A novel algorithm for wavelet neural networks with applicationto enhanced PID controller design$

Yuxin Zhao a, Xue Du a,n, Genglei Xia b, Ligang Wu a

a College of Automation, Harbin Engineering University, Harbin 150001, Chinab Fundamental Science on Nuclear Safety and Simulation Technology Laboratory, Harbin Engineering University, Harbin 150001, China

a r t i c l e i n f o

Article history:Received 7 October 2014Received in revised form24 November 2014Accepted 7 January 2015Communicated by Xiaojie Su.Available online 7 February 2015

Keywords:Wavelet neural network (WNN)Normalized least mean square (NLMS)Variable step-sizeParameters tuning

a b s t r a c t

This paper presents a variable step-size updating algorithm for wavelet neural network (WNN) in settingthe enhanced PID controller parameters. Compared to the iterative method with constant step-size, themost innovative character of the algorithm proposed is its capability of shortening tracking time andimproving the convergence in weights updating process for complex systems or large-scale networks. Bycombining the relationship among WNN, the Kalman filter and the normalized least mean square(NLMS), we introduce the T–S fuzzy inference mechanism for activation derived functions. Furthermore,a once-through steam generator (OTSG) model is established for validating the practicability andreliability in a real complicated system. Finally, simulation results are presented to exhibit theeffectiveness of the proposed variable step-size algorithm.

& 2015 Elsevier B.V. All rights reserved.

1. Introduction

Majority of marine nuclear power plants adopt integrateddesign, any of which once-through steam generator (OTSG)demands an efficient control system to ensure the safe operationof nuclear reactor. As a sophisticated advanced controller, PID andrelated controllers have played a significant role in industrialmanufacture with various complex dynamic systems [1,2], andmany of them involve nuclear reactor and OTSG [3–5]. Theessential question of PID controller is parameters tuning, whichis controlling system adaptively by setting parameters online. Inrecent years, artificial neural network (ANN) has developed newaspect to solve complex nonlinear uncertain problems by itsstrong learning ability and high ability of parallel computing.And neural network draws attention to process control, systemidentification and any other domains [6–9], thus PID parameterstuning problem also emerges in many different algorithms such asneural network [10–12]. In the realm of neural network, backpropagation (BP) network has the most widely application. How-ever, there are some open technical issues, and one of them is thatthe error coming from complex nonlinear function with highdimension vector causes local minimum easily.

Wavelet neural network (WNN) is structured as a novel neuralnetwork based on wavelet theory, which replaces the traditionalactivation function with wavelet function in hidden layer, andestablishes the connection between wavelet theory and networkcoefficients by affine transformation. This network, integratingadvantages of the wavelet and the ANN, is appropriate for dynamicprocessing by improving learning capacity and generalizationability, which enlarges WNN’ applications in many nonlinear andstrong coupling systems [13–15]. While WNN still faces theproblem of long tracking time in complex systems or large-scalenetwork. This paper points out that the problem of WNN existsgenerally in algorithms deriving from recursive algorithms such asstochastic gradient (SG) descent or recursive least squares (RLS),and main research achievements are produced in adaptive filteringand system identification [16–18]. It is well known that thestability and convergence of this type of algorithm is influencedby step-size parameter and the choice of this parameter reflects acompromise between the dual requirements of rapid convergenceand small misadjustment. To satisfy the conflicting requirementsabove, researchers have studied alternative means to adjust thestep-size parameter constantly, and many of studies come outfrom normalized least mean square (NLMS) algorithm [19–22] inadaptive filtering. Adaptive filtering has been frequently utilized insystem controlling, system identification, signal processing andother applications for its robustness and simplification of imple-mentation. As one of adaptive filter algorithms, the NLMS algo-rithm is a good candidate for applications with a high degree ofinput power uncertainty [23–25]. Actually adaptive filtering andsystem identification could be considered as a linear single neuron

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journal homepage: www.elsevier.com/locate/neucom

Neurocomputing

http://dx.doi.org/10.1016/j.neucom.2015.01.0150925-2312/& 2015 Elsevier B.V. All rights reserved.

☆This work was partially supported by the National Natural Science Foundationof China (60904087, 61174126, 51379049 and 61222301), and the FundamentalResearch Funds for the Central Universities (HEUCFX41302).

n Corresponding author.E-mail address: duxue2012cc@163.com (X. Du).

Neurocomputing 158 (2015) 257–267

of MISO unknown system [26], and WNN also could be regarded asa self-adaptive filter with more complex topological structure.Since the existing algorithm of NLMS has been applied in variousidentification and adaptive filter algorithms, it is the motivation ofthis paper to introduce NLMS algorithm into WNN variable step-size adjusting.

For the algorithm boundedness, NLMS is applied in linearstructure, which is difficult for WNN because of its nonlinearactivation functions. This simplification issue has received greatattention in the field of hardware implementation. In [27], a fuzzy-based activation function for artificial neural networks was pro-posed. This approach makes hardware implementation easy byinterpreting straightforward based on If–Then rules. Beside this,the performance of linear T–S fuzzy systems on approximation wasanalyzed in [28,29]. To further improve the performance of variablestep-size algorithm, we present a novel algorithm from the per-spective of state space. This thought is born in the state-spacerelationship among the NLMS, the WNN and the unscented Kalmanfilter (UKF) [30,31]. Through the application of UKF in [31] by reasonof its nonlinear application, Kalman filter is more convenient tocalculate neural network in T–S fuzzy linearization. Therefore, weconsider fully the relationship between the WNN, the Kalman filterand the NLMS, and provide the variable step-size algorithm.

The rest of this paper can be organized as follows. In Section 2, weconnect the wavelet function and the BP network, then analyze theweights updating in theory and illustrates the necessity to introducevariable step-size and T–S inference mechanism. Section 3 developsa sort of T–S fuzzy inference mechanism for activation derivedfunction, of which the membership function and relevant parameterssetting are described. In Section 4, variable step-size is inferred fromthe perspectives of output layer and hidden layer, and then wepropose a variable step-size updating algorithm and analyze theconvergence of this algorithm. The model of OTSG is structured byRELAP 5 program in Section 5, and the enhanced PID controller isalso designed. The performance of algorithm proposed in WNN isanalyzed in Section 6, which displays a faster convergence rate andless dynamic error to control dynamic nonlinear system. As for theefficacy losing problem in some real controllers, Section 6 alsoemulates and analyzes the control process in the basis of OTSGmodel. Finally, conclusion of this paper is drawn in Section 7.

