research article fuzzy counter propagation neural network...

Research ArticleFuzzy Counter Propagation Neural Network Control fora Class of Nonlinear Dynamical Systems

Vandana Sakhre1 Sanjeev Jain1 Vilas S Sapkal2 and Dev P Agarwal3

1Madhav Institute of Technology amp Science Gwalior 474005 India2SGB Amravati University Amravati 444062 India3IIT Delhi 110001 India

Correspondence should be addressed to Vandana Sakhre vssakhregmailcom

Received 3 May 2015 Revised 1 August 2015 Accepted 3 August 2015

Academic Editor Ezequiel Lopez-Rubio

Copyright copy 2015 Vandana Sakhre et alThis is an open access article distributed under theCreativeCommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Fuzzy Counter Propagation Neural Network (FCPN) controller design is developed for a class of nonlinear dynamical systems Inthis process the weight connecting between the instar and outstar that is input-hidden and hidden-output layer respectively isadjusted by using FuzzyCompetitive Learning (FCL) FCL paradigm adopts the principle of learning which is used to calculate BestMatched Node (BMN) which is proposedThis strategy offers a robust control of nonlinear dynamical systems FCPN is comparedwith the existing network like Dynamic Network (DN) and Back Propagation Network (BPN) on the basis of Mean Absolute Error(MAE) Mean Square Error (MSE) Best Fit Rate (BFR) and so forth It envisages that the proposed FCPN gives better results thanDNandBPNThe effectiveness of the proposed FCPN algorithms is demonstrated through simulations of four nonlinear dynamicalsystems and multiple input and single output (MISO) and a single input and single output (SISO) gas furnace Box-Jenkins timeseries data

1 Introduction

With the rapid advance of computers the digitization hasswept in all fields of science and technology with specialemphasize on modeling and identification Most of the phys-ical processes are continuous in nature but they are generallymodeled using discrete time with the use of derivative andintegral as they are easily realizable Taking these advantagesan enormous research has been focused on discrete timemethods andmany techniques based on linear and nonlineardiscrete systems have been developed [1 2] Neural networkhas been widely used for nonlinear dynamical systems [3ndash5] Various control systems like back stepping control [6 7]sliding mode control [8 9] and control using soft computing[10] are continuous source of interest The nonlinear dynam-ical system is a generic problem which finds its applicationin every field of engineering The technique based on softcomputing fuzzy logic and neural network has also foundits application in modeling of systems in various applicationdomains Feedforward Neural Network (FNN) is one of themost commonly used networks for this purpose FNN with

Back Propagation algorithm is another powerful networkNeural network ismost popular approach due to its capabilityofmodelingmost of the nonlinear functions approximately toany arbitrary degree of accuracy [11] Most of the systems aredesigned using feedforward network with gradient descentlearning but the gradient descent method encounters theproblem of slow convergence This problem of convergencecan be overcome by use of some Cauchy-Newton [12] Quasi-Newtonian method and Levenberg-Marquardt algorithms[13]

Optimal controllers for hybrid dynamical systems (HDS)developed using Hamilton-Jacobi-Bellman (HJB) solutionmethod [14 15] Novel Lyapunov-Krasovskii functional(LKF) with triple integral time constructed for exponentialsynchronization of complex dynamical networks will controlpacket loss and additive time varying delay [16] Control ofdiscrete time varying system using dynamical model isdifficult To overcome this condition new neural networkapproximation structure was developed to solve optimaltracking problem of nonlinear discrete time varying timesystem using reinforcement learning (RL) method [17] To

Hindawi Publishing CorporationComputational Intelligence and NeuroscienceVolume 2015 Article ID 719620 12 pageshttpdxdoiorg1011552015719620

2 Computational Intelligence and Neuroscience

enhance the performance of nonlinear multivariable systemfuzzy optimal controller based Takagi-Sugeno model wasdeveloped which was used to minimize a quadratic per-formance index [18] An ultimate stable training algorithminspired by adaptive observer for black box identificationof nonlinear discrete systems developed using state spacerecurrent neural networkThe algorithmdevelopedwas usingonly input and output measurement in discrete time [19]Indirect adaptive controller that uses neural network waspresented for the identification and control of experimentalpilot distillation column This system is multivariable withunknowndynamics and the neural networkwas trained usingLevenberg-Marquardt algorithm [20] Adaptive finite timestabilization of a class of switched nonlinear systems wasinvestigated with unknown nonlinear terms using neuralnetwork The finite law and adaptive law were constructedusing radial basis function neural network (RBFNN) toapproximate the unknown packaged function Stability anal-ysis of RBFNN was observed to evaluate the states of theclosed loop systems [21] An adaptive fuzzy output trackingcontrol approach was proposed in [22] for single input singleoutput (SISO) systemwhich is uncertain and nonlinear underarbitrary switching An iterative learning control schemewas proposed for a class of nonlinear dynamic systemswhich includes holonomic systems as its subsets with linearfeedback mechanism and feedforward learning strategies[23]

In this proposed work we have used instar-outstar struc-ture based CPN with Fuzzy Competitive Learning (FCL)We have designed Fuzzy Counter Propagation Networkdesign to control some nonlinear dynamical systems In theFCPN CPN model is trained by FCL algorithms The FCLlearning is used for adjusting weights and update of BestMatched Node (BMN) in discrete time nonlinear dynamicalsystem

The main contributions of this research are as follows

(1) This paper contributes the approximation for a classof nonlinear dynamical systems using Fuzzy Compet-itive Learning Based Counter Propagation Network(FCPN)

(2) FCPN is employed to optimize the Mean AbsoluteError Mean Square Error Best Fit Rate and so forth

(3) Performance criteria of proposed FCPN for nonlineardynamical systems are effectively improved by com-pensating reference signal and controller signal

The paper is organized as follows In Section 2 problemformulations for a class of nonlinear dynamical systems aregiven Section 3 contains description of Feedforward NeuralNetwork (FNN) with Back Propagation Network DynamicNetwork and Fuzzy Learning Based Counter PropagationNetwork Dynamical learning for CPN and optimal learningof dynamical system are presented in Section 4 The simula-tion results and comparison of different nonlinear dynamicalmodels are presented in Section 5 Section 6 gives conclusionof the study

2 Problem Formulation

Let us consider four models of discrete time nonlineardynamical systems [24] for single input single output (SISO)and multiple input and single output (MISO) system con-sidered in this paper and they are described by differenceequations (1)ndash(4) and Box-Jenkins time series data [10] Let119891 119877119899 rarr 119877 and 119892 119877119898 rarr 119877 be the nonlinear continuousdifferentiable function of Model IndashModel IV approximatedby FCPN to the desired degree of accuracy Once the systemhas been parameterized the performance evaluation has beencarried out by FCPN for (1) to (4)

Model I

119910 (119896 + 1) =119899minus1

sum119894=0

120572119894119910 (119896 minus 119894)

+ 119892 [119906 (119896) 119906 (119896 minus 119898 + 1)]

(1)

Model II

119910 (119896 + 1) = 119891 [119910 (119896) 119910 (119896 minus 119899 + 1)]

+119898minus1

sum119894=0

120573119894119906 (119896 minus 119894)

(2)

Model III

119910 (119896 + 1) = 119891 [119910 (119896) 119910 (119896 minus 119899 + 1)]

+ 119892 [119906 (119896) 119906 (119896 minus 119898 + 1)] (3)

Model IV

119910 (119896 + 1) = 119891 [119910 (119896) 119910 (119896 minus 119899 + 1) 119906 (119896)

119906 (119896 minus 119898 + 1)] (4)

where [119906(119896) 119910(119896)] represent the input-output pair of thesystem at time 119896 and their order is represented by (119899119898)FCL is used to learn the system defined for (1) to (4) and theperformance of the FCPN can be measured by error functiongiven in

119864 (119896) =1

2[119910 (119896) minus 119910 (119896)]

2

(5)

where 119906(119896) is the input 119910(119896) is the system output 119910(119896) isneural network output and (5) can be written for neuralcontroller as

119864119888 =1

119875

119875

sum119901=1

[119910 (119896) minus 119910119888

(119896)]2

(6)

where 119910119888(119896) is neural controller output and (6) is known asminimized error function and 119875 is total number of inputpatterns

3 Back Propagation Network (BPN)

A neural network is one of the most popular intelligentsystems which has capability to approximate calculated andtarget value at an arbitrary degree of accuracy BPN is one

Computational Intelligence and Neuroscience 3

x1

xi

xn

Z1

Zj

Zp

1 1

Y1

Yk

Ym

x1 y1

yk

ym

t1

tk

tm

X1

Xj

Xn

op

1i

1j

1p

i1

ij

ip

n1

np

nj

o1

oj

w1j

wj1

wp1

w1k

w1m

wjk

wjm

wpk

wpm

w01

w0m w0k

Figure 1 General architecture of BPN

of the most commonly used networks for this purpose TheBPN consists of input layer hidden layer and output layerand possesses weighted interconnections The BPN is basedon supervised learning This method is used where error ispropagated back to the hidden layer The aim of this neuralnetwork is to train the net to achieve a balance betweenthe netrsquos ability to respond (memorization) and its ability togive reasonable response to the input that is similar but notidentical to the one that is used in training (generalization)For a given set of training input-output pairs this networkprovides a procedure for changing weights to classify thegiven input patterns correctly The weight update algorithmis based on gradient descent method [25] The networkarchitecture is shown in Figure 1

31 Dynamic Network Neural network can be classifiedinto two categories (i) Static and (ii) Dynamic In StaticNeural Network output can be calculated directly from inputthrough feedforward network But in Dynamic Networkoutput depends on current or previous inputs outputs orstates of network Where the current output is function ofcurrent input and previous output it is known as recurrent(feedback) network Training process of Static and DynamicNetwork can be differentiated by use of gradient or Jacobianwhich is computed Dynamic Network contains delays andoperates on a sequence of inputs Dynamic Networks canhave purely feedforward connections or they can have somerecurrent connections They can be trained using DynamicBack Propagation Network [26]

The general equation of the net input 119899119898(119905) for119898th layeris given by

119899119898 (119905) = sum

119897isin119871119891119898

sum119889isinDL119898sdot119897

IW119898sdot119897 (119889) 119886119897 (119905 minus 119889)

+ sum119897isin119868119898

sum119889isinDI119898sdot119897

IW119898sdot119897 (119889) 119887119897 (119905 minus 119889) + 119887119898(7)

where 119887119897(119905) is the 119897th input vector at time 119905 IW119898119897 is the inputweight between 119897th and 119898th layer LW119898119897 is the layer weightbetween 119897th and119898th layer and 119887119898 is the bias vector for layer119898 DL

119898119897is the set of all delays in the Tapped Delay Line

(TDL) between 119897th and 119898th layer and DI119898119897

is the set of alldelays in the TDL between input 119897 and119898th layer 119868

119898is the set

of indices of input vectors that connect to layer119898 and 119871119891119898is

the set of indices of layers that connect to layer119898 The outputof119898th layer at time 119905 is computed as

119886119898 (119905) = 119891119898 (119899119898 (119905)) (8)

Network has a TDL on the input with DI11

= 0 1 2 and itsoutput is represented as

119886 (119905) = 119891 (119899 (119905)) = sum119889

IW (119889) lowast 119901 (119905 minus 119889)

119886 (119905) = 119891 [iw11(0) 119901 (119905) + iw

11(1) 119901 (119905 minus 1)

+ iw11(2) 119901 (119905 minus 2)]

(9)

One simple architecture with delay DI11

= 0 1 2 forfeedforward Dynamic Network is shown in Figure 2 knownas Dynamic Adaptive Linear Neuron

32 Counter Propagation Network (CPN) It is multilayerfeedforward network based on the combination of the inputcompetitive and output layers Model of CPN is instar-outstar It is three-layer neural network that performs input-output data mapping that is producing output in theresponse to an input vector on the basis of Fuzzy CompetitiveLearning In CPN the connection between input layer andcompetitive layer is instar structure and connection betweencompetitive and output layer is outstar structure CounterPropagation Network involves two-stage training process Infirst stage input vector is clustered on the basis of Euclidean


D

D

f

p(t)

a(t)

iw11(0)

iw11(1)

iw11(2)

sum

Figure 2 Simple architecture of Dynamic Adaptive Linear Neuron

distance and the performance of network is improved usinglinear topology Using Euclidean distance between input andweight vector we evaluated BMN In phase II the desiredresponse is obtained by adapting the weights from compet-itive layer to output layer [27] Let 119909 = [119909

11199092sdot sdot sdot 119909119899]

and 119910 = [119910lowast1119910lowast2sdot sdot sdot 119910lowast119898] be input and desired vector

respectively let V119894119895be weight of input layer and competitive

layer where 1 le 119894 le 119899 and 1 le 119895 le 119901 and let119908119895119896be the weight

between competitive layer and output layer where 1 le 119896 le 119898Euclidean distance between input vector 119909 and weight vectorV119894119895is

119863119895=119899

sum119894=1

(119909119894minus V119894119895)2

where 119895 = 1 2 119901 (10)

The architecture of CPN is shown in Figure 3

33 Fuzzy Competitive Learning (FCL) FCL is used foradjusting instar-outstar weights and update of BMN fordiscrete time nonlinear dynamical systems Learning of CPNis divided into two phases FuzzyCompetitive Learning phase(unsupervised) and Grossberg Learning phase (supervised)[28ndash31] CPN is efficient network for mapping the inputvector of size 119899 placed in clusters of size 119898 Like traditionalself-organizing map the weight vector for every node isconsidered as a sample vector to the input pattern Theprocess is to determine which weight vector (say 119869) is moresimilar to the input pattern chosen as BMN First phase isbased on closeness between weight vector and input vectorand second phase is weight update between competitive andoutput layer for desired response

Let V119894119895be weight of input node 119894 to neuron 119895 and let 119908

119895119896

be the weight between competitive node 119895 and neuron 119896 Wepropose a new learning rate calculation method for use ofweight update

prop (119869) =sum119899

119894=1(V119894119869minus 119909119894)2

sum119898

119895=1sum119899

119894=1(V119894119869minus 119909119894)2 (11)

where 119869 denotes the index of BMN andprop is known as FuzzyCompetitive Learning rate

Fuzzy Competitive Learning phase is described as fol-lows

331 Training Algorithm of FCL

Phase I (for determination of BMN) There are steps fortraining of CPN as follows

Step 1 Initialize instar-outstar weights

Step 2 Perform Steps 3ndash8 until stopping criteria for Phase Itraining fail The stopping condition may be fixed number ofepochs or learning rate has reduced to negligible value

Step 3 Perform Steps 4ndash6 for all input training vector119883

Step 4 Set the119883-input layer activation to vector119883

Step 5 Compute the Best Matched Node (BMN) (119869) usingEuclidean distance find the node119885

119869whose distance from the

input pattern is minimum Euclidean distance is calculated asfollows

119863119895=119899

sum119894=1

(119909119894minus V119894119895)2

where 119895 = 1 2 119901 (12)

Minimum net input

119911119894119899119869

=119899

sum119894=1

119909119894V119894119869 (13)

Step 6 Calculate Fuzzy Competitive Learning (14) rate forweight update using BMN

Step 7 We update weight for unit 119885119869

V119894119869(new) = V

119894119869(old) + prop ℎ (119869 119894 119895) (119909

119894minus V119894119869) (14)

where 119894 = 1 2 3 119899 and ℎ(sdot sdot) is neighborhood functionaround BMN Consider

ℎ (119869 119894 119895) = exp(minus10038171003817100381710038171003817V119894119895 minus V

119894119869

10038171003817100381710038171003817

21205902 (119905))

where ℎ (119869 119894 119895) isin [0 1]

(15)

120590 (119905) = 1205900exp(minus119905

119879) (16)

Step 8 Test the stopping criteria

Phase II (to obtain desired response)

Step 1 Set activation function to (119909 119910) input and output layerrespectively

