)




1198831= 1198791119875

119879

1+ 119864

1198841= 1198801119876

119879

1+ 119865

(21)


119875

119879= 119908119875

119879+ (1 minus 119908) 119875

119879

1

119876

119879= 119908119876

119879+ (1 minus 119908)119876

119879

1

(22)


2 and


1198832= 1198792119875

119879+ 119864 (23)

1198792


=


119884 = 119880119876

119879



1 119909

1198702

2 119909

119870119901


1 1198702 119870

119901


119882 =

1198992

sum

119894=1

10038161003816100381610038161003816

Y (119894) minus Y (119894)

10038161003816100381610038161003816

(24)






2 195 CO



4 28 weight carbon


2 and 44 N

2 The





1ndash1199096were


7ndash11990915













F1

F2

F3

F4

F5F6

Number 1 Number 2

Num

ber3

Number4

Number 5

Number 6

Charging

Discharging

T1T2

T3

T4

T5

T6

T8T7T9


Preh

eat z

one






1 1199091


2 1199092


3 1199093


4 1199094


5 1199095


6 1199096


7 1199097


∘C

8 1199098


∘C

9 1199099


∘C

10 11990910


∘C

11 11990911


∘C

12 11990912


∘C

13 11990913


∘C

14 11990914


∘C

15 11990915


∘C





of lag time of1198701 1198702 119870




[1198701 1198702 119870

15]

= [58 55 51 46 42 36 61 56 51 45 39 35 52 58 60]

(25)

The unit of [1198701 1198702 119870







119869 (119896) =

1

2

119873119901

sum

119901=0

(119903 (119896 + 119901) minus 119910 (119896 + 119901))

2

=

1

2

(119877 (119896) minus 119884 (119896))

119879(119877 (119896) minus 119884 (119896)) =

1

2

119864

119879(119896) 119864 (119896)

(27)

where

119877 (119896) = [119903 (119896) 119903 (119896 + 1) sdot sdot sdot 119903 (119896 + 119873119901)]

119879

119884 (119896) = [119910 (119896) 119910 (119896 + 1) sdot sdot sdot 119910 (119896 + 119873119901)]

119879


119901)]

119879

(28)



119901is selected


119901is min(11987011198702 11987015) =


119901)]


119906 (119896) = 119906 (119896 minus 1) + Δ119906 (119896) = 119906 (119896 minus 1) + 120578

120597119884

119879(119896)

120597119906 (119896)

119864 (119896)

= 119906 (119896 minus 1) + 120578119862

119879(119896) 119864 (119896)

(29)

where

119862 (119896) =

120597119884

119879(119896)

120597119906 (119896)

=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

120597119910 (119896)

120597119906 (119896)

120597119910 (119896)

120597119906 (119896 + 1)

sdot sdot sdot

120597119910 (119896)

120597119906 (119896 + 119873119901)

120597119910 (119896 + 1)

120597119906 (119896)

120597119910 (119896 + 1)

120597119906 (119896 + 1)

sdot sdot sdot

120597119910 (119896 + 1)

120597119906 (119896 + 119873119901)

d

120597119910 (119896 + 119873119901)

120597119906 (119896)

120597119910 (119896 + 119873119901)

120597119906 (119896 + 1)

sdot sdot sdot

120597119910 (119896 + 119873119901)

120597119906 (119896 + 119873119901)

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(30)


5 10 15 20 25 30 35 401200

1220

1240

1260

1280

1300

Batch


Tem

pera

ture

of t

ube(

∘ C)


5 10 15 20 25 301200

1220

1240

1260

1280

1300

Batch


Tem

pera

ture

of t

ube(

∘ C)




119906 (119896) = 119906 (119896 minus 1) + 120578119862

119879(119896) 119864 (119896) (31)

where

119862 (119896) = [

120597119910 (119896)

120597119906 (119896)

120597119910 (119896 + 1)

120597119906 (119896)

sdot sdot sdot

120597119910 (119896 + 119873119901)

120597119906 (119896)

]

119879

(32)


EPC Annularfurnace

EMP

r(k) u(k)

+ +minus

minusy(k + p)

y(k)

e(k)

y(k)


PLC1 PLC2

Printer

PROFIBUS-DP1

PROFIBUS-DP2

PROFIBUS-DP3

Industrial Ethernet








1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 380

100200300400500600700800900

1000110012001300

Position

Tem

pera

ture

of t

ube(

∘ C)




6 Conclusion


5 10 15 20 25 301255126012651270127512801285129012951300

Batch

Predicted control

Tem

pera

ture

of t

ube(

∘ C)