2. Problem description

This paper utilizes WNN with three layers. Input and outputaccount for In and Z, respectively; φð�Þ represents wavelet function

in hidden layer, and f ð�Þ denotes sigmoid function in output layer.Weight between input layer and hidden layer is regarded as wji, aswell as the one between hidden layer and output layer iswkjði¼ 1;2;…; I; j¼ 1;2;…; J; k¼ 1;2;…;KÞ, then the output Zk(n)of WNN could be expressed that

ZkðnÞ ¼ f vkðnÞð Þ; ð2:1Þ

vkðnÞ ¼ xjðnÞwjkðnÞ¼φ vjðnÞ

� �wjkðnÞ

¼φ xTi ðnÞwijðnÞ� �

wjkðnÞ; ð2:2Þ

where vkð�Þ and vjð�Þ are the induced local fields of output layer andhidden layer, respectively, then xk(n) and xj(n) are the input signalsof neuron k and j, respectively. Thus the online PID parameterstuning is the process of wkj and wji updating. Let w(n) representgeneralized weight such as wji(n) or wkj(n), and wðnÞ is theestimation of w at iteration of n. Then the process of weightsupdating could be displayed by wðnÞ thatwðnÞ ¼ wðn�1ÞþηδðnÞxðnÞ

¼ wkjðn�1Þþηf 0 vkðnÞð Þ norm� f vkðnÞð Þ� �xjðnÞ; ð2:3Þ

or

wðnÞ ¼ wjiðn�1Þþηφ0 vjðnÞ� � X

f 0 vkðnÞð Þ norm� f vkðnÞð Þ� �wkjðnÞ

h ixiðnÞ; ð2:4Þ

where (2.3) displays output layer updating, (2.4) presents hiddenlayer adjusting and η is the step-size. Define the network outputerror ewnn and the network output error energy Ewnn(n) that

EwnnðnÞ ¼ 12 ewnnð Þ2: ð2:5Þ

Consider z¼ z1; z2;…; zk;…� �

as the group of output signals,and there is a group of ideal outputs norm¼ norm1f ;norm2;

…;normk;…g in WNN which inspires the system to steady. Assumethat k is the sequence number of output neuron, then the errorsignal from kth output neuron could be generated based on (2.1)that

ewnnk ¼ normk�zk ¼ normk� f vkðnÞð Þ: ð2:6ÞOn the basis of literature [26], the instantaneous error energy

of output neuron k is defining by

Ewnnk ðnÞ ¼ 12 e

2wnnk

ðnÞ: ð2:7Þ

And total instantaneous error energy could be obtained by

EwnnðnÞ ¼ 12

XK

e2wnnkðnÞ: ð2:8Þ

Nomenclature

In WNN inputZ WNN outputφð�Þ wavelet functionf ð�Þ sigmoid functionf Lð�Þ the piecewise linear form of f ð�Þw WNN weightw the estimation of wv(n) induced local fieldx(n) input signal of neuron/state vectorη step sizeηðnÞ step size functionδðnÞ local gradientρðnÞ the linear form of δðnÞI the number of neurons in input layer/unit matrix

J the number of neurons in hidden layerK the number of neurons in output layerewnn the network output errorEwnn(n) the network output error energynorm ideal outputξðnÞ the fuzzy logic value of derived functionϕðnÞ state transition matrixH(n) observed matrixu(n) system input signaly(n) system observed signalz(n) observed vectorθðnÞ coefficient vectorνðnÞ the process noiseP(n) the estimation of correlation matrix of errorR(n) the correlation matrix of ν1ðnÞQ(n) the correlation matrix of ν2ðnÞ

Y. Zhao et al. / Neurocomputing 158 (2015) 257–267258

η is the step-size controlling convergence and the steady-statebehavior of network studying, so the choice of η represents abalance between the adaptation and misadjustment. The majorityof papers on neural networks examine the study algorithm with aconstant step-size. Thus variable step-size algorithm of neuralnetwork is required to adjust step-size with error adaptively. Thestep-size ηðnÞ of WNN consists of output layer ηoutðnÞ and hiddenlayer ηhidðnÞ. When i represents the neuron in output layer,g ηoutðnÞ� �

is defined by using the negative gradient search (stee-pest descent method) and minimizing the networks output errorenergy Eoutput(n)

g ηoutðnÞ� �¼ EoutputðnÞ

¼ 12

XKk ¼ 1

normk� f vkðnÞð Þ� �2: ð2:9Þ

When the best step-size exists, it is satisfied that g0 ηoutðnÞ� �¼ 0.

Given (2.9) and [26], it is obvious that (2.9) can be translated into thevariable step-size issue of NLMS when f ð�Þ is a linear function withgradient equaling to one. As both f ð�Þ and f 0ð�Þ are nonlinear functions,the computational process would be complicated. Moreover, thestructure is more complex when utilizing (2.4) in (2.9), which evenincreases the updating burden of WNN. As a result, f ð�Þ and φð�Þ aswell as their derived functions are difficult to implement in adaptivestep-size adjusting. Referring to the achievement of sigmoid in realmof hardware implementation [27], the first step in this paper issimplifying the derived functions of f ð�Þ and φð�Þ by T–S fuzzyinference, which also simplifies g ηoutðnÞ

� �and g ηhidðnÞ

� �, and makes

it possible to calculate variable step-size adaptively.

3. T–S fuzzy inference mechanism for activation derivedfunctions

Motivated by the reason stated above, thus a novel form ofactivation function can be drawn by means of the T–S fuzzy logicmethodology. To activation functions (sigmoid function or waveletfunction), their derived functions are continuous differentiability.And both types of activation functions could be described aspiecewise-linear continuous functions by [27]

f ðxÞ ¼A1 when x is lowξxðnÞþa0 when x is mediumA2 when x is high

8><>: ð3:1Þ

where x(n) is the function input, A1 and A2 are boundaries when xtends to positive or negative infinity. When f(x) is derivative, thederived function could be displayed that

f 0ðxÞ ¼0 when x is low or highξ when x is medium

(ð3:2Þ

where (3.2) illustrates that the value of ξ influences the results of(2.9) and later calculations directly, therefore T–S fuzzy inferencemechanism is introduced to solve ξ, and ξ could be considered as afuzzy logic output in the context of fuzzy logic, inspired by thetheories both of WNN and T–S fuzzy. T–S fuzzy model is proposedby Takagi and Sugeno in 1985 to deal with multi-variable complexnonlinear system effectively. Unlike the Mamdani model, theconsequence of T–S fuzzy model is linear functions of variableinputs. Thus every rule includes lots of information, which couldproduce the desired effect with fewer rules and achieve the linearregression of nonlinear function more easily.

Before fuzzy regression, step functions have been utilized tosection derived functions. Piecewise linear strategy is accepted toclose to the activation derived functions by fuzzy algorithm. Dividea derived function into M subsections, and regard bm as the leftborder of th subsection. Then the fuzzy algorithm could calculate

the gradient of every subsection based on actual shape of functionand match the derived function. In this paper, T–S fuzzy systemconsists of dual inputs x¼ x1; x2½ � and single output ξðnÞ, andinputs could be illustrated that

x¼ x1; x2½ � )x1 ¼ xðnÞ;x2 ¼ xðnÞ�bm:

(ð3:3Þ

Then a typical derived function value ξðnÞ could be described bya set of following T–S fuzzy rules Ri.

