Research ArticleA Novel Memristive Multilayer Feedforward Small-World NeuralNetwork with Its Applications in PID Control

Zhekang Dong1 Shukai Duan1 Xiaofang Hu2 Lidan Wang1 and Hai Li3

1 School of Electronics and Information Engineering Southwest University Chongqing 400715 China2Department of MBE City University of Hong Kong Hong Kong3Department of ECE University of Pittsburgh Pittsburgh PA 15261 USA

Correspondence should be addressed to Shukai Duan duanskswueducn

Received 16 June 2014 Accepted 17 July 2014 Published 14 August 2014

Academic Editor Jinde Cao

Copyright copy 2014 Zhekang Dong et alThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

In this paper we present an implementation scheme of memristor-based multilayer feedforward small-world neural network(MFSNN) inspirited by the lack of the hardware realization of the MFSNN on account of the need of a large number of electronicneurons and synapses More specially a mathematical closed-form charge-governed memristor model is presented with derivationprocedures and the corresponding Simulink model is presented which is an essential block for realizing the memristive synapseand the activation function in electronic neurons Furthermore we investigate a more intelligent memristive PID controllerby incorporating the proposed MFSNN into intelligent PID control based on the advantages of the memristive MFSNN oncomputation speed and accuracy Finally numerical simulations have demonstrated the effectiveness of the proposed scheme

1 Introduction

In 1971 Professor Chua theoretically formulated and definedthe memristor and described that the memristance (short forresistor of a memristor) is characterized by the relationshipbetween the electrical charge 119902 and flux 120593 passing througha device [1] However it was only after the first physicalrealization of the memristor in nanoscale at Hewlett-Packard(HP) Lab in 2008 that it immediately garnered extensiveinterests among numerous researchers [2ndash4] The reportedexperiments confirmed that the memristor possesses switch-ing characteristic memory capacity and continuous inputand output property Due to these unique properties mem-ristors are being explored for many potential applicationsin the areas of nonvolatile memory [5 6] very-large-scaleintegrated (VLSI) circuit [7] artificial neural networks [8ndash10]digital image processing [11ndash13] and signal processing andpattern recognition [14] At present a considerable numberof models of different complexity have been proposed in theliteratures such as Pickettrsquos model [15] spintronic memristormodel [16] nonlinear ionic drift model [17] boundary

condition-based model [18] and threshold adaptive mem-ristor model [19] These published models exhibit desirednonlinearity of nanoscale structures This paper still appliesthe TiO

2memristor model on account of its simplified

expressions and the same ideal physical behaviorsBrain neural network emerges from the interactions of

dozens perhaps hundreds of brain regions each containingmillions of neurons [20] They are highly evolved nervoussystems capable of high-speed information processing real-time integration of information across segregated sensorychannels and brain regions [20 21] In order to obtainthe similar intelligence of human brain artificial neuralnetwork is designed to imitate the human brain not merelyon architecture but also on work patterns The connectionstructure of artificial neural networks is generally divided intofeedforward feedback single-layer multilayer and so forthMost of these connection architectures are approximatelyregular However the bioneurological researches show thatbrain neural network has random features to a certain degreeand exhibits ldquosmall-worldrdquo effectiveness that is high levelsof clustering and short average path length [22] Therefore

Hindawi Publishing Corporatione Scientific World JournalVolume 2014 Article ID 394828 12 pageshttpdxdoiorg1011552014394828

2 The Scientific World Journal

Voltage

Dopingfront Pt

PtL

hz

TiO2 (width is Dminus w)

TiO2minusx (width is w)

Figure 1 Schematic model of the HP memristor

it becomes a hot issue to design bionics neural networkwith randomness in architecture based on the background ofneurobiology

Notably Watts and Strogatz revealed a significant effectthat is in common among complex networks They pointedout that the real architecture of network is nearly a middlemodel between regular connection and random connectionand defined it as small-world network (WS model) in 1998[23] Over the past several years a large number of inves-tigations on complex networks have provided new insightinto biological neural networks Bassett concluded thathuman brain functional networks have small-world networktopology derived from a series of magneto encephalographyexperiments [22] Douw et al found that the cognition isrelated to the resting-state small-world network topology[24] In literature [25] the authors applied small-world prop-erties into prefrontal cortex that correlate with predictors ofpsychopathology risk which holds promise as a potentialneurodiagnostic for young children Taylor has studied theprotein structures and binding based on small-world networkstrategies and has made great progress [26] Simard built upa small-world neural network through rewiring the regularconnections and found that the small-world neural networkhas faster learning speed and smaller error than that ofthe regular network and random network with the samesize [27] In this paper we incorporate the memristor intothe multilayer feedforward small-world neural network tobuild up a new type of memristive neural network thatis easy of VLSI implementation and closer to biologicalnetworks Furthermore based on the proposed memristiveneural network a novel memristive intelligent PID controlleris put forward The nanoscale memristor is beneficial foreasily adjusting the PID control parameters and the hardwarerealization of modern intelligent microcontrol system