5 10 15 20 25 301200

1220

1240

1260

1280

1300

Batch

PID control

Tem

pera

ture

of t

ube(

∘ C)





Acknowledgments



References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













X = [119909

1198701

1 119909

1198702

2 119909

119870119901

119901] (1)


1 1199092 119909

119901


119883 = 119879119875

119879+ 119864 =

119860

sum

119886=1

119905119886119901

119879

119886+ 119864

119884 = 119880119876

119879+ 119865 =

119860

sum

119886=1

119906119886119902

119879

119886+ 119865

(2)



119886(119898times




119886= 119891 (119905

119886) + 120576 (3)






119899times 119877


119894is



119894 119886119894 and


119891(119909119895) =

sum

119894=1

120573119894119866(119886119894 119887119894 119909119895) + 120576119895= 119905119895 (4)


119894are the






119867120573 = 119879 (5)

where

119867(1198861 119886

1198871 119887

1199091 119909

119873)

=

[

[

[

119866 (1198861 1198871 1199091) sdot sdot sdot 119866 (119886

119887 1199091)

sdot sdot sdot

119866 (1198861 1198871 119909119873) sdot sdot sdot 119866 (119886

119887 119909119873)

]

]

]119873times

120573 =

[

[

[

120573

119879

1

120573

119879

]

]

]times119898

119879 =

[

[

[

119905

119879

1

119905

119879

119873

]

]

]119873times119898

(6)


1 1199092 119909

119873 and the



119894and



120573 = 119867

+119879 (7)





120573 = 119867





119867

+= (119867

119879119867)

minus1

119867

119879 (8)

If 119867




120573 = (119867

119879119867)

minus1

119867

119879119879 (9)



1198730

119894=1and



0= 119870

minus1

0119867

119879

01198790

where1198700= 119867

119879



1198730+1198731

119894=1198730+1



10038171003817100381710038171003817100381710038171003817

[

1198670

1198671

] 120573 minus [

1198790

1198791

]

10038171003817100381710038171003817100381710038171003817

(10)



1205731= 119870

minus1

1[

1198670

1198671

]

119879

[

1198790

1198791

] where 1198701= [

1198670

1198671

]

119879

[

1198670

1198671

] (11)


function of 1205730 1198701 1198671 and 119879



1198701= [119867

119879

0119867

119879

1] [

1198670

1198671

] = 1198700+ 119867

119879

11198671 (12)

[

1198670

1198671

]

119879

[

1198790

1198791

] = 119867

119879

01198790+ 119867

119879

11198671= 1198700119870

minus1

0119867

119879

01198790+ 119867

119879

11198791

= 11987001205730+ 119867

119879

11198791= (1198701minus 119867

119879

11198671) 1205730+ 119867

119879

11198791

= 11987011205730minus 119867

119879

111986711205730+ 119867

119879

11198791

(13)


1205731= 119870

minus1

1[

1198670

1198671

]

119879

[

1198790

1198791

] = 119870

minus1

1(11987011205730minus 119867

119879

111986711205730+ 119867

119879

11198791)

= 1205730+ 119870

minus1

1119867

119879

1(1198791minus 11986711205730)

(14)

where1198701= 1198700+ 119867

119879

11198671


alefsym119896+1

= (119909119894 119905119894)

sum119896+1

119895=0119873119895

119894=(sum119896

119895=0119873119895)+1

(15)



119870119896+1

= 119870119896+ 119867

119879

119896+1119867119896+1

120573119896+1

= 120573119896+ 119870

minus1

119896+1119867

119879

119896+1(119879119896+1

minus 119867119896+1

120573119896)

(16)

119870

minus1

119896+1rather than 119870


119896+1from



Woodbury formula

119870

minus1

119896+1= (119870119896+ 119867

119879

119896+1119867119896+1

)

minus1

= 119870

minus1

119896minus 119870

minus1

119896119867

119879

119896+1(119868 + 119867

119896+1119870

minus1

119896119867

119879

119896+1)

minus1

times 119867119896+1

119870

minus1

119896

(17)

We let 119875119896+1

= 119870

minus1


119896+1can

be written as

119875119896+1

= 119875119896minus 119875119896119867

119879

119896+1(119868 + 119867

119896+1119875119896119867

119879

119896+1)

minus1

119867119896+1

119875119896

120573119896+1

= 120573119896+ 119875119896+1

119867

119879

119896+1(119879119896+1

minus 119867119896+1

120573119896)

(18)







119877


X = [119909

1198701

1 119909

1198702

2 119909

119870119901

119901] (19)