Ri: If x1ðnÞ is A1i and x2ðnÞ is A2

i , then bξ iðnÞ ¼ pix1ðnÞþqix2ðnÞþriði¼ 1;2;…;MÞ, where Ai

j ðj¼ 1;2Þ represents fuzzy set of ith rule. Itis obvious that Ai

2DAi1 and xðnÞ is the input signal of neuron,

namely, input of activation function. pi, qi and ri are constantsrelated to fuzzy set, and they are inherent characteristics offunction. As the fuzzy set, Aj

i's characterized by the followingGaussian-type membership function, and Ai

j xj� �

is the grade ofmembership of xj in Aj

i that

Aij xj� �¼ exp �

xj�cijσij

!224 35; j¼ 1;2; i¼ 1;2;…;M; ð3:4Þ

which contains two parameters cji and σji. Either of them has a

physical meaning: cji determines the center and σji is the half-

width. By applying T–S fuzzy inference mechanism, output ξðnÞ iscalculated as

ξðnÞ ¼XMi ¼ 1

bμ iðnÞbξiðnÞ; ð3:5Þ

where bμ iðnÞ ¼ μiðnÞ=PM

i ¼ 1 μiðnÞ, μiðnÞ ¼ Ai1ðnÞ � Ai

2ðnÞ, and (3.5)could be rewritten that

ξðnÞ ¼ bξðnÞTμðnÞ; ð3:6Þwhere bξðnÞ ¼ bξ1ðnÞ; bξ2ðnÞ;…; bξMðnÞ

h iT, μðnÞ ¼ bμ1ðnÞ; bμ2ðnÞ;…; bμM

�ðnÞ�T . In (3.6), bξðnÞ is the value of derived function and μðnÞdetermines the degree of contribution of each rule so that eachconsequence fuzzy inference mechanism corresponds to only oneadaptation value which needs to be adapted online. By T–S fuzzyinference mechanism, WNN that is established on BP networkcould be translated to a sort of piecewise-linear continuousstructure. Then combining with the weights updating alg-orithm in literature [26], f 0 vkðnÞð Þ in (2.3) and φ0 vjð

�nÞÞ P f 0�

vkðnÞð Þ norm� f vkðnÞð Þ� �wkjðnÞ� in (2.4) could be piecewise-

simplified as linear functions. Hence (2.3) could be rewritten that

bwðnÞ ¼ bwðn�1ÞþηρðnÞxðnÞ; ð3:7Þ

where ρðnÞ is linear form of δðnÞ. And this equation is proposed onlyfor step-size recursion, which has no effect on weights updating.

4. Proposed algorithm and convergence analysis

4.1. Relationship analysis

A state-space equation could be presented by

xðnþ1Þ ¼ϕðnÞxðnÞþν1ðnÞ;zðnÞ ¼HðnÞxðnÞþν2ðnÞ;

(ð4:1Þ

where xðnþ1Þ and xðnÞ are state vectors, zðnÞ is observed vector,ν1ðnÞ and ν2ðnÞ are process noise, ϕðnÞ and HðnÞ are state transitionmatrix and observed matrix respectively at time n. As a method toestimate the system state with noise, the Kalman filter equationscould be displayed that

eðnÞ ¼ zðnÞ�HðnÞxðnjn�1Þ; ð4:2Þ

Y. Zhao et al. / Neurocomputing 158 (2015) 257–267 259

KðnÞ ¼ Pðnjn�1ÞHT ðnÞ HðnÞPðnjn�1ÞHT ðnÞþRðnÞh i�1

; ð4:3Þ

xðnjnÞ ¼ xðnjn�1ÞþKðnÞ; ð4:4Þ

xðnþ1jnÞ ¼ϕðnÞxðnjnÞ; ð4:5Þ

PðnÞ ¼ I�KðnÞHðnÞ½ �Pðnjn�1Þ; ð4:6Þ

Pðnþ1jnÞ ¼ϕðnÞPðnÞϕT ðnÞþQ ðnÞ: ð4:7ÞIn the equations above, I is a unit matrix, xðnjn�1Þ is estimated

from xðnÞ based on sequence composed by zðnÞ, PðnÞ is theestimation of correlation matrix of error, and Pðnjn�1Þ is apreliminary estimation of PðnÞ, as well as RðnÞ and Q ðnÞ accountfor the correlation matrixes of process noise ν1ðnÞ and ν2ðnÞ.

Since the adaptive filter equation is that

yðnÞ ¼ uT ðnÞθðnÞþνðnÞ; ð4:8Þwhere uðnÞ means input signal, yðnÞ is the observed signal, θðnÞ isthe coefficient vector which is supposed to be updated, and νðnÞ isstill the process noise. Then consider the state vector in (4.8) ascoefficient vector, set PðnÞ ¼ I, and ignore the updating equation(4.6) and (4.7). Thus refer to the literatures [23,25], the updatingequation of θðnÞ could be displayed thatbθðnþ1Þ ¼ bθðnÞþηðnÞuðnÞ yðnÞ�uT ðnÞbθðnÞ�

; ð4:9Þ

ηðnÞ ¼ uT ðnÞuðnÞþε� ��1

; ð4:10Þwhere ηðnÞ is regarded as variable step-size of adaptive filter, and εis a regularization factor whose role is helping the equation avoidthe problem when uT ðnÞ is too small.

Compared (4.9) with (3.7), a phenomenon could be observedthat the process of parameters updating in these equations is thesame. While the situation of ρðnÞ in hidden layer will be analyzedfurther in the next section. Based on discussion above, therelationship among linear neural network, Kalman filter and NLMScould be obtained, and this conclusion will be utilized in variablestep-size algorithm for WNN.

4.2. Algorithm formulation

As stated in the previous section, assume the piecewise-linearform of f ð�Þ to be f Lð�Þ, then f LðnÞ ¼ ξðnÞvki ðnÞþbðnÞ, and bðnÞ is theintercept of f Lð�Þ. According to NLMS theory, when ξðnÞ ¼ 1 andbðnÞ ¼ 0, f Lð�Þ is equal to (4.8) without process noise. For a betterapproximation of ηoutðnÞ in (2.9), f Lð�Þ is applied to instead of f ðxÞwhose gradient ξðnÞ is calculated at each time instant n. Althoughthis treatment of f Lð�Þ is relatively poor compared to the sigmoid(wavelet) function, it provides a more efficient solution comparedto the WNN with constant step-size. NLMS is born in linear singleneuron with scalar step. However, WNN has nonlinear multi-layerstructure with vectorial step. Hence the step of WNN is supposedto calculate layer by layer. Assume WNN to be three-layerstructure, and the number of neurons in input layer, hidden layerand output layer is I, J and K respectively.