This paper is organized as follows In Section 2 we derivethe mathematical model of a nonlinear memristor whichtakes into account the nonlinear dopant drift effect nearby theterminals and the boundary conditions and give its Simulinkmodel correspondingly Following that the concepts anddesign algorithm of the memristive small-world neural net-work are described in detail in Section 3 Section 4 designs

a memristive PID controller by combining the proposedneural network with the standard PID control theory Inorder to guarantee the feasibility and effectiveness of theproposed scheme the computer simulations are performedin Section 5 Finally we give the conclusions in Section 6

2 The Nonlinear Memristor Model

21 The Mathematical Model of the Memristor A memristoror memristive device is essentially a two-terminal passiveelectronic element with memory capacity Its memristancestate depends on the amplitude polarity and duration ofthe external applied power The physical model of the HPmemristor from [28] shown in Figure 1 consists of a two-layer thin film (thickness 119863 asymp 10 nm) of TiO

2sandwiched

between two platinum electrodes One of the layers which isdescribed as TiO

2minus119909 is doped with oxygen vacancies (called

dopants) and thus it exhibits high conductivity The width 119908of the doped region is modulated depending on the amountof electric charge passing through the memristor The otherTiO2layer owning an insulating property has a perfect 2 1

oxygen-to-titanium ratio and this layer is referred to theundoped regionGenerally an external excitation V(119905) appliedacross the memristor may cause the charged dopants todrift and the boundary between the two regions would bemoved correspondingly with the total memristance changedeventually

The total resistance of the memristor119872 is a sum of theresistances of the doped and undoped regions

119872(119905) = 119877on (119908 (119905)

119863) + 119877off (1 minus

119908 (119905)

119863) (1)

where 119877on and 119877off are the limited values of the memristancefor 119908 = 119863 and 119908 = 0 respectively Setting the internal statevariable as 119909 = 119908119863 isin [0 1] (1) can be rewritten as

119872(119905) = 119877off + (119877on minus 119877off) 119909 (119905) (2)

When 119905 = 0 the initial memristance is

1198720= 119877off + (119877on minus 119877off) 1199090 (3)

The Scientific World Journal 3

0 05 10

02

04

06

08

1

f(x)

x

P = 1

P = 3

P = 5

P = 10

(a)

2 4 6 8

Linear model

0 2 40

1

2

2

15

1

05

0

times104

times104

times10minus6

times10minus6

P = 1

P = 3

P = 5

P = 10

M(Ω

)

Q (C)

(b)

Figure 2 The influence of different values of the integer 119875 on the memristor (a) Joglekar window function for 119875 = 1 119875 = 3 119875 = 5 and119875 = 10 (b) Relationship between memristance versus charge for the nonlinear memristor model As the integer 119875 increases the graphs tendto linearity

Themovement speed of the boundary between the dopedand undoped regions depends on the resistance of dopedarea the passing current and other factors according to thestate equation

119889119909

119889119905= 119896119894 (119905) 119891 (119909) 119896 =

120583V119877ON1198632

(4)

where 120583] asymp 10minus14m2 sminus1 Vminus1 is the average ionic mobil-

ity parameter As we all known small voltages can yieldenormous electric fields in nanoscale devices which cansecondarily produce significant nonlinearities in the ionictransport As for amemristive device these nonlinearities aremanifested particularly at the thin film edges especially at thetwo boundaries This phenomenon called nonlinear dopantdrift can be simulated by multiplying a proper windowfunction 119891(119909) on the right side of (4) Based on [28] thereare several kinds of classical window functions such asJoglekar window function and Biolek window function Thispaper chooses the Joglekar window function which can bedescribed by

119891 (119909) = 1 minus (2119909 minus 1)2119875 (5)

where 119875 is a positive integer called the control parameterFigure 2(a) exhibits the behavior of the Joglekar window

function for different values of 119875 Figure 2(b) shows thegraphs of the memristance versus charge of the memristorAs the value of119875 becomes smaller the nonlinearity increasesOn the other hand as the integer119875 increases themodel tendsto the linear model Based upon this as well as the literature

[3 28] we set the value of the integer 119875 = 1 in this windowfunction and obtain

119891 (119909) = 4119909 minus 41199092 (6)

Substituting (6) into (4)

int

119909(119905)

1199090

(1

119909 (120591)+

1

1 minus 119909 (120591)) 119889119909 (120591) = int

119905

0

4119896119894 (120591) 119889120591 (7)

where the internal state variable satisfies 119909(120591) isin [1199090 119909(119905)] and

the integration time is 0 le 120591 le 119905Assume 119902

0= 0 we can get

119909 (119905)

