research article a novel fractional-order pid controller...

Research ArticleA Novel Fractional-Order PID Controller forIntegrated Pressurized Water Reactor Based onWavelet Kernel Neural Network Algorithm

Yu-xin Zhao1 Xue Du1 and Geng-lei Xia2

1 College of Automation Harbin Engineering University Harbin Heilongjiang 150001 China2National Defense Key Subject Laboratory of Nuclear Safety and Simulation Technology Harbin Engineering UniversityHarbin Heilongjiang 150001 China

Correspondence should be addressed to Xue Du duxue2012cc163com

Received 9 June 2014 Accepted 15 July 2014 Published 12 August 2014

Academic Editor Ligang Wu

Copyright copy 2014 Yu-xin Zhao 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

This paper presents a novel wavelet kernel neural network (WKNN) with wavelet kernel function It is applicable in online learningwith adaptive parameters and is applied on parameters tuning of fractional-order PID (FOPID) controller which could handle timedelay problem of the complex control system Combining the wavelet function and the kernel function the wavelet kernel functionis adopted and validated the availability for neural network Compared to the conservative wavelet neural network the mostinnovative character of the WKNN is its rapid convergence and high precision in parameters updating process Furthermore theintegrated pressurized water reactor (IPWR) system is established by RELAP5 and a novel control strategy combiningWKNN andfuzzy logic rule is proposed for shortening controlling time and utilizing the experiential knowledge sufficiently Finally experimentresults verify that the control strategy and controller proposed have the practicability and reliability in actual complicated system

1 Introduction

With the tightening supplies of energy and deterioration ofenvironment pollution in the globe the reliability cleanli-ness and security help the nuclear technology gain increasingtraction The majority of marine nuclear power plants adoptintegrated design Since vessels request strong mobility andoperational condition changes dramatically the nuclear reac-tor and system device need to accommodate themselves tothe radical load changing The nuclear reactor power controldominates the whole system of nuclear power plant anddecides the security and reliability of the global deviceHencethe operation control strategy and controller design shouldaccord with its particularities which is convenient for powercontrolling and adjusting and ensures the steady operationThe reactor is a time delay system that is similar to otherindustrial systemsThe time delay effect influences the systemperformance which possibly brings oscillation and evendestroys the instability [1 2] Thus it is practically significantto research the time delay control of nuclear reactor

In numerous problems encountered in modern controlthe time delay issue is of central importance and also adifficult point to elaborate There is an increasing interest intime delay problem of industrial system from scientists andengineers Classical control theory utilizes plenty of Smithcontrol Dahlin control and so on [3ndash5] During recentyears some novel control algorithms also have obtainedachievements of time delay systems [6 7] for instanceYang and Zhang [8] studied a robust 119867

infinnetwork control

method for T-S fuzzy systems with uncertainty and timedelay Besides Shi and Yu proposed the 119867

2119867infin

controlfor the two-mode-dependent robust control synthesis ofnetworked control systems with random delay [9] Moreovera class of uncertain singular time delay systems is controlledby sliding mode [10 11] and an optimal sliding surface isdesigned for sliding model control in time delay uncertainsystems [12]The industrial system especially the large-scaledcomplex industrial systems possesses the feature of processdelay The control of these systems depends on both currentstate and past state which makes the control issue difficult

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 312541 13 pageshttpdxdoiorg1011552014312541

2 Mathematical Problems in Engineering

As a mature controller with strong availability PID andthe controllers have been applied in industrial processes forsophisticated dynamic systems which have large scale Theprimary reason is its relatively simple structure that couldbe easily comprehended and implemented [13ndash15] In recentyears considerable interests in time delay have been arousedin fractional-order PID controller andmany researchers havepresented beneficial methods to design stabilizing fractional-order PID (FOPID) controllers for varying time delay system[16ndash19] This paper adopts neural network (NN) to tunethe FOPID online to control the nuclear reactor powerwhich keeps the coolant average temperature constant onthe condition of load changing NN is an effective methodfor PID parameters tuning and the wavelet neural network(WNN) improves the efficiency of NN by transformingthe nonlinear excitation function to the wavelet functionHowever WNN still follows the gradient descent methodto adjust weights which inevitably causes the problemsof deficient generalization performance ill-posedness andoverfitting Motivated by the drawback above it should takeadvantage of the algorithm core LMS to improve WNNBoth LMS and KLMS are derived from mean squared errorcost function [20ndash22] LMS algorithm is famous for its easycomprehending and implementing but has no ability tobe utilized in nonlinear domain With the wide spread ofkernel theory the novel LMS algorithm with kernel trick hasdrawn considerable attention to learning machines includingneural network [23ndash26] Recently kernel functions (suchas Gaussian kernel) are applied in support vector machine(SVM) mostly which projects the linear nonseparable datato another space for being separable in the new space RBFnetwork is a sort of NNwhich useGaussian kernel frequentlyBut this kernel could not approach the target appropriatelywhen it encounters more complex signal in the calculationprocess There are some literatures indicating that Gaussiankernel in RBF network and SVM could not generate a set ofcomplete bases in the subspace ideally by translation [27]Some potential wavelet functions could be transformed askernel functions which are competitive to produce a set ofcomplete bases in 119871

2(119877) space (square and integrable space)

by stretching and translatingMeanwhile RBF network needsto identify the model of object and obtain the Jacobianinformation when it tunes the network weights and it ishard to gain the Jacobian information of complex industrialsystem But the conventional WNN could tune the PIDcontroller without such information and is more appropriateto be the parent body for improvement Thus this paperproposes a novel neural network based on wavelet kernelfunction and combines with fuzzy logic rules to achieve theonline tuning of FOPID Experiment results validate that thecontrol strategy and controller design improve the accuracyand speed which could control the nuclear reactor poweroperation and enhances the overall efficiency

The next content is composed of six sections Section 2provides a short introduction to LMS algorithm and kernelLMS algorithm Section 3 presents the related theory ofwavelet kernel function and proposes the method of waveletkernel neural network including the discussion of adaptiveparameters The model of nuclear reactor is structured by

RELAP 5 program in Section 4 Then the control strategyand the fractional-order PID controller are designed withfuzzy logic and wavelet kernel neural network in Section 5In Section 6 experiments illustrate the effectiveness of thewavelet kernel neural network and the fractional-order PIDcontroller As some controllers are faced with the problemof losing efficacy in practice Section 6 also emulates andanalyzes the control process in the basis of reactor modelFinally conclusion of this paper has been described inSection 7

2 The Description of Kernel LMS Algorithm

In this section the algorithm of kernel LMS is introduced as ashow review for wavelet kernel neural network Kernel LMSis an approach to perform the LMS algorithm in the kernelfeature space In the LMS algorithm the hypothesis space1198671is composed by all the linear operators on 119880 (mapping

119908 119880 rarr 119877) [28] which is embedded in a Hilbert spaceand the linear operator represents standard inner productThe LMS algorithm minimizes the total instantaneous errorof the system output

119864 (119899) =1

2sum e2 (119899) (1)

where e(119899) = norm(119899) minus x(119899)w(119899) norm is the desired valuew(119899) is the weight vector x(119899) is the input vector and 119899 is theinstant Using the straightforward stochastic gradient descentupdate rule the weight vector w is approximated based oninput vector x

w (119899 + 1) = w (119899) + 120578e (119899) x (119899) (2)

where 120578 is the step-size By training step by step the weightwis updated as a linear combination of the previous and presentinput vector

This LMS algorithm learns linear patterns satisfactorilywhile it could not approach the same sound effect in non-linear pattern Motivated by the drawback Pokharel withother scholars extended the LMS algorithm to RKHS by thekernel trick [28 29] Kernel methods map the input into ahigh dimensional space (HDS) [30] And any kernel functionbased on Mercerrsquos theorem has the follow mapping [31]

119896 (119909 1199091015840

) = ⟨120601 (119909) 120601 (1199091015840

)⟩ = 120601119879

(119909) 120601 (1199091015840

) (3)

When the input in (2) consists of 120601(x(1)) 120601(x(2)) 120601(x(119899)) but not x(1) x(2) x(119899) 120601(x(119899)) is the infinitefeature vector transformed from input vector x(119899) The LMSalgorithm is no longer applicable and the total instantaneouserror of the system output is as follows

119864 =1

2sum(norm minus 120601 (x (119899))w (119899))2 (4)

where 120601(x(119899)) is matrixes of input vector norm is desiredvalue vector and w(119899) is weight vector in RKHS

Then build the hypothesis that w(0) = 0 for convenienceand (2) could convert to

w (119899) = 120578119899minus1

sum

119894=0

e (119894) 120601 (x (119894)) (5)

Mathematical Problems in Engineering 3

and the final output in 119899 step updating of the learningalgorithm is

z (119899) = 120578119899minus1

sum

119894=0

e (119894) ⟨120601 (x (119894)) 120601 (x (119899))⟩ = 120578e119879 (119899 minus 1) k (6)

and k = [119896(x(1) x(119899)) 119896(x(2) x(119899)) 119896(x(119899 minus 1) x(119899))]From (4)ndash(6) the KLMS algorithm remedies the defectof the LMS algorithm of its relatively small dimensionalinput space and improves the well-posedness study in theinfinite dimensional space with finite training data [28] Asthe application of adaptive filter theory the neural networkcould take advantage of the KLMS algorithm which endowsthe learning of neural network with more efficiency andreliability

3 Wavelet Kernel Neural Network

Following the principle above this paper acquires the waveletkernel function by Mercer theorem and proposes a morecompetent method to update parameters on basis of con-ventional WNN With the kernel LMS above the waveletkernel neural network is proposed in this section on the basisof wavelet kernel function and conventional wavelet neuralnetwork First the wavelet kernel function is constructedwhich is admissible neural network kernel Second thewavelet kernel neural network is proposed including neuronexplanation and weight updating Then the modulationsof adaptive variable step-size and wavelet kernel width arediscussed in Sections 33 and 34

31 The Wavelet Kernel Function The wavelet analysis isexecuted to express or approximate functions or signalsgenerated family functions given by dilations and translationsfrom the mother wavelet 120595

119886119887(119909) isin 119871

2(119877) Hypothesize

that 1198711(119877) is linear integrable space and 1198712

(119877) is squareintegrable space When the wavelet function 120595(119909) isin 1198711(119877) cap1198712

(119877) ((0) = 0) the wavelet function group could bedefined as

120595119886119887(119909) = 119886

minus12

120595(119909 minus 119887

119886) (7)

where 119909 119886 119887 isin 119877 119886 is a dilation factor and 119887 is a translationfactor And 120595(119909) satisfy the condition [32 33]

119862120595= int119877

1003816100381610038161003816 (119908)1003816100381610038161003816

2

|119908|119889119908 lt infin (8)

where (119908) is the Fourier transform of 120595(119909) For a commonmultidimensional wavelet function if the wavelet function ofone dimension is 120595(119909) which satisfies the condition (8) theproduct of one-dimensional (1D) wavelet functions could bedescribed by tensor theory [29 31]

120595119889(119909) =

119889

prod

119894=1

120595 (119909119894) (9)

where x = (1199091 119909

119889) isin 119877119889

The conception above is the foundation of wavelet anal-ysis and theory Note that the wavelet kernel function forneural network is a special kernel function and comes fromwavelet theory it would be the horizontal function 119896(119909 1199091015840) =119896(119909 minus 119909

1015840

) and satisfied the condition of Mercer theorem [34]which could be summarized as follows

Lemma 1 A continuous symmetric kernel is 119896(119909 1199091015840) and 119909is bounded likewise for 1199091015840 Then 119896(119909 1199091015840) is the valid kernelfunction for wavelet neural network if and only if it holds forany nonzero function ℎ(119909)which makes int

119877119889ℎ2

(120585)119889120585 lt infin andthe following condition will be satisfied

∬119877119889otimes119877119889

119896 (119909 1199091015840

) ℎ (119909) ℎ (1199091015840

) 1198891199091198891199091015840

ge 0 (10)

With regard to the horizontal floating function thereis an analogous theory for support vector machine (SVM)[35] Allowing for the affiliation between SVM and NN [34]an approved horizontal floating kernel function of SVM iseffective in NN When 120595(119909) is a mother wavelet 119886 is adilation factor and 119887 is a translation factor 119909 119886 isin 119877the horizontal floating translation-invariant wavelet kernelscould be expressed as [36]

119896 (119909 1199091015840

) =

119889

prod

119894=1

120595(119909119894minus 1199091015840

119894

119886119894

) (11)

Since the number of wavelet kernel functions whichsatisfy the horizontal floating function condition [37] is notmuch up to now the neural network in this paper adopts anexistent wavelet kernel function Morlet wavelet function

120595 (119909) = cos (1199080119909) exp(minus119909

2

2) (12)

Then the Morlet wavelet kernel function is defined as

119896 (119909 1199091015840

)

=

119889

prod

119894=1

cos(1199080(119909119894minus 1199091015840

119894

119886119894

)) exp(minus(119909119894minus 1199091015840

119894)2

21198862

119894

)

(13)

Figure 1 displays the Morlet wavelet kernel function inthe four-dimensional space As a continuously differentiablefunction the derived function of Morlet kernel is illustratedin Figure 2

32 The Wavelet Kernel-LMS Neural Network The neuralnetwork on basis of wavelet kernel function focuses on thematter relating to the performance of a multilayer perceptiontrainedwith the backpropagation algorithm by adding KLMStheory to wavelet neural network This paper utilizes themultilayer perception (MLP) fully connected structure andchooses three layers for understandingThe network consistsof input layer 119868 hidden layer 119869 and output layer 119870 Theactivation functions in hidden layer and output layer are


0

2

4

minus4

minus2

Morlet kernel function

minus2 0 2 4

x

y

Figure 1 Morlet kernel function

minus2

0

2

4

x

y

minus8

minus6

minus4

minus2 0 2 4

6

8Morlet kernel derived function

Figure 2 Morlet kernel derived function

wavelet kernel function 120593(119909) and sigmoid function 119891(119909)respectively Weights between input layer and hidden layerare regarded as w

119869119868 and the one between hidden layer and

output layer is w119870119869

In the WKNN hypothesize that the input set is x119868(1)

x119868(2) x

119868(119899) practical output set and norm output set

are z119870(1) z119870(2) z

119870(119899) and norm

119870(1)norm

119870(2)

norm119870(119899) respectively and then the cost function is defined

as 120576(119899)

120576 (119899) =1

2e2 (119899) (14)

where e(119899) = norm(119899) minus z119870(119899)

As the hidden layer 119869 the signal of single neuron 119895 isprovided by input layer and the induced local field k

119869and

output of hidden layer x119869could be displayed as

k119869(119899) =

119868

sum

119894=1

119908119895119894(119899) 119909119894(119899) = x119879

119868(119899)w119869119868(119899)

x119869(119899) = 120593 (k

119869(119899))

(15)

where k119869(119899) is weights 119908

119894119895(119899) is the input of input layer

neuronsw119869119868(119899) and x

119868(119899) are thematrix or vector constituted

by119908119894119895(119899) and 119909

119894(119899) 119868 represents the sum of the neurons in the

input layer and 120593(sdot) is the wavelet kernel function in 119869 layerAs the output layer 119870 the signal of single neuron 119896 is

provided by hidden layer and the induced local field k119870and

output of output layer z119870could be displayed as

k119870(119899) =

119869

sum

119895=1

119908119896119895(119899) 119909119895(119899) = x119879

119869(119899)w119870119869(119899) (16)

z119870= 119891 (k

119870(119899)) = 119891 (120593

119879

(k119869(119899))w

119870119869(119899)) (17)

where119908119896119895(119899) is weights119909

119895(119899) is the input of hidden layer neu-

ronsw119870119869(119899) and x

119869(119899) are the matrix or vector constituted by

119908119896119895(119899) and 119909


input layer and 119891(sdot) is the sigmoid function in119870 layerSimilar to theWNN algorithm this paper achieves a gra-

dient search to solve the optimum weight For convenienceit chooses w

119870119869(0) to be zero Then based on (17) the weight

matrixw119870119869(119899) between output layer and hidden layer is given

as follows

w119870119869(119899) = 120578

119870(119899) 120593 (k

119869(119899))