1 1199092 119909

119901


119883 = 119879119875

119879+ 119864 =

119860

sum

119886=1

119905119886119901

119879

119886+ 119864

119884 = 119880119876

119879+ 119865 =

119860

sum

119886=1

119906119886119902

119879

119886+ 119865

(20)




1198831= 1198791119875

119879

1+ 119864

1198841= 1198801119876

119879

1+ 119865

(21)


119875

119879= 119908119875

119879+ (1 minus 119908) 119875

119879

1

119876

119879= 119908119876

119879+ (1 minus 119908)119876

119879

1

(22)


2 and


1198832= 1198792119875

119879+ 119864 (23)

1198792


=


119884 = 119880119876

119879



1 119909

1198702

2 119909

119870119901


1 1198702 119870

119901


119882 =

1198992

sum

119894=1

10038161003816100381610038161003816

Y (119894) minus Y (119894)

10038161003816100381610038161003816

(24)






2 195 CO



4 28 weight carbon


2 and 44 N

2 The





1ndash1199096were


7ndash11990915













F1

F2

F3

F4

F5F6

Number 1 Number 2

Num

ber3

Number4

Number 5

Number 6

Charging

Discharging

T1T2

T3

T4

T5

T6

T8T7T9


Preh

eat z

one






1 1199091


2 1199092


3 1199093


4 1199094


5 1199095


6 1199096


7 1199097


∘C

8 1199098


∘C

9 1199099


∘C

10 11990910


∘C

11 11990911


∘C

12 11990912


∘C

13 11990913


∘C

14 11990914


∘C

15 11990915


∘C





of lag time of1198701 1198702 119870




[1198701 1198702 119870

15]

= [58 55 51 46 42 36 61 56 51 45 39 35 52 58 60]

(25)

The unit of [1198701 1198702 119870







119869 (119896) =

1

2

119873119901

sum

119901=0

(119903 (119896 + 119901) minus 119910 (119896 + 119901))

2

=

1

2

(119877 (119896) minus 119884 (119896))

119879(119877 (119896) minus 119884 (119896)) =

1

2

119864

119879(119896) 119864 (119896)

(27)

where

119877 (119896) = [119903 (119896) 119903 (119896 + 1) sdot sdot sdot 119903 (119896 + 119873119901)]

119879

119884 (119896) = [119910 (119896) 119910 (119896 + 1) sdot sdot sdot 119910 (119896 + 119873119901)]

119879


119901)]

119879

(28)



119901is selected


119901is min(11987011198702 11987015) =


119901)]


119906 (119896) = 119906 (119896 minus 1) + Δ119906 (119896) = 119906 (119896 minus 1) + 120578

120597119884

119879(119896)

120597119906 (119896)

119864 (119896)

= 119906 (119896 minus 1) + 120578119862

119879(119896) 119864 (119896)

(29)

where

119862 (119896) =

120597119884

119879(119896)

120597119906 (119896)

=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

120597119910 (119896)

120597119906 (119896)

120597119910 (119896)

120597119906 (119896 + 1)

sdot sdot sdot

120597119910 (119896)

120597119906 (119896 + 119873119901)

120597119910 (119896 + 1)

120597119906 (119896)

120597119910 (119896 + 1)

120597119906 (119896 + 1)

sdot sdot sdot

120597119910 (119896 + 1)

120597119906 (119896 + 119873119901)

d

120597119910 (119896 + 119873119901)

120597119906 (119896)

120597119910 (119896 + 119873119901)

120597119906 (119896 + 1)

sdot sdot sdot

120597119910 (119896 + 119873119901)

120597119906 (119896 + 119873119901)

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(30)


5 10 15 20 25 30 35 401200

1220

1240

1260

1280

1300

Batch


Tem

pera

ture

of t

ube(

∘ C)


5 10 15 20 25 301200

1220

1240

1260

1280

1300

Batch


Tem

pera

ture

of t

ube(

∘ C)




119906 (119896) = 119906 (119896 minus 1) + 120578119862

119879(119896) 119864 (119896) (31)

where

119862 (119896) = [

120597119910 (119896)

120597119906 (119896)

120597119910 (119896 + 1)

120597119906 (119896)

sdot sdot sdot

120597119910 (119896 + 119873119901)

120597119906 (119896)

]

119879

(32)


EPC Annularfurnace

EMP

r(k) u(k)

+ +minus

minusy(k + p)

y(k)

e(k)

y(k)


PLC1 PLC2

Printer

PROFIBUS-DP1

PROFIBUS-DP2

PROFIBUS-DP3

Industrial Ethernet








1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 380

100200300400500600700800900

1000110012001300

Position

Tem

pera

ture

of t

ube(

∘ C)