1 minus 119909 (119905)=

1199090

1 minus 1199090

times 1198904119896119902(119905)

(8)

The initial value of the state variable can be expressed as

1199090=119877off minus 1198770Δ119877

(9)

Then the expression of 119909(119905) can be calculated as

119909 (119905) = 1 minus1

1198601198904119896119902(119905) + 1 (10)

where119860 is a constant and its value is determined by 119877off 119877onand 119877

0

119860 =119877off minus 11987701198770minus 119877on

(11)

4 The Scientific World Journal

+ minus

+ minus

Ronw

DRoff(1 minus

w

D)

(a)

30

25

20

15

10

5

0

minus5minus02 minus01 0 01 02

Positive to negativeNegative to positive

Q (mC)

M(kΩ

)

(b)

Figure 3 The nonlinear memristor model and its characteristic curves (a) The equivalent circuit of the memristor and its 3D symbol (b)The relationship between memristance and the charge

Integrator

Math function

+

+

+

Constant

Math function

ProductProduct

Constant1

Add

1Constant

+

+

+

+

Divide

D ivide

D

U

100

4

Ron

Roff

M0

U2 divide

divide

divide

minus

minus

minus

times times

times

times

times

times

times

times

eu

A q(t) 1

sI

100RonM(t)

Q minus M

VminusM

Vminus I

V20k

10k

Figure 4 The Simulink model of the nonlinear memristor

Combining (2)ndash(11) the resistance of memristor can berewritten as

119872(119905) = 119877on + Δ1198771

1198601198904119896119902(119905) + 1 (12)

where Δ119877 = 119877off minus 119877onGiving a sine stimulus to the memristor we get the

simulation results using MATLAB software It is noteworthythat the memristor is a two-terminal element with polaritywhich is shown in Figure 3(a) When the current flows intothe memristive device from the positive pole to the negativepole one can get the relationship curve (the blue line)between memristance and charge through it as shown inFigure 3(b) On the contrary when the current flows into thememristor from the negative pole to the positive pole therelationship curve is denoted by the red dashed line When

the charge is close to or exceeds the charge threshold valuesthe resistance of the memristor reaches and stays at 119877on and119877off respectively Notably the threshold value denotes thequantity of electric charge required when the memristancereaches the limit resistance The parameters of the model are119877on = 100Ω 119877off = 20 kΩ 119872

0= 10 kΩ 119863 = 10 nm

and 120583] asymp 10minus14m2sminus1Vminus1 Moreover the simulation results

in Figure 3(b) are consistent with the results concluded byAdhikari et al in [8 29]

22 The Simulink Model of the Memristor For the sakeof analyzing the characteristics of the memristor modelcomprehensively a Simulinkmodel is built upon (2)ndash(12) andillustrated in Figure 4 The model mainly consists of inputand outputmodules internal operationmodules (multipliers

The Scientific World Journal 5

0 05 1 15

Time (s)

I(m

A)

minus06

minus04

minus02

0

02

04

06

(a)

0

05

minus05

0 05minus05

I(m

A)

(V)

(b)

30

25

20

15

10

5

0

minus5minus02 minus01 0 01 02

Q (mC)

M(kΩ

)

(c)

M(kΩ

)

105

10

95

9

85

8

75

7

65minus04 minus02 0 02 04 06

(V)

(d)

Figure 5 The results of the memristor Simulink model (a) The input current source (b) Relationship between the current 119894 and the voltageV (c) Relationship between the memristance119872 and the charge 119902 (d) Relationship between the memristance119872 and the voltage V

adders and modules) and parameter control modules Themodel parameters are the same as those in Figure 3 Thesignal stimulus applied into the memristor is a sinusoidalcurrent source with amplitude of 05mA and frequency of1Hz

The simulation results are exhibited in Figure 5 The cur-rent flowing through the memristor is shown in Figure 5(a)The typical hysteresis loop in Figure 5(b) shows its switchingcharacteristic that is the memristance can switch betweenhigh resistance and low resistance Figure 5(c) illustrates thatthe memristance is a nonlinear function of the flow of chargeas discussed previously Figure 5(d) shows the relationshipbetween the memristance 119872 and the charge 119902 Notably inthe part of the higher memristance state the change ratioof the memristance is low while in the part of the lowermemristance state the change ratio of the memristance ishigh

3 The Memristive Multilayer FeedforwardSmall-World Neural Network

31 The Multilayer Feedforward Small-World Neural NetworkGenerally small-world phenomenon indicates that a networkhas highly concentrated local connections and also includes afew random long connections In real world a large numberof networks have the small-world effect such as diseasetransmission network social network and the food chainnetwork [22] As is known to all in the classical multilayerfeedforward neural network such as BP network the 119894thneuron in the 119897th layer V119897