119899minus1

sum

119901=0

e (119901) 1198911015840 (120593119879 (k119869(119901))w

119870119869(119901))

(18)

where 120578119870(119899) is adaptive variable step-size and the method to

find appropriate step-size will be illustrated in the followingsubsection By the kernel trick and (17) z

119870(119899) could be

determined as

z119870(119899) = 119891(120578

119870(119899)

119899minus1

sum

119901=0

120572119901119896 (k119869(119901) k

119869(119899)))

= 119891 (120578119870(119899)120572119879

(119899 minus 1) k119870)

(19)

where 120572(119899) = e(119899)1198911015840(120578119870(119899 minus 1)120572

119879

(119899 minus 1)k119870)

Then the weight matrix w119869119868(119899) between hidden layer and

input layer is obtained on the basis of (18) as follows

w119869119868(119899) = 120578

119869(119899) x119868(119899)

119899minus1

sum

119901=0

1205931015840

(x119879119868(119901)w119869119868(119901))

timessum

119870

e (119901) 1198911015840 (120593119879 (k119869(119901))w

119870119869(119901))w

119870119869(119901)

(20)


and the output of hidden neuron could be determined as

x119869= 120593 (⟨w

119869119868(119899) x119868(119899)⟩)

= 120593(120578119869(119899)

119899minus1

sum

119901=0

120573 (119901) 119896119869(x119868(119901) x

119868(119899)))

= 120593 (120578119869(119899)120573119879

(119899 minus 1) k119869)

(21)

where 120573(119899) = 1205931015840(120578119895(119899)120573119879

(119899 minus 1) k119869)120572(119899 minus 1)w

119870119869(119899 minus 1)

33 The Updating of Adaptive Variable Step According tothe algorithm presented above (see Section 2) it stresses thatthe adaptive variable step could be regarded as a significantcoefficient with the ability to influence the adaptivity andeffectivity of the algorithm The stability and convergenceof the algorithm are influenced by step-size parameter andthe choice of this parameter reflects a compromise betweenthe dual requirements of rapid convergence and small mis-adjustment A large prediction error would cause the step-size to increase to provide faster tracking while a smallprediction error will result in a decrease in the step-sizeto yield smaller misadjustment [37] Therefore this paperimproves the NLMS algorithm derived from LMS algorithmwhich is propitious to multidimensional nonlinear structureand avoids the limitation of linear structure

NLMS focuses on the main problem of LMS algorithmrsquosconstant step-size which designs an adaptive step-size toavoid instability and picks the convergence speedup

119908 (119899 + 1) = 119908 (119899) +1

⟨119906 (119899) 119906 (119899)⟩119890 (119899) times 119906 (119899) (22)

where 1⟨119906(119899) 119906(119899)⟩ is regarded as the variable step-sizecompared with the LMS algorithm But the NLMS has littleability to perform in the nonlinear space By the kernel trickthe kernel function could not only improve thewavelet neuralnetwork but also update the step-size of neural networkself-adaptively As what (17) reveals z

119870is a consequence

of nonlinear function 119891(sdot) Since 119891(sdot) is continuously dif-ferentiable this function could be matched by innumerablelinear function following the principle of the definition ofdifferential calculus

Theorem 2 The weights updating satisfies the NLMS condi-tion in the linear integrable space 1198711(119877) as follows

119908 (119899 + 1) = 119908 (119899) + 120578 (119899) times 119890 (119899) times 119906 (119899) (23)

where 120578(119899) = 1⟨119906(119899) 119906(119899)⟩ is the adaptive variable step-sizeThen weights are expanded to 119870 dimension matrix w

119870(119899) and

adjusted as

w119870(119899 + 1) = w

119870(119899) + 120578

119870(119899) times 119891

1015840

(w119870(119899) times u

119870(119899))

times e119870(119899) times u

119870(119899)

(24)

where the nonlinear function119891(sdot) is continuously differentiableeverywhere the 120578

119870(119899) will have119870 elements and each step-size

120578119896(119899) (119896 = 1 2 119870) is given as

120578119896(119899) =

1

1198911015840 (w119870(119899) times u

119870(119899)) ⟨u

119870(119899) u

119870(119899)⟩

(25)

Proof Hypothesize that 120576 is small enough and linear function119865(119899) = 120581(119899)u

119870(119899)+120574(119899) exists which has the relationship that

119865(119899) minus 119891(119899) lt 120576 in the closed interval [119886 119887] therefore regardthat 119865(119899) asymp 119891(119899) and 1198911015840(119899) asymp 120581 are satisfied

Consider the equations

ℎ (120578119870(119899)) = norm minus 119891 (w

119870(119899) times u

119870(119899))

asymp norm minus 120581 (119899) (w119870(119899) times u

119870(119899)) minus 119887 (119899)

= E119879119870(119899) [119868 minus 120578

119870(119899) 1205812

(119899) u119879119870(119899) u119870(119899)]E

119870

119892 (120578119870(119899)) =

1

2ℎ2

(120578119870(119899))

(26)

where 120578119870(119899) = [

1205781(119899) sdotsdotsdot 0

d

0 sdotsdotsdot 120578119870(119899)

] and minimize the functions

119892(120578119870(119899)) and 1198921015840(120578

119870(119899)) = 0 and then 120578

119870(119899) (119896 = 1 2 119870)

could be inferred as follows

120578119870(119899) =

[[

[

1205781(119899) sdot sdot sdot 0

d

0 sdot sdot sdot 120578119870(119899)

]]

]

(27)

120578119896(119899) =

1

1205812

119896(119899) ⟨u

119870(119899) u

119870(119899)⟩

(28)

Equations (27) and (28) reveal that the 120578119870(119899) is indepen-

dent of the linear function intercept 120574(119899) and only relatesto the input and slope 120581

119896(119899) Motivated by the drawback of

slight error 120576 from 119865(119899) minus 119891(119899) the 1205812119896(119899) in (28) is replaced

by 1198911015840(w119870(119899) times u

119870(119899)) which improves the accuracy and

completes the proof

On the basis of the discussion above (22) could berepresented as a special situation by Theorem 2 when theslope of function 119891(sdot) is 1 Theorem 2 expands the NLMS tothe multidimensional nonlinear space from the linear spaceand broadens the application of NLMS Compared with thevariable step-size in (22) (27) and (28) could be defined as theadaptive variable step-size of NLMS in the extended space

When ⟨u119870(119899) u119870(119899)⟩ is too small the problem caused by

numerical calculation has to be considered To overcome thisdifficulty (28) is altered to

120578119896(119899) =

1

1205812

119896(119899) ⟨u

119870(119899) u

119870(119899)⟩ + 120590

2

V (29)

where 1205902V gt 0 which is set as a pretty small value The weightsupdating in (20) is more complicated than that betweenoutput layer and hidden layer Because the updating in (20)is based on the output layer meanwhile the existence ofsum119870e(119901)1198911015840(120593119879(k

119869(119901))w

119870119869(119901))w

119870119869(119901) helps every neuron of

output layer have the relationship with the updating processbetween 119894th neuron of hidden layer and the previous layerin other words every neuron of output layer contributes to120578119869(119899)Therefore this paper adopts the same step-size in 120578

119869(119899)


Input Input signal x119868Ideal output norm

Output Observe output z119896(119899)Weights w(119899)

Initialize The number of layers in WNN structure119873layerThe number of neurons in every layer119872

11198722

The activation function of every layer 1198911(sdot) 1198912(sdot)

Loop count119873119906119898Begin

Initialize weights w(0)RepeatWavelet kernel neural network output z

119870(119899)

Output layer step-size 120578119870(119899)

Hidden layer step-size 120578119869(119899)

Synaptic weight updating w119870119869


Wavelet kernel Width modulating 119886(119899)Until 119899 gt 119873119906119898

End

Algorithm 1 Algorithm process

based on120578119870(119899) and the elements 120578

119895(119899) of120578

119869(119899) could be given

as follows

120578119895(119899) =

1

119870

119870

sum

119896=1

120578119896(119899) (30)

34 The Modulation of Wavelet Kernel Width The goalof minimizing the mean square error could be achievedby gradient search If kernel space structure cost functionis derivable the wavelet kernel of neural network couldalso be deduced by gradient algorithm In consideration ofconventional wavelet neural network the gradient searchmethod could be utilized for updating wavelet kernel weightdue to the derivability of the wavelet kernel function Besidesthe synaptic weights among various layers of network thereis another coefficientmdashdilation factor 119886mdashin (11) which alsoinfluences the mean square error With respect to the costfunction (14) the dilation factor 119886 is corrected by

119886 (119899 + 1) = 119886 (119899) minus 1205831015840

nabla119886(119899)

(e2 (119899)) (31)

where 1205831015840 is the constant step-size and nabla119886(119899)

is the partialderivative function that is gradient function According tothe chain rule of calculus this gradient is expressed as

nabla119886(119899)

= e (119899) ( 120597z119870

120597119886 (119899)) = e (119899) ( 120597z

119870

120597119906 (119899)

120597119906 (119899)

120597119886 (119899))

120597119906 (119899)

120597119886 (119899)=120597119906 (119899)

120597119896 (119899)

120597119896 (119899)

120597119886 (119899)= 120578119870(119899)

119899minus1

sum

119901=0

120572 (119901)119867119896 (119909 (119901) 119909 (119899))

(32)

where 119867 = 120597119896120597119886 and 119906 = 120578119870(119899)sum119899minus1

119901=0120572(119901)119896(119909(119901) 119909(119899))

Then nabla119886(119899)

could be represented as

nabla119886(119899)

= 120578119870(119899) e (119899) 1198911015840 (119906)

119899minus1

sum

119901=0

120572 (119901)119867119896 (119909 (119901) 119909 (119899)) (33)

Allowing for (19) and (33) the updating formulation ofdilation factor 119886(119899) is

119886 (119899 + 1) = 119886 (119899) minus 120583 (119899)120572 (119899)

119899minus1

sum

119901=0

120572 (119901)119867119896 (119909 (119901) 119909 (119899))

(34)

where 120583(119899) = 1205831015840120578119870(119899)

The WKNN process is presented as Algorithm 1 as fol-lows

4 The Model Design of IPWR

In order to reduce the volume of the pressure vessel theintegration reactor generally uses compactOTSG Since thereis no separator in the steam generator the superheated steamproduced by OTSG could improve the thermal efficiencyof the secondary loop systems Compared with naturalcirculation steam generator U-tube the OTSG is designedwith simpler structure but less water volume in heat transfertube Since it is difficult to measure the water level in OTSGespecially during variable load operation the OTSG needsa high-efficiency control system to ensure safe operationPrimary coolant average temperature is an important param-eter to ensure steam generator heat exchange The controlscheme with a constant average temperature of the coolantcan achieve accurate control of the reactor power and avoidgeneration of large volume coolant loop changes

In the integrated pressurized water reactor (IPWR) sys-tem the essential equation of thermal-hydraulic model islisted as follows

(1) Mass continuity equations

120597

120597119905(120572119892120588119892) +

1

119860

120597

120597119909(120572119892120588119892V119892119860) = Γ

119892

120597

120597119905(120572119891120588119891) +

1

119860

120597

120597119909(120572119891120588119891V119891119860) = Γ

119891

(35)


(2) Momentum conservation equations

120572119892120588119892119860

120597V119892

120597119905+1

2120572119892120588119892119860

120597V2119892

120597119909

= minus120572119892119860120597119875

120597119909+ 120572119892120588119892119861119909119860 minus (120572

119892120588119892119860) FWG (V

119892)

+ Γ119892119860(V119892119868minus V119892) minus (120572

119892120588119892119860) FIG (V

119892minus V119891)

minus 119862120572119892120572119891120588119898119860[

120597 (V119892minus V119891)

120597119905+ V119891

120597V119892

120597119909minus V119892

120597V119891

120597119909]

120572119891120588119891119860

120597V119891

120597119905+1

2120572119891120588119891119860

120597V2119891

120597119909

= minus120572119891119860120597119875

120597119909+ 120572119891120588119891119861119909119860 minus (120572

119891120588119891119860) FWF (V

119891)

minus Γ119892119860(V119891119868minus V119891) minus (120572

119891120588119891119860) FIF (V

119891minus V119892)

minus 119862120572119891120572119892120588119898119860[

120597 (V119891minus V119892)

120597119905+ V119892

120597V119891

120597119909minus V119891

120597V119892

120597119909]

(36)

(3) Energy conservation equations

120597

120597119905(120572119892120588119892119880119892) +

1

119860

120597

120597119909(120572119892120588119892119880119892V119892119860)

= minus119875

120597120572119892

120597119905minus119875

119860

120597

120597119909(120572119892V119892119860) + 119876

119908119892

+ 119876119894119892+ Γ119894119892ℎlowast

119892+ Γ119908ℎ1015840

119892+ DISS

119892

120597

120597119905(120572119891120588119891119880119891) +

1

119860

120597

120597119909(120572119891120588119891119880119891V119891119860)

= minus119875

120597120572119891

120597119905minus119875

119860

120597

120597119909(120572119891V119891119860) + 119876

119908119891

+ 119876119894119891minus Γ119894119892ℎlowast

119891minus Γ119908ℎ1015840

119891+ DISS

119891

(37)

(4) Noncondensables in the gas phase

120597

120597119905(120572119892120588119892119883119899) +

1

119860

120597

120597119909(120572119892120588119892V119892119883119899119860) = 0 (38)

(5) Boron concentration in the liquid field

120597120588119887

120597119905+1

119860

120597 (120588119887V119891119860)

120597119909= 0 (39)

(6) The point-reactor kinetics equations are

119889119899

119889119905=120588 minus 120573

119897119899 +

6

sum

119894=1

120582119894119862119894+ 119902

119889119862119894

119889119905=120573119894

119897119899 minus 120582119894119862119894

(40)

The RELAP5 thermal-hydraulic model solves eight fieldequations for eight primary dependent variablesTheprimary

dependent variables are pressure (119875) phasic specific internalenergies (119880

119892119880119891) vapor volume fraction (void fraction) (120572

119892)

phasic velocities (V119892 V119891) noncondensable quality (119883

119899) and

boron density (120588119887)The independent variables are time (119905) and

distance (119909) The secondary dependent variables used in theequations are phasic densities (120588

119892 120588119891) phasic temperatures

(119879119892 119879119891) saturation temperature (119879

119904) and noncondensable

mass fraction in noncondensable gas phase (119883119899119894) [38] The

point-reactor kinetics model in RELAP5 computes both theimmediate fission power and the power from decay of fissionfragments

5 The Control Strategy andFOPID Controller Design

New control law is demanded in nuclear device [39] Whenthe nuclear power plant operates stably on different con-ditions the variation of main parameters in primary andsecondary side is indeterminate The operation strategyemploys the coolant average temperature 119879av to be constantwith load changing The WKNN fuzzy fractional-order PIDsystem achieves coolant average temperature invariability bycontrolling reactor power system (Figure 4)