6 Conclusion


5 10 15 20 25 301255126012651270127512801285129012951300

Batch

Predicted control

Tem

pera

ture

of t

ube(

∘ C)


5 10 15 20 25 301200

1220

1240

1260

1280

1300

Batch

PID control

Tem

pera

ture

of t

ube(

∘ C)





Acknowledgments



References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












120573 = 119867





119867

+= (119867

119879119867)

minus1

119867

119879 (8)

If 119867




120573 = (119867

119879119867)

minus1

119867

119879119879 (9)



1198730

119894=1and



0= 119870

minus1

0119867

119879

01198790

where1198700= 119867

119879



1198730+1198731

119894=1198730+1



10038171003817100381710038171003817100381710038171003817

[

1198670

1198671

] 120573 minus [

1198790

1198791

]

10038171003817100381710038171003817100381710038171003817

(10)



1205731= 119870

minus1

1[

1198670

1198671

]

119879

[

1198790

1198791

] where 1198701= [

1198670

1198671

]

119879

[

1198670

1198671

] (11)


function of 1205730 1198701 1198671 and 119879



1198701= [119867

119879

0119867

119879

1] [

1198670

1198671

] = 1198700+ 119867

119879

11198671 (12)

[

1198670

1198671

]

119879

[

1198790

1198791

] = 119867

119879

01198790+ 119867

119879

11198671= 1198700119870

minus1

0119867

119879

01198790+ 119867

119879

11198791

= 11987001205730+ 119867

119879

11198791= (1198701minus 119867

119879

11198671) 1205730+ 119867

119879

11198791

= 11987011205730minus 119867

119879

111986711205730+ 119867

119879

11198791

(13)


1205731= 119870

minus1

1[

1198670

1198671

]

119879

[

1198790

1198791

] = 119870

minus1

1(11987011205730minus 119867

119879

111986711205730+ 119867

119879

11198791)

= 1205730+ 119870

minus1

1119867

119879

1(1198791minus 11986711205730)

(14)

where1198701= 1198700+ 119867

119879

11198671


alefsym119896+1

= (119909119894 119905119894)

sum119896+1

119895=0119873119895

119894=(sum119896

119895=0119873119895)+1

(15)



119870119896+1

= 119870119896+ 119867

119879

119896+1119867119896+1

120573119896+1

= 120573119896+ 119870

minus1

119896+1119867

119879

119896+1(119879119896+1

minus 119867119896+1

120573119896)

(16)

119870

minus1

119896+1rather than 119870


119896+1from



Woodbury formula

119870

minus1

119896+1= (119870119896+ 119867

119879

119896+1119867119896+1

)

minus1

= 119870

minus1

119896minus 119870

minus1

119896119867

119879

119896+1(119868 + 119867

119896+1119870

minus1

119896119867

119879

119896+1)

minus1

times 119867119896+1

119870

minus1

119896

(17)

We let 119875119896+1

= 119870

minus1


119896+1can

be written as

119875119896+1

= 119875119896minus 119875119896119867

119879

119896+1(119868 + 119867

119896+1119875119896119867

119879

119896+1)

minus1

119867119896+1

119875119896

120573119896+1

= 120573119896+ 119875119896+1

119867

119879

119896+1(119879119896+1

minus 119867119896+1

120573119896)

(18)







119877


X = [119909

1198701

1 119909

1198702

2 119909

119870119901

119901] (19)


1 1199092 119909

119901


119883 = 119879119875

119879+ 119864 =

119860

sum

119886=1

119905119886119901

119879

119886+ 119864

119884 = 119880119876

119879+ 119865 =

119860

sum

119886=1

119906119886119902

119879

119886+ 119865

(20)




1198831= 1198791119875

119879

1+ 119864

1198841= 1198801119876

119879

1+ 119865

(21)


119875

119879= 119908119875

119879+ (1 minus 119908) 119875

119879

1

119876

119879= 119908119876

119879+ (1 minus 119908)119876

119879

1

(22)


2 and


1198832= 1198792119875

119879+ 119864 (23)

1198792


=


119884 = 119880119876

119879



1 119909

1198702

2 119909

119870119901


1 1198702 119870

119901


119882 =

1198992

sum

119894=1

10038161003816100381610038161003816

Y (119894) minus Y (119894)

10038161003816100381610038161003816

(24)