Step 2 Update the BMN (119869) (Step 5 from Phase I) Alsoupdate the weights into unit 119885

119869as

V119894119869(new) = V

119894119869(old) + 120572ℎ (119869 119894 119895) [119909

119894minus V119894119869(old)] (17)


X1

Xi

Xn

x1

xi

xn

11

1p

i1ij

ij

ip

n1

np

nj

Z1

Zj

Zp

w11

wj1

wp1

wjk

wjm

wik

wim

wpk

wpm

Y1

Yk

Ym

yy1

yy

k

yym

Input layer Competitive layer Output layer Desired output

Learning rate prop Learning rate 120573

(Unsupervised)instar

(Supervised)outstar

BMN

Figure 3 Architecture of Counter Propagation Network

Step 3 Update the weights from node 119885119869to the output unit

as

119908119869119896(new) = 119908

119869119896(old) + 120573ℎ (119869 119895 119896) [119910

119895minus 119908119869119896(old)]

for 1 le 119896 le 119898(18)

Step 4 Learning rate 120573 is calculated as

120573 (119869) =sum119898

119896=1(119910119896minus 119908119869119896)2

sum119901

119895=1sum119898

119896=1(119910119896minus 119908119895119896)2 (19)

Step 5 Decrease the rate of learning st 120573 = 120573 minus 120576 where 120576 issmall positive number

Step 6 Test the stopping condition for Phase II (ie fixednumber of epochs or its learning rate has reduced to negli-gible value)

332 Testing Phase (to Test FCPN)

Step 1 Set the initial weights that is the weights obtainedduring training

Step 2 Apply FCPN to the input vector119883

Step 3 Find unit 119869 that is closest to vector119883

Step 4 Set activations of output units

Step 5 Apply activation function at 119910119896 where 119910

119896= sum119895119911119895119908119895119896

4 Dynamical Learning for CPN

Learning stabilities are fundamental issues for CPN howeverthere are few studies on the learning issue of FNN BPN forlearning is not always successful because of its sensitivity tolearning parameters Optimal learning rate always changesduring the training process Dynamical learning of CPN iscarried out using Lemmas 1 2 and 3

Assumption Let us assume 120601(119909) is a sigmoid function if itis bounded continuous and increasing function Since inputto the neural network in this model is bounded we considerLemmas 1 and 2 given below

Lemma 1 Let 120601(119909) be a sigmoid function and let Ω be acompact set in R119899 and 119891 R119899 rarr R on Ω is a continuousfunction and for arbitrary isingt 0 exist integer119873 and real constants119888119894 120579119894 and 119908

119894119895 119894 = 1 2 119873 119895 = 1 2 119899 such that

119891 (1199091 1199092 119909

119899) =119873

sum119894=1

1198881198940(119899

sum119895=1

119908119894119895119909119894minus 120579119894) (20)

satisfies

max119909isinΩ

10038171003817100381710038171003817119891 (1199091 1199092 119909119899) minus 119891 (1199091 1199092 119909119899)10038171003817100381710038171003817 ltisin (21)

Using Lemma 1 dynamical learning for a three-layer CPN canbe formulated where the hidden layer transfer functions are120601(119909) and transfer function at output layer is linear

Let all the vectors be column vectors and superscript 119889119896119901

refers to specific output vectors component


Let 119883 = [1199091 1199092 119909

119901] isin R119871times119875 119884 = [119910

1 1199102 119910

119901] isin

R119867times119875 119874 = [1199001 1199002 119900

119901] isin R119870times119875 119863 = [119889

1 1198892 119889

119901] isin

R119870times119875 be the input hidden output and desired vectorrespectively where 119871119867 and 119870 denote the number of inputhidden and output layer neurons Let 119881 and 119882 representthe input-hidden and hidden-output layer weight matrixrespectivelyThe objectives of the network training minimizeerror function 119869 where 119869 is

119869 =1

2119875119870

119875

sum119901=1

119870

sum119896=1

(119900119896119901minus 119889119896119901)2

(22)

41 Optimal Learning of Dynamical System Consider athree-layer CPN the network error matrix is defined by theerror between differences of desired and FCPN output at anyiteration and given as

119864119905= 119874119905minus 119863 = 119882

119905119884119905minus 119863 = 119882

119905119884119905119883 minus 119863 (23)

The objective of the network for minimization of error givenin (24) is defined as follows

st 119869 =1

2119875119870119879119903(119864119905119864119879119905) (24)

where119879119903represent the trace ofmatrix Using gradient descent

method updated weight is given by

119882119905+1

= 119882119905minus 120573119905

120597119869

120597119882119905

119881119905+1

= 119881119905minus 120573119905

120597119869

120597119881119905

(25)

or

119882119905+1

= 119882119905minus120573119905

119875119870119864119905119884119879119905

119881119905+1

= 119881119905minus120573119905

119875119870119882119879119905119864119905119883119879

(26)

Using (23)ndash(26) we have

119864119905+1119864119879119905+1

= (119882119905+1119884119905+1

minus 119863) (119882119905+1119884119905+1

minus 119863)119879

(27)

To obtain minimum error for multi-layer FNN in aboveequation (28) after simplification we have

119869119905+1

minus 119869119905=

1

2119875119870119892 (120573)

119869 =1

2119875119870

119875

sum119901=1

119870

sum119896=1

(119900119896119901minus 119889119896119901)2

119864119905+1119864119879119905+1

= (119882119905+1119881119905+1119883 minus 119863) (119882

119905+1119881119905+1119883 minus 119863)

119879

= [(119882119905minus120573119905

119875119870119864119905119884119879119905)(119881119905minus120573119905

119875119870119882119879119905119864119905119883119879)119883

minus 119863][(119882119905minus120573119905

119875119870119864119905119884119879119905)(119881119905minus120573119905

119875119870119882119879119905119864119905119883119879)119883

minus 119863]T= [119864119905minus120573119905

119875119870(119882119905119882119879119905119864119905119883119879119883 + 119864

119905119884119879119905119881119905119883)

+1205732119905

(119875119870)2119864119905119884119879119905119882119879119905119864119905119883119879119883] lowast [119864

119905

minus120573119905

119875119870(119882119905119882119879119905119864119905119883119879119883 + 119864

119905119884119879119905119881119905119883)

+1205732119905

(119875119870)2(119864119905119884119879119905119882119879119905119864119905119883119879119883)]

119879

= 119864119905119864119879119905

minus120573119905

119875119870[119864119905(119882119905119882119879119905119864119905119883119879119883)

119879

+ 119864119905(119864119905119884119879119905119881119905119883)119879

+ (119882119905119882119879119905119864119905119883119879119883119864119879

119905) + (119864

119905119884119879119905119881119905119883119864119879119905)]

+1205732119905

(119875119870)2[119864119905(119864119905119884119879119905119882119879119905119864119905119883119879119883)

119879

+ 119864119905119884119879119905119882119879119905119864119905119883119879119883119864119879

119905

+ 119864119905119884119879119905119881119905119883(119882119905119882119879119905119864119905119883119879119883)

119879

+119882119905119882119879119905119864119905119883119879119883(119864

119905119884119879119905119881119905119883)119879

+ 119864119905119884119879119905119881119905119883(119864119905119884119879119905119881119905119883)119879

+119882119905119882119879119905119864119905119883119879119883(119882

119905119882119879119905119864119905119883119879119883)

119879

]

minus1205733119905

(119875119870)3[119882119905119882119879119905119864119905119883119879119883(119864

119905119884119879119905119882119879119905119864119905119883119879119883)

+ 119864119905119884119879119905119881119905119883(119864119905119884119879119905119882119879119905119864119905119883119879119883)

119879

+ 119864119905119884119879119905119882119879119905119864119905119883119879119883(119882

119905119882119879119891119864119905119883119879119883)

119879

+ 119864119905119884119879119905119882119879119905119864119905119883119879119883(119864

119905119884119879119905119881119905119883)119879

]

+1205734119905

(119875119870)4[119864119905119884119879119905119882119879119905119864119905119883119879119883(119864

119905119884119879119905119882119879119905119864119905119883119879119883)

119879

]

(28)

For simplification omit subscript 119905 we have that is

119869119905+1

minus 119869119905=

1

2119875119870(1198601205734 + 1198611205734 + 1198621205734 +119872120573) (29)

where

119860 =1

(119875119870)4119879119903[119864119884119879119882119864119883119879119883(119864119884119879119882119879119864119883119879119883)

119879

]

119861 =1

(119875119870)3119879119903[119882119882119879119864119883119879119883(119864119884119879119882119879119864119883119879119883)

119879


+ 119864119884119879119881119883(119864119884119879119882119879119864119883119879119883)119879

+ 119864119884119879119882119879119864119883119879119883(119882119882119879119864119883119879119883)119879

+ 119864119884119879119882119879119864119883119879119883(119864119884119879119881119883)]

119862 =1

(119875119870)2119879119903[(119864119884119879119882119879119864119883119879119883)

119879

+ 119864119884119879119882119864119883119879119883119864119879 + 119864119884119879119881119883(119882119882119879119864119883119879119883)119879

+119882119882119879119864119883119879119883 + (119864119884119879119881119883)119879

+ 119864119884119879119881119883(119864119884119879119881119883)119879

+119882119882119879119864119883119879119883(119882119882119879119864119883119879119883)119879

]

119872 =1

119875119870119879119903[119864 (119882119882119879119864119883119879119883)

119879

+ 119864 (119864119884119879119881119883)119879

+119882119882119879119864119883119879119883 + 119864119884119879119881119883119864119879]

(30)

where

119892 (120573) = (1198601205734 + 1198611205733 + 1198621205732 +119872120573) (31)

120597119892

120597120573= 4119860 (1205733 + 1198861205732 + 119887120573 + 119888) (32)

where 119886 = 31198614119860 119887 = 21198624119860 and 119888 = 1198724119860

Lemma 2 For solution of general real cubic equation one usesthe following lemma

119891 (119909) = 1199093 + 1198861199092 + 119887119909 + 119888

119871119890119905 119863 = minus271198882 + 18119888119886119887 + 11988621198872 minus 411988631198873 minus 41198873(33)

where119863 is discriminant of 119891(119909)Then we have the following

(1) If119863 lt 0 119891(119909) has one real root

(2) If119863 ge 0 119891(119909) has three real roots

(a) 119863 gt 0 119891(119909) has three different real roots(b) 119863 = 0 6119887 minus 211988620 119891(119909) has one single root and

one multiple root(c) 119863 = 0 66 minus 21198862 = 0 119891(119909) has one root of three

multiplicities

Lemma 3 For given polynomial 119892(120573) given (31) if optimum120573 = 120573

119894| 119892(120573

119894) = min(119892(120573

1) 119892(1205732) 119892(1205733)) 119894 isin 1 2 3

where 120573119894is real root of 120597119892120597120573 the optimum (120573) is the optimal

learning rate and this learning process is stable

Proof To find stable learning range of 120573 consider thatLyapunov function is

119881119905= 1198692119905

Δ119881119905= 1198692119905+1

minus 1198692119905

if Δ119881119905lt 0

(34)

and then dynamical system is guaranteed to be stable ifΔ119881119905lt

0 that is 119869119905+1minus119869119905= (12119875119870)(1198601205734+1198611205734+1198621205734+119872120573) lt 0 (29)

where 120573 is learning rate since in the training process inputmatrix remains the same during the whole training processTo find the range of 120573 which satisfy 119892(120573) = (1198601205734 + 1198611205733 +

1198621205732 + 119872120573) lt 0 Since 120597119892120597120573 where has at least one realroot (Lemma 2) and one of them must give optimum 119892(120573)Obviously minimum value of 119892(120573) gives the largest reductionin 119869119905at each step of learning process Equation (31) shows

that 119892(120573) has two or four real roots one including 120573 = 0(Lemma 2) such that minimum value of 120573 shows largestreduction in error at two successive times and minimumvalue is obtained by differentiating (31) with respect to 120573 wehave from (32)

120597119892

120597120573= 4119860 (1205733 + 1198861205732 + 119887120573 + 119862) (35)

where 119886 = 31198614119860 119887 = 21198624119860 and 119888 = 1198724119860Solving 120597119892120597120573 = 0 gives 120573 which minimizes error in (5)

5 Simulation Results

To demonstrate the effectiveness and merit of proposedFCPN simulation results are presented and discussed Fourdifferent nonlinear dynamical systems and one generalbenchmark problem known as Box-Jenkins model with timeseries data are considered

51 Performance Criteria For accessing the performancecriteria of FCPN and comparing with Dynamic and BackPropagation Network we have evaluated various errors asgiven below In case of four dynamical models it is recom-mended to access criterion such as subset of the following

Given 119873 pair of data points (119910(119896) 119910(119896)) where 119910(119896)is output of system and 119910(119896) is output of controller theMaximum Absolute Error (MAE) is

119869MAE = 119869max = max1le119896le119873

1003816100381610038161003816119910 (119896) minus 119910 (119896)1003816100381610038161003816 (36)

the Sum of Squared Errors (SSE) is

119869SSE =119873

sum119896=1

(119910 (119896) minus 119910 (119896))2

(37)

the Mean Squared Error (MSE) is

119869MSE =1

119873

119873

sum119896=1

(119910 (119896) minus 119910 (119896))2

(38)


0 50 100 150 200 2500

1

2

3

4

5

6

7

Iteration

Mea

n Sq

uare

Err

ortimes10minus3

(a)

0 50 100 150 200 250

0

05

1

15

2

Iteration

Targ

et an

d FC

PN o

utpu

t

TargetFCPN

minus2

minus15

minus1

minus05

(b)

Figure 4 (a) Mean Square Error of system (Example 1) using FCPN (b) Performance of the controller using FCPN algorithm for Example 1

andor the Root Mean Squared Error (RMSE) is

119869RMSE = radic119869MSE (39)

The measure Variance Accounting Factor (VAF) is

119869VAF = (1 minusVar (119910 (119896) minus 119910 (119896))

Var (119910 (119896))) 100 (40)

A related measure is the Normalized Mean Squared Error(NMSE)

119869NMSE =sum119873

119896=1(119910 (119896) minus 119910 (119896))

2

sum119873

119896=1(119910 (119896) minus 119910)

2 (41)

and the Best Fit Rate (BFR) is

119869BFR = (1 minusradicsum119873

119896=1(119910 (119896) minus 119910 (119896))

2

sum119873

119896=1(119910 (119896) minus 119910)

2)100 (42)

Example 1 The nonlinear dynamical system [24] is governedby the following difference equation

119910 (119896 + 1) = 03119910 (119896) + 06119910 (119896 minus 1) + 119891 [119906 (119896)] (43)

The nonlinear function in the system in this case is 119891(119906) =1199063 + 031199062 minus 04119906 where 119910 and 119906 are uniformly distributedin [minus2 2] The objective of Example 1 is to control the systemto track reference output given as 250 sample data pointsThemodel has two inputs 119910(119896) and 119906(119896) and single output 119910(119896+1)and the system identificationwas initially performedwith thesystem input being uniformly distributed over [minus2 2] TheFCPN controller uses two inputs 119910(119896) and 119906(119896) to produceoutput 119910(119896 + 1) FCPN model governed by the differenceequation 119910(119896+1) = 03119910(119896)+06119910(119896minus1)+119873[119906(119896)]was usedFigures 4(a) and 4(b) show the error 119890(119896 + 1) = 119910(119896 + 1) minus119910(119896 + 1) and outputs of the dynamical system and the FCPN

Table 1 Calculated various errors for Example 1

NN model Different error calculationMAE SSE MSE RMSE NMSE

FCPN 13364 01143 45709119890 minus 004 00214 71627119890 minus 004

Dynamic 58231 31918 00128 01130 00028BPN 380666 481017 04810 06936 01035

As can be seen from the figure the identification error is smalleven when the input is changed to a sum of two sinusoids119906(119896) = sin(2120587119896250) + sin(212058711989625) at 119896 = 250