The five parameters including three coefficients andtwo exponents are acquired in two formsmdashfuzzy logic andwavelet kernel neural network The idea of this methodcombination is born from the fact that the power level ofnuclear reactor tends to alter dramatically in a short while andthen maintain slight variation within a certain period Dueto the boundedness of the WKNN excitation function theoutput range of FOPIDcoefficients could not bewide enoughThe initial parameters of FOPID are set byWKNN and fuzzylogic Moreover the exponents of FOPID have no need tovary with the small changing As a consequence the fuzzylogic rule is executed when the reactor power transformsremarkably to tune the five parameters of FOPID and theWKNN retains updating and tuning of the coefficients basedon the fuzzy logic all the while The process is illustrated inFigure 5

Based on theoretical analysis and design experiencefuzzy enhanced fractional-order PID controller is adoptedto achieve the reactor power changing with steam flowand maintain the coolant average temperature 119879av constantPower control system regards the average temperature devi-ation and steam flow as the main control signal and assistantcontrol signal respectively Since the thermal output ofreactor is proportional to the steamoutput of steamgeneratorthe consideration of steam flow variation could calculate thepower need immediately which impels the reactor powerto follow secondary loop load changing and improves themobility of equipment When the load alters reactor powercould be determined by the following equation

1198990= 1198701119866119904+ 119870119875119901Δ119879119898(119899) + 119870

119875119894119904120582

119899

sum

119895=0

119902119895Δ119879119898(119899 minus 119895)

+ 119870119875119889119904minus120583

119899

sum

119895=0

119889119895Δ119879119898(119899 minus 119895)

(41)


where 119902119895= (1 minus ((1 + 120582)119895))119902

119895minus1 119889119895= (1 minus ((1 minus 120583)

119895))119889119895minus1

119904 is time step 1198701is conversion coefficient 119870

119875119901 119870119875119894

and 119870119875119889

are design parameters 119866119904is new steam flow of

steam generator and Δ119879119898is the coolant average temperature

deviation Consider

Δ119879119898= 119879(0)

119898minus 119879119898 (42)

where 119879(0)119898

is norm value and 119879119898is measuring temperature

119879119898=119879119900+ 119879119894

2 (43)

where 119879119900and 119879

119894are the coolant temperature of core inlet

and outlet respectively The signal from signal processor isgenerated by

Δ119899 = 1198990minus 119899 (44)

Following the fuzzy logic principle the fuzzy FOPID con-troller considers the error and its rate of change as incomingsignals It adjusts 119870

119875119901minus119865 119870119875119894minus119865

119870119875119889minus119865

120582 and 120583 online andproduces them with the fractional factors satisfying condi-tions 0 lt 120582 120583 lt 2 The coordination principle is establishingthe relationship among 119890 119890119888 and parameters The fuzzy sub-set of fuzzy quantity chooses NBNMNSZOPSPMPBand the membership functions adopt normal distributionfunctions Larger 119870

119875119901minus119865 smaller 119870

119875119889minus119865 and 120583 could obtain

rapider system response and avoid the differential overflowby initial 119890 saltus step when 119890 is large Then adopt smaller119870119875119894minus119865

and 120582 to prevent exaggerated overshooting When 119890 ismedium 119870

119875119901minus119865should reduce properly meanwhile 119870

119875119894minus119865

119870119875119889minus119865

and 120582 should be moderate to decrease the systemovershooting and ensure the response fast If 119890 is small thelarger119870

119875119901minus119865119870119875119894minus119865

and 120582 have the ability to fade the steady-state error At the same time moderate 119870

119875119889minus119865and 120583 keep

output away from vibration around the norm value andguarantee anti-interference performance of the system Formore information one could refer to the literature [40]

The WKNN proposed above in Section 3 has beenadopted to improve the performance of FOPID parametertuning and three-layer structure has been applied TheWKNN input and output could be displayed as follows

x1(119899) = rin (119899)

x2(119899) = yout (119899)

x3(119899) = rin (119899) minus yout (119899)

x4(119899) = 1

z1(119899) = 119870

119875119901minus119882(119899)

z2(119899) = 119870

119875119894minus119882(119899)

z3(119899) = 119870

119875119889minus119882(119899)

(45)

As the boundedness of sigmoid function WKNN outputparameters for enhanced PID are maintained in a range of[minus1 1] To solve this problem the expertise improvementunit is added between WKNN and power require counter

x1

x2

xI

Hidden layer J

Input layer I

Output layer K

wIJ wKJ

zKe(n)

norm

f(middot)

120593(middot)

Figure 3 The structure of WKNN

in Figure 4 which corrects parameters order of magnitudesThis control strategy overcomes the limitation problemofNNactivation function and takes advantage of a priori knowledgefrom the researchers in reactor power domain which endowsthe algorithm with reliability and simplifies the fuzzy logicrules setting to reduce the control complexity

6 Simulation Results and Discussion

In this section simulations ofWKNNandWKNN fractional-order PID controller have been carried out aiming to examineperformance and assess accuracy of the algorithm proposedin the previous sections and it compares experimentally theperformance of the WKNN algorithm and WNN algorithmTo display the results clearly simulations are divided intotwo parts One is achieved as numerical examples in Matlabprogramwhich includesWKNNalgorithm and its fractional-order PID application in time delay system Another iswritten in FORTRAN language and connected with RELAP5 which implements the reactor power controlling

61 Numerical Examples To demonstrate the performance ofWKNNalgorithm two simulations are conducted comparingwith the WNN algorithm The structure of neural networkcould be consulted in Figure 3 and the initial weights aregenerated randomlyThe first experiment shows the behaviorof WKNN and WNN in a stationary environment of whichnorm output is 06 (Figures 5 and 6) And the second oneoperates in the variable time process as 03 sin(002119905) + 05(Figures 7 and 8)

Figures 6ndash9 adopt WKNN and WNN to calculate theregression process in the constant and time varying environ-ment respectively And both methods bring relatively largesystem error in initial period In Figures 6 and 7 the constantnorm output and straight regression process emphasize thefaster convergence of algorithms In Figures 8 and 9 the timevarying impels the regression process more difficult than theprevious experiment Firstly there is overshoot from 0 s to50 s in theWNN which impels the error ofWNN to decreaseslowly When the error of WNN starts to reduce the oneof WKNN has approached zero Secondly comparing thetwo error curves the phenomenon could not be ignored inwhich the WKNN curve has sharp points while the WNNcurve has the relative smoothness Thirdly although theinitial parameters have randomness the regression process isperiodical The figures imply that the convergence rate and


de

dt

de

dt

Temperaturecomparer

Power requirecounter

Signalprocessor Actuator Nuclear

reactor OTSG

Fuzzy logicrules

KWNN

Expertiseimprovement

T0m

T0m

Tm

Tm

Tm

e

e

eTo

Ti

n0

120588Δn

Gs

120582

KPpminusW

KPiminusW

KPdminusW

KPpminusW

KPiminusW

KPdminusW

KPpminusF

KPiminusF

KPdminusF

120583

intedt

Figure 4 Control system of nuclear reactor power

WKNN

If the 1st loop

FOPID

Fuzzy logic

KPpminusW

KPiminusW

KPdminusW

KPpminusF

KPiminusF

KPdminusF

120582120583

KPpminusF

KPiminusF

KPdminusF

120582120583

Tm T0m error If error gt 1205751

dedt gt 1205752

Figure 5 The control strategy of FOPID

regression quality of WKNN are better than that of WNNand initial condition brings no benefit to WKNN Fromthe argument above it indicates preliminary that WKNNalgorithm reduces the amplitude of error variation and hasbetter performance than WNN However it remains to beverified whether or not the WKNN is more appropriatefor FOPID controller tuning in time delay system Then inthis paper the algorithm proposed is designed to examine

the effectiveness of tuning the FOPID controller in time delaycontrol To illustrate clearly a known dynamic time delaycontrol system is proposed as follows

119866 (119904) =4

5119904 + 1119890minus3119904

(46)

and then analyze the performance of unit step response Thecontrol process is displayed in Figures 10 and 11


0 10 20 30035

04

045

05

055

06

065

07

Time (s)

Out

put

WKNNWNN

norm

Figure 6 The comparison of output

0 10 20 30Time (s)

Erro

r

WKNNWNN

03

025

02

015

01

005

minus005

0

norm

Figure 7 The comparison of error

0 200 400 60002

03

04

05

06

07

08

09

1

Time (s)

Out

put

WKNNWNN

norm


0 200 400 600Time (s)

Erro

r

minus001

minus002

minus003

minus004

minus005

003

002

001

0

WKNNWNN

norm



0 50 100 150 200Time (s)

Out

put

norm outputWNN-FOPIDWKNN-FOPID

minus15

minus1

minus05

0

05

1

15

2

25

Figure 10 The comparison of controller output

From the information of Figures 10 and 11 when thetime delay system utilizes FOPID controller with the sameorder the overshooting of WNN-FOPID controller and theoutput oscillation frequency and amplitude are all higherthanWKNN-FOPID obviouslyThe control curves and errorcurves of different algorithm confirm that the whole tuningperformance of WKNN is better than that of WNN Tovalidate the algorithm practicability and independence frommodel the next subsection will apply and control a morecomplex nuclear reactor model with the strategy of Section 5

62 RELAP5 Operation Example This paper establishesthe integrated pressurized water reactor (IPWR) model byRELAP5 transient analysis program and the control strategyin Figure 5 is applied to control the reactor power As theboundedness of sigmoid function WNN output parametersfor enhanced PID remained in a range of [minus1 1] To solve thisproblem the expertise improvement unit is added betweenKWNN and power require counter in Figure 1 which cor-rects parameters order of magnitudes Through the analysisof load shedding limiting condition of nuclear power unitthe reliability and availability of the control method proposedhave been demonstrated In the process of load sheddingsecondary feedwater reduces from 100 FP to 60 FP in10 s and then reduces to 20 FP The operating characteristiccould be displayed in Figure 7

Figure 12(a) displays the variation of once-through steamgenerator (OTSG) secondary feedwater flow and steamflow Due to the small water capacity in OTSG the steamflow decreased rapidly when water flow decreases Thenthe decrease in steam flow leads to the reduction of the

0 50 100 150 200Time (s)

Erro

rnorm outputWNN-FOPIDWKNN-FOPID

minus15

minus1

minus05

0

05

1

15

2

25

Figure 11 The comparison of controller error

heat by feedwater and primary coolant average temperatureincreases The reactor power control system accords withthe deviation of average temperature and introduces negativereactivity to bring the power down and maintain the temper-ature constant The variation of outlet and inlet temperatureis shown in Figure 12(c) The primary coolant flow staysthe same and reduces the core temperature difference toguarantee the heat derivation In the process of rapid loadvariation the change of coolant temperature causes themotion of coolant loading and then leads to the variationof voltage regulator pressure (Figure 12(d)) The pressuremaximum approaches 1562MPa however it drops rapidlywith the power decrease and volume compression Thenpressure comes to the stable state when coolant temperaturestabilizes

7 Conclusions

The goal of this paper is to present a novel wavelet kernelneural network Before the proposed WKNN the waveletkernel function has been confirmed to be valid in neuralnetwork The WKNN takes advantage of KLMS and NLMSwhich endow the algorithm with reliability stability and wellconvergence On basis of theWKNN a novel control strategyfor FOPID controller design has been proposed It overcomesthe boundary problem of activation functions in network andutilizes the experiential knowledge of researchers sufficientlyFinally by establishing the model of IPWR the methodabove is applied in a certain simulation of load shedding andsimulation results validate the fact that the method proposedhas the practicability and reliability in actual complicatedsystem


100 200 300 400 500 60000

02

04

06

08

10

12M

ass fl

ow ra

te (k

gs)

Time (s)

Feedwater flowSteam flow

(a)

100 200 300 400 500 60000

02

04

06

08

10

12

Time (s)

Reac

tor p

ower

(b)

100 200 300 400 500 600Time (s)

550

560

570

580

590

600

Tem

pera

ture

(K)

OTSG outlet temperatureOTSG inlet temperaturePrimary average temperature

(c)

100 200 300 400 500 600Time (s)

150

152

154

156

158Pr

essu

re (M

Pa)

Primary coolant pressure

(d)

Figure 12 Operating characteristic

Conflict of Interests

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

Acknowledgments

The authors are grateful for the funding by Grants from theNational Natural Science Foundation of China (Project nos51379049 and 51109045) the Fundamental Research Fundsfor the Central Universities (Project nos HEUCFX41302 andHEUCF110419) the Scientific Research Foundation for theReturned Overseas Chinese Scholars Heilongjiang Province(no LC2013C21) and the Young College Academic Backboneof Heilongjiang Province (no 1254G018) in support of the

present research They also thank the anonymous reviewersfor improving the quality of this paper

References

[1] M V D Oliveira and J C S D Almeida ldquoApplication ofartificial intelligence techniques in modeling and control of anuclear power plant pressurizer systemrdquo Progress in NuclearEnergy vol 63 pp 71ndash85 2013

[2] C Liu J Peng F Zhao and C Li ldquoDesign and optimizationof fuzzy-PID controller for the nuclear reactor power controlrdquoNuclear Engineering and Design vol 239 no 11 pp 2311ndash23162009

[3] W Zhang Y Sun and X Xu ldquoTwo degree-of-freedom smithpredictor for processes with time delayrdquoAutomatica vol 34 no10 pp 1279ndash1282 1998


[4] S Niculescu and A M Annaswamy ldquoAn adaptive Smith-controller for time-delay systems with relative degree 119899lowast ≦ 2rdquoSystems and Control Letters vol 49 no 5 pp 347ndash358 2003

[5] W D Zhang Y X Sun and XM Xu ldquoDahlin controller designfor multivariable time-delay systemsrdquo Acta Automatica Sinicavol 24 no 1 pp 64ndash72 1998

[6] L Wu Z Feng and J Lam ldquoStability and synchronization ofdiscrete-time neural networks with switching parameters andtime-varying delaysrdquo IEEE Transactions on Neural Networksand Learning Systems vol 24 no 12 pp 1957ndash1972 2013

[7] LWu Z Feng andW X Zheng ldquoExponential stability analysisfor delayed neural networks with switching parameters averagedwell time approachrdquo IEEE Transactions on Neural Networksvol 21 no 9 pp 1396ndash1407 2010

[8] D Yang and H Zhang ldquoRobust 119867infin

networked control foruncertain fuzzy systems with time-delayrdquo Acta AutomaticaSinica vol 33 no 7 pp 726ndash730 2007

[9] Y Shi and B Yu ldquoRobust mixed 1198672119867infin

control of networkedcontrol systems with random time delays in both forward andbackward communication linksrdquo Automatica vol 47 no 4 pp754ndash760 2011

[10] L Wu and W X Zheng ldquoPassivity-based sliding mode controlof uncertain singular time-delay systemsrdquo Automatica vol 45no 9 pp 2120ndash2127 2009

[11] L Wu X Su and P Shi ldquoSliding mode control with boundedL2gain performance of Markovian jump singular time-delay

systemsrdquo Automatica vol 48 no 8 pp 1929ndash1933 2012[12] Z Qin S Zhong and J Sun ldquoSlidingmode control experiments

of uncertain dynamical systems with time delayrdquo Communica-tions in Nonlinear Science and Numerical Simulation vol 18 no12 pp 3558ndash3566 2013

[13] L M Eriksson and M Johansson ldquoPID controller tuning rulesfor varying time-delay systemsrdquo in American Control Con-ference pp 619ndash625 IEEE July 2007

[14] I Pan S Das and A Gupta ldquoTuning of an optimal fuzzyPID controller with stochastic algorithms for networked controlsystems with random time delayrdquo ISA Transactions vol 50 no1 pp 28ndash36 2011

[15] H Tran Z Guan X Dang X Cheng and F Yuan ldquoAnormalized PID controller in networked control systems withvarying time delaysrdquo ISA Transactions vol 52 no 5 pp 592ndash599 2013