2 195 CO



4 28 weight carbon


2 and 44 N

2 The





1ndash1199096were


7ndash11990915













F1

F2

F3

F4

F5F6

Number 1 Number 2

Num

ber3

Number4

Number 5

Number 6

Charging

Discharging

T1T2

T3

T4

T5

T6

T8T7T9


Preh

eat z

one






1 1199091


2 1199092


3 1199093


4 1199094


5 1199095


6 1199096


7 1199097


∘C

8 1199098


∘C

9 1199099


∘C

10 11990910


∘C

11 11990911


∘C

12 11990912


∘C

13 11990913


∘C

14 11990914


∘C

15 11990915


∘C





of lag time of1198701 1198702 119870




[1198701 1198702 119870

15]

= [58 55 51 46 42 36 61 56 51 45 39 35 52 58 60]

(25)

The unit of [1198701 1198702 119870







119869 (119896) =

1

2

119873119901

sum

119901=0

(119903 (119896 + 119901) minus 119910 (119896 + 119901))

2

=

1

2

(119877 (119896) minus 119884 (119896))

119879(119877 (119896) minus 119884 (119896)) =

1

2

119864

119879(119896) 119864 (119896)

(27)

where

119877 (119896) = [119903 (119896) 119903 (119896 + 1) sdot sdot sdot 119903 (119896 + 119873119901)]

119879

119884 (119896) = [119910 (119896) 119910 (119896 + 1) sdot sdot sdot 119910 (119896 + 119873119901)]

119879


119901)]

119879

(28)



119901is selected


119901is min(11987011198702 11987015) =


119901)]


119906 (119896) = 119906 (119896 minus 1) + Δ119906 (119896) = 119906 (119896 minus 1) + 120578

120597119884

119879(119896)

120597119906 (119896)

119864 (119896)

= 119906 (119896 minus 1) + 120578119862

119879(119896) 119864 (119896)

(29)

where

119862 (119896) =

120597119884

119879(119896)

120597119906 (119896)

=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

120597119910 (119896)

120597119906 (119896)

120597119910 (119896)

120597119906 (119896 + 1)

sdot sdot sdot

120597119910 (119896)

120597119906 (119896 + 119873119901)

120597119910 (119896 + 1)

120597119906 (119896)

120597119910 (119896 + 1)

120597119906 (119896 + 1)

sdot sdot sdot

120597119910 (119896 + 1)

120597119906 (119896 + 119873119901)

d

120597119910 (119896 + 119873119901)

120597119906 (119896)

120597119910 (119896 + 119873119901)

120597119906 (119896 + 1)

sdot sdot sdot

120597119910 (119896 + 119873119901)

120597119906 (119896 + 119873119901)

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(30)


5 10 15 20 25 30 35 401200

1220

1240

1260

1280

1300

Batch


Tem

pera

ture

of t

ube(

∘ C)


5 10 15 20 25 301200

1220

1240

1260

1280

1300

Batch


Tem

pera

ture

of t

ube(

∘ C)




119906 (119896) = 119906 (119896 minus 1) + 120578119862

119879(119896) 119864 (119896) (31)

where

119862 (119896) = [

120597119910 (119896)

120597119906 (119896)

120597119910 (119896 + 1)

120597119906 (119896)

sdot sdot sdot

120597119910 (119896 + 119873119901)

120597119906 (119896)

]

119879

(32)


EPC Annularfurnace

EMP

r(k) u(k)

+ +minus

minusy(k + p)

y(k)

e(k)

y(k)


PLC1 PLC2

Printer

PROFIBUS-DP1

PROFIBUS-DP2

PROFIBUS-DP3

Industrial Ethernet








1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 380

100200300400500600700800900

1000110012001300

Position

Tem

pera

ture

of t

ube(

∘ C)




6 Conclusion


5 10 15 20 25 301255126012651270127512801285129012951300

Batch

Predicted control

Tem

pera

ture

of t

ube(

∘ C)


5 10 15 20 25 301200

1220

1240

1260

1280

1300

Batch

PID control

Tem

pera

ture

of t

ube(

∘ C)





Acknowledgments



References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119877


X = [119909

1198701

1 119909

1198702

2 119909

119870119901

119901] (19)


1 1199092 119909

119901


119883 = 119879119875

119879+ 119864 =

119860

sum

119886=1

119905119886119901

119879

119886+ 119864

119884 = 119880119876

119879+ 119865 =

119860

sum

119886=1

119906119886119902

119879

119886+ 119865

(20)




1198831= 1198791119875

119879

1+ 119864

1198841= 1198801119876

119879

1+ 119865

(21)


119875

119879= 119908119875

119879+ (1 minus 119908) 119875

119879

1

119876

119879= 119908119876

119879+ (1 minus 119908)119876

119879

1

(22)