4.2.1. Output layerBy utilizing f Lð�Þ with (3.6), (2.9) could be inferred as follows:

h ηf ðnÞ�

¼ norm�ξf WTJK ðnÞXJðnÞ

� �bf ðnÞ; ð4:11Þ

g ηf ðnÞ�

¼ 12 h ηf ðnÞ

� h i2; ð4:12Þ

where norm¼ norm1;norm2;…;normK½ �T is the desired outputof network, WJK ðJ � KÞ is the weight matrix between output layerand hidden layer, XJ ¼ x1; x2;…; xJ

� �T is the output vector of hidden

layer (amount to the input of NLMS algorithm),ξf is a diagonalmatrix including K gradients of f ð�Þ,and bf ¼ bf1; bf2;…; bfK

� �T is theintercept vector. Define that EK ðnÞ ¼ ewnn1 ; ewnn2 ;…; ewnnK

� �T , EK ðnÞis a diagonal matrix consisting of ewnn1 , ewnn2 ,…, ewnnK ,and ηf ðnÞ is adiagonal matrix composing of ηf 1 ðnÞ, ηf 2 ðnÞ;…;ηf K ðnÞ. According to(2.3),it could be found that every wkjðnÞ which to be updatedcorresponds to a xjðnÞ as the input of activation. To satisfy themultiplication principle of matrix and updating condition of (2.3),XJðnÞ is required to be expanded to XKJðnÞ with J � K , whose everyrow is constituted by XJðnÞ. Then substitute (2.4) into (4.11), thefollowing equation could be obtained that

h ηf ðnÞ�

¼ norm�ξf ðnÞ cWKJðn�1Þþ bXKJðnÞηf ðnÞξf ðnÞEK ðnÞ� T

XJðnÞ�bf ðnÞ

¼ EK ðnÞ�ηf ðnÞξ2f ðnÞEK ðnÞbXT

KJðnÞXJðnÞ

¼

h1ðnÞh2ðnÞ⋮

hK ðnÞ

266664377775: ð4:13Þ

Substitute (4.13) into (4.12), and rewrite g ηf ðnÞ�

as G1 ηf ðnÞ�

,then G1 ηf ðnÞ

� is defined that

G1 ηf ðnÞ�

¼ 12

h21ðnÞ ⋯ 0⋮ ⋱ ⋮0 ⋯ h2K ðnÞ

26643775: ð4:14Þ

Let G1 ηf ðnÞ� 0

¼ 0, then give the best step-size

ηf ðnÞ ¼ηf1ðnÞ ⋯ 0

⋮ ⋱ ⋮0 ⋯ ηfK ðnÞ

264375; ð4:15Þ

where ηfkðnÞ ¼ 1=ξ2fkðnÞ x21ðnÞþx22ðnÞþ⋯þx2J ðnÞ�

.When

PJj ¼ 1 x

2j ðnÞ is too small, the problem caused by numer-

ical calculation has to be considered. To overcome this difficulty,(4.14) is altered to

ηf k ðnÞ ¼1

ξ2f k ðnÞ x21ðnÞþx22ðnÞþ⋯þx2J ðnÞ�

þσ2v

¼ PðnÞPðnÞξ2f k ðnÞ x21ðnÞþx22ðnÞþ⋯þx2J ðnÞ

� þσ2

v

; ð4:16Þ

where PðnÞ and σ2v40 which is set as a pretty small value in (4.16).

Based on (4.16), when it is satisfied that ξðnÞ ¼ 1 and PðnÞ ¼ 1,(4.16) is the typical NLMS algorithm, when ξðnÞ ¼ 1 and PðnÞa1,(4.16) is a form of Kalman filter. From the conclusion above, (4.16)could be further optimized from the perspectives of NLMS andKalman filter. As inferred in the previous section, assuming PðnÞ ¼ Ifor all instant n is a rough approximation of PðnÞ ¼ 1. For evaluat-ing the ηoutðnÞ more precise, PðnÞ ¼ 1 is set to be λðnÞ. λðnÞ isupdating at every instant n. Since the precision of SG is less thanrecursive least squares (RLS) [27], the introduction of λðnÞ drivesthe precision to close to RLS, moreover, the proposed treatmenthas less computational complexity than the RLS algorithm. Byutilizing λðnÞ, the following update equation could be obtainedbased on (2.3) that

bwjkðnÞ ¼ bwjkðn�1Þþ f 0 vkðnÞð Þ norm� f vkðnÞð Þ� �xjðnÞ

λkðnÞDλ

; ð4:17Þ

where Dλ ¼ λkðnÞξ2f k ðnÞ x21ðnÞþx22ðnÞþ⋯þx2J ðnÞ�

þσ2v .

Given the relationship between the proposed algorithm andKalman filter theory, λðnÞ could be determined that

E norm� f vkðnÞð Þ� �2n o¼ xTk ðnÞPkðnÞxkþσ2

v

¼ λkðnÞxTk ðnÞxkðnÞþσ2v ; ð4:18Þ

Y. Zhao et al. / Neurocomputing 158 (2015) 257–267260

and λðnÞ could be produced by

λðnÞ ¼E norm� f vkðnÞð Þ� �2n o

�σ2v

xTk ðnÞxkðnÞ: ð4:19Þ

4.2.2. Hidden layerAssume that ηφðnÞ is a diagonal matrix composing of

ηφ1ðnÞ;ηφ2

ðnÞ;…;ηφJðnÞ

n o, and XJðnÞ is required to be expanded

to with XKJðnÞ, whose every row is constituted by XJðnÞ. Then thefollowing equation could be obtained that

h ηφðnÞ�

¼ norm�ξf cWT

JK ðnÞ ξφ cWIJðnÞXIðnÞ�

�bφðnÞh i �

�bf ðnÞ;

ð4:20Þ

Assuming CðnÞ ¼WJK ðnÞξf ðnÞEK ðnÞ, the consequence could beconcluded according to (4.18) that

CðnÞ ¼ ξfK ðnÞwJK ðnÞewnnK ðnÞihJ�K

¼

C1ðnÞC2ðnÞ⋮

CJðnÞ

266664377775: ð4:21Þ

Assume

DðnÞ ¼D1ðnÞ ⋯ 0⋮ ⋱ ⋮0 ⋯ DJðnÞ

264375;

where

DjðnÞ ¼XK

CjkðnÞ ¼XKk ¼ 1

ξfkðnÞwjkðnÞewnnk ðnÞ:

Combine with (2.4), and substitute the results above into (4.20)that

h ηφðnÞ�

¼ EK ðnÞ�AðnÞηφðnÞBðnÞ ¼ hn

1ðnÞ hn

2ðnÞ ⋯ hn

K ðnÞ� �T

;

ð4:22Þ

where

AðnÞ ¼ AkjðnÞ� �

K�J ¼ ξφjðnÞξf k ðnÞwjkðnÞ

h iK�J

;

BðnÞ ¼ BjðnÞ� �

J�1 ¼ DjX ij� �

J�I ½x�I�1:

Then g ηφðnÞ�

could be rewritten as G2 ηφðnÞ�

according to(4.12) and (4.22) that

G2 ηφðnÞ�

¼ 12 EK ðnÞ�AðnÞηφðnÞBðnÞh i

EK ðnÞ�AðnÞηφðnÞBðnÞh iT

:

ð4:23ÞLet G2 ηφðnÞ

� 0¼ 0, and the following equation could be

obtained that

AðnÞηφðnÞBðnÞBT ðnÞAT ðnÞ ¼ 12 EK ðnÞBT ðnÞAT ðnÞþ EK ðnÞBT ðnÞAT ðnÞ

� T� ð4:24Þ

When K¼ J, AðnÞ is a square matrix and following steps could beimplement as output linear. When Ka J, AðnÞ is not invertible.Thus the solution of ηφðnÞ could be estimated by the solutionidentification theorem, and categorization describes the situationof infinitely many solutions or no solution. If ηφðnÞ is no solution, itjust states that the best step-size could not be determined by thisrecursion algorithm. In conclusion, the result of hidden layer is stillmore complex than traditional Kalman filter even after piecewise-linearization, thus it could not optimize step-size as Kalman filterfurther.