[16] C Hwang and Y Cheng ldquoA numerical algorithm for stabilitytesting of fractional delay systemsrdquo Automatica vol 42 no 5pp 825ndash831 2006

[17] S E Hamamci ldquoAn algorithm for stabilization of fractional-order time delay systems using fractional-order PID con-trollersrdquo IEEE Transactions on Automatic Control vol 52 no10 pp 1964ndash1968 2007

[18] S Das I Pan S Das and A Gupta ldquoA novel fractional orderfuzzy PID controller and its optimal time domain tuning basedon integral performance indicesrdquo Engineering Applications ofArtificial Intelligence vol 25 no 2 pp 430ndash442 2012

[19] H Ozbay C Bonnet and A R Fioravanti ldquoPID controllerdesign for fractional-order systems with time delaysrdquo Systemsand Control Letters vol 61 no 1 pp 18ndash23 2012

[20] S Haykin Adaptive Filter Theory Prentice-Hall Upper SaddleRiver NJ USA 4th edition 2002

[21] J Kivinen A J Smola and R Williamson ldquoOnline learningwith kernelsrdquo IEEE Transactions on Signal Processing vol 52no 8 pp 2165ndash2176 2004

[22] W Liu J C Principe and S Haykin Kernel Adaptive FilteringA Comprehensive Introduction John Wiley amp Sons 1st edition2010

[23] W D Parreira J C M Bermudez and J Tourneret ldquoStochasticbehavior analysis of the Gaussian kernel least-mean-squarealgorithmrdquo IEEE Transactions on Signal Processing vol 60 no5 pp 2208ndash2222 2012

[24] H Fan and Q Song ldquoA linear recurrent kernel online learningalgorithm with sparse updatesrdquo Neural Neworks vol 50 pp142ndash153 2014

[25] W Gao J Chen C Richard J Huang and R Flamary ldquoKernelLMS algorithm with forward-backward splitting for dictionarylearningrdquo in Proceedings of the IEEE International Conference onAcoustics Speech and Signal Processing (ICASSP rsquo13) pp 5735ndash5739 Vancouver BC Canada 2013

[26] J Chen F Ma and J Chen ldquoA new scheme to learn a kernel inregularization networksrdquoNeurocomputing vol 74 no 12-13 pp2222ndash2227 2011

[27] F Wu and Y Zhao ldquoNovel reduced support vector machine onMorlet wavelet kernel functionrdquo Control and Decision vol 21no 8 pp 848ndash856 2006

[28] W Liu P P Pokharel and J C Principe ldquoThekernel least-mean-square algorithmrdquo IEEE Transactions on Signal Processing vol56 no 2 pp 543ndash554 2008

[29] P P Pokharel L Weifeng and J C Principe ldquoKernel LMSrdquo inProceedings of the IEEE International Conference on AcousticsSpeech and Signal Processing (ICASSP rsquo07) pp III1421ndashIII1424April 2007

[30] Z Khandan andH S Yazdi ldquoA novel neuron in kernel domainrdquoISRN Signal Processing vol 2013 Article ID 748914 11 pages2013

[31] N Aronszajn ldquoTheory of reproducing kernelsrdquo Transactions ofthe American Mathematical Society vol 68 pp 337ndash404 1950

[32] Q Zhang ldquoUsing wavelet network in nonparametric estima-tionrdquo IEEE Transactions on Neural Networks vol 8 no 2 pp227ndash236 1997

[33] Q Zhang and A Benveniste ldquoWavelet networksrdquo IEEE Trans-actions on Neural Networks vol 3 no 6 pp 889ndash898 1992

[34] S Haykin Neural Networks and Learning Machines PrenticHall Upper Saddle River NJ USA 3rd edition 2008

[35] B Scholkopf C J C Burges andA J SmolaAdvances in KernelMethods Support Vector Learning MIT Press 1999

[36] L Zhang W Zhou and L Jiao ldquoWavelet support vectormachinerdquo IEEE Transactions on Systems Man and CyberneticsPart B Cybernetics vol 34 pp 34ndash39 2004

[37] R H Kwong and E W Johnston ldquoA variable step size LMSalgorithmrdquo IEEE Transactions on Signal Processing vol 40 no7 pp 1633ndash1642 1992

[38] S M Sloan R R Schultz and G E Wilson RELAP5MOD3Code Manual United States Nuclear Regulatory CommissionInstituto Nacional Electoral 1998

[39] G Xia J Su and W Zhang ldquoMultivariable integrated modelpredictive control of nuclear power plantrdquo in Proceedings ofthe IEEE 2nd International Conference on Future GenerationCommunication and Networking Symposia (FGCNS rsquo08) vol 4pp 8ndash11 2008

[40] S Das I Pan and S Das ldquoFractional order fuzzy controlof nuclear reactor power with thermal-hydraulic effects inthe presence of random network induced delay and sensornoise having long range dependencerdquo Energy Conversion andManagement vol 68 pp 200ndash218 2013

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


As a mature controller with strong availability PID andthe controllers have been applied in industrial processes forsophisticated dynamic systems which have large scale Theprimary reason is its relatively simple structure that couldbe easily comprehended and implemented [13ndash15] In recentyears considerable interests in time delay have been arousedin fractional-order PID controller andmany researchers havepresented beneficial methods to design stabilizing fractional-order PID (FOPID) controllers for varying time delay system[16ndash19] This paper adopts neural network (NN) to tunethe FOPID online to control the nuclear reactor powerwhich keeps the coolant average temperature constant onthe condition of load changing NN is an effective methodfor PID parameters tuning and the wavelet neural network(WNN) improves the efficiency of NN by transformingthe nonlinear excitation function to the wavelet functionHowever WNN still follows the gradient descent methodto adjust weights which inevitably causes the problemsof deficient generalization performance ill-posedness andoverfitting Motivated by the drawback above it should takeadvantage of the algorithm core LMS to improve WNNBoth LMS and KLMS are derived from mean squared errorcost function [20ndash22] LMS algorithm is famous for its easycomprehending and implementing but has no ability tobe utilized in nonlinear domain With the wide spread ofkernel theory the novel LMS algorithm with kernel trick hasdrawn considerable attention to learning machines includingneural network [23ndash26] Recently kernel functions (suchas Gaussian kernel) are applied in support vector machine(SVM) mostly which projects the linear nonseparable datato another space for being separable in the new space RBFnetwork is a sort of NNwhich useGaussian kernel frequentlyBut this kernel could not approach the target appropriatelywhen it encounters more complex signal in the calculationprocess There are some literatures indicating that Gaussiankernel in RBF network and SVM could not generate a set ofcomplete bases in the subspace ideally by translation [27]Some potential wavelet functions could be transformed askernel functions which are competitive to produce a set ofcomplete bases in 119871

2(119877) space (square and integrable space)

by stretching and translatingMeanwhile RBF network needsto identify the model of object and obtain the Jacobianinformation when it tunes the network weights and it ishard to gain the Jacobian information of complex industrialsystem But the conventional WNN could tune the PIDcontroller without such information and is more appropriateto be the parent body for improvement Thus this paperproposes a novel neural network based on wavelet kernelfunction and combines with fuzzy logic rules to achieve theonline tuning of FOPID Experiment results validate that thecontrol strategy and controller design improve the accuracyand speed which could control the nuclear reactor poweroperation and enhances the overall efficiency

The next content is composed of six sections Section 2provides a short introduction to LMS algorithm and kernelLMS algorithm Section 3 presents the related theory ofwavelet kernel function and proposes the method of waveletkernel neural network including the discussion of adaptiveparameters The model of nuclear reactor is structured by

RELAP 5 program in Section 4 Then the control strategyand the fractional-order PID controller are designed withfuzzy logic and wavelet kernel neural network in Section 5In Section 6 experiments illustrate the effectiveness of thewavelet kernel neural network and the fractional-order PIDcontroller As some controllers are faced with the problemof losing efficacy in practice Section 6 also emulates andanalyzes the control process in the basis of reactor modelFinally conclusion of this paper has been described inSection 7

2 The Description of Kernel LMS Algorithm

In this section the algorithm of kernel LMS is introduced as ashow review for wavelet kernel neural network Kernel LMSis an approach to perform the LMS algorithm in the kernelfeature space In the LMS algorithm the hypothesis space1198671is composed by all the linear operators on 119880 (mapping

119908 119880 rarr 119877) [28] which is embedded in a Hilbert spaceand the linear operator represents standard inner productThe LMS algorithm minimizes the total instantaneous errorof the system output

119864 (119899) =1

2sum e2 (119899) (1)

where e(119899) = norm(119899) minus x(119899)w(119899) norm is the desired valuew(119899) is the weight vector x(119899) is the input vector and 119899 is theinstant Using the straightforward stochastic gradient descentupdate rule the weight vector w is approximated based oninput vector x

w (119899 + 1) = w (119899) + 120578e (119899) x (119899) (2)

where 120578 is the step-size By training step by step the weightwis updated as a linear combination of the previous and presentinput vector

This LMS algorithm learns linear patterns satisfactorilywhile it could not approach the same sound effect in non-linear pattern Motivated by the drawback Pokharel withother scholars extended the LMS algorithm to RKHS by thekernel trick [28 29] Kernel methods map the input into ahigh dimensional space (HDS) [30] And any kernel functionbased on Mercerrsquos theorem has the follow mapping [31]

119896 (119909 1199091015840

) = ⟨120601 (119909) 120601 (1199091015840

)⟩ = 120601119879

(119909) 120601 (1199091015840

) (3)

When the input in (2) consists of 120601(x(1)) 120601(x(2)) 120601(x(119899)) but not x(1) x(2) x(119899) 120601(x(119899)) is the infinitefeature vector transformed from input vector x(119899) The LMSalgorithm is no longer applicable and the total instantaneouserror of the system output is as follows

119864 =1

2sum(norm minus 120601 (x (119899))w (119899))2 (4)

where 120601(x(119899)) is matrixes of input vector norm is desiredvalue vector and w(119899) is weight vector in RKHS

Then build the hypothesis that w(0) = 0 for convenienceand (2) could convert to

w (119899) = 120578119899minus1

sum

119894=0

e (119894) 120601 (x (119894)) (5)



z (119899) = 120578119899minus1

sum

119894=0

e (119894) ⟨120601 (x (119894)) 120601 (x (119899))⟩ = 120578e119879 (119899 minus 1) k (6)





119886119887(119909) isin 119871





120595119886119887(119909) = 119886

minus12

120595(119909 minus 119887

119886) (7)


119862120595= int119877

1003816100381610038161003816 (119908)1003816100381610038161003816

2

|119908|119889119908 lt infin (8)


120595119889(119909) =

119889

prod

119894=1

120595 (119909119894) (9)

where x = (1199091 119909

119889) isin 119877119889


1015840



119877119889ℎ2


∬119877119889otimes119877119889

119896 (119909 1199091015840

) ℎ (119909) ℎ (1199091015840

) 1198891199091198891199091015840

ge 0 (10)


119896 (119909 1199091015840

) =

119889

prod

119894=1

120595(119909119894minus 1199091015840

119894

119886119894

) (11)


120595 (119909) = cos (1199080119909) exp(minus119909

2

2) (12)


119896 (119909 1199091015840

)

=

119889

prod

119894=1

cos(1199080(119909119894minus 1199091015840

119894

119886119894

)) exp(minus(119909119894minus 1199091015840

119894)2

21198862

119894

)

(13)




0

2

4

minus4

minus2


minus2 0 2 4

x

y


minus2

0

2

4

x

y

minus8

minus6

minus4

minus2 0 2 4

6







x119868(2) x


are z119870(1) z119870(2) z

119870(119899) and norm

119870(1)norm

119870(2)


as 120576(119899)

120576 (119899) =1

2e2 (119899) (14)



119869and


k119869(119899) =

119868

sum

119894=1

119908119895119894(119899) 119909119894(119899) = x119879

119868(119899)w119869119868(119899)

x119869(119899) = 120593 (k

119869(119899))

(15)



neuronsw119869119868(119899) and x


by119908119894119895(119899) and 119909





k119870(119899) =

119869

sum

119895=1

119908119896119895(119899) 119909119895(119899) = x119879

119869(119899)w119870119869(119899) (16)

z119870= 119891 (k

119870(119899)) = 119891 (120593

119879

(k119869(119899))w

119870119869(119899)) (17)

where119908119896119895(119899) is weights119909


ronsw119870119869(119899) and x


119908119896119895(119899) and 119909






as follows

w119870119869(119899) = 120578

119870(119899) 120593 (k

119869(119899))

119899minus1

sum

119901=0

e (119901) 1198911015840 (120593119879 (k119869(119901))w

119870119869(119901))

(18)



119870(119899) could be

determined as

z119870(119899) = 119891(120578

119870(119899)

119899minus1

sum

119901=0

120572119901119896 (k119869(119901) k

119869(119899)))

= 119891 (120578119870(119899)120572119879

(119899 minus 1) k119870)

(19)

where 120572(119899) = e(119899)1198911015840(120578119870(119899 minus 1)120572

119879

(119899 minus 1)k119870)



w119869119868(119899) = 120578

119869(119899) x119868(119899)

119899minus1

sum

119901=0

1205931015840

(x119879119868(119901)w119869119868(119901))

timessum

119870

e (119901) 1198911015840 (120593119879 (k119869(119901))w

119870119869(119901))w

119870119869(119901)

(20)



x119869= 120593 (⟨w

119869119868(119899) x119868(119899)⟩)

= 120593(120578119869(119899)

119899minus1

sum

119901=0

120573 (119901) 119896119869(x119868(119901) x

119868(119899)))

= 120593 (120578119869(119899)120573119879

(119899 minus 1) k119869)

(21)

where 120573(119899) = 1205931015840(120578119895(119899)120573119879

(119899 minus 1) k119869)120572(119899 minus 1)w

119870119869(119899 minus 1)



119908 (119899 + 1) = 119908 (119899) +1

⟨119906 (119899) 119906 (119899)⟩119890 (119899) times 119906 (119899) (22)





119908 (119899 + 1) = 119908 (119899) + 120578 (119899) times 119890 (119899) times 119906 (119899) (23)


119870(119899) and

adjusted as

w119870(119899 + 1) = w

119870(119899) + 120578

119870(119899) times 119891

1015840

(w119870(119899) times u

119870(119899))


119870(119899)

(24)



120578119896(119899) (119896 = 1 2 119870) is given as

120578119896(119899) =

1

1198911015840 (w119870(119899) times u

119870(119899)) ⟨u

119870(119899) u

119870(119899)⟩

(25)





ℎ (120578119870(119899)) = norm minus 119891 (w

119870(119899) times u

119870(119899))


119870(119899)) minus 119887 (119899)

= E119879119870(119899) [119868 minus 120578

119870(119899) 1205812

(119899) u119879119870(119899) u119870(119899)]E

119870

119892 (120578119870(119899)) =

1

2ℎ2

(120578119870(119899))

(26)

where 120578119870(119899) = [


d

0 sdotsdotsdot 120578119870(119899)


119892(120578119870(119899)) and 1198921015840(120578

119870(119899)) = 0 and then 120578

119870(119899) (119896 = 1 2 119870)


120578119870(119899) =

[[

[

1205781(119899) sdot sdot sdot 0

d

0 sdot sdot sdot 120578119870(119899)

]]

]

(27)

120578119896(119899) =

1

1205812

119896(119899) ⟨u

119870(119899) u

119870(119899)⟩

(28)





by 1198911015840(w119870(119899) times u


completes the proof




120578119896(119899) =

1

1205812

119896(119899) ⟨u

119870(119899) u

119870(119899)⟩ + 120590

2

V (29)