2 and


1198832= 1198792119875

119879+ 119864 (23)

1198792


=


119884 = 119880119876

119879



1 119909

1198702

2 119909

119870119901


1 1198702 119870

119901


119882 =

1198992

sum

119894=1

10038161003816100381610038161003816

Y (119894) minus Y (119894)

10038161003816100381610038161003816

(24)






2 195 CO



4 28 weight carbon


2 and 44 N

2 The





1ndash1199096were


7ndash11990915













F1

F2

F3

F4

F5F6

Number 1 Number 2

Num

ber3

Number4

Number 5

Number 6

Charging

Discharging

T1T2

T3

T4

T5

T6

T8T7T9


Preh

eat z

one






1 1199091


2 1199092


3 1199093


4 1199094


5 1199095


6 1199096


7 1199097


∘C

8 1199098


∘C

9 1199099


∘C

10 11990910


∘C

11 11990911


∘C

12 11990912


∘C

13 11990913


∘C

14 11990914


∘C

15 11990915


∘C





of lag time of1198701 1198702 119870




[1198701 1198702 119870

15]

= [58 55 51 46 42 36 61 56 51 45 39 35 52 58 60]

(25)

The unit of [1198701 1198702 119870







119869 (119896) =

1

2

119873119901

sum

119901=0

(119903 (119896 + 119901) minus 119910 (119896 + 119901))

2

=

1

2

(119877 (119896) minus 119884 (119896))

119879(119877 (119896) minus 119884 (119896)) =

1

2

119864

119879(119896) 119864 (119896)

(27)

where

119877 (119896) = [119903 (119896) 119903 (119896 + 1) sdot sdot sdot 119903 (119896 + 119873119901)]

119879

119884 (119896) = [119910 (119896) 119910 (119896 + 1) sdot sdot sdot 119910 (119896 + 119873119901)]

119879


119901)]

119879

(28)



119901is selected


119901is min(11987011198702 11987015) =


119901)]


119906 (119896) = 119906 (119896 minus 1) + Δ119906 (119896) = 119906 (119896 minus 1) + 120578

120597119884

119879(119896)

120597119906 (119896)

119864 (119896)

= 119906 (119896 minus 1) + 120578119862

119879(119896) 119864 (119896)

(29)

where

119862 (119896) =

120597119884

119879(119896)

120597119906 (119896)

=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

120597119910 (119896)

120597119906 (119896)

120597119910 (119896)

120597119906 (119896 + 1)

sdot sdot sdot

120597119910 (119896)

120597119906 (119896 + 119873119901)

120597119910 (119896 + 1)

120597119906 (119896)

120597119910 (119896 + 1)

120597119906 (119896 + 1)

sdot sdot sdot

120597119910 (119896 + 1)

120597119906 (119896 + 119873119901)

d

120597119910 (119896 + 119873119901)

120597119906 (119896)

120597119910 (119896 + 119873119901)

120597119906 (119896 + 1)

sdot sdot sdot

120597119910 (119896 + 119873119901)

120597119906 (119896 + 119873119901)

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(30)


5 10 15 20 25 30 35 401200

1220

1240

1260

1280

1300

Batch


Tem

pera

ture

of t

ube(

∘ C)


5 10 15 20 25 301200

1220

1240

1260

1280

1300

Batch


Tem

pera

ture

of t

ube(

∘ C)




119906 (119896) = 119906 (119896 minus 1) + 120578119862

119879(119896) 119864 (119896) (31)

where

119862 (119896) = [

120597119910 (119896)

120597119906 (119896)

120597119910 (119896 + 1)

120597119906 (119896)

sdot sdot sdot

120597119910 (119896 + 119873119901)

120597119906 (119896)

]

119879

(32)


EPC Annularfurnace

EMP

r(k) u(k)

+ +minus

minusy(k + p)

y(k)

e(k)

y(k)


PLC1 PLC2

Printer

PROFIBUS-DP1

PROFIBUS-DP2

PROFIBUS-DP3

Industrial Ethernet








1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 380

100200300400500600700800900

1000110012001300

Position

Tem

pera

ture

of t

ube(

∘ C)




6 Conclusion


5 10 15 20 25 301255126012651270127512801285129012951300

Batch

Predicted control

Tem

pera

ture

of t

ube(

∘ C)


5 10 15 20 25 301200

1220

1240

1260

1280

1300

Batch

PID control

Tem

pera

ture

of t

ube(

∘ C)