119894only connects its neighboring

neuron sets 119881119897minus1 and 119881119897+1 In addition all connections

are feedforward and no connections exist between neuronswithin the same layerThis kind of network can be consideredas a regular network Based on [27] and the constructionprocess of WS small world model we introduce Algorithm 1

6 The Scientific World Journal

Input Output

(a) (b)

Figure 6The construction progress of the multilayer feedforwardWS small-world neural network (a)The regular network (119875 = 0) (b)Themultilayer feedforward WS small-world neural network (0 lt 119875 lt 1)

Set the network has 119871 layersSet each layer has 119899

119897nodes

For ℎ = 1 To 119871 minus 21199011(ℎ) = 120572119890

minus120573(ℎ+2)

For 1198971= 1 To 119871 minus 2

For 119899119897= 1 To 119899

119897

For 1198992= 1 To 119899

119897

If rand lt 1199011199081198971(1198971+1)

11989911198992= 0

1198993= randint(1 119899

119897)

119905 = 1

While 119905 = 11199012= rand

For 1198972= 119871 minus 2 To 119897

1+ 2

If 1199012lt 1199011(1198972)

1198871198971 1198972

11989911198993= rand

119905 = 0

BreakFinish

Algorithm 1 Constructive procedure of the small-world neuralnetwork

which is used to construct multilayer feedforward neuralnetwork model according to the rewiring probability Thespecific construction process is given as follows

Step 1 Initialization assuming the number of the networklayers is 119871 each layer has 119899

119897neuron nodes and the rewiring

probability is 119875

Step 2 Generate the multilayer feedforward regular neuralnetwork as shown in Figure 6(a)

Step 3As shown in Algorithm 1 where 1199011(ℎ) is the probabil-

ity to select reconnection layer selection probability betweentwo neurons decreases exponentially 120572 and 120573 are the distancecoefficients rand and randint both are MATLAB functionsthe former is used to generate a number between (0 1)

randomly and the latter can be used to randomly generatean integer from 1 to 119899

119897 Since the connections of the (119871 minus

1)th layer cannot generate new long-connections if they aredisconnected the connections of the last two layers are notreconnected in the network

As shown in Figure 6(a) when the rewiring probability119875 = 0 the connection of the network maintains completelyregular mode Nonetheless when 119875 ranges from 0 to 1the long cross-layer connections are generated accordingto the rewiring probability 119875 and the probability of recon-nection layer selecting The resulting structure is betweencompletely regular and random connection mode as shownin Figure 6(b)

More specially we set the network connection matrix as119882 where119882119897denotes the connection submatrix between the119897th layer and the (119897 + 1)th layer 119908119897

119894119895isin 119877 is the connection

weight between the neuron 119894 of the 119897th layer and the neuron 119895of the (119897 + 1)th layer If there exists connection between thesetwo neurons then 119908119897

119894119895= 0 otherwise 119908119897

119894119895= 0 Therefore the

regular network connection matrix can be expressed as thefollowing equation

119882 =(((

(

0 1198821

0 0 0 0 0

0 0 1198822

0 0 0 0

0 0 0 1198823

0 0 0

0 0 0 0 1198824

0 0

0 0 0 0 0 1198825

0

0 0 0 0 0 0 1198826

0 0 0 0 0 0 0

)))

)

(13)

in which the number zero means no connection existsbetween the corresponding layers As for multilayer feedfor-ward small-world neural network because of the reconnec-tion performance the connection matrix changes into as

1198821015840=(((

(

0 11988211198613

11198614

11198615

11198616

11198617

1

0 0 11988221198614

21198615

21198616

21198617

2

0 0 0 11988231198615

31198616

31198617

3

0 0 0 0 11988241198616

41198617

4

0 0 0 0 0 11988251198617

5

0 0 0 0 0 0 1198826

0 0 0 0 0 0 0

)))

)

(14)

where119882119897 represents the reconnection submatrix between the119897th layer and the (119897 + 1)th layer and the 119861119897

1015840

119897is the submatrix

between two nonadjacent layers which 119897 isin 1 2 5 1198971015840 isin3 4 7

The Scientific World Journal 7

32 The Combination of the MFSNN and the Memristor

321 The Memristive Synapse The nanoscale memristor hashigh potential of information storage on account of the non-volatility with respect to long periods of power-down soit can be used as electric synapse in the artificial neuralnetworks and the primary reasons are manifold Firstly asa kind of analog component this device can realize weightupdating continuously Moreover the memristor possessesthe capacity of information storage due to the nonvolatilityThis feature is consistent with the memory ability of the neu-rons in humanrsquos brain Additionally the memristive neuralnetwork can be further integrated in crossbar array whichhas significant advantages in better information processingcapacity and huger storage

According to the nonlinear memristor model inSection 2 the memristive conductance can be calculatedfrom (12) as

119866 (119905) =1

119872 (119905)=

1

119877on + Δ119877 (1 (1198601198904119896119902(119905) + 1))