119869(119901))w

119870119869(119901))w



119869(119899)





11198722




119870(119899)






End



119895(119899) of120578


as follows

120578119895(119899) =

1

119870

119870

sum

119896=1

120578119896(119899) (30)


119886 (119899 + 1) = 119886 (119899) minus 1205831015840

nabla119886(119899)

(e2 (119899)) (31)



nabla119886(119899)

= e (119899) ( 120597z119870

120597119886 (119899)) = e (119899) ( 120597z

119870

120597119906 (119899)

120597119906 (119899)

120597119886 (119899))

120597119906 (119899)

120597119886 (119899)=120597119906 (119899)

120597119896 (119899)

120597119896 (119899)

120597119886 (119899)= 120578119870(119899)

119899minus1

sum

119901=0

120572 (119901)119867119896 (119909 (119901) 119909 (119899))

(32)


119901=0120572(119901)119896(119909(119901) 119909(119899))

Then nabla119886(119899)


nabla119886(119899)

= 120578119870(119899) e (119899) 1198911015840 (119906)

119899minus1

sum

119901=0

120572 (119901)119867119896 (119909 (119901) 119909 (119899)) (33)


119886 (119899 + 1) = 119886 (119899) minus 120583 (119899)120572 (119899)

119899minus1

sum

119901=0

120572 (119901)119867119896 (119909 (119901) 119909 (119899))

(34)

where 120583(119899) = 1205831015840120578119870(119899)






120597

120597119905(120572119892120588119892) +

1

119860

120597

120597119909(120572119892120588119892V119892119860) = Γ

119892

120597

120597119905(120572119891120588119891) +

1

119860

120597

120597119909(120572119891120588119891V119891119860) = Γ

119891

(35)



120572119892120588119892119860

120597V119892

120597119905+1

2120572119892120588119892119860

120597V2119892

120597119909

= minus120572119892119860120597119875

120597119909+ 120572119892120588119892119861119909119860 minus (120572

119892120588119892119860) FWG (V

119892)

+ Γ119892119860(V119892119868minus V119892) minus (120572

119892120588119892119860) FIG (V

119892minus V119891)

minus 119862120572119892120572119891120588119898119860[

120597 (V119892minus V119891)

120597119905+ V119891

120597V119892

120597119909minus V119892

120597V119891

120597119909]

120572119891120588119891119860

120597V119891

120597119905+1

2120572119891120588119891119860

120597V2119891

120597119909

= minus120572119891119860120597119875

120597119909+ 120572119891120588119891119861119909119860 minus (120572

119891120588119891119860) FWF (V

119891)


119891120588119891119860) FIF (V

119891minus V119892)

minus 119862120572119891120572119892120588119898119860[

120597 (V119891minus V119892)

120597119905+ V119892

120597V119891

120597119909minus V119891

120597V119892

120597119909]

(36)


120597

120597119905(120572119892120588119892119880119892) +

1

119860

120597

120597119909(120572119892120588119892119880119892V119892119860)

= minus119875

120597120572119892

120597119905minus119875

119860

120597

120597119909(120572119892V119892119860) + 119876

119908119892

+ 119876119894119892+ Γ119894119892ℎlowast

119892+ Γ119908ℎ1015840

119892+ DISS

119892

120597

120597119905(120572119891120588119891119880119891) +

1

119860

120597

120597119909(120572119891120588119891119880119891V119891119860)

= minus119875

120597120572119891

120597119905minus119875

119860

120597

120597119909(120572119891V119891119860) + 119876

119908119891

+ 119876119894119891minus Γ119894119892ℎlowast

119891minus Γ119908ℎ1015840

119891+ DISS

119891

(37)


120597

120597119905(120572119892120588119892119883119899) +

1

119860

120597

120597119909(120572119892120588119892V119892119883119899119860) = 0 (38)


120597120588119887

120597119905+1

119860

120597 (120588119887V119891119860)

120597119909= 0 (39)


119889119899

119889119905=120588 minus 120573

119897119899 +

6

sum

119894=1

120582119894119862119894+ 119902

119889119862119894

119889119905=120573119894

119897119899 minus 120582119894119862119894

(40)




119892)


119899) and












1198990= 1198701119866119904+ 119870119875119901Δ119879119898(119899) + 119870

119875119894119904120582

119899

sum

119895=0

119902119895Δ119879119898(119899 minus 119895)

+ 119870119875119889119904minus120583

119899

sum

119895=0

119889119895Δ119879119898(119899 minus 119895)

(41)


where 119902119895= (1 minus ((1 + 120582)119895))119902


119895))119889119895minus1


119875119901 119870119875119894

and 119870119875119889



deviation Consider

Δ119879119898= 119879(0)

119898minus 119879119898 (42)

where 119879(0)119898


119879119898=119879119900+ 119879119894

2 (43)

where 119879119900and 119879



Δ119899 = 1198990minus 119899 (44)


119875119901minus119865 119870119875119894minus119865

119870119875119889minus119865


119875119901minus119865 smaller 119870





119875119894minus119865

119870119875119889minus119865


119875119901minus119865119870119875119894minus119865


119875119889minus119865and 120583 keep



x1(119899) = rin (119899)

x2(119899) = yout (119899)

x3(119899) = rin (119899) minus yout (119899)

x4(119899) = 1

z1(119899) = 119870

119875119901minus119882(119899)

z2(119899) = 119870

119875119894minus119882(119899)

z3(119899) = 119870

119875119889minus119882(119899)

(45)


x1

x2

xI

Hidden layer J

Input layer I

Output layer K

wIJ wKJ

zKe(n)

norm

f(middot)

120593(middot)








de

dt

de

dt

Temperaturecomparer



reactor OTSG

Fuzzy logicrules

KWNN


T0m

T0m

Tm

Tm

Tm

e

e

eTo

Ti

n0

120588Δn

Gs

120582

KPpminusW

KPiminusW

KPdminusW

KPpminusW

KPiminusW

KPdminusW

KPpminusF

KPiminusF

KPdminusF

120583

intedt


WKNN

If the 1st loop

FOPID

Fuzzy logic

KPpminusW

KPiminusW

KPdminusW

KPpminusF

KPiminusF

KPdminusF

120582120583

KPpminusF

KPiminusF

KPdminusF

120582120583


dedt gt 1205752




119866 (119904) =4

5119904 + 1119890minus3119904

(46)



0 10 20 30035

04

045

05

055

06

065

07

Time (s)

Out

put

WKNNWNN

norm


0 10 20 30Time (s)

Erro

r

WKNNWNN

03

025

02

015

01

005

minus005

0

norm


0 200 400 60002

03

04

05

06

07

08

09

1

Time (s)

Out

put

WKNNWNN

norm


0 200 400 600Time (s)

Erro

r

minus001

minus002

minus003

minus004

minus005

003

002

001

0

WKNNWNN

norm



0 50 100 150 200Time (s)

Out

put


minus15

minus1

minus05

0

05

1

15

2

25





0 50 100 150 200Time (s)

Erro


minus15

minus1

minus05

0

05

1

15

2

25



7 Conclusions



100 200 300 400 500 60000

02

04

06

08

10

12M

ass fl

ow ra

te (k

gs)

Time (s)


(a)

100 200 300 400 500 60000

02

04

06

08

10

12

Time (s)

Reac

tor p

ower

(b)

100 200 300 400 500 600Time (s)

550

560

570

580

590

600

Tem

pera

ture

(K)


(c)

100 200 300 400 500 600Time (s)

150

152

154

156

158Pr

essu

re (M

Pa)


(d)




Acknowledgments



References




















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











z (119899) = 120578119899minus1

sum

119894=0

e (119894) ⟨120601 (x (119894)) 120601 (x (119899))⟩ = 120578e119879 (119899 minus 1) k (6)





119886119887(119909) isin 119871





120595119886119887(119909) = 119886

minus12

120595(119909 minus 119887

119886) (7)


119862120595= int119877

1003816100381610038161003816 (119908)1003816100381610038161003816

2

|119908|119889119908 lt infin (8)


120595119889(119909) =

119889

prod

119894=1

120595 (119909119894) (9)

where x = (1199091 119909

119889) isin 119877119889


1015840



119877119889ℎ2


∬119877119889otimes119877119889

119896 (119909 1199091015840

) ℎ (119909) ℎ (1199091015840

) 1198891199091198891199091015840

ge 0 (10)


119896 (119909 1199091015840

) =

119889

prod

119894=1

120595(119909119894minus 1199091015840

119894

119886119894

) (11)


120595 (119909) = cos (1199080119909) exp(minus119909

2

2) (12)


119896 (119909 1199091015840

)

=

119889

prod

119894=1

cos(1199080(119909119894minus 1199091015840

119894

119886119894

)) exp(minus(119909119894minus 1199091015840

119894)2

21198862

119894

)

(13)




0

2

4

minus4

minus2


minus2 0 2 4

x

y


minus2

0

2

4

x

y

minus8

minus6

minus4

minus2 0 2 4

6







x119868(2) x


are z119870(1) z119870(2) z

119870(119899) and norm

119870(1)norm

119870(2)


as 120576(119899)

120576 (119899) =1

2e2 (119899) (14)



119869and


k119869(119899) =

119868

sum

119894=1

119908119895119894(119899) 119909119894(119899) = x119879

119868(119899)w119869119868(119899)

x119869(119899) = 120593 (k

119869(119899))

(15)



neuronsw119869119868(119899) and x


by119908119894119895(119899) and 119909





k119870(119899) =

119869

sum

119895=1

119908119896119895(119899) 119909119895(119899) = x119879

119869(119899)w119870119869(119899) (16)

z119870= 119891 (k

119870(119899)) = 119891 (120593

119879

(k119869(119899))w

119870119869(119899)) (17)

where119908119896119895(119899) is weights119909


ronsw119870119869(119899) and x


119908119896119895(119899) and 119909






as follows

w119870119869(119899) = 120578

119870(119899) 120593 (k

119869(119899))

119899minus1

sum

119901=0

e (119901) 1198911015840 (120593119879 (k119869(119901))w

119870119869(119901))

(18)



119870(119899) could be

determined as

z119870(119899) = 119891(120578

119870(119899)

119899minus1

sum

119901=0

120572119901119896 (k119869(119901) k

119869(119899)))

= 119891 (120578119870(119899)120572119879

(119899 minus 1) k119870)

(19)

where 120572(119899) = e(119899)1198911015840(120578119870(119899 minus 1)120572

119879

(119899 minus 1)k119870)



w119869119868(119899) = 120578

119869(119899) x119868(119899)

119899minus1

sum

119901=0

1205931015840

(x119879119868(119901)w119869119868(119901))

timessum

119870

e (119901) 1198911015840 (120593119879 (k119869(119901))w

119870119869(119901))w

119870119869(119901)

(20)



x119869= 120593 (⟨w

119869119868(119899) x119868(119899)⟩)

= 120593(120578119869(119899)

119899minus1

sum

119901=0

120573 (119901) 119896119869(x119868(119901) x

119868(119899)))

= 120593 (120578119869(119899)120573119879

(119899 minus 1) k119869)

(21)

where 120573(119899) = 1205931015840(120578119895(119899)120573119879

(119899 minus 1) k119869)120572(119899 minus 1)w

119870119869(119899 minus 1)



119908 (119899 + 1) = 119908 (119899) +1

⟨119906 (119899) 119906 (119899)⟩119890 (119899) times 119906 (119899) (22)





119908 (119899 + 1) = 119908 (119899) + 120578 (119899) times 119890 (119899) times 119906 (119899) (23)


119870(119899) and

adjusted as

w119870(119899 + 1) = w

119870(119899) + 120578

119870(119899) times 119891

1015840

(w119870(119899) times u

119870(119899))


119870(119899)

(24)



120578119896(119899) (119896 = 1 2 119870) is given as

120578119896(119899) =

1

1198911015840 (w119870(119899) times u

119870(119899)) ⟨u

119870(119899) u

119870(119899)⟩

(25)





ℎ (120578119870(119899)) = norm minus 119891 (w

119870(119899) times u

119870(119899))


119870(119899)) minus 119887 (119899)

= E119879119870(119899) [119868 minus 120578

119870(119899) 1205812

(119899) u119879119870(119899) u119870(119899)]E

119870

119892 (120578119870(119899)) =

1

2ℎ2

(120578119870(119899))

(26)

where 120578119870(119899) = [


d

0 sdotsdotsdot 120578119870(119899)


119892(120578119870(119899)) and 1198921015840(120578

119870(119899)) = 0 and then 120578

119870(119899) (119896 = 1 2 119870)


120578119870(119899) =

[[

[

1205781(119899) sdot sdot sdot 0

d

0 sdot sdot sdot 120578119870(119899)

]]

]

(27)

120578119896(119899) =

1

1205812

119896(119899) ⟨u

119870(119899) u

119870(119899)⟩

(28)





by 1198911015840(w119870(119899) times u


completes the proof




120578119896(119899) =

1

1205812

119896(119899) ⟨u

119870(119899) u

119870(119899)⟩ + 120590

2

V (29)


119869(119901))w

119870119869(119901))w



119869(119899)





11198722




119870(119899)






End



119895(119899) of120578


as follows

120578119895(119899) =

1

119870

119870

sum

119896=1

120578119896(119899) (30)


119886 (119899 + 1) = 119886 (119899) minus 1205831015840

nabla119886(119899)

(e2 (119899)) (31)



nabla119886(119899)

= e (119899) ( 120597z119870

120597119886 (119899)) = e (119899) ( 120597z

119870

120597119906 (119899)

120597119906 (119899)

120597119886 (119899))

120597119906 (119899)

120597119886 (119899)=120597119906 (119899)

120597119896 (119899)

120597119896 (119899)

120597119886 (119899)= 120578119870(119899)

119899minus1

sum

119901=0

120572 (119901)119867119896 (119909 (119901) 119909 (119899))

(32)


119901=0120572(119901)119896(119909(119901) 119909(119899))

Then nabla119886(119899)


nabla119886(119899)

= 120578119870(119899) e (119899) 1198911015840 (119906)

119899minus1

sum

119901=0

120572 (119901)119867119896 (119909 (119901) 119909 (119899)) (33)


119886 (119899 + 1) = 119886 (119899) minus 120583 (119899)120572 (119899)

119899minus1

sum

119901=0

120572 (119901)119867119896 (119909 (119901) 119909 (119899))

(34)

where 120583(119899) = 1205831015840120578119870(119899)






120597

120597119905(120572119892120588119892) +

1

119860

120597

120597119909(120572119892120588119892V119892119860) = Γ

119892

120597

120597119905(120572119891120588119891) +

1

119860

120597

120597119909(120572119891120588119891V119891119860) = Γ

119891

(35)



120572119892120588119892119860

120597V119892

120597119905+1

2120572119892120588119892119860

120597V2119892

120597119909

= minus120572119892119860120597119875

120597119909+ 120572119892120588119892119861119909119860 minus (120572

119892120588119892119860) FWG (V

119892)

+ Γ119892119860(V119892119868minus V119892) minus (120572

119892120588119892119860) FIG (V

119892minus V119891)

minus 119862120572119892120572119891120588119898119860[

120597 (V119892minus V119891)

120597119905+ V119891

120597V119892

120597119909minus V119892

120597V119891

120597119909]

120572119891120588119891119860

120597V119891

120597119905+1

2120572119891120588119891119860

120597V2119891

120597119909

= minus120572119891119860120597119875

120597119909+ 120572119891120588119891119861119909119860 minus (120572

119891120588119891119860) FWF (V

119891)


119891120588119891119860) FIF (V

119891minus V119892)

minus 119862120572119891120572119892120588119898119860[

120597 (V119891minus V119892)