Table 1 includes various calculated errors of differentNN models for Example 1 It can be seen from the tablethat various errors calculated for FCPN are minimum ascompared to the DN and BPN

Example 2 Thenonlinear dynamical system [24] is governedby the following difference equation

119910 (119896 + 1) = 119891 [119910119901(119896) 119910

119901(119896 minus 1)] + 119906 (119896) (44)

where

119891 [119910119901(119896) 119910

119901(119896 minus 1)]

= [119910119901(119896) 119910119901(119896 minus 1) [119910

119901(119896) + 25]

1 + 1199102119901(119896) + 1199102

119901(119896 minus 1)

]

(45)

Theobjective of (44) is to control the system to track referenceoutput given 100 sample data points The model has twoinputs119910(119896) and 119906(119896) and single output119910(119896+1) and the systemidentification was initially performed with the system inputbeing uniformly distributed over [minus2 2] Training and testingsamples contain 100 sample data pointsThe FCPN controlleruses two inputs 119910(119896) and 119906(119896) to output 119910(119896 + 1) FCPN


0 20 40 60 80 1000

1

2

3

Iteration

Mea

n Sq

uare

Err

ortimes10minus4

(a)

0 20 40 60 80 100

0

05

1

15

2

25

Iteration

Targ

et an

d FC

PN o

utpu

t

TargetFCPN

minus1

minus05

(b)Figure 5 (a) Mean Square Error of system (Example 2) using FCPN (b) Performance of the controller using FCPN algorithm for Example 2


NN models Different error calculationMAE SSE MSE RMSE NMSE

FCPN 01956 00032 32328119890 minus 005 00057 27164119890 minus 005

Dynamic 95152 29856 00299 01728 00308BP 101494 45163 00452 02125 00323

network discussed earlier is used to identify the system frominput-output data and is described by the equation

119910 (119896 + 1) = 119873 [119910 (119896) 119910 (119896 minus 1)] + 119906 (119896) (46)

For FCPN the identification process involves the adjustmentof the weights of 119873 using FCL FCPN identifier needs someprior information concerning the input-output behavior ofthe system before identification can be undertaken FCL isused to adjust the weights of the neural network so that theerror 119890(119896 + 1) = 119910(119896 + 1) minus 119910(119896 + 1) is minimized as shownin Figure 5(a) The behavior of the FCPN model for (16) isshown in Figure 5(b) The input 119906(119896) was assumed to be arandom signal uniformly distributed in the interval [minus2 2]The weights in the neural network were adjusted



119910 (119896 + 1) =119910 (119896)

1 + 119910 (119896)2+ 1199063 (119896) (47)

which corresponds to 119891[119910(119896)] = 119910(119896)(1 + 119910(119896)2) and119892[119906(119896)] = 1199063(119896) FCPN network equation (47) is used toidentify the system from input-output data Weights in the

Table 3 Calculated various errors and BFR for Example 3

NN model MAE SSE MSE RMSE NMSEFCPN 03519 00032 31709119890 minus 005 00056 56371119890 minus 006

Dynamic 55652 36158 00362 01902 00223BP 426524 469937 04699 06855 02892

neural networks were adjusted for every instant of discretetime steps

The objective of (47) is to control the system to trackreference output given 100 sample data pointsThemodel hastwo inputs 119910(119896) and 119906(119896) and single output 119910(119896 + 1) and thesystem identification was initially performed with the systeminput being uniformly distributed over [minus2 2] Training andtesting samples contain 250 sample data points The FCPNcontroller uses two inputs 119910(119896) and 119906(119896) to output 119910(119896 + 1)The input data is generated using uniform distribution ininterval [minus2 2] The function 119891 obtained by FCPN is usedto adjust the weights of the neural network so that the error119890(119896 + 1) = 119910(119896 + 1) minus 119910(119896 + 1) is minimized as shown inFigure 6(a) In Figure 6(b) the desired outputs of the systemas well as the FCPN model are shown and are seen to beindistinguishable


Example 4 Thenonlinear dynamical system [24] is governedby the difference equation (48) Generalized form of thisequation is Model IV In this example the system is assumedto be of the form

119910119901(119896 + 1)

= 119891 [119910119901(119896) 119910

119901(119896 minus 1) 119910

119901(119896 minus 2) 119906 (119896) 119906 (119896 minus 1)]

(48)


0 20 40 60 80 1000

1

2

Iteration

Mea

n Sq

uare

Err

ortimes10minus4

(a)

0 20 40 60 80 100

0

1

2

3

Iteration

Targ

et an

d FC

PN o

utpu

t

TargetFCPN

minus3

minus2

minus1

(b)


0 100 200 300 400 500 600 700 8000

1

2

3

4

5

6

Iteration

Mea

n Sq

uare

Err

or

times10minus4

(a)

0 100 200 300 400 500 600 700 800

0

05

1

Iteration

Targ

et an

d FC

PN o

utpu

t

TargetFCPN

minus1

minus05

(b)

Figure 7 (a)Mean Square Error of system (Example 4) using FCPN (b) Performance of the controller using FCPN algorithm for Example 4

where the unknown function 119891 has the form

119891 [1199091 1199092 1199093 1199094 1199095] =

1199091119909211990931199095(1199093minus 1) + 119909

4

1 + 11990923+ 11990922

(49)

In the identification model which is approximated by FCPNFigure 7(b) shows the output of the system and the modelwhen the identification procedure carried random inputsignal uniformly distributed in the interval [minus1 1] Theperformance of the model is studied and the error 119890(119896 +1) = 119873[119910

119901(119896 + 1)] is minimized as shown in Figure 7(a)

In Figure 7(b) the outputs of the system as well as outputof FCPN model are shown Input to the system and theidentified model is given by 119906(119896) = sin(2120587119896250) for 119896 le 500and 119906(119896) = 08 sin(2120587119896250) + 02 sin(212058711989625) for 119896 gt 500



NN models MAE SSE MSE RMSE NMSEFCPN 304348 21931 00027 00524 00186Dynamic 328952 64059 00080 00895 00276BP 531988 134174 00168 01295 00844

Example 5 In this example we have used Box-Jenkins timeseries [32] of 296 pairs of data measured from a gas furnacesystem with single input 119906(119905) being gas flow rate and singleoutput 119910(119905) being CO

2concentration in outlet gas Training

samples and testing samples contained 196 and 100 datapoints respectively The FCPN uses the two inputs thecurrent state 119910(119896) and the desired state 119910119889(119896) to produce anoutput which is 119910(119896) [33]

Figure 8(a) shows the mean squared control errors ofFCPN methods Figure 8(b) shows the performance of thecontroller with FCPN algorithm Result shows that FCPNalgorithm enable us to appropriate the approximation using


0 10 20 30 40 50 60 70 80 90 100044046048

05052054056058

06

Iteration

Mea

n Sq

uare

Err

or

(a)

0 10 20 30 40 50 60 70 80 90 10045

50

55

60

Iteration

Targ

et an

d FC

PN o

utpu

t

DesiredFCPN


Table 5 BFR () values of NN models for various examples

NN models BFR ()Example 1 Example 2 Example 3 Example 4

FCPN 9732 9948 9976 8635DN 9469 7932 8508 8339BPN 6783 7928 462234 7094

fuzzy learning as given in equation (11) of Box Jenkins timeseries data based on calculation of BMN

Table 5 shows BFR () various NN models for all Exam-ples 1ndash5 respectively It can be observed from Table 5 that theBest Fit Rate found for FCPN is maximum as compared toDN and BPN which shows better performance of the FCPNnetwork for nonlinear system

6 Conclusions

FCPN is a neural network control method which was devel-oped and presented It is based on the concept of combiningFCL algorithm and CPN The key ideas explored are theuse of the FCL algorithm for training the weights of theinstar-outstar The performances of the FCPN controllerbased training are tested using four nonlinear dynamicalsystems and on time series (Box-Jenkins) data and comparedwith the Dynamic Network and standard Back Propagationalgorithm The comparative performances of the FCPNalgorithm and Dynamic Network such as the number ofiterations and performance functions like MAE MSE SSENMSE BFR and so forth of FCPN and Dynamic Networkerror are summarized in Tables 1ndash4 It can be seen that theFCPN algorithm gives minimum errors as compared to theDynamic Network and standard Back Propagation Networkfor all four models of nonlinear dynamical systems and Box-Jenkins time series data Results obtained from FCPN werecompared for various errors and it is clearly shown that FCPNworks much better for the control of nonlinear dynamicalsystems

Conflict of Interests

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

Acknowledgment

The authors gratefully acknowledged the financial assistanceprovided by the All India Council of Technical Education(AICTE) in the form of Research Promotion Scheme (RPS)project in 2012

References

[1] T Soderstrom and P Stoica System Identification Prentice HallNew York NY USA 1989

[2] S A Billings Nonlinear System Identification NARMAXMeth-ods in the Time Frequency and Spatio-Temporal DomainsWiley Chichester UK 2013

[3] M Liu ldquoDecentralized control of robotmanipulators nonlinearand adaptive approachesrdquo IEEE Transactions on AutomaticControl vol 44 no 2 pp 357ndash363 1999

[4] C-M Lin A-B Ting M-C Li and T-Y Chen ldquoNeural-network-based robust adaptive control for a class of nonlinearsystemsrdquo Neural Computing and Applications vol 20 no 4 pp557ndash563 2011

[5] I Rivals and L Personnaz ldquoNonlinear internal model controlusing neural networks application to processes with delay anddesign issuesrdquo IEEETransactions onNeural Networks vol 11 no1 pp 80ndash90 2000

[6] I Kanellakopoulos P V Kokotovic and A S Morse ldquoSys-tematic design of adaptive controllers for feedback linearizablesystemsrdquo IEEE Transactions on Automatic Control vol 36 no11 pp 1241ndash1253 1991

[7] P Kokotovic ldquoThe joy of feedback nonlinear and adaptiverdquoIEEE Control Systems vol 12 no 3 pp 7ndash17 1992

[8] H Elmali andNOlgac ldquoRobust output tracking control of non-linear MIMO systems via sliding mode techniquerdquoAutomaticavol 28 no 1 pp 145ndash151 1992


[9] N Sadati and R Ghadami ldquoAdaptivemulti-model slidingmodecontrol of robotic manipulators using soft computingrdquo Neuro-computing vol 71 no 13ndash15 pp 2702ndash2710 2008

[10] A Kroll and H Schulte ldquoBenchmark problems for nonlinearsystem identification and control using Soft Computing meth-ods need and overviewrdquo Applied Soft Computing Journal vol25 pp 496ndash513 2014

[11] K Hornik M Stinchcombe and HWhite ldquoMultilayer feedfor-ward networks are universal approximatorsrdquo Neural Networksvol 2 no 5 pp 359ndash366 1989

[12] A Bortoletti C Di Fiore S Fanelli and P Zellini ldquoA new classof Quasi-Newtonian methods for optimal learning in MLP-networksrdquo IEEE Transactions on Neural Networks vol 14 no2 pp 263ndash273 2003

[13] G Lera and M Pinzolas ldquoNeighborhood based Levenberg-Marquardt algorithm for neural network trainingrdquo IEEE Trans-actions on Neural Networks vol 13 no 5 pp 1200ndash1203 2002

[14] A K Shah and D M Adhyaru ldquoClustering based multiplemodel control of hybrid dynamical systems usingHJB solutionrdquoApplied Soft Computing vol 31 pp 103ndash117 2015

[15] Y Cui H Zhang Y Wang and Z Zhang ldquoAdaptive neuraldynamic surface control for a class of uncertain nonlinearsystems with disturbancesrdquo Neurocomputing vol 165 pp 152ndash158 2015

[16] R Rakkiyappan N Sakthivel and J Cao ldquoStochastic sampled-data control for synchronization of complex dynamical net-works with control packet loss and additive time-varyingdelaysrdquo Neural Networks vol 66 pp 46ndash63 2015

[17] B Kiumarsi F L Lewis and D S Levine ldquoOptimal controlof nonlinear discrete time-varying systems using a new neuralnetwork approximation structurerdquo Neurocomputing vol 156pp 157ndash165 2015

[18] B M Al-Hadithi A Jimenez and R G Lopez ldquoFuzzy optimalcontrol using generalized Takagi-Sugeno model for multivari-able nonlinear systemsrdquoApplied SoftComputing Journal vol 30pp 205ndash213 2015

[19] MAGonzalez-Olvera andY Tang ldquoIdentification of nonlineardiscrete systems by a state-space recurrent neurofuzzy networkwith a convergent algorithmrdquoNeurocomputing vol 148 pp 318ndash325 2015

[20] J F de Canete P Del Saz-Orozco I Garcia-Moral and SGonzalez-Perez ldquoIndirect adaptive structure for multivariableneural identification and control of a pilot distillation plantrdquoApplied Soft Computing vol 12 no 9 pp 2728ndash2739 2012

[21] M Cai and Z Xiang ldquoAdaptive neural finite-time control for aclass of switched nonlinear systemsrdquo Neurocomputing vol 155pp 177ndash185 2015

[22] Y Li S Tong and T Li ldquoAdaptive fuzzy backstepping controldesign for a class of pure-feedback switched nonlinear systemsrdquoNonlinear Analysis Hybrid Systems vol 16 pp 72ndash80 2015

[23] T-Y Kuc J S Lee and K Nam ldquoAn iterative learning controltheory for a class of nonlinear dynamic systemsrdquo Automaticavol 28 no 6 pp 1215ndash1221 1992

[24] K S Narendra and K Parthasarathy ldquoNeural networks anddynamical systemsrdquo International Journal of Approximate Rea-soning vol 6 no 2 pp 109ndash131 1992

[25] R Hecht-Nielsen ldquoTheory of the back propagation neuralnetworkrdquo Neural Networks vol 1 pp 593ndash605 1989

[26] M T Hagan H B Demuth M H Beale and O De JesusNeural Network Design Cengage Learning 2nd edition 2014

[27] F-J Chang and Y-C Chen ldquoA counterpropagation fuzzy-neural network modeling approach to real time streamflowpredictionrdquo Journal of Hydrology vol 245 no 1ndash4 pp 153ndash1642001

[28] A Dwivedi N S C Bose A Kumar P Kandula D Mishraand P K Kalra ldquoA novel hybrid image compression techniquewavelet-MFOCPNrdquo in Proceedings of the 9th Asian Symposiumon Information Display (ASID rsquo06) pp 492ndash495 2006

[29] C J C Burges P Simard and H S Malvar Improving WaveletImage Compression with Neural Networks Microsoft ResearchRedmond Wash USA 2001

[30] D Woods ldquoBack and counter propagation aberrationsrdquo in Pro-ceedings of the IEEE International Conference on Neural Net-works pp 473ndash479 1988

[31] DMishra N S C Bose A Tolambiya et al ldquoColor image com-pression with modified forward-only counterpropagation neu-ral network improvement of the quality using different distancemeasuresrdquo in Proceedings of the 9th International Conferenceon Information Technology (ICIT rsquo06) pp 139ndash140 IEEEBhubaneswar India December 2006

[32] G E Box and G M Jenkins Time Series Analysis Forecastingand Control Holden Day San Francisco Calif USA 1970

[33] J Kim and N Kasabov ldquoHyFIS adaptive neuro-fuzzy inferencesystems and their application to nonlinear dynamical systemsrdquoNeural Networks vol 12 no 9 pp 1301ndash1319 1999