(15)

Differentiating (15) with respect to time 119905 we can beobtain

119889119866 (119905)

119889119905=

41198961198601198904119896119902(119905)

Δ119877

(119877on1198601198904119896119902(119905) + 119877off)

2times119889119902 (119905)

119889119905 (16)

where the current 119894(119905) = 119889119902(119905)119889119905 Notably when Δ119905 rarr 0119889119866(119905) asymp Δ119866 Hence the rate of the memristive conductanceΔ119866 can be described as the synapse weight update ruleThe relationship curve between the rate of the memristiveconductance change and the current is shown in Figure 7When the current is tiny the memristive conductance isalmost invariant While the current tends to plusmn4mA thememristive conductance changes suddenly So the currentthreshold value of thememristive synapse can be set as |119868th| =4mA

322 The Memristive Activation Function In the standardMFSNN the activation function for each neuron is usuallythe nonlinear Sigmoid function Particularly the activationfunction of the hidden layer adopts bipolar Sigmoid functionbut the output layer activation function is unipolar Sigmoidfunction

Based on the constitutive relationship of the memristor alot of nonlinear curves can be simulated and substituted [5 611 12] Based on the Simulink model of the nonlinear mem-ristor described in Section 2 we get its simplified Simulinkmodel accordingly as shown in Figure 8(a) Furthermore wedesign a package of the memristive device (in Figure 8(b))which can be considered as a system with single-input anddouble-output In this system the input variable is the current119868 and the output variable is thememristance119872(119905) and charge119902(119905) respectively

Then the behavior of the output curve can be adjustedefficiently by the parameter control module gain moduleand internal operationmodule which is crucial to implementthe activation function in the neural network Here we set

ΔG

8

6

4

2

0

minus2

minus4

minus6

minus8

times10minus3

minus4 minus2 0 2 4

I (mA)

Figure 7The relationship curve between the rate of the memristiveconductance change and the current

the activation functions for the hidden layer and outputlayer neurons of the memristive MFSNN as ℎ(119909) and 119910(119909)respectively

Figure 9(a) exhibits the constructing principle diagramof the memristive activation function in the hidden layerin which the red dotted line frame represents the parameteradjustment area 119870

1is the adjustable gain which is used for

controlling the shape of the activation function and1198702is the

fixed gain whose value is 1198702= 10minus4 The suitable parameters

of the memristor are chosen as 119877on = 100Ω 119877off = 20 kΩ1198720= 10 kΩ 119863 = 10 nm and 120583] asymp 10

minus14m2sminus1Vminus1 Theinput signal is a sinusoidal current with an amplitude of05mA and a frequency of 1Hz Notably the polarity of thevoltage applied into the memristor is opposite to the polarityof the memristor itself that is the current flows throughthe memristor from the negative polar to the positive polarFigure 9(b) shows the memristive activation function of thehidden layer and its shape varies with different values of119870

1

Similarly Figure 10(a) is the constructing principle dia-gram of the activation function of the output layer In theparameter adjustment part (the red dotted line frame) 119870

3

is an adjustable gain and 1198704is the fixed gain whose value

is 1198704= 2 times 10

minus4 The parameters are the same with thesimulation in Figure 9 Figure 10(b) shows the memristiveactivation function of the output layer Obviously as the valueof 1198703increases the graphs tend to flatten

4 The Memristive Intelligent PID Controller

So far the PID control has found widespread applicationsin the modern control field By adjusting the control actionof the proportion integration and differentiation we getan interactive nonlinear relationship among these controlvariables The neural network has the ability of expressing

8 The Scientific World Journal

Integrator Product ProductAdd

Divide

Constant Constant

Math function

Ron

q(t)

M(t)

+

+

+1

+

divide

times

times

times eu

A 100

ΔR4k

1

s

I

(a)

InOutOutI

q(t)

M(t)

(b)

Figure 8 The simplified memristor model (a) The simplified Simulink model of the memristor (b) The package of the memristor

InOut

OutI q(t)

M(t)

K1

K2

1

h(x)

minus

+

(a)

2

1

0

minus1

h(x)

minus4 minus2 0 2 4

K1 = 20000

K1 = 30000

K1 = 40000x

(b)

Figure 9Theprinciple diagramof thememristive activation function in the hidden layer (a)TheSimulinkmodel of thememristive activationfunction in the hidden layer (b) The curve of the memristive activation function ℎ(119909)

InOut

OutI q(t)

M(t)

K4

K3y(x)

(a)

minus4 minus2 0 2 4

K3 = 20000

K3 = 30000

K3 = 40000

x

2

15

1

05

0

minus05

y(x)

(b)

Figure 10 The principle diagram of the memristive activation function of the output layer (a) The Simulink model of the memristiveactivation function of the output layer (b) The curve of the memristive activation function 119910(119909)