120597119905+ V119892

120597V119891

120597119909minus V119891

120597V119892

120597119909]

(36)


120597

120597119905(120572119892120588119892119880119892) +

1

119860

120597

120597119909(120572119892120588119892119880119892V119892119860)

= minus119875

120597120572119892

120597119905minus119875

119860

120597

120597119909(120572119892V119892119860) + 119876

119908119892

+ 119876119894119892+ Γ119894119892ℎlowast

119892+ Γ119908ℎ1015840

119892+ DISS

119892

120597

120597119905(120572119891120588119891119880119891) +

1

119860

120597

120597119909(120572119891120588119891119880119891V119891119860)

= minus119875

120597120572119891

120597119905minus119875

119860

120597

120597119909(120572119891V119891119860) + 119876

119908119891

+ 119876119894119891minus Γ119894119892ℎlowast

119891minus Γ119908ℎ1015840

119891+ DISS

119891

(37)


120597

120597119905(120572119892120588119892119883119899) +

1

119860

120597

120597119909(120572119892120588119892V119892119883119899119860) = 0 (38)


120597120588119887

120597119905+1

119860

120597 (120588119887V119891119860)

120597119909= 0 (39)


119889119899

119889119905=120588 minus 120573

119897119899 +

6

sum

119894=1

120582119894119862119894+ 119902

119889119862119894

119889119905=120573119894

119897119899 minus 120582119894119862119894

(40)




119892)


119899) and












1198990= 1198701119866119904+ 119870119875119901Δ119879119898(119899) + 119870

119875119894119904120582

119899

sum

119895=0

119902119895Δ119879119898(119899 minus 119895)

+ 119870119875119889119904minus120583

119899

sum

119895=0

119889119895Δ119879119898(119899 minus 119895)

(41)


where 119902119895= (1 minus ((1 + 120582)119895))119902


119895))119889119895minus1


119875119901 119870119875119894

and 119870119875119889



deviation Consider

Δ119879119898= 119879(0)

119898minus 119879119898 (42)

where 119879(0)119898


119879119898=119879119900+ 119879119894

2 (43)

where 119879119900and 119879



Δ119899 = 1198990minus 119899 (44)


119875119901minus119865 119870119875119894minus119865

119870119875119889minus119865


119875119901minus119865 smaller 119870





119875119894minus119865

119870119875119889minus119865


119875119901minus119865119870119875119894minus119865


119875119889minus119865and 120583 keep



x1(119899) = rin (119899)

x2(119899) = yout (119899)

x3(119899) = rin (119899) minus yout (119899)

x4(119899) = 1

z1(119899) = 119870

119875119901minus119882(119899)

z2(119899) = 119870

119875119894minus119882(119899)

z3(119899) = 119870

119875119889minus119882(119899)

(45)


x1

x2

xI

Hidden layer J

Input layer I

Output layer K

wIJ wKJ

zKe(n)

norm

f(middot)

120593(middot)








de

dt

de

dt

Temperaturecomparer



reactor OTSG

Fuzzy logicrules

KWNN


T0m

T0m

Tm

Tm

Tm

e

e

eTo

Ti

n0

120588Δn

Gs

120582

KPpminusW

KPiminusW

KPdminusW

KPpminusW

KPiminusW

KPdminusW

KPpminusF

KPiminusF

KPdminusF

120583

intedt


WKNN

If the 1st loop

FOPID

Fuzzy logic

KPpminusW

KPiminusW

KPdminusW

KPpminusF

KPiminusF

KPdminusF

120582120583

KPpminusF

KPiminusF

KPdminusF

120582120583


dedt gt 1205752




119866 (119904) =4

5119904 + 1119890minus3119904

(46)



0 10 20 30035

04

045

05

055

06

065

07

Time (s)

Out

put

WKNNWNN

norm


0 10 20 30Time (s)

Erro

r

WKNNWNN

03

025

02

015

01

005

minus005

0

norm


0 200 400 60002

03

04

05

06

07

08

09

1

Time (s)

Out

put

WKNNWNN

norm


0 200 400 600Time (s)

Erro

r

minus001

minus002

minus003

minus004

minus005

003

002

001

0

WKNNWNN

norm



0 50 100 150 200Time (s)

Out

put


minus15

minus1

minus05

0

05

1

15

2

25





0 50 100 150 200Time (s)

Erro


minus15

minus1

minus05

0

05

1

15

2

25



7 Conclusions



100 200 300 400 500 60000

02

04

06

08

10

12M

ass fl

ow ra

te (k

gs)

Time (s)


(a)

100 200 300 400 500 60000

02

04

06

08

10

12

Time (s)

Reac

tor p

ower

(b)

100 200 300 400 500 600Time (s)

550

560

570

580

590

600

Tem

pera

ture

(K)


(c)

100 200 300 400 500 600Time (s)

150

152

154

156

158Pr

essu

re (M

Pa)


(d)




Acknowledgments



References




















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

2

4

minus4

minus2


minus2 0 2 4

x

y


minus2

0

2

4

x

y

minus8

minus6

minus4

minus2 0 2 4

6







x119868(2) x


are z119870(1) z119870(2) z

119870(119899) and norm

119870(1)norm

119870(2)


as 120576(119899)

120576 (119899) =1

2e2 (119899) (14)



119869and


k119869(119899) =

119868

sum

119894=1

119908119895119894(119899) 119909119894(119899) = x119879

119868(119899)w119869119868(119899)

x119869(119899) = 120593 (k

119869(119899))

(15)



neuronsw119869119868(119899) and x


by119908119894119895(119899) and 119909





k119870(119899) =

119869

sum

119895=1

119908119896119895(119899) 119909119895(119899) = x119879

119869(119899)w119870119869(119899) (16)

z119870= 119891 (k

119870(119899)) = 119891 (120593

119879

(k119869(119899))w

119870119869(119899)) (17)

where119908119896119895(119899) is weights119909


ronsw119870119869(119899) and x


119908119896119895(119899) and 119909






as follows

w119870119869(119899) = 120578

119870(119899) 120593 (k

119869(119899))

119899minus1

sum

119901=0

e (119901) 1198911015840 (120593119879 (k119869(119901))w

119870119869(119901))

(18)



119870(119899) could be

determined as

z119870(119899) = 119891(120578

119870(119899)

119899minus1

sum

119901=0

120572119901119896 (k119869(119901) k

119869(119899)))

= 119891 (120578119870(119899)120572119879

(119899 minus 1) k119870)

(19)

where 120572(119899) = e(119899)1198911015840(120578119870(119899 minus 1)120572

119879

(119899 minus 1)k119870)



w119869119868(119899) = 120578

119869(119899) x119868(119899)

119899minus1

sum

119901=0

1205931015840

(x119879119868(119901)w119869119868(119901))

timessum

119870

e (119901) 1198911015840 (120593119879 (k119869(119901))w

119870119869(119901))w

119870119869(119901)

(20)



x119869= 120593 (⟨w

119869119868(119899) x119868(119899)⟩)

= 120593(120578119869(119899)

119899minus1

sum

119901=0

120573 (119901) 119896119869(x119868(119901) x

119868(119899)))

= 120593 (120578119869(119899)120573119879

(119899 minus 1) k119869)

(21)

where 120573(119899) = 1205931015840(120578119895(119899)120573119879

(119899 minus 1) k119869)120572(119899 minus 1)w

119870119869(119899 minus 1)



119908 (119899 + 1) = 119908 (119899) +1

⟨119906 (119899) 119906 (119899)⟩119890 (119899) times 119906 (119899) (22)





119908 (119899 + 1) = 119908 (119899) + 120578 (119899) times 119890 (119899) times 119906 (119899) (23)


119870(119899) and

adjusted as

w119870(119899 + 1) = w

119870(119899) + 120578

119870(119899) times 119891

1015840

(w119870(119899) times u

119870(119899))


119870(119899)

(24)



120578119896(119899) (119896 = 1 2 119870) is given as

120578119896(119899) =

1

1198911015840 (w119870(119899) times u

119870(119899)) ⟨u

119870(119899) u

119870(119899)⟩

(25)





ℎ (120578119870(119899)) = norm minus 119891 (w

119870(119899) times u

119870(119899))


119870(119899)) minus 119887 (119899)

= E119879119870(119899) [119868 minus 120578

119870(119899) 1205812

(119899) u119879119870(119899) u119870(119899)]E

119870

119892 (120578119870(119899)) =

1

2ℎ2

(120578119870(119899))

(26)

where 120578119870(119899) = [


d

0 sdotsdotsdot 120578119870(119899)


119892(120578119870(119899)) and 1198921015840(120578

119870(119899)) = 0 and then 120578

119870(119899) (119896 = 1 2 119870)


120578119870(119899) =

[[

[

1205781(119899) sdot sdot sdot 0

d

0 sdot sdot sdot 120578119870(119899)

]]

]

(27)

120578119896(119899) =

1

1205812

119896(119899) ⟨u

119870(119899) u

119870(119899)⟩

(28)





by 1198911015840(w119870(119899) times u


completes the proof




120578119896(119899) =

1

1205812

119896(119899) ⟨u

119870(119899) u

119870(119899)⟩ + 120590

2

V (29)


119869(119901))w

119870119869(119901))w



119869(119899)





11198722




119870(119899)






End



119895(119899) of120578


as follows

120578119895(119899) =

1

119870

119870

sum

119896=1

120578119896(119899) (30)


119886 (119899 + 1) = 119886 (119899) minus 1205831015840

nabla119886(119899)

(e2 (119899)) (31)



nabla119886(119899)

= e (119899) ( 120597z119870

120597119886 (119899)) = e (119899) ( 120597z

119870

120597119906 (119899)

120597119906 (119899)

120597119886 (119899))

120597119906 (119899)

120597119886 (119899)=120597119906 (119899)

120597119896 (119899)

120597119896 (119899)

120597119886 (119899)= 120578119870(119899)

119899minus1

sum

119901=0

120572 (119901)119867119896 (119909 (119901) 119909 (119899))

(32)


119901=0120572(119901)119896(119909(119901) 119909(119899))

Then nabla119886(119899)


nabla119886(119899)

= 120578119870(119899) e (119899) 1198911015840 (119906)

119899minus1

sum

119901=0

120572 (119901)119867119896 (119909 (119901) 119909 (119899)) (33)


119886 (119899 + 1) = 119886 (119899) minus 120583 (119899)120572 (119899)

119899minus1

sum

119901=0

120572 (119901)119867119896 (119909 (119901) 119909 (119899))

(34)

where 120583(119899) = 1205831015840120578119870(119899)






120597

120597119905(120572119892120588119892) +

1

119860

120597

120597119909(120572119892120588119892V119892119860) = Γ

119892

120597

120597119905(120572119891120588119891) +

1

119860

120597

120597119909(120572119891120588119891V119891119860) = Γ

119891

(35)



120572119892120588119892119860

120597V119892

120597119905+1

2120572119892120588119892119860

120597V2119892

120597119909

= minus120572119892119860120597119875

120597119909+ 120572119892120588119892119861119909119860 minus (120572

119892120588119892119860) FWG (V

119892)

+ Γ119892119860(V119892119868minus V119892) minus (120572

119892120588119892119860) FIG (V

119892minus V119891)

minus 119862120572119892120572119891120588119898119860[

120597 (V119892minus V119891)

120597119905+ V119891

120597V119892

120597119909minus V119892

120597V119891

120597119909]

120572119891120588119891119860

120597V119891

120597119905+1

2120572119891120588119891119860

120597V2119891

120597119909

= minus120572119891119860120597119875

120597119909+ 120572119891120588119891119861119909119860 minus (120572

119891120588119891119860) FWF (V

119891)


119891120588119891119860) FIF (V

119891minus V119892)

minus 119862120572119891120572119892120588119898119860[

120597 (V119891minus V119892)

120597119905+ V119892

120597V119891

120597119909minus V119891

120597V119892

120597119909]

(36)


120597

120597119905(120572119892120588119892119880119892) +

1

119860

120597

120597119909(120572119892120588119892119880119892V119892119860)

= minus119875

120597120572119892

120597119905minus119875

119860

120597

120597119909(120572119892V119892119860) + 119876

119908119892

+ 119876119894119892+ Γ119894119892ℎlowast

119892+ Γ119908ℎ1015840

119892+ DISS

119892

120597

120597119905(120572119891120588119891119880119891) +

1

119860

120597

120597119909(120572119891120588119891119880119891V119891119860)

= minus119875

120597120572119891

120597119905minus119875

119860

120597

120597119909(120572119891V119891119860) + 119876

119908119891

+ 119876119894119891minus Γ119894119892ℎlowast

119891minus Γ119908ℎ1015840

119891+ DISS

119891

(37)


120597

120597119905(120572119892120588119892119883119899) +

1

119860

120597

120597119909(120572119892120588119892V119892119883119899119860) = 0 (38)


120597120588119887

120597119905+1

119860

120597 (120588119887V119891119860)

120597119909= 0 (39)


119889119899

119889119905=120588 minus 120573

119897119899 +

6

sum

119894=1

120582119894119862119894+ 119902

119889119862119894

119889119905=120573119894

119897119899 minus 120582119894119862119894

(40)




119892)


119899) and












1198990= 1198701119866119904+ 119870119875119901Δ119879119898(119899) + 119870

119875119894119904120582

119899

sum

119895=0

119902119895Δ119879119898(119899 minus 119895)

+ 119870119875119889119904minus120583

119899

sum

119895=0

119889119895Δ119879119898(119899 minus 119895)

(41)


where 119902119895= (1 minus ((1 + 120582)119895))119902


119895))119889119895minus1


119875119901 119870119875119894

and 119870119875119889



deviation Consider

Δ119879119898= 119879(0)

119898minus 119879119898 (42)

where 119879(0)119898


119879119898=119879119900+ 119879119894

2 (43)

where 119879119900and 119879



Δ119899 = 1198990minus 119899 (44)


119875119901minus119865 119870119875119894minus119865

119870119875119889minus119865


119875119901minus119865 smaller 119870





119875119894minus119865

119870119875119889minus119865


119875119901minus119865119870119875119894minus119865


119875119889minus119865and 120583 keep



x1(119899) = rin (119899)

x2(119899) = yout (119899)

x3(119899) = rin (119899) minus yout (119899)

x4(119899) = 1

z1(119899) = 119870

119875119901minus119882(119899)

z2(119899) = 119870

119875119894minus119882(119899)

z3(119899) = 119870

119875119889minus119882(119899)

(45)


x1

x2

xI

Hidden layer J

Input layer I

Output layer K

wIJ wKJ

zKe(n)

norm

f(middot)

120593(middot)








de

dt

de

dt

Temperaturecomparer



reactor OTSG

Fuzzy logicrules

KWNN


T0m

T0m

Tm

Tm

Tm

e

e

eTo

Ti

n0

120588Δn

Gs

120582

KPpminusW

KPiminusW

KPdminusW

KPpminusW

KPiminusW

KPdminusW

KPpminusF

KPiminusF

KPdminusF

120583

intedt


WKNN

If the 1st loop

FOPID

Fuzzy logic

KPpminusW

KPiminusW

KPdminusW

KPpminusF

KPiminusF

KPdminusF

120582120583

KPpminusF

KPiminusF

KPdminusF

120582120583


dedt gt 1205752




119866 (119904) =4

5119904 + 1119890minus3119904

(46)



0 10 20 30035

04

045

05

055

06

065

07

Time (s)

Out

put

WKNNWNN

norm


0 10 20 30Time (s)

Erro

r

WKNNWNN

03

025

02

015

01

005

minus005

0

norm


0 200 400 60002

03

04

05

06

07

08

09

1

Time (s)

Out

put

WKNNWNN

norm


0 200 400 600Time (s)