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Human-ComputerInteraction

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enhance the performance of nonlinear multivariable systemfuzzy optimal controller based Takagi-Sugeno model wasdeveloped which was used to minimize a quadratic per-formance index [18] An ultimate stable training algorithminspired by adaptive observer for black box identificationof nonlinear discrete systems developed using state spacerecurrent neural networkThe algorithmdevelopedwas usingonly input and output measurement in discrete time [19]Indirect adaptive controller that uses neural network waspresented for the identification and control of experimentalpilot distillation column This system is multivariable withunknowndynamics and the neural networkwas trained usingLevenberg-Marquardt algorithm [20] Adaptive finite timestabilization of a class of switched nonlinear systems wasinvestigated with unknown nonlinear terms using neuralnetwork The finite law and adaptive law were constructedusing radial basis function neural network (RBFNN) toapproximate the unknown packaged function Stability anal-ysis of RBFNN was observed to evaluate the states of theclosed loop systems [21] An adaptive fuzzy output trackingcontrol approach was proposed in [22] for single input singleoutput (SISO) systemwhich is uncertain and nonlinear underarbitrary switching An iterative learning control schemewas proposed for a class of nonlinear dynamic systemswhich includes holonomic systems as its subsets with linearfeedback mechanism and feedforward learning strategies[23]

In this proposed work we have used instar-outstar struc-ture based CPN with Fuzzy Competitive Learning (FCL)We have designed Fuzzy Counter Propagation Networkdesign to control some nonlinear dynamical systems In theFCPN CPN model is trained by FCL algorithms The FCLlearning is used for adjusting weights and update of BestMatched Node (BMN) in discrete time nonlinear dynamicalsystem

The main contributions of this research are as follows

(1) This paper contributes the approximation for a classof nonlinear dynamical systems using Fuzzy Compet-itive Learning Based Counter Propagation Network(FCPN)

(2) FCPN is employed to optimize the Mean AbsoluteError Mean Square Error Best Fit Rate and so forth

(3) Performance criteria of proposed FCPN for nonlineardynamical systems are effectively improved by com-pensating reference signal and controller signal

The paper is organized as follows In Section 2 problemformulations for a class of nonlinear dynamical systems aregiven Section 3 contains description of Feedforward NeuralNetwork (FNN) with Back Propagation Network DynamicNetwork and Fuzzy Learning Based Counter PropagationNetwork Dynamical learning for CPN and optimal learningof dynamical system are presented in Section 4 The simula-tion results and comparison of different nonlinear dynamicalmodels are presented in Section 5 Section 6 gives conclusionof the study

2 Problem Formulation

Let us consider four models of discrete time nonlineardynamical systems [24] for single input single output (SISO)and multiple input and single output (MISO) system con-sidered in this paper and they are described by differenceequations (1)ndash(4) and Box-Jenkins time series data [10] Let119891 119877119899 rarr 119877 and 119892 119877119898 rarr 119877 be the nonlinear continuousdifferentiable function of Model IndashModel IV approximatedby FCPN to the desired degree of accuracy Once the systemhas been parameterized the performance evaluation has beencarried out by FCPN for (1) to (4)

Model I

119910 (119896 + 1) =119899minus1

sum119894=0

120572119894119910 (119896 minus 119894)

+ 119892 [119906 (119896) 119906 (119896 minus 119898 + 1)]

(1)

Model II

119910 (119896 + 1) = 119891 [119910 (119896) 119910 (119896 minus 119899 + 1)]

+119898minus1

sum119894=0

120573119894119906 (119896 minus 119894)

(2)

Model III

119910 (119896 + 1) = 119891 [119910 (119896) 119910 (119896 minus 119899 + 1)]

+ 119892 [119906 (119896) 119906 (119896 minus 119898 + 1)] (3)

Model IV

119910 (119896 + 1) = 119891 [119910 (119896) 119910 (119896 minus 119899 + 1) 119906 (119896)

119906 (119896 minus 119898 + 1)] (4)

where [119906(119896) 119910(119896)] represent the input-output pair of thesystem at time 119896 and their order is represented by (119899119898)FCL is used to learn the system defined for (1) to (4) and theperformance of the FCPN can be measured by error functiongiven in

119864 (119896) =1

2[119910 (119896) minus 119910 (119896)]

2

(5)

where 119906(119896) is the input 119910(119896) is the system output 119910(119896) isneural network output and (5) can be written for neuralcontroller as

119864119888 =1

119875

119875

sum119901=1

[119910 (119896) minus 119910119888

(119896)]2

(6)

where 119910119888(119896) is neural controller output and (6) is known asminimized error function and 119875 is total number of inputpatterns

3 Back Propagation Network (BPN)

A neural network is one of the most popular intelligentsystems which has capability to approximate calculated andtarget value at an arbitrary degree of accuracy BPN is one


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119899119898 (119905) = sum

119897isin119871119891119898


IW119898sdot119897 (119889) 119886119897 (119905 minus 119889)

+ sum119897isin119868119898


IW119898sdot119897 (119889) 119887119897 (119905 minus 119889) + 119887119898(7)





119898is the set



119886119898 (119905) = 119891119898 (119899119898 (119905)) (8)



119886 (119905) = 119891 (119899 (119905)) = sum119889

IW (119889) lowast 119901 (119905 minus 119889)

119886 (119905) = 119891 [iw11(0) 119901 (119905) + iw

11(1) 119901 (119905 minus 1)

+ iw11(2) 119901 (119905 minus 2)]

(9)





D

D

f

p(t)

a(t)

iw11(0)

iw11(1)

iw11(2)

sum



11199092sdot sdot sdot 119909119899]





119863119895=119899

sum119894=1

(119909119894minus V119894119895)2

where 119895 = 1 2 119901 (10)




119895119896


prop (119869) =sum119899

119894=1(V119894119869minus 119909119894)2

sum119898

119895=1sum119899

119894=1(V119894119869minus 119909119894)2 (11)












119863119895=119899

sum119894=1

(119909119894minus V119894119895)2

where 119895 = 1 2 119901 (12)

Minimum net input

119911119894119899119869

=119899

sum119894=1

119909119894V119894119869 (13)



V119894119869(new) = V

119894119869(old) + prop ℎ (119869 119894 119895) (119909

119894minus V119894119869) (14)



119894119869

10038171003817100381710038171003817

21205902 (119905))

where ℎ (119869 119894 119895) isin [0 1]

(15)

120590 (119905) = 1205900exp(minus119905

119879) (16)





119869as

V119894119869(new) = V

119894119869(old) + 120572ℎ (119869 119894 119895) [119909

119894minus V119894119869(old)] (17)


X1

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(Supervised)outstar

BMN



as

119908119869119896(new) = 119908

119869119896(old) + 120573ℎ (119869 119895 119896) [119910

119895minus 119908119869119896(old)]

for 1 le 119896 le 119898(18)


120573 (119869) =sum119898

119896=1(119910119896minus 119908119869119896)2

sum119901

119895=1sum119898

119896=1(119910119896minus 119908119895119896)2 (19)









119896= sum119895119911119895119908119895119896





119894119895 119894 = 1 2 119873 119895 = 1 2 119899 such that

119891 (1199091 1199092 119909

119899) =119873

sum119894=1

1198881198940(119899

sum119895=1

119908119894119895119909119894minus 120579119894) (20)

satisfies

max119909isinΩ

10038171003817100381710038171003817119891 (1199091 1199092 119909119899) minus 119891 (1199091 1199092 119909119899)10038171003817100381710038171003817 ltisin (21)





Let 119883 = [1199091 1199092 119909

119901] isin R119871times119875 119884 = [119910

1 1199102 119910

119901] isin

R119867times119875 119874 = [1199001 1199002 119900

119901] isin R119870times119875 119863 = [119889

1 1198892 119889

119901] isin


119869 =1

2119875119870

119875

sum119901=1

119870

sum119896=1

(119900119896119901minus 119889119896119901)2

(22)


119864119905= 119874119905minus 119863 = 119882

119905119884119905minus 119863 = 119882

119905119884119905119883 minus 119863 (23)


st 119869 =1

2119875119870119879119903(119864119905119864119879119905) (24)



119882119905+1

= 119882119905minus 120573119905

120597119869

120597119882119905

119881119905+1

= 119881119905minus 120573119905

120597119869

120597119881119905

(25)

or

119882119905+1

= 119882119905minus120573119905

119875119870119864119905119884119879119905

119881119905+1

= 119881119905minus120573119905

119875119870119882119879119905119864119905119883119879

(26)


119864119905+1119864119879119905+1

= (119882119905+1119884119905+1

minus 119863) (119882119905+1119884119905+1

minus 119863)119879

(27)


119869119905+1

minus 119869119905=

1

2119875119870119892 (120573)

119869 =1

2119875119870

119875

sum119901=1

119870

sum119896=1

(119900119896119901minus 119889119896119901)2

119864119905+1119864119879119905+1

= (119882119905+1119881119905+1119883 minus 119863) (119882

119905+1119881119905+1119883 minus 119863)

119879

= [(119882119905minus120573119905

119875119870119864119905119884119879119905)(119881119905minus120573119905

119875119870119882119879119905119864119905119883119879)119883

minus 119863][(119882119905minus120573119905

119875119870119864119905119884119879119905)(119881119905minus120573119905

119875119870119882119879119905119864119905119883119879)119883

minus 119863]T= [119864119905minus120573119905

119875119870(119882119905119882119879119905119864119905119883119879119883 + 119864

119905119884119879119905119881119905119883)

+1205732119905

(119875119870)2119864119905119884119879119905119882119879119905119864119905119883119879119883] lowast [119864

119905

minus120573119905

119875119870(119882119905119882119879119905119864119905119883119879119883 + 119864

119905119884119879119905119881119905119883)

+1205732119905

(119875119870)2(119864119905119884119879119905119882119879119905119864119905119883119879119883)]

119879

= 119864119905119864119879119905

minus120573119905

119875119870[119864119905(119882119905119882119879119905119864119905119883119879119883)

119879

+ 119864119905(119864119905119884119879119905119881119905119883)119879

+ (119882119905119882119879119905119864119905119883119879119883119864119879

119905) + (119864

119905119884119879119905119881119905119883119864119879119905)]

+1205732119905

(119875119870)2[119864119905(119864119905119884119879119905119882119879119905119864119905119883119879119883)

119879

+ 119864119905119884119879119905119882119879119905119864119905119883119879119883119864119879

119905

+ 119864119905119884119879119905119881119905119883(119882119905119882119879119905119864119905119883119879119883)

119879

+119882119905119882119879119905119864119905119883119879119883(119864

119905119884119879119905119881119905119883)119879

+ 119864119905119884119879119905119881119905119883(119864119905119884119879119905119881119905119883)119879

+119882119905119882119879119905119864119905119883119879119883(119882

119905119882119879119905119864119905119883119879119883)

119879

]

minus1205733119905

(119875119870)3[119882119905119882119879119905119864119905119883119879119883(119864

119905119884119879119905119882119879119905119864119905119883119879119883)

+ 119864119905119884119879119905119881119905119883(119864119905119884119879119905119882119879119905119864119905119883119879119883)

119879

+ 119864119905119884119879119905119882119879119905119864119905119883119879119883(119882

119905119882119879119891119864119905119883119879119883)

119879

+ 119864119905119884119879119905119882119879119905119864119905119883119879119883(119864

119905119884119879119905119881119905119883)119879

]

+1205734119905

(119875119870)4[119864119905119884119879119905119882119879119905119864119905119883119879119883(119864

119905119884119879119905119882119879119905119864119905119883119879119883)

119879

]

(28)


119869119905+1

minus 119869119905=

1

2119875119870(1198601205734 + 1198611205734 + 1198621205734 +119872120573) (29)

where

119860 =1

(119875119870)4119879119903[119864119884119879119882119864119883119879119883(119864119884119879119882119879119864119883119879119883)

119879

]

119861 =1

(119875119870)3119879119903[119882119882119879119864119883119879119883(119864119884119879119882119879119864119883119879119883)

119879


+ 119864119884119879119881119883(119864119884119879119882119879119864119883119879119883)119879

+ 119864119884119879119882119879119864119883119879119883(119882119882119879119864119883119879119883)119879

+ 119864119884119879119882119879119864119883119879119883(119864119884119879119881119883)]

119862 =1

(119875119870)2119879119903[(119864119884119879119882119879119864119883119879119883)

119879

+ 119864119884119879119882119864119883119879119883119864119879 + 119864119884119879119881119883(119882119882119879119864119883119879119883)119879

+119882119882119879119864119883119879119883 + (119864119884119879119881119883)119879

+ 119864119884119879119881119883(119864119884119879119881119883)119879

+119882119882119879119864119883119879119883(119882119882119879119864119883119879119883)119879

]

119872 =1

119875119870119879119903[119864 (119882119882119879119864119883119879119883)

119879

+ 119864 (119864119884119879119881119883)119879

+119882119882119879119864119883119879119883 + 119864119884119879119881119883119864119879]

(30)

where

119892 (120573) = (1198601205734 + 1198611205733 + 1198621205732 +119872120573) (31)

120597119892

120597120573= 4119860 (1205733 + 1198861205732 + 119887120573 + 119888) (32)

where 119886 = 31198614119860 119887 = 21198624119860 and 119888 = 1198724119860


119891 (119909) = 1199093 + 1198861199092 + 119887119909 + 119888







multiplicities


119894| 119892(120573

119894) = min(119892(120573

1) 119892(1205732) 119892(1205733)) 119894 isin 1 2 3




119881119905= 1198692119905

Δ119881119905= 1198692119905+1

minus 1198692119905

if Δ119881119905lt 0

(34)


0 that is 119869119905+1minus119869119905= (12119875119870)(1198601205734+1198611205734+1198621205734+119872120573) lt 0 (29)




120597119892

120597120573= 4119860 (1205733 + 1198861205732 + 119887120573 + 119862) (35)






119869MAE = 119869max = max1le119896le119873

1003816100381610038161003816119910 (119896) minus 119910 (119896)1003816100381610038161003816 (36)


119869SSE =119873

sum119896=1

(119910 (119896) minus 119910 (119896))2

(37)


119869MSE =1

119873

119873

sum119896=1

(119910 (119896) minus 119910 (119896))2

(38)


0 50 100 150 200 2500

1

2

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4

5

6

7

Iteration

Mea

n Sq

uare

Err

ortimes10minus3

(a)

0 50 100 150 200 250

0

05

1

15

2

Iteration

Targ

et an

d FC

PN o

utpu

t

TargetFCPN

minus2

minus15

minus1

minus05

(b)



119869RMSE = radic119869MSE (39)



Var (119910 (119896))) 100 (40)


119869NMSE =sum119873

119896=1(119910 (119896) minus 119910 (119896))

2

sum119873

119896=1(119910 (119896) minus 119910)

2 (41)



119896=1(119910 (119896) minus 119910 (119896))

2

sum119873

119896=1(119910 (119896) minus 119910)

2)100 (42)


119910 (119896 + 1) = 03119910 (119896) + 06119910 (119896 minus 1) + 119891 [119906 (119896)] (43)




FCPN 13364 01143 45709119890 minus 004 00214 71627119890 minus 004

Dynamic 58231 31918 00128 01130 00028BPN 380666 481017 04810 06936 01035




119910 (119896 + 1) = 119891 [119910119901(119896) 119910

119901(119896 minus 1)] + 119906 (119896) (44)

where

119891 [119910119901(119896) 119910

119901(119896 minus 1)]

= [119910119901(119896) 119910119901(119896 minus 1) [119910

119901(119896) + 25]

1 + 1199102119901(119896) + 1199102

119901(119896 minus 1)

]

(45)



0 20 40 60 80 1000

1

2

3

Iteration

Mea

n Sq

uare

Err

ortimes10minus4

(a)

0 20 40 60 80 100

0

05

1

15

2

25

Iteration

Targ

et an

d FC

PN o

utpu

t

TargetFCPN

minus1

minus05




FCPN 01956 00032 32328119890 minus 005 00057 27164119890 minus 005

Dynamic 95152 29856 00299 01728 00308BP 101494 45163 00452 02125 00323


119910 (119896 + 1) = 119873 [119910 (119896) 119910 (119896 minus 1)] + 119906 (119896) (46)




119910 (119896 + 1) =119910 (119896)

1 + 119910 (119896)2+ 1199063 (119896) (47)




Dynamic 55652 36158 00362 01902 00223BP 426524 469937 04699 06855 02892





119910119901(119896 + 1)