The Scientific World Journal 9

the nonlinearity which can be used in the PID controlfor implementing the optimal nonlinear relationship amongcontrol variables In this work we build up a more intelligentPID controller with the parameters (119896

119901 119896119894 and 119896

119889) self-tune

based on the presented memristive multilayer feedforwardsmall-world neural network

According to the literature [30] the classical incrementaldigital PID control algorithm can be described as

119906 (119896) = 119906 (119896 minus 1) + 119896119901(119890 (119896) minus 119890 (119896 minus 1))

+ 119896119894119890 (119896) + 119896

119889(119890 (119896) minus 2119890 (119896 minus 1) + 119890 (119896 minus 2))

(17)

where the 119896119901 119896119894 and 119896

119889are the coefficient of the proportion

integration and differentiation respectivelyIn Figure 11 the ANN is the memristive multilayer feed-

forward small-world neural network Its learning algorithmconsisted of the backward error propagation and the forwardinput signal propagation Different from the traditional mul-tilayer feedforward neural network the state of the neurons ineach layer not only affects the state of the neurons in the nextlayer but also affects the state of the neurons in the cross-layer

Based on the novel neural network presented in Section 3we set the 119895 119894 and 119897 that represent the input layer hiddenlayer and output layer respectively The number of the inputlayer is 1 which is same with that of the output layer andthe number of the hidden layer is 119878 119909

119894represents the input

vector of the network then the set of the input samples is1198741

119895= [1199091 1199092 1199093 119909

119873] The number of the input vectors

is dependent on the complexity of the system Notably thesuperscript 1 represents the first layer in the whole neuralnetwork

The input and output vectors of the first hidden layer canbe expressed as

net1198941119894(119896) =

119873

sum

119895=0

1199081198941

1198941198951198741

119895

1198741198941

119894(119896) = ℎ (net1198941

119894(119896))

119894 = 1 2 3 119876

(18)

where the superscript 1198941denotes the first hidden layer of the

network and ℎ(119909) is the memristive bipolar sigmoid functionproposed in Section 3

By that analogy the input and output vectors of the 119878thhidden layer can be written as

net119894119904119894(119896) =

119873

sum

119895=0

119908119894119904

1198941198951198741

119895+

119904minus1

sum

119886=1

119904

sum

119887=1

119908119894119904

119894119886119894119904net119894119904119887(119896)

119874119894119904

119894(119896) = ℎ (net119894119904

119894(119896))

119894 = 1 2 3 119876

(19)

Finally the input and output vectors of the output layercan be obtained as

net119904+2119897(119896) =

119873

sum

119895=0

119908119904+2

1198941198971198741

119895+

119904

sum

119886=1

119904

sum

119887=1

119908119904+2

119894119886119897net119904+2119887(119896)

119874119904+2

119897(119896) = 119910 (net119904+2

119897(119896))

119897 = 1 2 3

(20)

where 119910(119909) denotes the memristive unipolar sigmoid func-tion The three nodes of the output layer are correspondingwith the nonnegative adjustable parameters 119896

119901 119896119894 and 119896

119889of

the PID controller respectivelyFrom [27] we conclude the weight update algorithm of

the memristive multilayer feedforward small-world neuralnetwork as below

Δ119908119904+2

119897119894(119896) = 120572Δ119908

119904+2

119897119894(119896 minus 1) + 120578120575

119904+2

119897119874119904+1

119894(119896)

119908119904+2

119897119894(119896 + 1) = 119908

119904+2

119897119894(119896) + Δ119908

119904+2

119897119894(119896)

(21)

where 120572 is the inertial coefficient whose scope ranges from 0to 1 and 120578 isin (0 1) is the learning rate

5 Computer Simulations and Results

In this section somenumerical simulations of thememristivemultilayer feedforward small-world neural network PIDcontroller have been executed on MATLAB software Themathematical model of the controlled plant is given as

119910out (119896) =119886 (119896) 119910out (119896 minus 1)

1 + 1199102out (119896 minus 1)+ 119906 (119896 minus 1) (22)

where the 119886(119896) is slow time-variant and its expression is119886(119896) = 12(1 minus 08119890

01119896)

The memristive neural network under investigation isconstituted by seven layers with four neurons in the inputlayer three neurons in the output layer and five in each of thefive hidden layers The learning rate of the network 120578 = 04and the inertial coefficient 120572 = 005 The initial weighs asrandom values fall in [minus05 05] and the value of the rewiringprobability is chosen as 119875 = 0 119875 = 008 119875 = 01 and 119875 = 02respectively The parameters are 119877on = 100Ω 119877off = 20 kΩ1198720= 10 kΩ 119863 = 10 nm and 120583] asymp 10

minus14m2sminus1Vminus1 1198701and

1198703are user-specified parameters whose value both are 20000

and the action time is 119905119904 = 0001 s When the system workssteadily the tracking results can be gotten as follows