The algorithm proposed is on the basis of WNN and infersactivation derived functions by T–S fuzzy method, then solves thestep-size ordinally. Since the weights updating in hidden layer ismore complicated than that in output layer, even though AðnÞ issquare matrix, the solution procedure would increase muchburden. According to (2.3) and (2.4), the updating of hidden layeris based on the output layer, meanwhile the existence ofP

f 0 vkðnÞð Þ norm� f vkðnÞð Þ� �wkjðnÞ helps every neuron of output

layer have the relationship with the updating process betweenith neuron of hidden layer and the previous layer, in other words,every neuron of output layer contributes to ηφðnÞ. Therefore thispaper adopts the same step-size in ηφðnÞ based on ηf ðnÞ that

ηφðnÞ ¼1K

XKk ¼ 1

ηf ðnÞ: ð4:25Þ

The WNN process is presented as Table 1.

4.3. Convergence of the algorithm proposed

WNN is utilized to solve problems by updating weights, and thecovariance of the weight matrixes is directly related to the mean-square error. When the output of WNN varies slightly and the stateremains steady, a relationship equation could be met that

limn-1

E eðnÞ2h i

¼ limn-1

E eaðnÞ2h i

þσ2v : ð4:26Þ

In the equation above, the expectation squared eaðnÞ is a priorierror vector which could be obtained [32] by

eaðnÞ ¼ uðnÞT W�cW ðnÞ�

: ð4:27Þ

W donates the optimal weight which cW ðnÞ is estimated toapproach, thus limi-1E eðnÞ2

h irepresents the excess mean square

error. Unlike the NLMS algorithm and Kalman filter algorithm,step-size has been calculated more than once in WNN, andweights also have been updated in different layers. Given thesignificant of main calculated steps in output layer and theintimation between error and output layer weights for WNNback-propagation, the convergence of steps in output layer comesto be decisive for the whole step set. Therefore the convergence ofsteps in output layer is discussed principally in this section.Referring to [32], assume the step-size of the proposed algorithm

Table 1Algorithm process.

Input: Input signal In Ideal output normOutput: Observe output ZðnÞ Weights WðnÞ

Initialize: The number of layers in WNN structure Nlayer

The number of neurons in every layer M1, M2…The activation function of every layer f 1ð�Þ, f 2ð�Þ…Loop count Num

Begin:Initialize weights W0

RepeatOutput layer ZðnÞ ¼ f vkðnÞð ÞT–S fuzzy inference: activation derived functionAlgorithm proposed: output layer ηf ðnÞ

hidden layer ηφ ¼PK

k ¼ 1ηf ðnÞ

� �K

Update WðnÞUntil n4Num

End

Y. Zhao et al. / Neurocomputing 158 (2015) 257–267 261

in the steady state to be ηf ð1Þ which could be illustrated that [25]

ηð1Þ ¼ ηf ð1ÞJXJ J2 ¼ 1� σ2v

limn-1E eðnÞ2h i

0@ 1A: ð4:28Þ

Based on literature [32] and assume RXJ as the correlationmatrix of XJ, the following equation could be satisfied that

limn-1

E eðnÞ2h i

¼ ηð1Þσ2v

2�ηð1ÞTr RXJ

� E

1JXJ J2

� �þσ2

v : ð4:29Þ

From the point of [32], (4.29) is summarized by fixed step-size,while it is introduced in this section for variable step-size, thereason of this conduct is that ηf ð1Þwill remain unchanged when napproaches infinity and state maintains stable. Then the followingequation could be inferred that

limn-1

E eðnÞ2h i

¼E eðnÞ2h i

�σ2v

E eðnÞ2h i

þσ2v

0@ 1Aσ2vTr RXJ

� E

1JXJ J2

� �þσ2

v : ð4:30Þ

And it could be rearranged that

limn-1

E eðnÞ2h i

�σ2v ¼ E eðnÞ2

h i�σ2

v

� σ2vTr RXJ

� E eðnÞ2h i

þσ2v

0@ 1AE1

JXJ J2

� �:

ð4:31Þ

Based on statistical characteristics of XJ, σ2vTr RXJ

� =ðE

eðnÞ2h i

þσ2v Þ could be considered as an arbitrary value. Thus to

establish the condition of (4.31), limn-1E eðnÞ2h i

is supposed toequal to σv2, and it suggests that E eðnÞ2

h iis influenced by σv2. When

the WNN approaches stable and σv2 decrease to zero, ηf ð1Þ andηð1Þ could be calculated that

ηf ð1Þ ¼ 1JXJ J2

ηð1Þ ¼ 1JXJ J2

1� σ2v

limn-1E eðnÞ2h i

0@ 1A� 0: ð4:32Þ

As ηf ð1Þ comes from the error of WNN, the results abovereflect a class of desired weights for the neural network, and implythe stable state of WNN. Hence, the convergence of ηf ð1Þ couldinfluence ηφð1Þ and step-size in other layers if any.

5. The model and PID controller of OTSG

The OTSG with double-size heat transfer has more compactstructure and stronger heat exchange capability, thus this OTSG isa sort of efficient heat transfer steam generator, whose heatexchange capability could reach 2.6 times as high as spiral tubesteam generator. The OTSG in concentric annuli tube could reducethe scale of reactor pressure vessel (RPV) and improve mobility ofthe device. Compared with natural circulation steam generatorU-tube, the OTSG is designed with simpler structure but less watervolume in heat transfer tube. Since it is difficult to measure thewater level in OTSG especially during variable load operation, theOTSG needs a high-efficiency control system to ensure safeoperation. From the OTSG research, the steam pressure of outletcould be influenced by feed-water and steam flow. The exchangeof steam pressure has an effect on the transfer of reactor heat fromprimary loop to second loop, which impacts the average tempera-ture of primary coolant. This transformation is a synchronous andcomplicated process. Thus the argument above illustrates that theoutlet steam pressure of OTSG is a significant variable, and astrategy to control outlet steam pressure should be utilized in thispressure. With small water capacity, steam pressure could changeeasily in the period of steam flow transformation, and a high-efficiency feed-water control system is demanded to remain steampressure stable. In this section, the OTSG model is established byRELAP5 transient analysis program. The WNN enhanced PIDsystem achieves outlet steam pressure invariability by controllingsecondary feed water system (Fig. 1).