= 119891 [119910119901(119896) 119910

119901(119896 minus 1) 119910

119901(119896 minus 2) 119906 (119896) 119906 (119896 minus 1)]

(48)


0 20 40 60 80 1000

1

2

Iteration

Mea

n Sq

uare

Err

ortimes10minus4

(a)

0 20 40 60 80 100

0

1

2

3

Iteration

Targ

et an

d FC

PN o

utpu

t

TargetFCPN

minus3

minus2

minus1

(b)


0 100 200 300 400 500 600 700 8000

1

2

3

4

5

6

Iteration

Mea

n Sq

uare

Err

or

times10minus4

(a)

0 100 200 300 400 500 600 700 800

0

05

1

Iteration

Targ

et an

d FC

PN o

utpu

t

TargetFCPN

minus1

minus05

(b)



119891 [1199091 1199092 1199093 1199094 1199095] =

1199091119909211990931199095(1199093minus 1) + 119909

4

1 + 11990923+ 11990922

(49)












0 10 20 30 40 50 60 70 80 90 100044046048

05052054056058

06

Iteration

Mea

n Sq

uare

Err

or

(a)

0 10 20 30 40 50 60 70 80 90 10045

50

55

60

Iteration

Targ

et an

d FC

PN o

utpu

t

DesiredFCPN




FCPN 9732 9948 9976 8635DN 9469 7932 8508 8339BPN 6783 7928 462234 7094



6 Conclusions




Acknowledgment


References










































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




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119899119898 (119905) = sum

119897isin119871119891119898


IW119898sdot119897 (119889) 119886119897 (119905 minus 119889)

+ sum119897isin119868119898


IW119898sdot119897 (119889) 119887119897 (119905 minus 119889) + 119887119898(7)





119898is the set



119886119898 (119905) = 119891119898 (119899119898 (119905)) (8)



119886 (119905) = 119891 (119899 (119905)) = sum119889

IW (119889) lowast 119901 (119905 minus 119889)

119886 (119905) = 119891 [iw11(0) 119901 (119905) + iw

11(1) 119901 (119905 minus 1)

+ iw11(2) 119901 (119905 minus 2)]

(9)





D

D

f

p(t)

a(t)

iw11(0)

iw11(1)

iw11(2)

sum



11199092sdot sdot sdot 119909119899]





119863119895=119899

sum119894=1

(119909119894minus V119894119895)2

where 119895 = 1 2 119901 (10)




119895119896


prop (119869) =sum119899

119894=1(V119894119869minus 119909119894)2

sum119898

119895=1sum119899

119894=1(V119894119869minus 119909119894)2 (11)












119863119895=119899

sum119894=1

(119909119894minus V119894119895)2

where 119895 = 1 2 119901 (12)

Minimum net input

119911119894119899119869

=119899

sum119894=1

119909119894V119894119869 (13)



V119894119869(new) = V

119894119869(old) + prop ℎ (119869 119894 119895) (119909

119894minus V119894119869) (14)



119894119869

10038171003817100381710038171003817

21205902 (119905))

where ℎ (119869 119894 119895) isin [0 1]

(15)

120590 (119905) = 1205900exp(minus119905

119879) (16)





119869as

V119894119869(new) = V

119894119869(old) + 120572ℎ (119869 119894 119895) [119909

119894minus V119894119869(old)] (17)


X1

Xi

Xn

x1

xi

xn

11

1p

i1ij

ij

ip

n1

np

nj

Z1

Zj

Zp

w11

wj1

wp1

wjk

wjm

wik

wim

wpk

wpm

Y1

Yk

Ym

yy1

yy

k

yym




(Supervised)outstar

BMN



as

119908119869119896(new) = 119908

119869119896(old) + 120573ℎ (119869 119895 119896) [119910

119895minus 119908119869119896(old)]

for 1 le 119896 le 119898(18)


120573 (119869) =sum119898

119896=1(119910119896minus 119908119869119896)2

sum119901

119895=1sum119898

119896=1(119910119896minus 119908119895119896)2 (19)









119896= sum119895119911119895119908119895119896





119894119895 119894 = 1 2 119873 119895 = 1 2 119899 such that

119891 (1199091 1199092 119909

119899) =119873

sum119894=1

1198881198940(119899

sum119895=1

119908119894119895119909119894minus 120579119894) (20)

satisfies

max119909isinΩ

10038171003817100381710038171003817119891 (1199091 1199092 119909119899) minus 119891 (1199091 1199092 119909119899)10038171003817100381710038171003817 ltisin (21)





Let 119883 = [1199091 1199092 119909

119901] isin R119871times119875 119884 = [119910

1 1199102 119910

119901] isin

R119867times119875 119874 = [1199001 1199002 119900

119901] isin R119870times119875 119863 = [119889

1 1198892 119889

119901] isin


119869 =1

2119875119870

119875

sum119901=1

119870

sum119896=1

(119900119896119901minus 119889119896119901)2

(22)


119864119905= 119874119905minus 119863 = 119882

119905119884119905minus 119863 = 119882

119905119884119905119883 minus 119863 (23)


st 119869 =1

2119875119870119879119903(119864119905119864119879119905) (24)



119882119905+1

= 119882119905minus 120573119905

120597119869

120597119882119905

119881119905+1

= 119881119905minus 120573119905

120597119869

120597119881119905

(25)

or

119882119905+1

= 119882119905minus120573119905

119875119870119864119905119884119879119905

119881119905+1

= 119881119905minus120573119905

119875119870119882119879119905119864119905119883119879

(26)


119864119905+1119864119879119905+1

= (119882119905+1119884119905+1

minus 119863) (119882119905+1119884119905+1

minus 119863)119879

(27)


119869119905+1

minus 119869119905=

1

2119875119870119892 (120573)

119869 =1

2119875119870

119875

sum119901=1

119870

sum119896=1

(119900119896119901minus 119889119896119901)2

119864119905+1119864119879119905+1

= (119882119905+1119881119905+1119883 minus 119863) (119882

119905+1119881119905+1119883 minus 119863)

119879

= [(119882119905minus120573119905

119875119870119864119905119884119879119905)(119881119905minus120573119905

119875119870119882119879119905119864119905119883119879)119883

minus 119863][(119882119905minus120573119905

119875119870119864119905119884119879119905)(119881119905minus120573119905

119875119870119882119879119905119864119905119883119879)119883

minus 119863]T= [119864119905minus120573119905

119875119870(119882119905119882119879119905119864119905119883119879119883 + 119864

119905119884119879119905119881119905119883)

+1205732119905

(119875119870)2119864119905119884119879119905119882119879119905119864119905119883119879119883] lowast [119864

119905

minus120573119905

119875119870(119882119905119882119879119905119864119905119883119879119883 + 119864

119905119884119879119905119881119905119883)

+1205732119905

(119875119870)2(119864119905119884119879119905119882119879119905119864119905119883119879119883)]

119879

= 119864119905119864119879119905

minus120573119905

119875119870[119864119905(119882119905119882119879119905119864119905119883119879119883)

119879

+ 119864119905(119864119905119884119879119905119881119905119883)119879

+ (119882119905119882119879119905119864119905119883119879119883119864119879

119905) + (119864

119905119884119879119905119881119905119883119864119879119905)]

+1205732119905

(119875119870)2[119864119905(119864119905119884119879119905119882119879119905119864119905119883119879119883)

119879

+ 119864119905119884119879119905119882119879119905119864119905119883119879119883119864119879

119905

+ 119864119905119884119879119905119881119905119883(119882119905119882119879119905119864119905119883119879119883)

119879

+119882119905119882119879119905119864119905119883119879119883(119864

119905119884119879119905119881119905119883)119879

+ 119864119905119884119879119905119881119905119883(119864119905119884119879119905119881119905119883)119879

+119882119905119882119879119905119864119905119883119879119883(119882

119905119882119879119905119864119905119883119879119883)

119879

]

minus1205733119905

(119875119870)3[119882119905119882119879119905119864119905119883119879119883(119864

119905119884119879119905119882119879119905119864119905119883119879119883)

+ 119864119905119884119879119905119881119905119883(119864119905119884119879119905119882119879119905119864119905119883119879119883)

119879

+ 119864119905119884119879119905119882119879119905119864119905119883119879119883(119882

119905119882119879119891119864119905119883119879119883)

119879

+ 119864119905119884119879119905119882119879119905119864119905119883119879119883(119864

119905119884119879119905119881119905119883)119879

]

+1205734119905

(119875119870)4[119864119905119884119879119905119882119879119905119864119905119883119879119883(119864

119905119884119879119905119882119879119905119864119905119883119879119883)

119879

]

(28)


119869119905+1

minus 119869119905=

1

2119875119870(1198601205734 + 1198611205734 + 1198621205734 +119872120573) (29)

where

119860 =1

(119875119870)4119879119903[119864119884119879119882119864119883119879119883(119864119884119879119882119879119864119883119879119883)

119879

]

119861 =1

(119875119870)3119879119903[119882119882119879119864119883119879119883(119864119884119879119882119879119864119883119879119883)

119879


+ 119864119884119879119881119883(119864119884119879119882119879119864119883119879119883)119879

+ 119864119884119879119882119879119864119883119879119883(119882119882119879119864119883119879119883)119879

+ 119864119884119879119882119879119864119883119879119883(119864119884119879119881119883)]

119862 =1

(119875119870)2119879119903[(119864119884119879119882119879119864119883119879119883)

119879

+ 119864119884119879119882119864119883119879119883119864119879 + 119864119884119879119881119883(119882119882119879119864119883119879119883)119879

+119882119882119879119864119883119879119883 + (119864119884119879119881119883)119879

+ 119864119884119879119881119883(119864119884119879119881119883)119879

+119882119882119879119864119883119879119883(119882119882119879119864119883119879119883)119879

]

119872 =1

119875119870119879119903[119864 (119882119882119879119864119883119879119883)

119879

+ 119864 (119864119884119879119881119883)119879

+119882119882119879119864119883119879119883 + 119864119884119879119881119883119864119879]

(30)

where

119892 (120573) = (1198601205734 + 1198611205733 + 1198621205732 +119872120573) (31)

120597119892

120597120573= 4119860 (1205733 + 1198861205732 + 119887120573 + 119888) (32)

where 119886 = 31198614119860 119887 = 21198624119860 and 119888 = 1198724119860


119891 (119909) = 1199093 + 1198861199092 + 119887119909 + 119888







multiplicities


119894| 119892(120573

119894) = min(119892(120573

1) 119892(1205732) 119892(1205733)) 119894 isin 1 2 3




119881119905= 1198692119905

Δ119881119905= 1198692119905+1

minus 1198692119905

if Δ119881119905lt 0

(34)


0 that is 119869119905+1minus119869119905= (12119875119870)(1198601205734+1198611205734+1198621205734+119872120573) lt 0 (29)




120597119892

120597120573= 4119860 (1205733 + 1198861205732 + 119887120573 + 119862) (35)






119869MAE = 119869max = max1le119896le119873

1003816100381610038161003816119910 (119896) minus 119910 (119896)1003816100381610038161003816 (36)


119869SSE =119873

sum119896=1

(119910 (119896) minus 119910 (119896))2

(37)


119869MSE =1

119873

119873

sum119896=1

(119910 (119896) minus 119910 (119896))2

(38)


0 50 100 150 200 2500

1

2

3

4

5

6

7

Iteration

Mea

n Sq

uare

Err

ortimes10minus3

(a)

0 50 100 150 200 250

0

05

1

15

2

Iteration

Targ

et an

d FC

PN o

utpu

t

TargetFCPN

minus2

minus15

minus1

minus05

(b)



119869RMSE = radic119869MSE (39)



Var (119910 (119896))) 100 (40)


119869NMSE =sum119873

119896=1(119910 (119896) minus 119910 (119896))

2

sum119873

119896=1(119910 (119896) minus 119910)

2 (41)



119896=1(119910 (119896) minus 119910 (119896))

2

sum119873

119896=1(119910 (119896) minus 119910)

2)100 (42)


119910 (119896 + 1) = 03119910 (119896) + 06119910 (119896 minus 1) + 119891 [119906 (119896)] (43)




FCPN 13364 01143 45709119890 minus 004 00214 71627119890 minus 004

Dynamic 58231 31918 00128 01130 00028BPN 380666 481017 04810 06936 01035




119910 (119896 + 1) = 119891 [119910119901(119896) 119910

119901(119896 minus 1)] + 119906 (119896) (44)

where

119891 [119910119901(119896) 119910

119901(119896 minus 1)]

= [119910119901(119896) 119910119901(119896 minus 1) [119910

119901(119896) + 25]

1 + 1199102119901(119896) + 1199102

119901(119896 minus 1)

]

(45)



0 20 40 60 80 1000

1

2

3

Iteration

Mea

n Sq

uare

Err

ortimes10minus4

(a)

0 20 40 60 80 100

0

05

1

15

2

25

Iteration

Targ

et an

d FC

PN o

utpu

t

TargetFCPN

minus1

minus05




FCPN 01956 00032 32328119890 minus 005 00057 27164119890 minus 005

Dynamic 95152 29856 00299 01728 00308BP 101494 45163 00452 02125 00323


119910 (119896 + 1) = 119873 [119910 (119896) 119910 (119896 minus 1)] + 119906 (119896) (46)




119910 (119896 + 1) =119910 (119896)

1 + 119910 (119896)2+ 1199063 (119896) (47)




Dynamic 55652 36158 00362 01902 00223BP 426524 469937 04699 06855 02892





119910119901(119896 + 1)

= 119891 [119910119901(119896) 119910

119901(119896 minus 1) 119910

119901(119896 minus 2) 119906 (119896) 119906 (119896 minus 1)]

(48)


0 20 40 60 80 1000

1

2

Iteration

Mea

n Sq

uare

Err

ortimes10minus4

(a)

0 20 40 60 80 100

0

1

2

3

Iteration

Targ

et an

d FC

PN o

utpu

t

TargetFCPN

minus3

minus2

minus1

(b)


0 100 200 300 400 500 600 700 8000

1

2

3

4

5

6

Iteration

Mea

n Sq

uare

Err

or

times10minus4

(a)

0 100 200 300 400 500 600 700 800

0

05

1

Iteration

Targ

et an

d FC

PN o

utpu

t

TargetFCPN

minus1

minus05

(b)



119891 [1199091 1199092 1199093 1199094 1199095] =

1199091119909211990931199095(1199093minus 1) + 119909

4

1 + 11990923+ 11990922

(49)












0 10 20 30 40 50 60 70 80 90 100044046048

05052054056058

06

Iteration

Mea

n Sq

uare

Err

or

(a)

0 10 20 30 40 50 60 70 80 90 10045

50

55

60

Iteration

Targ

et an

d FC

PN o

utpu

t

DesiredFCPN




FCPN 9732 9948 9976 8635DN 9469 7932 8508 8339BPN 6783 7928 462234 7094



6 Conclusions




Acknowledgment


References










































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




D

D

f

p(t)

a(t)

iw11(0)

iw11(1)

iw11(2)

sum



11199092sdot sdot sdot 119909119899]





119863119895=119899

sum119894=1

(119909119894minus V119894119895)2

where 119895 = 1 2 119901 (10)




119895119896


prop (119869) =sum119899

119894=1(V119894119869minus 119909119894)2

sum119898

119895=1sum119899

119894=1(V119894119869minus 119909119894)2 (11)












119863119895=119899

sum119894=1

(119909119894minus V119894119895)2

where 119895 = 1 2 119901 (12)

Minimum net input

119911119894119899119869

=119899

sum119894=1

119909119894V119894119869 (13)



V119894119869(new) = V

119894119869(old) + prop ℎ (119869 119894 119895) (119909

119894minus V119894119869) (14)



119894119869

10038171003817100381710038171003817

21205902 (119905))

where ℎ (119869 119894 119895) isin [0 1]

(15)