Erro

r

minus001

minus002

minus003

minus004

minus005

003

002

001

0

WKNNWNN

norm



0 50 100 150 200Time (s)

Out

put


minus15

minus1

minus05

0

05

1

15

2

25





0 50 100 150 200Time (s)

Erro


minus15

minus1

minus05

0

05

1

15

2

25



7 Conclusions



100 200 300 400 500 60000

02

04

06

08

10

12M

ass fl

ow ra

te (k

gs)

Time (s)


(a)

100 200 300 400 500 60000

02

04

06

08

10

12

Time (s)

Reac

tor p

ower

(b)

100 200 300 400 500 600Time (s)

550

560

570

580

590

600

Tem

pera

ture

(K)


(c)

100 200 300 400 500 600Time (s)

150

152

154

156

158Pr

essu

re (M

Pa)


(d)




Acknowledgments



References




















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











x119869= 120593 (⟨w

119869119868(119899) x119868(119899)⟩)

= 120593(120578119869(119899)

119899minus1

sum

119901=0

120573 (119901) 119896119869(x119868(119901) x

119868(119899)))

= 120593 (120578119869(119899)120573119879

(119899 minus 1) k119869)

(21)

where 120573(119899) = 1205931015840(120578119895(119899)120573119879

(119899 minus 1) k119869)120572(119899 minus 1)w

119870119869(119899 minus 1)



119908 (119899 + 1) = 119908 (119899) +1

⟨119906 (119899) 119906 (119899)⟩119890 (119899) times 119906 (119899) (22)





119908 (119899 + 1) = 119908 (119899) + 120578 (119899) times 119890 (119899) times 119906 (119899) (23)


119870(119899) and

adjusted as

w119870(119899 + 1) = w

119870(119899) + 120578

119870(119899) times 119891

1015840

(w119870(119899) times u

119870(119899))


119870(119899)

(24)



120578119896(119899) (119896 = 1 2 119870) is given as

120578119896(119899) =

1

1198911015840 (w119870(119899) times u

119870(119899)) ⟨u

119870(119899) u

119870(119899)⟩

(25)





ℎ (120578119870(119899)) = norm minus 119891 (w

119870(119899) times u

119870(119899))


119870(119899)) minus 119887 (119899)

= E119879119870(119899) [119868 minus 120578

119870(119899) 1205812

(119899) u119879119870(119899) u119870(119899)]E

119870

119892 (120578119870(119899)) =

1

2ℎ2

(120578119870(119899))

(26)

where 120578119870(119899) = [


d

0 sdotsdotsdot 120578119870(119899)


119892(120578119870(119899)) and 1198921015840(120578

119870(119899)) = 0 and then 120578

119870(119899) (119896 = 1 2 119870)


120578119870(119899) =

[[

[

1205781(119899) sdot sdot sdot 0

d

0 sdot sdot sdot 120578119870(119899)

]]

]

(27)

120578119896(119899) =

1

1205812

119896(119899) ⟨u

119870(119899) u

119870(119899)⟩

(28)





by 1198911015840(w119870(119899) times u


completes the proof




120578119896(119899) =

1

1205812

119896(119899) ⟨u

119870(119899) u

119870(119899)⟩ + 120590

2

V (29)


119869(119901))w

119870119869(119901))w



119869(119899)





11198722




119870(119899)






End



119895(119899) of120578


as follows

120578119895(119899) =

1

119870

119870

sum

119896=1

120578119896(119899) (30)


119886 (119899 + 1) = 119886 (119899) minus 1205831015840

nabla119886(119899)

(e2 (119899)) (31)



nabla119886(119899)

= e (119899) ( 120597z119870

120597119886 (119899)) = e (119899) ( 120597z

119870

120597119906 (119899)

120597119906 (119899)

120597119886 (119899))

120597119906 (119899)

120597119886 (119899)=120597119906 (119899)

120597119896 (119899)

120597119896 (119899)

120597119886 (119899)= 120578119870(119899)

119899minus1

sum

119901=0

120572 (119901)119867119896 (119909 (119901) 119909 (119899))

(32)


119901=0120572(119901)119896(119909(119901) 119909(119899))

Then nabla119886(119899)


nabla119886(119899)

= 120578119870(119899) e (119899) 1198911015840 (119906)

119899minus1

sum

119901=0

120572 (119901)119867119896 (119909 (119901) 119909 (119899)) (33)


119886 (119899 + 1) = 119886 (119899) minus 120583 (119899)120572 (119899)

119899minus1

sum

119901=0

120572 (119901)119867119896 (119909 (119901) 119909 (119899))

(34)

where 120583(119899) = 1205831015840120578119870(119899)






120597

120597119905(120572119892120588119892) +

1

119860

120597

120597119909(120572119892120588119892V119892119860) = Γ

119892

120597

120597119905(120572119891120588119891) +

1

119860

120597

120597119909(120572119891120588119891V119891119860) = Γ

119891

(35)



120572119892120588119892119860

120597V119892

120597119905+1

2120572119892120588119892119860

120597V2119892

120597119909

= minus120572119892119860120597119875

120597119909+ 120572119892120588119892119861119909119860 minus (120572

119892120588119892119860) FWG (V

119892)

+ Γ119892119860(V119892119868minus V119892) minus (120572

119892120588119892119860) FIG (V

119892minus V119891)

minus 119862120572119892120572119891120588119898119860[

120597 (V119892minus V119891)

120597119905+ V119891

120597V119892

120597119909minus V119892

120597V119891

120597119909]

120572119891120588119891119860

120597V119891

120597119905+1

2120572119891120588119891119860

120597V2119891

120597119909

= minus120572119891119860120597119875

120597119909+ 120572119891120588119891119861119909119860 minus (120572

119891120588119891119860) FWF (V

119891)


119891120588119891119860) FIF (V

119891minus V119892)

minus 119862120572119891120572119892120588119898119860[

120597 (V119891minus V119892)

120597119905+ V119892

120597V119891

120597119909minus V119891

120597V119892

120597119909]

(36)


120597

120597119905(120572119892120588119892119880119892) +

1

119860

120597

120597119909(120572119892120588119892119880119892V119892119860)

= minus119875

120597120572119892

120597119905minus119875

119860

120597

120597119909(120572119892V119892119860) + 119876

119908119892

+ 119876119894119892+ Γ119894119892ℎlowast

119892+ Γ119908ℎ1015840

119892+ DISS

119892

120597

120597119905(120572119891120588119891119880119891) +

1

119860

120597

120597119909(120572119891120588119891119880119891V119891119860)

= minus119875

120597120572119891

120597119905minus119875

119860

120597

120597119909(120572119891V119891119860) + 119876

119908119891

+ 119876119894119891minus Γ119894119892ℎlowast

119891minus Γ119908ℎ1015840

119891+ DISS

119891

(37)


120597

120597119905(120572119892120588119892119883119899) +

1

119860

120597

120597119909(120572119892120588119892V119892119883119899119860) = 0 (38)


120597120588119887

120597119905+1

119860

120597 (120588119887V119891119860)

120597119909= 0 (39)


119889119899

119889119905=120588 minus 120573

119897119899 +

6

sum

119894=1

120582119894119862119894+ 119902

119889119862119894

119889119905=120573119894

119897119899 minus 120582119894119862119894

(40)




119892)


119899) and












1198990= 1198701119866119904+ 119870119875119901Δ119879119898(119899) + 119870

119875119894119904120582

119899

sum

119895=0

119902119895Δ119879119898(119899 minus 119895)

+ 119870119875119889119904minus120583

119899

sum

119895=0

119889119895Δ119879119898(119899 minus 119895)

(41)


where 119902119895= (1 minus ((1 + 120582)119895))119902


119895))119889119895minus1


119875119901 119870119875119894

and 119870119875119889



deviation Consider

Δ119879119898= 119879(0)

119898minus 119879119898 (42)

where 119879(0)119898


119879119898=119879119900+ 119879119894

2 (43)

where 119879119900and 119879



Δ119899 = 1198990minus 119899 (44)


119875119901minus119865 119870119875119894minus119865

119870119875119889minus119865


119875119901minus119865 smaller 119870





119875119894minus119865

119870119875119889minus119865


119875119901minus119865119870119875119894minus119865


119875119889minus119865and 120583 keep



x1(119899) = rin (119899)

x2(119899) = yout (119899)

x3(119899) = rin (119899) minus yout (119899)

x4(119899) = 1

z1(119899) = 119870

119875119901minus119882(119899)

z2(119899) = 119870

119875119894minus119882(119899)

z3(119899) = 119870

119875119889minus119882(119899)

(45)


x1

x2

xI

Hidden layer J

Input layer I

Output layer K

wIJ wKJ

zKe(n)

norm

f(middot)

120593(middot)








de

dt

de

dt

Temperaturecomparer



reactor OTSG

Fuzzy logicrules

KWNN


T0m

T0m

Tm

Tm

Tm

e

e

eTo

Ti

n0

120588Δn

Gs

120582

KPpminusW

KPiminusW

KPdminusW

KPpminusW

KPiminusW

KPdminusW

KPpminusF

KPiminusF

KPdminusF

120583

intedt


WKNN

If the 1st loop

FOPID

Fuzzy logic

KPpminusW

KPiminusW

KPdminusW

KPpminusF

KPiminusF

KPdminusF

120582120583

KPpminusF

KPiminusF

KPdminusF

120582120583


dedt gt 1205752




119866 (119904) =4

5119904 + 1119890minus3119904

(46)



0 10 20 30035

04

045

05

055

06

065

07

Time (s)

Out

put

WKNNWNN

norm


0 10 20 30Time (s)

Erro

r

WKNNWNN

03

025

02

015

01

005

minus005

0

norm


0 200 400 60002

03

04

05

06

07

08

09

1

Time (s)

Out

put

WKNNWNN

norm


0 200 400 600Time (s)

Erro

r

minus001

minus002

minus003

minus004

minus005

003

002

001

0

WKNNWNN

norm



0 50 100 150 200Time (s)

Out

put


minus15

minus1

minus05

0

05

1

15

2

25





0 50 100 150 200Time (s)

Erro


minus15

minus1

minus05

0

05

1

15

2

25



7 Conclusions



100 200 300 400 500 60000

02

04

06

08

10

12M

ass fl

ow ra

te (k

gs)

Time (s)


(a)

100 200 300 400 500 60000

02

04

06

08

10

12

Time (s)

Reac

tor p

ower

(b)

100 200 300 400 500 600Time (s)

550

560

570

580

590

600

Tem

pera

ture

(K)


(c)

100 200 300 400 500 600Time (s)

150

152

154

156

158Pr

essu

re (M

Pa)


(d)




Acknowledgments



References




















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













11198722




119870(119899)






End



119895(119899) of120578


as follows

120578119895(119899) =

1

119870

119870

sum

119896=1

120578119896(119899) (30)


119886 (119899 + 1) = 119886 (119899) minus 1205831015840

nabla119886(119899)

(e2 (119899)) (31)



nabla119886(119899)

= e (119899) ( 120597z119870

120597119886 (119899)) = e (119899) ( 120597z

119870

120597119906 (119899)

120597119906 (119899)

120597119886 (119899))

120597119906 (119899)

120597119886 (119899)=120597119906 (119899)

120597119896 (119899)

120597119896 (119899)

120597119886 (119899)= 120578119870(119899)

119899minus1

sum

119901=0

120572 (119901)119867119896 (119909 (119901) 119909 (119899))

(32)


119901=0120572(119901)119896(119909(119901) 119909(119899))

Then nabla119886(119899)


nabla119886(119899)

= 120578119870(119899) e (119899) 1198911015840 (119906)

119899minus1

sum

119901=0

120572 (119901)119867119896 (119909 (119901) 119909 (119899)) (33)


119886 (119899 + 1) = 119886 (119899) minus 120583 (119899)120572 (119899)

119899minus1

sum

119901=0

120572 (119901)119867119896 (119909 (119901) 119909 (119899))

(34)

where 120583(119899) = 1205831015840120578119870(119899)






120597

120597119905(120572119892120588119892) +

1

119860

120597

120597119909(120572119892120588119892V119892119860) = Γ

119892

120597

120597119905(120572119891120588119891) +

1

119860

120597

120597119909(120572119891120588119891V119891119860) = Γ

119891

(35)



120572119892120588119892119860

120597V119892

120597119905+1

2120572119892120588119892119860

120597V2119892

120597119909

= minus120572119892119860120597119875

120597119909+ 120572119892120588119892119861119909119860 minus (120572

119892120588119892119860) FWG (V

119892)

+ Γ119892119860(V119892119868minus V119892) minus (120572

119892120588119892119860) FIG (V

119892minus V119891)

minus 119862120572119892120572119891120588119898119860[

120597 (V119892minus V119891)

120597119905+ V119891

120597V119892

120597119909minus V119892

120597V119891

120597119909]

120572119891120588119891119860

120597V119891

120597119905+1

2120572119891120588119891119860

120597V2119891

120597119909

= minus120572119891119860120597119875

120597119909+ 120572119891120588119891119861119909119860 minus (120572

119891120588119891119860) FWF (V

119891)


119891120588119891119860) FIF (V

119891minus V119892)

minus 119862120572119891120572119892120588119898119860[

120597 (V119891minus V119892)

120597119905+ V119892

120597V119891

120597119909minus V119891

120597V119892

120597119909]

(36)


120597

120597119905(120572119892120588119892119880119892) +

1

119860

120597

120597119909(120572119892120588119892119880119892V119892119860)

= minus119875

120597120572119892

120597119905minus119875

119860

120597

120597119909(120572119892V119892119860) + 119876

119908119892

+ 119876119894119892+ Γ119894119892ℎlowast

119892+ Γ119908ℎ1015840

119892+ DISS

119892

120597

120597119905(120572119891120588119891119880119891) +

1

119860

120597

120597119909(120572119891120588119891119880119891V119891119860)

= minus119875

120597120572119891

120597119905minus119875

119860

120597

120597119909(120572119891V119891119860) + 119876

119908119891

+ 119876119894119891minus Γ119894119892ℎlowast

119891minus Γ119908ℎ1015840

119891+ DISS

119891

(37)


120597

120597119905(120572119892120588119892119883119899) +

1

119860

120597

120597119909(120572119892120588119892V119892119883119899119860) = 0 (38)


120597120588119887

120597119905+1

119860

120597 (120588119887V119891119860)

120597119909= 0 (39)


119889119899

119889119905=120588 minus 120573

119897119899 +

6

sum

119894=1

120582119894119862119894+ 119902

119889119862119894

119889119905=120573119894

119897119899 minus 120582119894119862119894

(40)




119892)


119899) and












1198990= 1198701119866119904+ 119870119875119901Δ119879119898(119899) + 119870

119875119894119904120582

119899

sum

119895=0

119902119895Δ119879119898(119899 minus 119895)

+ 119870119875119889119904minus120583

119899

sum

119895=0

119889119895Δ119879119898(119899 minus 119895)

(41)


where 119902119895= (1 minus ((1 + 120582)119895))119902


119895))119889119895minus1


119875119901 119870119875119894

and 119870119875119889



deviation Consider

Δ119879119898= 119879(0)

119898minus 119879119898 (42)

where 119879(0)119898


119879119898=119879119900+ 119879119894

2 (43)

where 119879119900and 119879



Δ119899 = 1198990minus 119899 (44)


119875119901minus119865 119870119875119894minus119865

119870119875119889minus119865


119875119901minus119865 smaller 119870





119875119894minus119865

119870119875119889minus119865


119875119901minus119865119870119875119894minus119865


119875119889minus119865and 120583 keep



x1(119899) = rin (119899)

x2(119899) = yout (119899)

x3(119899) = rin (119899) minus yout (119899)

x4(119899) = 1

z1(119899) = 119870

119875119901minus119882(119899)

z2(119899) = 119870

119875119894minus119882(119899)

z3(119899) = 119870

119875119889minus119882(119899)