Acknowledgments



References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra




















F1

F2

F3

F4

F5F6

Number 1 Number 2

Num

ber3

Number4

Number 5

Number 6

Charging

Discharging

T1T2

T3

T4

T5

T6

T8T7T9


Preh

eat z

one






1 1199091


2 1199092


3 1199093


4 1199094


5 1199095


6 1199096


7 1199097


∘C

8 1199098


∘C

9 1199099


∘C

10 11990910


∘C

11 11990911


∘C

12 11990912


∘C

13 11990913


∘C

14 11990914


∘C

15 11990915


∘C





of lag time of1198701 1198702 119870




[1198701 1198702 119870

15]

= [58 55 51 46 42 36 61 56 51 45 39 35 52 58 60]

(25)

The unit of [1198701 1198702 119870







119869 (119896) =

1

2

119873119901

sum

119901=0

(119903 (119896 + 119901) minus 119910 (119896 + 119901))

2

=

1

2

(119877 (119896) minus 119884 (119896))

119879(119877 (119896) minus 119884 (119896)) =

1

2

119864

119879(119896) 119864 (119896)

(27)

where

119877 (119896) = [119903 (119896) 119903 (119896 + 1) sdot sdot sdot 119903 (119896 + 119873119901)]

119879

119884 (119896) = [119910 (119896) 119910 (119896 + 1) sdot sdot sdot 119910 (119896 + 119873119901)]

119879


119901)]

119879

(28)



119901is selected


119901is min(11987011198702 11987015) =


119901)]


119906 (119896) = 119906 (119896 minus 1) + Δ119906 (119896) = 119906 (119896 minus 1) + 120578

120597119884

119879(119896)

120597119906 (119896)

119864 (119896)

= 119906 (119896 minus 1) + 120578119862

119879(119896) 119864 (119896)

(29)

where

119862 (119896) =

120597119884

119879(119896)

120597119906 (119896)

=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

120597119910 (119896)

120597119906 (119896)

120597119910 (119896)

120597119906 (119896 + 1)

sdot sdot sdot

120597119910 (119896)

120597119906 (119896 + 119873119901)

120597119910 (119896 + 1)

120597119906 (119896)

120597119910 (119896 + 1)

120597119906 (119896 + 1)

sdot sdot sdot

120597119910 (119896 + 1)

120597119906 (119896 + 119873119901)

d

120597119910 (119896 + 119873119901)

120597119906 (119896)

120597119910 (119896 + 119873119901)

120597119906 (119896 + 1)

sdot sdot sdot

120597119910 (119896 + 119873119901)

120597119906 (119896 + 119873119901)

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(30)


5 10 15 20 25 30 35 401200

1220

1240

1260

1280

1300

Batch


Tem

pera

ture

of t

ube(

∘ C)


5 10 15 20 25 301200

1220

1240

1260

1280

1300

Batch


Tem

pera

ture

of t

ube(

∘ C)




119906 (119896) = 119906 (119896 minus 1) + 120578119862

119879(119896) 119864 (119896) (31)

where

119862 (119896) = [

120597119910 (119896)

120597119906 (119896)

120597119910 (119896 + 1)

120597119906 (119896)

sdot sdot sdot

120597119910 (119896 + 119873119901)

120597119906 (119896)

]

119879

(32)


EPC Annularfurnace

EMP

r(k) u(k)

+ +minus

minusy(k + p)

y(k)

e(k)

y(k)


PLC1 PLC2

Printer

PROFIBUS-DP1

PROFIBUS-DP2

PROFIBUS-DP3

Industrial Ethernet








1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 380

100200300400500600700800900

1000110012001300

Position

Tem

pera

ture

of t

ube(

∘ C)




6 Conclusion


5 10 15 20 25 301255126012651270127512801285129012951300

Batch

Predicted control

Tem

pera

ture

of t

ube(

∘ C)


5 10 15 20 25 301200

1220

1240

1260

1280

1300

Batch

PID control

Tem

pera

ture

of t

ube(

∘ C)





Acknowledgments



References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












of lag time of1198701 1198702 119870




[1198701 1198702 119870

15]

= [58 55 51 46 42 36 61 56 51 45 39 35 52 58 60]

(25)

The unit of [1198701 1198702 119870







119869 (119896) =

1

2

119873119901

sum

119901=0

(119903 (119896 + 119901) minus 119910 (119896 + 119901))

2

=

1

2

(119877 (119896) minus 119884 (119896))

119879(119877 (119896) minus 119884 (119896)) =

1

2

119864

119879(119896) 119864 (119896)

(27)

where

119877 (119896) = [119903 (119896) 119903 (119896 + 1) sdot sdot sdot 119903 (119896 + 119873119901)]