In OTSG system, the essential equation of thermal-hydraulicmodel is listed that

(1) Mass continuity equations

∂∂t

αgρg

� þ1A∂∂x

αgρgvgA�

¼Γg ; ð5:1Þ

∂∂t

αfρf

� þ1A∂∂x

αfρf vf A�

¼Γf : ð5:2Þ

(2) Momentum conservation equations

αgρgA∂vg∂t

þ12αgρgA

∂v2g∂x

¼ �αgA∂P∂x

þαgρgBxA�ðαgρgAÞFWGðvgÞ

þΓgAðvgI�vgÞ�ðαgρgAÞFIGðvg�vf Þ

Fig. 1. Control system of reactor load following.

Y. Zhao et al. / Neurocomputing 158 (2015) 257–267262

�CαgαfρmA∂ðvg�vf Þ

∂tþvf

∂vg∂x

�vg∂vf∂x

� ; ð5:3Þ

αfρf A∂vf∂t

þ12αfρf A

∂v2f∂x

¼ �αf A∂P∂x

þαfρf BxA�ðαfρf AÞFWFðvf Þ�ΓgAðvfI�vf Þ�ðαfρf AÞFIFðvf �vgÞ

�CαfαgρmA∂ðvf �vgÞ

∂tþvg

∂vf∂x

�vf∂vg∂x

� : ð5:4Þ

(3) Energy conservation equations

∂∂t

αgρgUg

� þ1A∂∂x

αgρgUgvgA�

¼ �P∂αg

∂t�PA∂∂x

αgvgA� �þQwgþQigþΓigh

n

gþΓwh0gþDISSg ;

ð5:5Þ

∂∂t

αfρf Uf

� þ1A∂∂x

αfρf Uf vf A�

¼ �P∂αf

∂t�PA∂∂x

αf vf A� �þQwf þQif �Γigh

n

f �Γwh0f þDISSf :

ð5:6Þ(4) Noncondensables in the gas phase

∂∂t

αgρgXn

� þ1A∂∂x

αgρgvgXnA�

¼ 0: ð5:7Þ

(5) Boron concentration in the liquid fieldThe RELAP5 thermal-hydraulic model solves eight field equa-

tions for eight primary dependent variables. The primary depen-dent variables are pressure (P), phasic specific internal energies(Ug,Uf), vapor volume fraction (void fraction) (αg), phasic velocities(vg,vf), noncondensable quality (Xn), and boron density (ρb). Theindependent variables are time (t) and distance (x). The secondarydependent variables used in the equations are phasic densities (ρg,ρf), phasic temperatures (Tg,Tf), saturation temperature (Ts), andnoncondensable mass fraction in noncondensable gas phase (Xni)[33].

To meet the operation demand of nuclear power equipment,steam power should be maintained within a certain range by OTSGfeed-water control. Based on theoretical analysis and designexperience, enhanced PID controller is adopted which comprisesreactor power control system and secondary feed-water controlsystem, and the two subsystems run in the meantime. The controlof outlet steam power of OTSG is generated by secondary feed-water system, and the one of reactor power could be achieved bythe average temperature of primary coolant. When the load alters,feed-water flow and reactor power could be determined byequations below

Gw ¼ K1GsþKPp ΔPsðnÞ�ΔPsðn�1Þ� �þKPi ΔPsðnÞ� �

þKPd ΔPsðnÞ�2ΔPsðn�1ÞþΔPs n�2ð Þ� �: ð5:8Þ

In the function, Gs represents new steam flow. K1 is theconversion coefficient. KPp, KPi and KPd are control coefficients.And ΔP is the pressure divergence of OTSG. To achieve PIDparameter tuning adaptively, the algorithm proposed above inthe previous section has adopted to improve the performance ofWNN, and three layer structure has been applied with systemerror esys, and the WNN input could be displayed that

in1ðnÞ ¼ esysðnÞ�esysðn�1Þ;in2ðnÞ ¼ esysðnÞ;in3ðnÞ ¼ esysðnÞ�2esysðn�1Þþesysðn�2Þ;

8><>: ð5:9Þ

uðnÞ ¼ uðn�1ÞþΔu; ð5:10Þ

Δu¼ KPp ΔPsðnÞ�ΔPsðn�1Þ� �þKPi ΔPsðnÞ� �

þKPd ΔPsðnÞ�2ΔPsðn�1ÞþΔPsðn�2Þ� �; ð5:11Þ

where the number of output neurons and input neurons is thesame. As the boundedness of sigmoid function, WNN outputparameters for enhanced PID are remained in a range of [�1, 1].To solve this problem, the expertise improvement unit is addedbetween WNN and flow require counter in Fig. 1, which correctsparameters order of magnitudes. And the WNN weights for settingPID are altered tobwðnÞ ¼ bwðn�1ÞþηδðnÞxðnÞ

¼ bwjkðn�1Þþηsgn∂Ps

∂Δu

� �f 0 vkðnÞð ÞesysðnÞinkðnÞxjðnÞ ð5:12Þ

or

bwðnÞ ¼ bwijðn�1ÞþX

sgn∂Ps

∂Δu

� �f 0 vkðnÞð ÞesysðnÞinkðnÞwjkðnÞ

� �ηφ0 vjðnÞ

� �xiðnÞ: ð5:13Þ

By substituting (2.3) into (5.12), the equation could be dis-played as follows:

h ηf ðnÞ�

¼ norm�ξf ðnÞ cWJK ðn�1Þþsgn∂Ps

∂Δu

� ���XJK ðnÞηf ðnÞξf ðnÞInðnÞEK ðnÞ

TXJðnÞ�bf ðnÞ; ð5:14Þ

where Ps is secondary outlet pressure of OTSG, and InðnÞ is adiagonal matrix composed of in1ðnÞ, in2ðnÞ and in3ðnÞ.

6. Simulation results and discussion

In this section, simulations of T–S fuzzy inference mechanismand enhanced PID controller have been carried out aiming toexamine performance and assess accuracy of the algorithm pro-posed in this paper. To display the results clearly, simulations aredivided into two parts. One is achieved as numerical examples inMatlab program which includes T–S fuzzy method and controllerutilized in nonlinear control systems. Another is written in FOR-TRAN language, and connected with RELAP 5 which implementsthe OTSG modeling.

6.1. Numerical examples

This paper applies T–S fuzzy inference mechanism for calculat-ing nonlinear continuously differentiable function by piecewise-linear, and dispose sigmoid function and Morlet function by linearregression respectively. To illustrate this method clearly, Table 2 isoffered to display fuzzy sets and related rules. As sigmoid functionis axisymmetric and Morlet function is centrosymmetric, Table 2only lists the parameters of functions in the positive violent field.

Sigmoid function and Morlet function are continuously differ-entiable and bounded, and their derived functions have the sameproperties, as Figs. 2–5 shows. The properties provide the possi-bility for T–S fuzzy inference mechanism. Based on the member-ship functions in Section 3 and the rules in Table 2, the derivedfunction of sigmoid and Morlet could be calculated in Figs. 6 and 7.