Figure 12(a) shows the input signal (step response curve119903in(119896) = 10) and the output curves under a different rewiringprobability 119875 As can be seen from the figure when thetime 119905 = 05 s the whole system reaches the steady stateMaking a further analysis we can conclude that when therewiring probability 119875 = 0 the memristive neural networkkeeps regularly in architecture Its respond speed is slowerthan that of network when the rewiring probability 119875 =

008 and 119875 = 01 Moreover Figure 12(b) exhibits theerror curves between the input signal and the output signal

10 The Scientific World Journal

rin Error

aeat

ANN

PID Plant yout

kp ki kd

Figure 11 The structure diagram of the memristive neural network PID controller

0

05

1

15

20 05 1 15

Time (s)

Inpu

t ou

tput

Regularnetwork

InputP = 008

P = 01

P = 0

P = 02

(a)

0

08

06

04

02

1

20 05 1 15

Time (s)

Erro

rRegular network

P = 008

P = 01

P = 0

P = 02

(b)

001

002

003

0

005

01

0 2 4 6Time (s)

0 2 4 6Time (s)

0 2 4 60

2

4

Time (s)

P = 008

P = 008

P = 008

times10minus3

k pk i

k d

(c)

Figure 12The simulation results of the memristive neural network PID controller (119903in(119896) = 10) under a different rewiring probability 119875 (a)The step response curve (b) The error curves (c) The curves of the control parameters when the rewiring probability 119875 = 008

correspondingly When the rewiring probability 119875 = 008

and 119875 = 01 the network spends less time on approachingthe predefined approximation error than the regular network(when 119875 = 0) Figure 12(c) shows the output variables ofthe memristive multilayer feedforward small-world neural

network when 119875 = 008 which are the control parameters 119896119901

119896119894 and 119896

119889 correspondingly

In order to verify the superior performance of thememristive small-world neuronal networks andfigure out theoptimal structure we conducted a series of simulations to

The Scientific World Journal 11

4500

5000

5500

6000

6500

7000

0 02 04 06 08 1

Aver

age i

nter

actio

ns

Sampling points

P

(a)

35

37

39

41

43

45

47

0 02 04 06 08 1

Succ

essfu

l int

erac

tions

Sampling points

P

(b)

Figure 13 The convergence performance of the memristive neural network under different 119875 (a) The relationship between iteration andrewiring probability (b) The effective approximation number in 50 times simulations under varying rewiring probability

observe the convergence performance of the proposed net-work under different119875 Figure 13(a) shows the approximationspeed (iteration times) of different network structures thatis the smallest iteration number for reaching the predefinedapproximation error 120576 = 00001 Each drawn point is theaverage value of 50 times runs It can be observed thatthe small-world networks (0 lt 119875 lt 1) need much lessiteration times than the regular neural network (when 119875 =

0) which demonstrates its advantage in processing speedFurthermore when 119875 = 008 the network has the fastapproximation speed

Notably the mathematical function of this system has thelocal minimum for getting out of the local minimum wedefine the maximum allowable iteration times to be 10000as previously mentioned for each 119875 and we performed thesimulation for 50 times where the effective approximationtimes that is error lt 00001 within 10000 iterations arepresented in Figure 13(b) It can be found that the small-worldnetworks have higher accuracy rate than the regular network

6 Conclusions

A mathematical closed-form charge-governed memristormodel is recalled firstly and the corresponding Simulinkmodel is presented Using the change rule of memconduc-tance a memristive realization scheme for synaptic weightis proposed Moreover the activation functions in electricneurons are also implemented based on the single-inputand double-output package of the memristor Combiningthe proposed memristive synapse and activation functions amemristor-basedMFSNN is addressed It exhibits advantagesin computation speed and accuracy over the traditionalmultilayer neural networks by considering the small-worldeffect Meanwhile it has potential of hardware realizationof the neural network because of the nanoscale size of thememristive synapse These superior properties can furtherimprove the application of the neural networks such as inthe intelligent controller design Motivated by this we apply

the memristor-based MFSNN to classical PID control andthe proposed memristive PID controller may possess thefollowing superiorities (i) Its nanoscale physical implemen-tation could promote the development of themicrocontroller(ii) Because of the participation of the memristive neuralnetwork the proposed PID controller can realize the param-eters self-adjustment (iii) The control speed and accuracyare improved Eventually extensive numerical simulationsjustify the effectiveness and efficiency of the memristive PIDcontroller over the regular neural network PID controllerThis work may provide a theoretical reference to physicallyrealize the small-world neural networks and further promotethe development of modern intelligent control technology