119879

119884 (119896) = [119910 (119896) 119910 (119896 + 1) sdot sdot sdot 119910 (119896 + 119873119901)]

119879


119901)]

119879

(28)



119901is selected


119901is min(11987011198702 11987015) =


119901)]


119906 (119896) = 119906 (119896 minus 1) + Δ119906 (119896) = 119906 (119896 minus 1) + 120578

120597119884

119879(119896)

120597119906 (119896)

119864 (119896)

= 119906 (119896 minus 1) + 120578119862

119879(119896) 119864 (119896)

(29)

where

119862 (119896) =

120597119884

119879(119896)

120597119906 (119896)

=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

120597119910 (119896)

120597119906 (119896)

120597119910 (119896)

120597119906 (119896 + 1)

sdot sdot sdot

120597119910 (119896)

120597119906 (119896 + 119873119901)

120597119910 (119896 + 1)

120597119906 (119896)

120597119910 (119896 + 1)

120597119906 (119896 + 1)

sdot sdot sdot

120597119910 (119896 + 1)

120597119906 (119896 + 119873119901)

d

120597119910 (119896 + 119873119901)

120597119906 (119896)

120597119910 (119896 + 119873119901)

120597119906 (119896 + 1)

sdot sdot sdot

120597119910 (119896 + 119873119901)

120597119906 (119896 + 119873119901)

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(30)


5 10 15 20 25 30 35 401200

1220

1240

1260

1280

1300

Batch


Tem

pera

ture

of t

ube(

∘ C)


5 10 15 20 25 301200

1220

1240

1260

1280

1300

Batch


Tem

pera

ture

of t

ube(

∘ C)




119906 (119896) = 119906 (119896 minus 1) + 120578119862

119879(119896) 119864 (119896) (31)

where

119862 (119896) = [

120597119910 (119896)

120597119906 (119896)

120597119910 (119896 + 1)

120597119906 (119896)

sdot sdot sdot

120597119910 (119896 + 119873119901)

120597119906 (119896)

]

119879

(32)


EPC Annularfurnace

EMP

r(k) u(k)

+ +minus

minusy(k + p)

y(k)

e(k)

y(k)


PLC1 PLC2

Printer

PROFIBUS-DP1

PROFIBUS-DP2

PROFIBUS-DP3

Industrial Ethernet








1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 380

100200300400500600700800900

1000110012001300

Position

Tem

pera

ture

of t

ube(

∘ C)




6 Conclusion


5 10 15 20 25 301255126012651270127512801285129012951300

Batch

Predicted control

Tem

pera

ture

of t

ube(

∘ C)


5 10 15 20 25 301200

1220

1240

1260

1280

1300

Batch

PID control

Tem

pera

ture

of t

ube(

∘ C)





Acknowledgments



References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










5 10 15 20 25 30 35 401200

1220

1240

1260

1280

1300

Batch


Tem

pera

ture

of t

ube(

∘ C)


5 10 15 20 25 301200

1220

1240

1260

1280

1300

Batch


Tem

pera

ture

of t

ube(

∘ C)




119906 (119896) = 119906 (119896 minus 1) + 120578119862

119879(119896) 119864 (119896) (31)

where

119862 (119896) = [

120597119910 (119896)

120597119906 (119896)

120597119910 (119896 + 1)

120597119906 (119896)

sdot sdot sdot

120597119910 (119896 + 119873119901)

120597119906 (119896)

]

119879

(32)


EPC Annularfurnace

EMP

r(k) u(k)

+ +minus

minusy(k + p)

y(k)

e(k)

y(k)


PLC1 PLC2

Printer

PROFIBUS-DP1

PROFIBUS-DP2

PROFIBUS-DP3

Industrial Ethernet








1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 380

100200300400500600700800900

1000110012001300

Position

Tem

pera

ture

of t

ube(

∘ C)




6 Conclusion


5 10 15 20 25 301255126012651270127512801285129012951300

Batch

Predicted control

Tem

pera

ture

of t

ube(

∘ C)


5 10 15 20 25 301200

1220

1240

1260

1280

1300

Batch

PID control

Tem

pera

ture

of t

ube(

∘ C)





Acknowledgments



References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 380

100200300400500600700800900

1000110012001300

Position

Tem

pera

ture

of t

ube(

∘ C)




6 Conclusion


5 10 15 20 25 301255126012651270127512801285129012951300

Batch

Predicted control

Tem

pera

ture

of t

ube(

∘ C)


5 10 15 20 25 301200

1220

1240

1260

1280

1300

Batch

PID control

Tem

pera

ture

of t

ube(

∘ C)





Acknowledgments



References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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