(45)


x1

x2

xI

Hidden layer J

Input layer I

Output layer K

wIJ wKJ

zKe(n)

norm

f(middot)

120593(middot)








de

dt

de

dt

Temperaturecomparer



reactor OTSG

Fuzzy logicrules

KWNN


T0m

T0m

Tm

Tm

Tm

e

e

eTo

Ti

n0

120588Δn

Gs

120582

KPpminusW

KPiminusW

KPdminusW

KPpminusW

KPiminusW

KPdminusW

KPpminusF

KPiminusF

KPdminusF

120583

intedt


WKNN

If the 1st loop

FOPID

Fuzzy logic

KPpminusW

KPiminusW

KPdminusW

KPpminusF

KPiminusF

KPdminusF

120582120583

KPpminusF

KPiminusF

KPdminusF

120582120583


dedt gt 1205752




119866 (119904) =4

5119904 + 1119890minus3119904

(46)



0 10 20 30035

04

045

05

055

06

065

07

Time (s)

Out

put

WKNNWNN

norm


0 10 20 30Time (s)

Erro

r

WKNNWNN

03

025

02

015

01

005

minus005

0

norm


0 200 400 60002

03

04

05

06

07

08

09

1

Time (s)

Out

put

WKNNWNN

norm


0 200 400 600Time (s)

Erro

r

minus001

minus002

minus003

minus004

minus005

003

002

001

0

WKNNWNN

norm



0 50 100 150 200Time (s)

Out

put


minus15

minus1

minus05

0

05

1

15

2

25





0 50 100 150 200Time (s)

Erro


minus15

minus1

minus05

0

05

1

15

2

25



7 Conclusions



100 200 300 400 500 60000

02

04

06

08

10

12M

ass fl

ow ra

te (k

gs)

Time (s)


(a)

100 200 300 400 500 60000

02

04

06

08

10

12

Time (s)

Reac

tor p

ower

(b)

100 200 300 400 500 600Time (s)

550

560

570

580

590

600

Tem

pera

ture

(K)


(c)

100 200 300 400 500 600Time (s)

150

152

154

156

158Pr

essu

re (M

Pa)


(d)




Acknowledgments



References




















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











120572119892120588119892119860

120597V119892

120597119905+1

2120572119892120588119892119860

120597V2119892

120597119909

= minus120572119892119860120597119875

120597119909+ 120572119892120588119892119861119909119860 minus (120572

119892120588119892119860) FWG (V

119892)

+ Γ119892119860(V119892119868minus V119892) minus (120572

119892120588119892119860) FIG (V

119892minus V119891)

minus 119862120572119892120572119891120588119898119860[

120597 (V119892minus V119891)

120597119905+ V119891

120597V119892

120597119909minus V119892

120597V119891

120597119909]

120572119891120588119891119860

120597V119891

120597119905+1

2120572119891120588119891119860

120597V2119891

120597119909

= minus120572119891119860120597119875

120597119909+ 120572119891120588119891119861119909119860 minus (120572

119891120588119891119860) FWF (V

119891)


119891120588119891119860) FIF (V

119891minus V119892)

minus 119862120572119891120572119892120588119898119860[

120597 (V119891minus V119892)

120597119905+ V119892

120597V119891

120597119909minus V119891

120597V119892

120597119909]

(36)


120597

120597119905(120572119892120588119892119880119892) +

1

119860

120597

120597119909(120572119892120588119892119880119892V119892119860)

= minus119875

120597120572119892

120597119905minus119875

119860

120597

120597119909(120572119892V119892119860) + 119876

119908119892

+ 119876119894119892+ Γ119894119892ℎlowast

119892+ Γ119908ℎ1015840

119892+ DISS

119892

120597

120597119905(120572119891120588119891119880119891) +

1

119860

120597

120597119909(120572119891120588119891119880119891V119891119860)

= minus119875

120597120572119891

120597119905minus119875

119860

120597

120597119909(120572119891V119891119860) + 119876

119908119891

+ 119876119894119891minus Γ119894119892ℎlowast

119891minus Γ119908ℎ1015840

119891+ DISS

119891

(37)


120597

120597119905(120572119892120588119892119883119899) +

1

119860

120597

120597119909(120572119892120588119892V119892119883119899119860) = 0 (38)


120597120588119887

120597119905+1

119860

120597 (120588119887V119891119860)

120597119909= 0 (39)


119889119899

119889119905=120588 minus 120573

119897119899 +

6

sum

119894=1

120582119894119862119894+ 119902

119889119862119894

119889119905=120573119894

119897119899 minus 120582119894119862119894

(40)




119892)


119899) and












1198990= 1198701119866119904+ 119870119875119901Δ119879119898(119899) + 119870

119875119894119904120582

119899

sum

119895=0

119902119895Δ119879119898(119899 minus 119895)

+ 119870119875119889119904minus120583

119899

sum

119895=0

119889119895Δ119879119898(119899 minus 119895)

(41)


where 119902119895= (1 minus ((1 + 120582)119895))119902


119895))119889119895minus1


119875119901 119870119875119894

and 119870119875119889



deviation Consider

Δ119879119898= 119879(0)

119898minus 119879119898 (42)

where 119879(0)119898


119879119898=119879119900+ 119879119894

2 (43)

where 119879119900and 119879



Δ119899 = 1198990minus 119899 (44)


119875119901minus119865 119870119875119894minus119865

119870119875119889minus119865


119875119901minus119865 smaller 119870





119875119894minus119865

119870119875119889minus119865


119875119901minus119865119870119875119894minus119865


119875119889minus119865and 120583 keep



x1(119899) = rin (119899)

x2(119899) = yout (119899)

x3(119899) = rin (119899) minus yout (119899)

x4(119899) = 1

z1(119899) = 119870

119875119901minus119882(119899)

z2(119899) = 119870

119875119894minus119882(119899)

z3(119899) = 119870

119875119889minus119882(119899)

(45)


x1

x2

xI

Hidden layer J

Input layer I

Output layer K

wIJ wKJ

zKe(n)

norm

f(middot)

120593(middot)








de

dt

de

dt

Temperaturecomparer



reactor OTSG

Fuzzy logicrules

KWNN


T0m

T0m

Tm

Tm

Tm

e

e

eTo

Ti

n0

120588Δn

Gs

120582

KPpminusW

KPiminusW

KPdminusW

KPpminusW

KPiminusW

KPdminusW

KPpminusF

KPiminusF

KPdminusF

120583

intedt


WKNN

If the 1st loop

FOPID

Fuzzy logic

KPpminusW

KPiminusW

KPdminusW

KPpminusF

KPiminusF

KPdminusF

120582120583

KPpminusF

KPiminusF

KPdminusF

120582120583


dedt gt 1205752




119866 (119904) =4

5119904 + 1119890minus3119904

(46)



0 10 20 30035

04

045

05

055

06

065

07

Time (s)

Out

put

WKNNWNN

norm


0 10 20 30Time (s)

Erro

r

WKNNWNN

03

025

02

015

01

005

minus005

0

norm


0 200 400 60002

03

04

05

06

07

08

09

1

Time (s)

Out

put

WKNNWNN

norm


0 200 400 600Time (s)

Erro

r

minus001

minus002

minus003

minus004

minus005

003

002

001

0

WKNNWNN

norm



0 50 100 150 200Time (s)

Out

put


minus15

minus1

minus05

0

05

1

15

2

25





0 50 100 150 200Time (s)

Erro


minus15

minus1

minus05

0

05

1

15

2

25



7 Conclusions



100 200 300 400 500 60000

02

04

06

08

10

12M

ass fl

ow ra

te (k

gs)

Time (s)


(a)

100 200 300 400 500 60000

02

04

06

08

10

12

Time (s)

Reac

tor p

ower

(b)

100 200 300 400 500 600Time (s)

550

560

570

580

590

600

Tem

pera

ture

(K)


(c)

100 200 300 400 500 600Time (s)

150

152

154

156

158Pr

essu

re (M

Pa)


(d)




Acknowledgments



References




















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










where 119902119895= (1 minus ((1 + 120582)119895))119902


119895))119889119895minus1


119875119901 119870119875119894

and 119870119875119889



deviation Consider

Δ119879119898= 119879(0)

119898minus 119879119898 (42)

where 119879(0)119898


119879119898=119879119900+ 119879119894

2 (43)

where 119879119900and 119879



Δ119899 = 1198990minus 119899 (44)


119875119901minus119865 119870119875119894minus119865

119870119875119889minus119865


119875119901minus119865 smaller 119870





119875119894minus119865

119870119875119889minus119865


119875119901minus119865119870119875119894minus119865


119875119889minus119865and 120583 keep



x1(119899) = rin (119899)

x2(119899) = yout (119899)

x3(119899) = rin (119899) minus yout (119899)

x4(119899) = 1

z1(119899) = 119870

119875119901minus119882(119899)

z2(119899) = 119870

119875119894minus119882(119899)

z3(119899) = 119870

119875119889minus119882(119899)

(45)


x1

x2

xI

Hidden layer J

Input layer I

Output layer K

wIJ wKJ

zKe(n)

norm

f(middot)

120593(middot)








de

dt

de

dt

Temperaturecomparer



reactor OTSG

Fuzzy logicrules

KWNN


T0m

T0m

Tm

Tm

Tm

e

e

eTo

Ti

n0

120588Δn

Gs

120582

KPpminusW

KPiminusW

KPdminusW

KPpminusW

KPiminusW

KPdminusW

KPpminusF

KPiminusF

KPdminusF

120583

intedt


WKNN

If the 1st loop

FOPID

Fuzzy logic

KPpminusW

KPiminusW

KPdminusW

KPpminusF

KPiminusF

KPdminusF

120582120583

KPpminusF

KPiminusF

KPdminusF

120582120583


dedt gt 1205752




119866 (119904) =4

5119904 + 1119890minus3119904

(46)



0 10 20 30035

04

045

05

055

06

065

07

Time (s)

Out

put

WKNNWNN

norm


0 10 20 30Time (s)

Erro

r

WKNNWNN

03

025

02

015

01

005

minus005

0

norm


0 200 400 60002

03

04

05

06

07

08

09

1

Time (s)

Out

put

WKNNWNN

norm


0 200 400 600Time (s)

Erro

r

minus001

minus002

minus003

minus004

minus005

003

002

001

0

WKNNWNN

norm



0 50 100 150 200Time (s)

Out

put


minus15

minus1

minus05

0

05

1

15

2

25





0 50 100 150 200Time (s)

Erro


minus15

minus1

minus05

0

05

1

15

2

25



7 Conclusions



100 200 300 400 500 60000

02

04

06

08

10

12M

ass fl

ow ra

te (k

gs)

Time (s)


(a)

100 200 300 400 500 60000

02

04

06

08

10

12

Time (s)

Reac

tor p

ower

(b)

100 200 300 400 500 600Time (s)

550

560

570

580

590

600

Tem

pera

ture

(K)


(c)

100 200 300 400 500 600Time (s)

150

152

154

156

158Pr

essu

re (M

Pa)


(d)




Acknowledgments



References




















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










de

dt

de

dt

Temperaturecomparer



reactor OTSG

Fuzzy logicrules

KWNN


T0m

T0m

Tm

Tm

Tm

e

e

eTo

Ti

n0

120588Δn

Gs

120582

KPpminusW

KPiminusW

KPdminusW

KPpminusW

KPiminusW

KPdminusW

KPpminusF

KPiminusF

KPdminusF

120583

intedt


WKNN

If the 1st loop

FOPID

Fuzzy logic

KPpminusW

KPiminusW

KPdminusW

KPpminusF

KPiminusF

KPdminusF

120582120583

KPpminusF

KPiminusF

KPdminusF

120582120583


dedt gt 1205752




119866 (119904) =4

5119904 + 1119890minus3119904

(46)



0 10 20 30035

04

045

05

055

06

065

07

Time (s)

Out

put

WKNNWNN

norm


0 10 20 30Time (s)

Erro

r

WKNNWNN

03

025

02

015

01

005

minus005

0

norm


0 200 400 60002

03

04

05

06

07

08

09

1

Time (s)

Out

put

WKNNWNN

norm


0 200 400 600Time (s)

Erro

r

minus001

minus002

minus003

minus004

minus005

003

002

001

0

WKNNWNN

norm



0 50 100 150 200Time (s)

Out

put


minus15

minus1

minus05

0

05

1

15

2

25





0 50 100 150 200Time (s)

Erro


minus15

minus1

minus05

0

05

1

15

2

25



7 Conclusions



100 200 300 400 500 60000

02

04

06

08

10

12M

ass fl

ow ra

te (k

gs)

Time (s)


(a)

100 200 300 400 500 60000

02

04

06

08

10

12

Time (s)

Reac

tor p

ower

(b)

100 200 300 400 500 600Time (s)

550

560

570

580

590

600

Tem

pera

ture

(K)


(c)

100 200 300 400 500 600Time (s)

150

152

154

156

158Pr

essu

re (M

Pa)


(d)




Acknowledgments



References




















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 10 20 30035

04

045

05

055

06

065

07

Time (s)

Out

put

WKNNWNN

norm


0 10 20 30Time (s)

Erro

r

WKNNWNN

03

025

02

015

01

005

minus005

0

norm


0 200 400 60002

03

04

05

06

07

08

09

1

Time (s)

Out

put

WKNNWNN

norm


0 200 400 600Time (s)

Erro

r

minus001

minus002

minus003

minus004

minus005

003

002

001

0

WKNNWNN

norm



0 50 100 150 200Time (s)

Out

put


minus15

minus1

minus05

0

05

1

15

2

25





0 50 100 150 200Time (s)

Erro


minus15

minus1

minus05

0

05

1

15

2

25



7 Conclusions



100 200 300 400 500 60000

02

04

06

08

10

12M

ass fl

ow ra

te (k

gs)

Time (s)


(a)

100 200 300 400 500 60000

02

04

06

08

10

12

Time (s)

Reac

tor p

ower

(b)

100 200 300 400 500 600Time (s)

550

560

570

580

590

600

Tem

pera

ture

(K)


(c)

100 200 300 400 500 600Time (s)

150

152

154

156

158Pr

essu

re (M

Pa)


(d)




Acknowledgments



References




















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 50 100 150 200Time (s)

Out

put


minus15

minus1

minus05

0

05

1

15

2

25





0 50 100 150 200Time (s)

Erro


minus15

minus1

minus05

0

05

1

15

2

25



7 Conclusions



100 200 300 400 500 60000

02

04

06

08

10

12M

ass fl

ow ra

te (k

gs)

Time (s)


(a)

100 200 300 400 500 60000

02

04

06

08

10

12

Time (s)

Reac

tor p

ower

(b)

100 200 300 400 500 600Time (s)

550

560

570

580

590

600

Tem

pera

ture

(K)


(c)

100 200 300 400 500 600Time (s)

150

152

154

156

158Pr

essu

re (M

Pa)


(d)




Acknowledgments



References




















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










100 200 300 400 500 60000

02

04

06

08

10

12M

ass fl

ow ra

te (k

gs)

Time (s)


(a)

100 200 300 400 500 60000

02

04

06

08

10

12

Time (s)

Reac

tor p

ower

(b)

100 200 300 400 500 600Time (s)

550

560

570

580

590

600

Tem

pera

ture

(K)


(c)

100 200 300 400 500 600Time (s)

150

152

154

156

158Pr

essu

re (M

Pa)


(d)




Acknowledgments



References




















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra

























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article a novel fractional-order pid controller...

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