120590 (119905) = 1205900exp(minus119905

119879) (16)





119869as

V119894119869(new) = V

119894119869(old) + 120572ℎ (119869 119894 119895) [119909

119894minus V119894119869(old)] (17)


X1

Xi

Xn

x1

xi

xn

11

1p

i1ij

ij

ip

n1

np

nj

Z1

Zj

Zp

w11

wj1

wp1

wjk

wjm

wik

wim

wpk

wpm

Y1

Yk

Ym

yy1

yy

k

yym




(Supervised)outstar

BMN



as

119908119869119896(new) = 119908

119869119896(old) + 120573ℎ (119869 119895 119896) [119910

119895minus 119908119869119896(old)]

for 1 le 119896 le 119898(18)


120573 (119869) =sum119898

119896=1(119910119896minus 119908119869119896)2

sum119901

119895=1sum119898

119896=1(119910119896minus 119908119895119896)2 (19)









119896= sum119895119911119895119908119895119896





119894119895 119894 = 1 2 119873 119895 = 1 2 119899 such that

119891 (1199091 1199092 119909

119899) =119873

sum119894=1

1198881198940(119899

sum119895=1

119908119894119895119909119894minus 120579119894) (20)

satisfies

max119909isinΩ

10038171003817100381710038171003817119891 (1199091 1199092 119909119899) minus 119891 (1199091 1199092 119909119899)10038171003817100381710038171003817 ltisin (21)





Let 119883 = [1199091 1199092 119909

119901] isin R119871times119875 119884 = [119910

1 1199102 119910

119901] isin

R119867times119875 119874 = [1199001 1199002 119900

119901] isin R119870times119875 119863 = [119889

1 1198892 119889

119901] isin


119869 =1

2119875119870

119875

sum119901=1

119870

sum119896=1

(119900119896119901minus 119889119896119901)2

(22)


119864119905= 119874119905minus 119863 = 119882

119905119884119905minus 119863 = 119882

119905119884119905119883 minus 119863 (23)


st 119869 =1

2119875119870119879119903(119864119905119864119879119905) (24)



119882119905+1

= 119882119905minus 120573119905

120597119869

120597119882119905

119881119905+1

= 119881119905minus 120573119905

120597119869

120597119881119905

(25)

or

119882119905+1

= 119882119905minus120573119905

119875119870119864119905119884119879119905

119881119905+1

= 119881119905minus120573119905

119875119870119882119879119905119864119905119883119879

(26)


119864119905+1119864119879119905+1

= (119882119905+1119884119905+1

minus 119863) (119882119905+1119884119905+1

minus 119863)119879

(27)


119869119905+1

minus 119869119905=

1

2119875119870119892 (120573)

119869 =1

2119875119870

119875

sum119901=1

119870

sum119896=1

(119900119896119901minus 119889119896119901)2

119864119905+1119864119879119905+1

= (119882119905+1119881119905+1119883 minus 119863) (119882

119905+1119881119905+1119883 minus 119863)

119879

= [(119882119905minus120573119905

119875119870119864119905119884119879119905)(119881119905minus120573119905

119875119870119882119879119905119864119905119883119879)119883

minus 119863][(119882119905minus120573119905

119875119870119864119905119884119879119905)(119881119905minus120573119905

119875119870119882119879119905119864119905119883119879)119883

minus 119863]T= [119864119905minus120573119905

119875119870(119882119905119882119879119905119864119905119883119879119883 + 119864

119905119884119879119905119881119905119883)

+1205732119905

(119875119870)2119864119905119884119879119905119882119879119905119864119905119883119879119883] lowast [119864

119905

minus120573119905

119875119870(119882119905119882119879119905119864119905119883119879119883 + 119864

119905119884119879119905119881119905119883)

+1205732119905

(119875119870)2(119864119905119884119879119905119882119879119905119864119905119883119879119883)]

119879

= 119864119905119864119879119905

minus120573119905

119875119870[119864119905(119882119905119882119879119905119864119905119883119879119883)

119879

+ 119864119905(119864119905119884119879119905119881119905119883)119879

+ (119882119905119882119879119905119864119905119883119879119883119864119879

119905) + (119864

119905119884119879119905119881119905119883119864119879119905)]

+1205732119905

(119875119870)2[119864119905(119864119905119884119879119905119882119879119905119864119905119883119879119883)

119879

+ 119864119905119884119879119905119882119879119905119864119905119883119879119883119864119879

119905

+ 119864119905119884119879119905119881119905119883(119882119905119882119879119905119864119905119883119879119883)

119879

+119882119905119882119879119905119864119905119883119879119883(119864

119905119884119879119905119881119905119883)119879

+ 119864119905119884119879119905119881119905119883(119864119905119884119879119905119881119905119883)119879

+119882119905119882119879119905119864119905119883119879119883(119882

119905119882119879119905119864119905119883119879119883)

119879

]

minus1205733119905

(119875119870)3[119882119905119882119879119905119864119905119883119879119883(119864

119905119884119879119905119882119879119905119864119905119883119879119883)

+ 119864119905119884119879119905119881119905119883(119864119905119884119879119905119882119879119905119864119905119883119879119883)

119879

+ 119864119905119884119879119905119882119879119905119864119905119883119879119883(119882

119905119882119879119891119864119905119883119879119883)

119879

+ 119864119905119884119879119905119882119879119905119864119905119883119879119883(119864

119905119884119879119905119881119905119883)119879

]

+1205734119905

(119875119870)4[119864119905119884119879119905119882119879119905119864119905119883119879119883(119864

119905119884119879119905119882119879119905119864119905119883119879119883)

119879

]

(28)


119869119905+1

minus 119869119905=

1

2119875119870(1198601205734 + 1198611205734 + 1198621205734 +119872120573) (29)

where

119860 =1

(119875119870)4119879119903[119864119884119879119882119864119883119879119883(119864119884119879119882119879119864119883119879119883)

119879

]

119861 =1

(119875119870)3119879119903[119882119882119879119864119883119879119883(119864119884119879119882119879119864119883119879119883)

119879


+ 119864119884119879119881119883(119864119884119879119882119879119864119883119879119883)119879

+ 119864119884119879119882119879119864119883119879119883(119882119882119879119864119883119879119883)119879

+ 119864119884119879119882119879119864119883119879119883(119864119884119879119881119883)]

119862 =1

(119875119870)2119879119903[(119864119884119879119882119879119864119883119879119883)

119879

+ 119864119884119879119882119864119883119879119883119864119879 + 119864119884119879119881119883(119882119882119879119864119883119879119883)119879

+119882119882119879119864119883119879119883 + (119864119884119879119881119883)119879

+ 119864119884119879119881119883(119864119884119879119881119883)119879

+119882119882119879119864119883119879119883(119882119882119879119864119883119879119883)119879

]

119872 =1

119875119870119879119903[119864 (119882119882119879119864119883119879119883)

119879

+ 119864 (119864119884119879119881119883)119879

+119882119882119879119864119883119879119883 + 119864119884119879119881119883119864119879]

(30)

where

119892 (120573) = (1198601205734 + 1198611205733 + 1198621205732 +119872120573) (31)

120597119892

120597120573= 4119860 (1205733 + 1198861205732 + 119887120573 + 119888) (32)

where 119886 = 31198614119860 119887 = 21198624119860 and 119888 = 1198724119860


119891 (119909) = 1199093 + 1198861199092 + 119887119909 + 119888







multiplicities


119894| 119892(120573

119894) = min(119892(120573

1) 119892(1205732) 119892(1205733)) 119894 isin 1 2 3




119881119905= 1198692119905

Δ119881119905= 1198692119905+1

minus 1198692119905

if Δ119881119905lt 0

(34)


0 that is 119869119905+1minus119869119905= (12119875119870)(1198601205734+1198611205734+1198621205734+119872120573) lt 0 (29)




120597119892

120597120573= 4119860 (1205733 + 1198861205732 + 119887120573 + 119862) (35)






119869MAE = 119869max = max1le119896le119873

1003816100381610038161003816119910 (119896) minus 119910 (119896)1003816100381610038161003816 (36)


119869SSE =119873

sum119896=1

(119910 (119896) minus 119910 (119896))2

(37)


119869MSE =1

119873

119873

sum119896=1

(119910 (119896) minus 119910 (119896))2

(38)


0 50 100 150 200 2500

1

2

3

4

5

6

7

Iteration

Mea

n Sq

uare

Err

ortimes10minus3

(a)

0 50 100 150 200 250

0

05

1

15

2

Iteration

Targ

et an

d FC

PN o

utpu

t

TargetFCPN

minus2

minus15

minus1

minus05

(b)



119869RMSE = radic119869MSE (39)



Var (119910 (119896))) 100 (40)


119869NMSE =sum119873

119896=1(119910 (119896) minus 119910 (119896))

2

sum119873

119896=1(119910 (119896) minus 119910)

2 (41)



119896=1(119910 (119896) minus 119910 (119896))

2

sum119873

119896=1(119910 (119896) minus 119910)

2)100 (42)


119910 (119896 + 1) = 03119910 (119896) + 06119910 (119896 minus 1) + 119891 [119906 (119896)] (43)




FCPN 13364 01143 45709119890 minus 004 00214 71627119890 minus 004

Dynamic 58231 31918 00128 01130 00028BPN 380666 481017 04810 06936 01035




119910 (119896 + 1) = 119891 [119910119901(119896) 119910

119901(119896 minus 1)] + 119906 (119896) (44)

where

119891 [119910119901(119896) 119910

119901(119896 minus 1)]

= [119910119901(119896) 119910119901(119896 minus 1) [119910

119901(119896) + 25]

1 + 1199102119901(119896) + 1199102

119901(119896 minus 1)

]

(45)



0 20 40 60 80 1000

1

2

3

Iteration

Mea

n Sq

uare

Err

ortimes10minus4

(a)

0 20 40 60 80 100

0

05

1

15

2

25

Iteration

Targ

et an

d FC

PN o

utpu

t

TargetFCPN

minus1

minus05




FCPN 01956 00032 32328119890 minus 005 00057 27164119890 minus 005

Dynamic 95152 29856 00299 01728 00308BP 101494 45163 00452 02125 00323


119910 (119896 + 1) = 119873 [119910 (119896) 119910 (119896 minus 1)] + 119906 (119896) (46)




119910 (119896 + 1) =119910 (119896)

1 + 119910 (119896)2+ 1199063 (119896) (47)




Dynamic 55652 36158 00362 01902 00223BP 426524 469937 04699 06855 02892





119910119901(119896 + 1)

= 119891 [119910119901(119896) 119910

119901(119896 minus 1) 119910

119901(119896 minus 2) 119906 (119896) 119906 (119896 minus 1)]

(48)


0 20 40 60 80 1000

1

2

Iteration

Mea

n Sq

uare

Err

ortimes10minus4

(a)

0 20 40 60 80 100

0

1

2

3

Iteration

Targ

et an

d FC

PN o

utpu

t

TargetFCPN

minus3

minus2

minus1

(b)


0 100 200 300 400 500 600 700 8000

1

2

3

4

5

6

Iteration

Mea

n Sq

uare

Err

or

times10minus4

(a)

0 100 200 300 400 500 600 700 800

0

05

1

Iteration

Targ

et an

d FC

PN o

utpu

t

TargetFCPN

minus1

minus05

(b)



119891 [1199091 1199092 1199093 1199094 1199095] =

1199091119909211990931199095(1199093minus 1) + 119909

4

1 + 11990923+ 11990922

(49)












0 10 20 30 40 50 60 70 80 90 100044046048

05052054056058

06

Iteration

Mea

n Sq

uare

Err

or

(a)

0 10 20 30 40 50 60 70 80 90 10045

50

55

60

Iteration

Targ

et an

d FC

PN o

utpu

t

DesiredFCPN




FCPN 9732 9948 9976 8635DN 9469 7932 8508 8339BPN 6783 7928 462234 7094



6 Conclusions




Acknowledgment


References










































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




X1

Xi

Xn

x1

xi

xn

11

1p

i1ij

ij

ip

n1

np

nj

Z1

Zj

Zp

w11

wj1

wp1

wjk

wjm

wik

wim

wpk

wpm

Y1

Yk

Ym

yy1

yy

k

yym




(Supervised)outstar

BMN



as

119908119869119896(new) = 119908

119869119896(old) + 120573ℎ (119869 119895 119896) [119910

119895minus 119908119869119896(old)]

for 1 le 119896 le 119898(18)


120573 (119869) =sum119898

119896=1(119910119896minus 119908119869119896)2

sum119901

119895=1sum119898

119896=1(119910119896minus 119908119895119896)2 (19)









119896= sum119895119911119895119908119895119896





119894119895 119894 = 1 2 119873 119895 = 1 2 119899 such that

119891 (1199091 1199092 119909

119899) =119873

sum119894=1

1198881198940(119899

sum119895=1

119908119894119895119909119894minus 120579119894) (20)

satisfies

max119909isinΩ

10038171003817100381710038171003817119891 (1199091 1199092 119909119899) minus 119891 (1199091 1199092 119909119899)10038171003817100381710038171003817 ltisin (21)





Let 119883 = [1199091 1199092 119909

119901] isin R119871times119875 119884 = [119910

1 1199102 119910

119901] isin

R119867times119875 119874 = [1199001 1199002 119900

119901] isin R119870times119875 119863 = [119889

1 1198892 119889

119901] isin


119869 =1

2119875119870

119875

sum119901=1

119870

sum119896=1

(119900119896119901minus 119889119896119901)2

(22)


119864119905= 119874119905minus 119863 = 119882

119905119884119905minus 119863 = 119882

119905119884119905119883 minus 119863 (23)


st 119869 =1

2119875119870119879119903(119864119905119864119879119905) (24)



119882119905+1

= 119882119905minus 120573119905

120597119869

120597119882119905

119881119905+1

= 119881119905minus 120573119905

120597119869

120597119881119905

(25)

or

119882119905+1

= 119882119905minus120573119905

119875119870119864119905119884119879119905

119881119905+1

= 119881119905minus120573119905

119875119870119882119879119905119864119905119883119879

(26)


119864119905+1119864119879119905+1

= (119882119905+1119884119905+1

minus 119863) (119882119905+1119884119905+1

minus 119863)119879

(27)


119869119905+1

minus 119869119905=

1

2119875119870119892 (120573)

119869 =1

2119875119870

119875

sum119901=1

119870

sum119896=1

(119900119896119901minus 119889119896119901)2

119864119905+1119864119879119905+1

= (119882119905+1119881119905+1119883 minus 119863) (119882

119905+1119881119905+1119883 minus 119863)

119879

= [(119882119905minus120573119905

119875119870119864119905119884119879119905)(119881119905minus120573119905

119875119870119882119879119905119864119905119883119879)119883

minus 119863][(119882119905minus120573119905

119875119870119864119905119884119879119905)(119881119905minus120573119905

119875119870119882119879119905119864119905119883119879)119883

minus 119863]T= [119864119905minus120573119905

119875119870(119882119905119882119879119905119864119905119883119879119883 + 119864

119905119884119879119905119881119905119883)

+1205732119905

(119875119870)2119864119905119884119879119905119882119879119905119864119905119883119879119883] lowast [119864

119905

minus120573119905

119875119870(119882119905119882119879119905119864119905119883119879119883 + 119864

119905119884119879119905119881119905119883)

+1205732119905

(119875119870)2(119864119905119884119879119905119882119879119905119864119905119883119879119883)]

119879

= 119864119905119864119879119905

minus120573119905

119875119870[119864119905(119882119905119882119879119905119864119905119883119879119883)

119879

+ 119864119905(119864119905119884119879119905119881119905119883)119879

+ (119882119905119882119879119905119864119905119883119879119883119864119879

119905) + (119864

119905119884119879119905119881119905119883119864119879119905)]

+1205732119905

(119875119870)2[119864119905(119864119905119884119879119905119882119879119905119864119905119883119879119883)