Figs. 6 and 7 display the linear regression of derived functionsby T–S fuzzy inferring. T–S fuzzy algorithm proposed is comparedwith the piecewise step approach algorithm which shows that T–Sfuzzy appears a better effect, especially in the dramatic realms(detailed drawing). T–S fuzzy solves the rule consequences adap-tively by fuzzy set, which could obtain more precise results. Theinference of derived functions is the theoretical basis of step-sizeupdating, thus the consequence would influence the value of kf in

Y. Zhao et al. / Neurocomputing 158 (2015) 257–267 263

later calculation. It is indicated in Figs. 6 and 7 that the errorbetween original function and consequence of T–S is slight, and T–S fuzzy is linear to produce less computational burden, therefore

the T–S fuzzy inference proves to be available. Then this paper thealgorithm proposed is designed to examine the effectiveness, andthe enhanced PID controller with updating parameters is applied

Table 2Consequent parameters of T–S fuzzy rules.

x1 x2 Sigmoid derived function Morlet derived function

p(i) q(i) r(i) p(i) q(i) r(i)

0–0.5 0–0.167 �0.2136 0.15 0.5 �2.9364 �1.4 0.000010–0.5 0.167–0.333 �0.2136 0.00001 0.524 �2.9364 0.00001 �0.230–0.5 0.333–0.5 �0.2136 �0.11 0.555 �2.9364 1.4 �0.70.5–1.0 0–0.167 �0.3664 �0.015 0.5764 �1.0638 �1.35 �2.00010.5–1.0 0.167–0.333 �0.3664 0.00001 0.5814 �1.0638 0.00001 �2.20.5–1.0 0.333–0.5 �0.3664 0.02 0.5664 �1.0638 1.2 �2.61.0–1.5 0–0.167 �0.2392 �0.04 0.4492 2.1582 0.4 �3.09451.0–1.5 0.167–0.333 �0.2392 0.00001 0.4412 2.1582 0.00001 3.01451.0–1.5 0.333–0.5 �0.2392 0.04 0.4292 2.1582 �0.4 3.29451.5–2.0 0–0.167 �0.1102 �0.04 0.2557 0.3874 0.6 �0.43831.5–2.0 0.167–0.333 �0.1102 0.00001 0.2497 0.3874 0.00001 �0.33831.5–2.0 0.333–0.5 �0.1102 0.04 0.2357 0.3874 �0.6 �0.13832.0–2.5 0–0.167 �0.044 �0.02 0.1233 �0.4552 0.05 1.24692.0–2.5 0.167–0.333 �0.044 0.00001 0.1203 �0.4552 0.00001 1.24992.0–2.5 0.333–0.5 �0.044 0.02 0.1133 �0.4552 0.01 1.2422.5–3.0 0–0.167 �0.0168 �0.002 0.0553 �0.2186 �0.12 0.65542.5–3.0 0.167–0.333 �0.0168 0.00001 0.0548 �0.2186 0.00001 0.63542.5–3.0 0.333–0.5 �0.0168 0.002 0.0543 �0.2186 0.12 0.59543.0–3.5 0–0.167 �0.0062 �0.002 0.0235 �0.0132 �0.04 0.03923.0–3.5 0.167–0.333 �0.0062 0.00001 0.023 �0.0132 0.00001 0.03423.0–3.5 0.333–0.5 �0.0062 0.002 0.0225 �0.0132 0.02 0.02923.5–4.0 0–0.167 �0.0023 0.0005 0.0097 0.0112 0.0005 �0.04623.5–4.0 0.167–0.333 �0.0023 0.00001 0.0097 0.0112 0.00001 �0.04623.5–4.0 0.333–0.5 �0.0023 �0.0005 0.00995 0.0112 �0.0005 0.0464.0–4.5 0–0.167 �0.0008 0.0005 0.0041 0.0027 0.00001 �0.01214.0–4.5 0.167–0.333 �0.0008 0.00001 0.0041 0.0027 0.00001 �0.01214.0–4.5 0.333–0.5 �0.0008 �0.0005 0.00435 0.0027 0.00001 �0.0121

−5 0 5−0.2

0

0.2

0.4

0.6

0.8

1

x position

Sigmoid Function

y po

sitio

n

Fig. 2. Sigmoid function.

−10 −5 0 5 10−0.5

0

0.5

1

1.5

x position

Morlet Function

y po

sitio

n

Fig. 3. Morlet function.

−5 0 5−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

x position

Sigmoid Derived Function

y po

sitio

n

Fig. 4. Sigmoid derived function.

−10 −5 0 5 10−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

x position

Morlet Derived Function

y po

sitio

n

Fig. 5. Morlet derived function.

Y. Zhao et al. / Neurocomputing 158 (2015) 257–267264

to control known nonlinear system. The nonlinear systems aredisplayed that

norm¼ a1 sin b1πnð Þþc1 log dn; ð6:1Þ

norm¼ a2 cos b2πnð Þþc2n: ð6:2ÞFigs. 8–11 show the control performance between variable

step-size and constant step-size. They adopt constant step-sizein WNN and the algorithm proposed to control nonlinear system,and both methods bring relatively large system error in initialperiod. In Figs. 8 and 9, the two algorithm shows roughly sameerror in 0–30 s. After 30 s, the curve of variable step-size is lowerthan constant step-sizes. Error amplitude is also smaller thanconstant step-sizes after 50 s. Figs. 10 and 11 show that thenonlinear system could approach stable state in 30 s based onthe algorithm proposed, while system could not be steady until50 s on the condition of constant step-size in WNN. In the timerealm of system error varying dramatically, the two algorithmspresent different performances. In 10–12 s, error of both methodsis roughly same, then the adaptively adjusting of step-size beginsto work at 12 s and the amplitude of error is smaller than theconstant step-sizes. This comparison indicates that enhanced PIDcontroller with variable step-size in WNN could following thenonlinear system with better performance. From the argumentabove, the conclusion could be summarized that variable step-sizein WNN is adjusted actively by system error, which could reducethe amplitude of variation, advance the weights updating rage,and control nonlinear system adaptively.

6.2. Relap5 operation example

This paper establishes the Integrated Pressurized Water Reactor(IPWR) model by RELAP5 transient analysis program, and thecontrol strategy in Fig. 1 is applied to control outlet pressure of

−4 −2 0 2 4−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

x position

Sigmoid Derived Functiony

posi

tion

Origianal FunctionPiecewise StepT−S Fuzzy

−0.5 0 0.50.4

0.45

0.5

1 1.5 2

0.1

0.2

Fig. 6. Sigmoid derived function regression.

−5 0 5−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

x position

Morlet Derived Function

y po

sitio

n

Original FunctionPiecewise StepT−S Fuzzy 0.4 0.6 0.8

−1.4

−1.2

1.5 20.10.20.3

Fig. 7. Morlet derived function regression.