119879

+ 119864119905119884119879119905119882119879119905119864119905119883119879119883119864119879

119905

+ 119864119905119884119879119905119881119905119883(119882119905119882119879119905119864119905119883119879119883)

119879

+119882119905119882119879119905119864119905119883119879119883(119864

119905119884119879119905119881119905119883)119879

+ 119864119905119884119879119905119881119905119883(119864119905119884119879119905119881119905119883)119879

+119882119905119882119879119905119864119905119883119879119883(119882

119905119882119879119905119864119905119883119879119883)

119879

]

minus1205733119905

(119875119870)3[119882119905119882119879119905119864119905119883119879119883(119864

119905119884119879119905119882119879119905119864119905119883119879119883)

+ 119864119905119884119879119905119881119905119883(119864119905119884119879119905119882119879119905119864119905119883119879119883)

119879

+ 119864119905119884119879119905119882119879119905119864119905119883119879119883(119882

119905119882119879119891119864119905119883119879119883)

119879

+ 119864119905119884119879119905119882119879119905119864119905119883119879119883(119864

119905119884119879119905119881119905119883)119879

]

+1205734119905

(119875119870)4[119864119905119884119879119905119882119879119905119864119905119883119879119883(119864

119905119884119879119905119882119879119905119864119905119883119879119883)

119879

]

(28)


119869119905+1

minus 119869119905=

1

2119875119870(1198601205734 + 1198611205734 + 1198621205734 +119872120573) (29)

where

119860 =1

(119875119870)4119879119903[119864119884119879119882119864119883119879119883(119864119884119879119882119879119864119883119879119883)

119879

]

119861 =1

(119875119870)3119879119903[119882119882119879119864119883119879119883(119864119884119879119882119879119864119883119879119883)

119879


+ 119864119884119879119881119883(119864119884119879119882119879119864119883119879119883)119879

+ 119864119884119879119882119879119864119883119879119883(119882119882119879119864119883119879119883)119879

+ 119864119884119879119882119879119864119883119879119883(119864119884119879119881119883)]

119862 =1

(119875119870)2119879119903[(119864119884119879119882119879119864119883119879119883)

119879

+ 119864119884119879119882119864119883119879119883119864119879 + 119864119884119879119881119883(119882119882119879119864119883119879119883)119879

+119882119882119879119864119883119879119883 + (119864119884119879119881119883)119879

+ 119864119884119879119881119883(119864119884119879119881119883)119879

+119882119882119879119864119883119879119883(119882119882119879119864119883119879119883)119879

]

119872 =1

119875119870119879119903[119864 (119882119882119879119864119883119879119883)

119879

+ 119864 (119864119884119879119881119883)119879

+119882119882119879119864119883119879119883 + 119864119884119879119881119883119864119879]

(30)

where

119892 (120573) = (1198601205734 + 1198611205733 + 1198621205732 +119872120573) (31)

120597119892

120597120573= 4119860 (1205733 + 1198861205732 + 119887120573 + 119888) (32)

where 119886 = 31198614119860 119887 = 21198624119860 and 119888 = 1198724119860


119891 (119909) = 1199093 + 1198861199092 + 119887119909 + 119888







multiplicities


119894| 119892(120573

119894) = min(119892(120573

1) 119892(1205732) 119892(1205733)) 119894 isin 1 2 3




119881119905= 1198692119905

Δ119881119905= 1198692119905+1

minus 1198692119905

if Δ119881119905lt 0

(34)


0 that is 119869119905+1minus119869119905= (12119875119870)(1198601205734+1198611205734+1198621205734+119872120573) lt 0 (29)




120597119892

120597120573= 4119860 (1205733 + 1198861205732 + 119887120573 + 119862) (35)






119869MAE = 119869max = max1le119896le119873

1003816100381610038161003816119910 (119896) minus 119910 (119896)1003816100381610038161003816 (36)


119869SSE =119873

sum119896=1

(119910 (119896) minus 119910 (119896))2

(37)


119869MSE =1

119873

119873

sum119896=1

(119910 (119896) minus 119910 (119896))2

(38)


0 50 100 150 200 2500

1

2

3

4

5

6

7

Iteration

Mea

n Sq

uare

Err

ortimes10minus3

(a)

0 50 100 150 200 250

0

05

1

15

2

Iteration

Targ

et an

d FC

PN o

utpu

t

TargetFCPN

minus2

minus15

minus1

minus05

(b)



119869RMSE = radic119869MSE (39)



Var (119910 (119896))) 100 (40)


119869NMSE =sum119873

119896=1(119910 (119896) minus 119910 (119896))

2

sum119873

119896=1(119910 (119896) minus 119910)

2 (41)



119896=1(119910 (119896) minus 119910 (119896))

2

sum119873

119896=1(119910 (119896) minus 119910)

2)100 (42)


119910 (119896 + 1) = 03119910 (119896) + 06119910 (119896 minus 1) + 119891 [119906 (119896)] (43)




FCPN 13364 01143 45709119890 minus 004 00214 71627119890 minus 004

Dynamic 58231 31918 00128 01130 00028BPN 380666 481017 04810 06936 01035




119910 (119896 + 1) = 119891 [119910119901(119896) 119910

119901(119896 minus 1)] + 119906 (119896) (44)

where

119891 [119910119901(119896) 119910

119901(119896 minus 1)]

= [119910119901(119896) 119910119901(119896 minus 1) [119910

119901(119896) + 25]

1 + 1199102119901(119896) + 1199102

119901(119896 minus 1)

]

(45)



0 20 40 60 80 1000

1

2

3

Iteration

Mea

n Sq

uare

Err

ortimes10minus4

(a)

0 20 40 60 80 100

0

05

1

15

2

25

Iteration

Targ

et an

d FC

PN o

utpu

t

TargetFCPN

minus1

minus05




FCPN 01956 00032 32328119890 minus 005 00057 27164119890 minus 005

Dynamic 95152 29856 00299 01728 00308BP 101494 45163 00452 02125 00323


119910 (119896 + 1) = 119873 [119910 (119896) 119910 (119896 minus 1)] + 119906 (119896) (46)




119910 (119896 + 1) =119910 (119896)

1 + 119910 (119896)2+ 1199063 (119896) (47)




Dynamic 55652 36158 00362 01902 00223BP 426524 469937 04699 06855 02892





119910119901(119896 + 1)

= 119891 [119910119901(119896) 119910

119901(119896 minus 1) 119910

119901(119896 minus 2) 119906 (119896) 119906 (119896 minus 1)]

(48)


0 20 40 60 80 1000

1

2

Iteration

Mea

n Sq

uare

Err

ortimes10minus4

(a)

0 20 40 60 80 100

0

1

2

3

Iteration

Targ

et an

d FC

PN o

utpu

t

TargetFCPN

minus3

minus2

minus1

(b)


0 100 200 300 400 500 600 700 8000

1

2

3

4

5

6

Iteration

Mea

n Sq

uare

Err

or

times10minus4

(a)

0 100 200 300 400 500 600 700 800

0

05

1

Iteration

Targ

et an

d FC

PN o

utpu

t

TargetFCPN

minus1

minus05

(b)



119891 [1199091 1199092 1199093 1199094 1199095] =

1199091119909211990931199095(1199093minus 1) + 119909

4

1 + 11990923+ 11990922

(49)












0 10 20 30 40 50 60 70 80 90 100044046048

05052054056058

06

Iteration

Mea

n Sq

uare

Err

or

(a)

0 10 20 30 40 50 60 70 80 90 10045

50

55

60

Iteration

Targ

et an

d FC

PN o

utpu

t

DesiredFCPN




FCPN 9732 9948 9976 8635DN 9469 7932 8508 8339BPN 6783 7928 462234 7094



6 Conclusions




Acknowledgment


References










































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




Let 119883 = [1199091 1199092 119909

119901] isin R119871times119875 119884 = [119910

1 1199102 119910

119901] isin

R119867times119875 119874 = [1199001 1199002 119900

119901] isin R119870times119875 119863 = [119889

1 1198892 119889

119901] isin


119869 =1

2119875119870

119875

sum119901=1

119870

sum119896=1

(119900119896119901minus 119889119896119901)2

(22)


119864119905= 119874119905minus 119863 = 119882

119905119884119905minus 119863 = 119882

119905119884119905119883 minus 119863 (23)


st 119869 =1

2119875119870119879119903(119864119905119864119879119905) (24)



119882119905+1

= 119882119905minus 120573119905

120597119869

120597119882119905

119881119905+1

= 119881119905minus 120573119905

120597119869

120597119881119905

(25)

or

119882119905+1

= 119882119905minus120573119905

119875119870119864119905119884119879119905

119881119905+1

= 119881119905minus120573119905

119875119870119882119879119905119864119905119883119879

(26)


119864119905+1119864119879119905+1

= (119882119905+1119884119905+1

minus 119863) (119882119905+1119884119905+1

minus 119863)119879

(27)


119869119905+1

minus 119869119905=

1

2119875119870119892 (120573)

119869 =1

2119875119870

119875

sum119901=1

119870

sum119896=1

(119900119896119901minus 119889119896119901)2

119864119905+1119864119879119905+1

= (119882119905+1119881119905+1119883 minus 119863) (119882

119905+1119881119905+1119883 minus 119863)

119879

= [(119882119905minus120573119905

119875119870119864119905119884119879119905)(119881119905minus120573119905

119875119870119882119879119905119864119905119883119879)119883

minus 119863][(119882119905minus120573119905

119875119870119864119905119884119879119905)(119881119905minus120573119905

119875119870119882119879119905119864119905119883119879)119883

minus 119863]T= [119864119905minus120573119905

119875119870(119882119905119882119879119905119864119905119883119879119883 + 119864

119905119884119879119905119881119905119883)

+1205732119905

(119875119870)2119864119905119884119879119905119882119879119905119864119905119883119879119883] lowast [119864

119905

minus120573119905

119875119870(119882119905119882119879119905119864119905119883119879119883 + 119864

119905119884119879119905119881119905119883)

+1205732119905

(119875119870)2(119864119905119884119879119905119882119879119905119864119905119883119879119883)]

119879

= 119864119905119864119879119905

minus120573119905

119875119870[119864119905(119882119905119882119879119905119864119905119883119879119883)

119879

+ 119864119905(119864119905119884119879119905119881119905119883)119879

+ (119882119905119882119879119905119864119905119883119879119883119864119879

119905) + (119864

119905119884119879119905119881119905119883119864119879119905)]

+1205732119905

(119875119870)2[119864119905(119864119905119884119879119905119882119879119905119864119905119883119879119883)

119879

+ 119864119905119884119879119905119882119879119905119864119905119883119879119883119864119879

119905

+ 119864119905119884119879119905119881119905119883(119882119905119882119879119905119864119905119883119879119883)

119879

+119882119905119882119879119905119864119905119883119879119883(119864

119905119884119879119905119881119905119883)119879

+ 119864119905119884119879119905119881119905119883(119864119905119884119879119905119881119905119883)119879

+119882119905119882119879119905119864119905119883119879119883(119882

119905119882119879119905119864119905119883119879119883)

119879

]

minus1205733119905

(119875119870)3[119882119905119882119879119905119864119905119883119879119883(119864

119905119884119879119905119882119879119905119864119905119883119879119883)

+ 119864119905119884119879119905119881119905119883(119864119905119884119879119905119882119879119905119864119905119883119879119883)

119879

+ 119864119905119884119879119905119882119879119905119864119905119883119879119883(119882

119905119882119879119891119864119905119883119879119883)

119879

+ 119864119905119884119879119905119882119879119905119864119905119883119879119883(119864

119905119884119879119905119881119905119883)119879

]

+1205734119905

(119875119870)4[119864119905119884119879119905119882119879119905119864119905119883119879119883(119864

119905119884119879119905119882119879119905119864119905119883119879119883)

119879

]

(28)


119869119905+1

minus 119869119905=

1

2119875119870(1198601205734 + 1198611205734 + 1198621205734 +119872120573) (29)

where

119860 =1

(119875119870)4119879119903[119864119884119879119882119864119883119879119883(119864119884119879119882119879119864119883119879119883)

119879

]

119861 =1

(119875119870)3119879119903[119882119882119879119864119883119879119883(119864119884119879119882119879119864119883119879119883)

119879


+ 119864119884119879119881119883(119864119884119879119882119879119864119883119879119883)119879

+ 119864119884119879119882119879119864119883119879119883(119882119882119879119864119883119879119883)119879

+ 119864119884119879119882119879119864119883119879119883(119864119884119879119881119883)]

119862 =1

(119875119870)2119879119903[(119864119884119879119882119879119864119883119879119883)

119879

+ 119864119884119879119882119864119883119879119883119864119879 + 119864119884119879119881119883(119882119882119879119864119883119879119883)119879

+119882119882119879119864119883119879119883 + (119864119884119879119881119883)119879

+ 119864119884119879119881119883(119864119884119879119881119883)119879

+119882119882119879119864119883119879119883(119882119882119879119864119883119879119883)119879

]

119872 =1

119875119870119879119903[119864 (119882119882119879119864119883119879119883)

119879

+ 119864 (119864119884119879119881119883)119879

+119882119882119879119864119883119879119883 + 119864119884119879119881119883119864119879]

(30)

where

119892 (120573) = (1198601205734 + 1198611205733 + 1198621205732 +119872120573) (31)

120597119892

120597120573= 4119860 (1205733 + 1198861205732 + 119887120573 + 119888) (32)

where 119886 = 31198614119860 119887 = 21198624119860 and 119888 = 1198724119860


119891 (119909) = 1199093 + 1198861199092 + 119887119909 + 119888







multiplicities


119894| 119892(120573

119894) = min(119892(120573

1) 119892(1205732) 119892(1205733)) 119894 isin 1 2 3




119881119905= 1198692119905

Δ119881119905= 1198692119905+1

minus 1198692119905

if Δ119881119905lt 0

(34)


0 that is 119869119905+1minus119869119905= (12119875119870)(1198601205734+1198611205734+1198621205734+119872120573) lt 0 (29)




120597119892

120597120573= 4119860 (1205733 + 1198861205732 + 119887120573 + 119862) (35)






119869MAE = 119869max = max1le119896le119873

1003816100381610038161003816119910 (119896) minus 119910 (119896)1003816100381610038161003816 (36)


119869SSE =119873

sum119896=1

(119910 (119896) minus 119910 (119896))2

(37)


119869MSE =1

119873

119873

sum119896=1

(119910 (119896) minus 119910 (119896))2

(38)


0 50 100 150 200 2500

1

2

3

4

5

6

7

Iteration

Mea

n Sq

uare

Err

ortimes10minus3

(a)

0 50 100 150 200 250

0

05

1

15

2

Iteration

Targ

et an

d FC

PN o

utpu

t

TargetFCPN

minus2

minus15

minus1

minus05

(b)



119869RMSE = radic119869MSE (39)



Var (119910 (119896))) 100 (40)


119869NMSE =sum119873

119896=1(119910 (119896) minus 119910 (119896))

2

sum119873

119896=1(119910 (119896) minus 119910)

2 (41)



119896=1(119910 (119896) minus 119910 (119896))

2

sum119873

119896=1(119910 (119896) minus 119910)

2)100 (42)


119910 (119896 + 1) = 03119910 (119896) + 06119910 (119896 minus 1) + 119891 [119906 (119896)] (43)




FCPN 13364 01143 45709119890 minus 004 00214 71627119890 minus 004

Dynamic 58231 31918 00128 01130 00028BPN 380666 481017 04810 06936 01035




119910 (119896 + 1) = 119891 [119910119901(119896) 119910

119901(119896 minus 1)] + 119906 (119896) (44)

where

119891 [119910119901(119896) 119910

119901(119896 minus 1)]

= [119910119901(119896) 119910119901(119896 minus 1) [119910

119901(119896) + 25]

1 + 1199102119901(119896) + 1199102

119901(119896 minus 1)

]

(45)



0 20 40 60 80 1000

1

2

3

Iteration

Mea

n Sq

uare

Err

ortimes10minus4

(a)

0 20 40 60 80 100

0

05

1

15

2

25

Iteration

Targ

et an

d FC

PN o

utpu

t

TargetFCPN

minus1

minus05




FCPN 01956 00032 32328119890 minus 005 00057 27164119890 minus 005

Dynamic 95152 29856 00299 01728 00308BP 101494 45163 00452 02125 00323


119910 (119896 + 1) = 119873 [119910 (119896) 119910 (119896 minus 1)] + 119906 (119896) (46)




119910 (119896 + 1) =119910 (119896)

1 + 119910 (119896)2+ 1199063 (119896) (47)




Dynamic 55652 36158 00362 01902 00223BP 426524 469937 04699 06855 02892





119910119901(119896 + 1)

= 119891 [119910119901(119896) 119910

119901(119896 minus 1) 119910

119901(119896 minus 2) 119906 (119896) 119906 (119896 minus 1)]

(48)


0 20 40 60 80 1000

1

2

Iteration

Mea

n Sq

uare

Err

ortimes10minus4

(a)

0 20 40 60 80 100

0

1

2

3

Iteration

Targ

et an

d FC

PN o

utpu

t

TargetFCPN

minus3

minus2

minus1

(b)


0 100 200 300 400 500 600 700 8000

1

2

3

4

5

6

Iteration

Mea

n Sq

uare

Err

or

times10minus4

(a)

0 100 200 300 400 500 600 700 800

0

05

1

Iteration

Targ

et an

d FC

PN o

utpu

t

TargetFCPN

minus1

minus05

(b)



119891 [1199091 1199092 1199093 1199094 1199095] =

1199091119909211990931199095(1199093minus 1) + 119909

4

1 + 11990923+ 11990922

(49)












0 10 20 30 40 50 60 70 80 90 100044046048

05052054056058

06

Iteration

Mea

n Sq

uare

Err

or

(a)

0 10 20 30 40 50 60 70 80 90 10045

50

55

60

Iteration

Targ

et an

d FC

PN o

utpu

t

DesiredFCPN




FCPN 9732 9948 9976 8635DN 9469 7932 8508 8339BPN 6783 7928 462234 7094



6 Conclusions




Acknowledgment


References










































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




+ 119864119884119879119881119883(119864119884119879119882119879119864119883119879119883)119879

+ 119864119884119879119882119879119864119883119879119883(119882119882119879119864119883119879119883)119879

+ 119864119884119879119882119879119864119883119879119883(119864119884119879119881119883)]

119862 =1

(119875119870)2119879119903[(119864119884119879119882119879119864119883119879119883)

119879

+ 119864119884119879119882119864119883119879119883119864119879 + 119864119884119879119881119883(119882119882119879119864119883119879119883)119879

+119882119882119879119864119883119879119883 + (119864119884119879119881119883)119879

+ 119864119884119879119881119883(119864119884119879119881119883)119879

+119882119882119879119864119883119879119883(119882119882119879119864119883119879119883)119879

]

119872 =1

119875119870119879119903[119864 (119882119882119879119864119883119879119883)

119879

+ 119864 (119864119884119879119881119883)119879

+119882119882119879119864119883119879119883 + 119864119884119879119881119883119864119879]

(30)

where

119892 (120573) = (1198601205734 + 1198611205733 + 1198621205732 +119872120573) (31)

120597119892

120597120573= 4119860 (1205733 + 1198861205732 + 119887120573 + 119888) (32)

where 119886 = 31198614119860 119887 = 21198624119860 and 119888 = 1198724119860


119891 (119909) = 1199093 + 1198861199092 + 119887119909 + 119888







multiplicities


119894| 119892(120573

119894) = min(119892(120573

1) 119892(1205732) 119892(1205733)) 119894 isin 1 2 3




119881119905= 1198692119905

Δ119881119905= 1198692119905+1

minus 1198692119905

if Δ119881119905lt 0

(34)


0 that is 119869119905+1minus119869119905= (12119875119870)(1198601205734+1198611205734+1198621205734+119872120573) lt 0 (29)




120597119892

120597120573= 4119860 (1205733 + 1198861205732 + 119887120573 + 119862) (35)






119869MAE = 119869max = max1le119896le119873

1003816100381610038161003816119910 (119896) minus 119910 (119896)1003816100381610038161003816 (36)


119869SSE =119873

sum119896=1

(119910 (119896) minus 119910 (119896))2

(37)


119869MSE =1

119873

119873

sum119896=1

(119910 (119896) minus 119910 (119896))2

(38)


0 50 100 150 200 2500

1

2

3

4

5

6

7

Iteration

Mea

n Sq

uare

Err

ortimes10minus3

(a)

0 50 100 150 200 250

0

05

1

15

2

Iteration

Targ

et an

d FC

PN o

utpu

t

TargetFCPN

minus2

minus15

minus1

minus05

(b)



119869RMSE = radic119869MSE (39)



Var (119910 (119896))) 100 (40)


119869NMSE =sum119873

119896=1(119910 (119896) minus 119910 (119896))

2

sum119873

119896=1(119910 (119896) minus 119910)

2 (41)



119896=1(119910 (119896) minus 119910 (119896))

2

sum119873

119896=1(119910 (119896) minus 119910)

2)100 (42)


119910 (119896 + 1) = 03119910 (119896) + 06119910 (119896 minus 1) + 119891 [119906 (119896)] (43)




FCPN 13364 01143 45709119890 minus 004 00214 71627119890 minus 004

Dynamic 58231 31918 00128 01130 00028BPN 380666 481017 04810 06936 01035




119910 (119896 + 1) = 119891 [119910119901(119896) 119910

119901(119896 minus 1)] + 119906 (119896) (44)

where

119891 [119910119901(119896) 119910

119901(119896 minus 1)]

= [119910119901(119896) 119910119901(119896 minus 1) [119910

119901(119896) + 25]

1 + 1199102119901(119896) + 1199102

119901(119896 minus 1)

]

(45)



0 20 40 60 80 1000

1

2

3

Iteration

Mea

n Sq

uare

Err

ortimes10minus4

(a)

0 20 40 60 80 100

0

05

1

15

2

25

Iteration

Targ

et an

d FC

PN o

utpu

t

TargetFCPN

minus1

minus05




FCPN 01956 00032 32328119890 minus 005 00057 27164119890 minus 005

Dynamic 95152 29856 00299 01728 00308BP 101494 45163 00452 02125 00323


119910 (119896 + 1) = 119873 [119910 (119896) 119910 (119896 minus 1)] + 119906 (119896) (46)




119910 (119896 + 1) =119910 (119896)

1 + 119910 (119896)2+ 1199063 (119896) (47)




Dynamic 55652 36158 00362 01902 00223BP 426524 469937 04699 06855 02892





119910119901(119896 + 1)

= 119891 [119910119901(119896) 119910

119901(119896 minus 1) 119910

119901(119896 minus 2) 119906 (119896) 119906 (119896 minus 1)]

(48)


0 20 40 60 80 1000

1

2

Iteration

Mea

n Sq

uare

Err

ortimes10minus4

(a)

0 20 40 60 80 100

0

1

2

3

Iteration

Targ

et an

d FC

PN o

utpu

t

TargetFCPN

minus3

minus2

minus1

(b)


0 100 200 300 400 500 600 700 8000

1

2

3

4

5

6

Iteration

Mea

n Sq

uare

Err

or

times10minus4

(a)

0 100 200 300 400 500 600 700 800

0

05

1

Iteration

Targ

et an

d FC

PN o

utpu

t

TargetFCPN

minus1

minus05

(b)



119891 [1199091 1199092 1199093 1199094 1199095] =

1199091119909211990931199095(1199093minus 1) + 119909

4

1 + 11990923+ 11990922

(49)












0 10 20 30 40 50 60 70 80 90 100044046048

05052054056058

06

Iteration

Mea

n Sq

uare

Err

or

(a)

0 10 20 30 40 50 60 70 80 90 10045

50

55

60

Iteration

Targ

et an

d FC

PN o

utpu

t

DesiredFCPN




FCPN 9732 9948 9976 8635DN 9469 7932 8508 8339BPN 6783 7928 462234 7094



6 Conclusions




Acknowledgment


References










































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




0 50 100 150 200 2500

1

2

3

4

5

6

7

Iteration

Mea

n Sq

uare

Err

ortimes10minus3

(a)

0 50 100 150 200 250

0

05

1

15

2

Iteration

Targ

et an

d FC

PN o

utpu

t

TargetFCPN

minus2

minus15

minus1

minus05

(b)



119869RMSE = radic119869MSE (39)



Var (119910 (119896))) 100 (40)


119869NMSE =sum119873

119896=1(119910 (119896) minus 119910 (119896))

2

sum119873

119896=1(119910 (119896) minus 119910)

2 (41)



119896=1(119910 (119896) minus 119910 (119896))

2

sum119873

119896=1(119910 (119896) minus 119910)

2)100 (42)


119910 (119896 + 1) = 03119910 (119896) + 06119910 (119896 minus 1) + 119891 [119906 (119896)] (43)




FCPN 13364 01143 45709119890 minus 004 00214 71627119890 minus 004

Dynamic 58231 31918 00128 01130 00028BPN 380666 481017 04810 06936 01035




119910 (119896 + 1) = 119891 [119910119901(119896) 119910

119901(119896 minus 1)] + 119906 (119896) (44)

where

119891 [119910119901(119896) 119910

119901(119896 minus 1)]

= [119910119901(119896) 119910119901(119896 minus 1) [119910

119901(119896) + 25]

1 + 1199102119901(119896) + 1199102

119901(119896 minus 1)

]

(45)



0 20 40 60 80 1000

1

2

3

Iteration

Mea

n Sq

uare

Err

ortimes10minus4

(a)

0 20 40 60 80 100

0

05

1

15

2

25

Iteration

Targ

et an

d FC

PN o

utpu

t

TargetFCPN

minus1

minus05




FCPN 01956 00032 32328119890 minus 005 00057 27164119890 minus 005

Dynamic 95152 29856 00299 01728 00308BP 101494 45163 00452 02125 00323


119910 (119896 + 1) = 119873 [119910 (119896) 119910 (119896 minus 1)] + 119906 (119896) (46)




119910 (119896 + 1) =119910 (119896)

1 + 119910 (119896)2+ 1199063 (119896) (47)




Dynamic 55652 36158 00362 01902 00223BP 426524 469937 04699 06855 02892





119910119901(119896 + 1)

= 119891 [119910119901(119896) 119910

119901(119896 minus 1) 119910

119901(119896 minus 2) 119906 (119896) 119906 (119896 minus 1)]

(48)


0 20 40 60 80 1000

1

2

Iteration

Mea

n Sq

uare

Err

ortimes10minus4

(a)

0 20 40 60 80 100

0

1

2

3

Iteration

Targ

et an

d FC

PN o

utpu

t

TargetFCPN

minus3

minus2

minus1

(b)


0 100 200 300 400 500 600 700 8000

1

2

3

4

5

6

Iteration

Mea

n Sq

uare

Err

or

times10minus4

(a)

0 100 200 300 400 500 600 700 800

0

05

1

Iteration

Targ

et an

d FC

PN o

utpu

t

TargetFCPN

minus1

minus05

(b)



119891 [1199091 1199092 1199093 1199094 1199095] =

1199091119909211990931199095(1199093minus 1) + 119909

4

1 + 11990923+ 11990922

(49)












0 10 20 30 40 50 60 70 80 90 100044046048

05052054056058

06

Iteration

Mea

n Sq

uare

Err

or

(a)

0 10 20 30 40 50 60 70 80 90 10045

50

55

60

Iteration

Targ

et an

d FC

PN o

utpu

t

DesiredFCPN




FCPN 9732 9948 9976 8635DN 9469 7932 8508 8339BPN 6783 7928 462234 7094



6 Conclusions




Acknowledgment


References










































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




0 20 40 60 80 1000

1

2

3

Iteration

Mea

n Sq

uare

Err

ortimes10minus4

(a)

0 20 40 60 80 100

0

05

1

15

2

25

Iteration

Targ

et an

d FC

PN o

utpu

t

TargetFCPN

minus1

minus05




FCPN 01956 00032 32328119890 minus 005 00057 27164119890 minus 005

Dynamic 95152 29856 00299 01728 00308BP 101494 45163 00452 02125 00323


119910 (119896 + 1) = 119873 [119910 (119896) 119910 (119896 minus 1)] + 119906 (119896) (46)




119910 (119896 + 1) =119910 (119896)

1 + 119910 (119896)2+ 1199063 (119896) (47)




Dynamic 55652 36158 00362 01902 00223BP 426524 469937 04699 06855 02892





119910119901(119896 + 1)

= 119891 [119910119901(119896) 119910

119901(119896 minus 1) 119910

119901(119896 minus 2) 119906 (119896) 119906 (119896 minus 1)]

(48)


0 20 40 60 80 1000

1

2

Iteration

Mea

n Sq

uare

Err

ortimes10minus4

(a)

0 20 40 60 80 100

0

1

2

3

Iteration

Targ

et an

d FC

PN o

utpu

t

TargetFCPN

minus3

minus2

minus1

(b)


0 100 200 300 400 500 600 700 8000

1

2

3

4

5

6

Iteration

Mea

n Sq

uare

Err

or

times10minus4

(a)

0 100 200 300 400 500 600 700 800

0

05

1

Iteration

Targ

et an

d FC

PN o

utpu

t

TargetFCPN

minus1

minus05

(b)



119891 [1199091 1199092 1199093 1199094 1199095] =

1199091119909211990931199095(1199093minus 1) + 119909

4

1 + 11990923+ 11990922

(49)












0 10 20 30 40 50 60 70 80 90 100044046048

05052054056058

06

Iteration

Mea

n Sq

uare

Err

or

(a)

0 10 20 30 40 50 60 70 80 90 10045

50

55

60

Iteration

Targ

et an

d FC

PN o

utpu

t

DesiredFCPN




FCPN 9732 9948 9976 8635DN 9469 7932 8508 8339BPN 6783 7928 462234 7094



6 Conclusions




Acknowledgment


References










































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




0 20 40 60 80 1000

1

2

Iteration

Mea

n Sq

uare

Err

ortimes10minus4

(a)

0 20 40 60 80 100

0

1

2

3

Iteration

Targ

et an

d FC

PN o

utpu

t

TargetFCPN

minus3

minus2

minus1

(b)


0 100 200 300 400 500 600 700 8000

1

2

3

4

5

6

Iteration

Mea

n Sq

uare

Err

or

times10minus4

(a)

0 100 200 300 400 500 600 700 800

0

05

1

Iteration

Targ

et an

d FC

PN o

utpu

t

TargetFCPN

minus1

minus05

(b)



119891 [1199091 1199092 1199093 1199094 1199095] =

1199091119909211990931199095(1199093minus 1) + 119909

4

1 + 11990923+ 11990922

(49)












0 10 20 30 40 50 60 70 80 90 100044046048

05052054056058

06

Iteration

Mea

n Sq

uare

Err

or

(a)

0 10 20 30 40 50 60 70 80 90 10045

50

55

60

Iteration

Targ

et an

d FC

PN o

utpu

t

DesiredFCPN




FCPN 9732 9948 9976 8635DN 9469 7932 8508 8339BPN 6783 7928 462234 7094



6 Conclusions




Acknowledgment


References










































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




0 10 20 30 40 50 60 70 80 90 100044046048

05052054056058

06

Iteration

Mea

n Sq

uare

Err

or

(a)

0 10 20 30 40 50 60 70 80 90 10045

50

55

60

Iteration

Targ

et an

d FC

PN o

utpu

t

DesiredFCPN




FCPN 9732 9948 9976 8635DN 9469 7932 8508 8339BPN 6783 7928 462234 7094



6 Conclusions




Acknowledgment


References










































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in



research article fuzzy counter propagation neural network...

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