0 100 200 300 400 500 6000

100

200

300

400

500

600

700

time(s)

norm,yout

Constant Step SizeVariable Step Size

20 40 60150

200

250

300

350400

Fig. 8. The output results in the first system.

0 100 200 300 400 500 600−50

0

50

100

150

time(s)

erro

r

Constant Step SizeVariable Step Size

20 40 600

50

100

150

Fig. 9. The error in the first system.

0 100 200 300 400 500−20

0

20

40

60

80

100

time(s)

norm

,you

t

Constant Step SizeVariable Step Size

0 50

0

20

40

Fig. 10. The output results in the second system.

0 100 200 300 400 500−30

−20

−10

0

10

20

30

40

time(s)

erro

r

Constant Step SizeVariable Step Size

0 50−20

0

20

40

Fig. 11. The error in the second system.

Y. Zhao et al. / Neurocomputing 158 (2015) 257–267 265

OTSG. As the boundedness of sigmoid function, WNN outputparameters for enhanced PID are remained in a range of [�1, 1].To solve this problem, the expertise improvement unit is addedbetween WNN and flow require counter in Fig. 1, which correctsparameters order of magnitudes. Through the analysis of loadshedding limiting condition of nuclear power unit, the reliabilityand availability of the control method proposed have beendemonstrated. In the process of load shedding, secondary steamdemand reduces from 100%FP to 20%FP in 5 s, then the operatingcharacteristic could be displayed in Figs. 12–16.

Fig. 12 illustrates that when load reduces, steam descendsquickly. Since the hysteresis exists in feed water control system,OTSG produces more steam than secondary steam demand in theinitial stage of load shedding, which cause the rapid increasing ofOTSG outlet steam pressure and the maximum even reaches

4.7 MPa. On the condition of large deviation, the feed water flowof control system is adjusted to reducing, then outlet steampressure reduces steam production decreasing.

Since the variation process of steam pressure appears quickly, alarge deviation would be produced and creates certain of overshoot.By adjusting for 20 s around, the feed water flow could match steamflow eventually, and maintain steam pressure steady as Fig. 13.

Fig. 14 displays the variation of outlet and inlet temperature ofOTSG based on average temperature of primary coolant in theprocess of load shedding. The reducing of heat absorption in OTSGsecond side causes the decreasing of secondary feed water flow.On the condition of coolant flow remaining stable, temperaturedifference lessens while primary coolant average temperature rise.By adjusting the reactor power down, power has been achieved tomatch second side feed water flow, and the variation of reactorpower in load shedding is displayed in Fig. 15.

In the process of load shedding, the steam pressure has beenremained steady and secondary saturation temperature keepsstable, which accomplishes the following control and ensuresthe superheat degree of OTSG outlet steam.

7. Conclusions

In this paper, a variable step-size updating algorithm has beeninvestigated for wavelet neural network. This algorithm takesadvantage of T–S fuzzy inference and NLMS method and improvesthem to be suitable for the variable step-size of WNN, whichendows the algorithmwith availability and reliability. Based on thealgorithm proposed, the enhanced PID with updating parameterscould control nonlinear system with less error and faster conver-gence rate. Motivated by the practicality of algorithm, the model ofOTSG is established. Meanwhile, the algorithm is utilized in a

Fig. 12. Operating characteristic A.

Fig. 13. Operating characteristic B.

Fig. 14. Operating characteristic C.

Fig. 15. Operating characteristic D.

Fig. 16. Operating characteristic E.

Y. Zhao et al. / Neurocomputing 158 (2015) 257–267266

certain simulation of load shedding which could be regarded as anunknown complicated nonlinear process. Simulation results havevalidated the effectiveness of the proposed variable step-sizealgorithm.

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Yuxin Zhao received the B.S. degree in Automationfrom Harbin Engineering University, China, in 2001; thePh.D. degree in Navigation Guidance and Control fromHarbin Engineering University, China, in 2005; andcompleted the post-doctorial research in ControlScience and Engineering from Harbin Institute of Tech-nology, China, in 2008. From September 2004 toJanuary 2005, he was a Visiting Scholar in the StateUniversity of New York, USA. From March 2012 toMarch 2013, he was a Research Associate in the Centrefor Transport Studies, Department of Civil and Environ-mental Engineering, Imperial College London, London,UK. In 2001, he joined the Harbin Engineering Uni-

versity, China, and was promoted to a Professor in 2013.Dr. Zhao is the member of Royal Institute of Navigation, the senior member of

China Navigation Institute, and the member of Mission Planning Committee ofChinese Society of Astronautics. His current research interests include complexhybrid dynamical systems, optimal filtering, multi-objective optimization techni-ques, and intelligent vehicle navigation systems.

Xue Du received the B.S. degree and M.S. degree from theCollege of Automation, Harbin Engineering University in2010 and 2012 respectively. And she is pursuing for herPh.D. degree in Control Science and Engineering fromHarbin Engineering University. Her research interestsinclude adaptive and learning control, neural networkcontrol, industrial systems control and the applications.

Genglei Xia received the B.S. degree from the Depart-ment of Thermal Engineering, Shandong Jianzhu Uni-versity, Jinan, China, in 2007, and the M.S. degree fromthe College of Nuclear Science and Technology, HarbinEngineering University, Harbin, China, in 2010. He iscurrently pursuing for Ph.D. degree in Nuclear Scienceand Technology from Harbin Engineering University.His research interests include thermal analysis ofnuclear reactors, two phase flow instability researchand the simulation of nuclear power plant.

LigangWu received the B.S. degree in Automation fromHarbin University of Science and Technology, China, in2001; the M.E. degree in Navigation Guidance andControl from Harbin Institute of Technology, China, in2003; the Ph.D. degree in Control Theory and ControlEngineering from Harbin Institute of Technology, China,in 2006. From January 2006 to April 2007, he was aResearch Associate in the Department of MechanicalEngineering, The University of Hong Kong, Hong Kong.From September 2007 to June 2008, he was a SeniorResearch Associate in the Department of Mathematics,City University of Hong Kong, Hong Kong. From Decem-ber 2012 to December 2013, he was a Research Associ-

ate in the Department of Electrical and Electronic Engineering, Imperial CollegeLondon, London, UK. In 2008, he joined the Harbin Institute of Technology, China,as an Associate Professor, and was then promoted to a Professor in 2012.

Dr. Wu currently serves as an Associate Editor for a number of journals, includingIEEE Transactions on Automatic Control, Information Sciences, Signal Processing,and IET Control Theory and Applications. He is also an Associate Editor for theConference Editorial Board, IEEE Control Systems Society. Dr. Wu has publishedmore than 100 research papers in international referred journals. He is the authorof the monographs Sliding Mode Control of Uncertain Parameter-Switching HybridSystems (John Wiley & Sons, 2014), and Fuzzy Control Systems with Time-Delay andStochastic Perturbation: Analysis and Synthesis (Springer, 2015). His current researchinterests include switched hybrid systems, computational and intelligent systems,sliding mode control, optimal filtering, and model reduction.

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