Conflict of Interests

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

Acknowledgments

The work was supported by Program for New CenturyExcellent Talents in University (Grant nos [2013] 47)National Natural Science Foundation of China (Grant nos61372139 61101233 and 60972155) ldquoSpring Sunshine PlanrdquoResearch Project of Ministry of Education of China (Grantno z2011148) Technology Foundation for Selected OverseasChinese Scholars Ministry of Personnel in China (Grantno 2012-186) University Excellent Talents Supporting Foun-dations in of Chongqing (Grant no 2011-65) UniversityKey Teacher Supporting Foundations of Chongqing (Grantno 2011-65) Fundamental Research Funds for the CentralUniversities (Grant nos XDJK2014A009 XDJK2013B011)

References

[1] L O Chua ldquoMemristors the missing circuit elementrdquo IEEETransactions on Circuits and Systems vol 18 pp 507ndash519 1971

12 The Scientific World Journal

[2] J M Tour and H Tao ldquoElectronics the fourth elementrdquoNaturevol 453 no 7191 pp 42ndash43 2008

[3] D B Strukov G S Snider D R Stewart and R S WilliamsldquoThe missing memristor foundrdquo Nature vol 453 pp 80ndash832008

[4] R S Williams ldquoHow we found the missing Memristorrdquo IEEESpectrum vol 45 no 12 pp 28ndash35 2008

[5] S K Duan X F Hu L D Wang C D Li and P MazumderldquoMemristor-based RRAM with applicationsrdquo Science ChinaInformation Sciences vol 55 no 6 pp 1446ndash1460 2012

[6] X F Hu S K Duan L DWang and C D Li ldquoAnalog memorybased on pulse controlled memristorrdquo Journal of the Universityof Electronic Science and Technology of China vol 40 no 5 pp642ndash647 2011

[7] S Shin K Kim and S Kang ldquoResistive computingmemristors-enabled signal multiplicationrdquo IEEE Transactions on Circuitsand Systems I Regular Papers vol 60 no 5 pp 1241ndash1249 2013

[8] S P Adhikari C Yang H Kim and L O Chua ldquoMemristorbridge synapse-based neural network and its learningrdquo IEEETransactions on Neural Networks and Learning Systems vol 23no 9 pp 1426ndash1435 2012

[9] S H Jo T Chang I Ebong B B Bhadviya P Mazumder andW Lu ldquoNanoscale memristor device as synapse in neuromor-phic systemsrdquo Nano Letters vol 10 no 4 pp 1297ndash1301 2010

[10] Y Dai C Li and H Wang ldquoExpanded HP memristor modeland simulation in STDP learningrdquo Neural Computing andApplications vol 24 no 1 pp 51ndash57 2014

[11] S K Duan X F Hu L DWang S Y Gao andCD Li ldquoHybridmemristorRTD structure-based cellular neural networks withapplications in image processingrdquoNeural Computing and Appli-cations vol 25 no 2 pp 291ndash296 2014

[12] L Chen C D Li T W Huang and Y R Chen ldquoMemristorcrossbar-based unsupervised image learningrdquo Neural Comput-ing and Applications vol 25 no 2 pp 393ndash400 2013

[13] X F Hu S K Duan L D Wang and X F Liao ldquoMemristivecrossbar array with applications in image processingrdquo ScienceChina Information Sciences vol 41 pp 500ndash512 2011

[14] B Mouttet ldquoProposal for memristors in signal processingrdquo inNano-Net 3rd International ICST Conference NanoNet BostonMS USA September 14ndash16 2008 Revised Selected Papersvol 3 of Lecture Notes of the Institute for Computer SciencesSocial Informatics and Telecommunications Engineering pp 11ndash13 2009

[15] H Abdalla andM D Pickett ldquoSPICEmodeling of memristorsrdquoin Proceedings of the 2011 IEEE International Symposium ofCircuits and Systems (ISCAS rsquo11) pp 1832ndash1835 May 2011

[16] X B Wang Y R Chen H W Xi and D Dimitrov ldquoSpin-tronic memristor through spin-thorque-induced magnetiza-tion motionrdquo IEEE Electron Device Letters vol 30 no 3 pp294ndash297 2009

[17] D B Strukov and R S Williams ldquoExponential ionic drift fastswitching and low volatility of thin-film memristorsrdquo AppliedPhysics A Materials Science amp Processing vol 94 pp 515ndash5192009

[18] F Corinto A Ascoli and M Gilli ldquoMathematical models andcircuit implementations of memristive systemsrdquo in Proceedingsof the 13th International Workshop on Cellular Nanoscale Net-works and their Applications (CNNA rsquo12) pp 1ndash6 August 2012

[19] S Kvatinsky E G Friedman A Kolodny and U C WeiserldquoTEAMThrEshold adaptive memristor modelrdquo IEEE Transac-tions on Circuits and Systems I Regular Papers vol 60 no 1 pp211ndash221 2013

[20] O Sporns and C J Honey ldquoSmall worlds inside big brainsrdquoProceedings of the National Academy of Sciences of the UnitedStates of America vol 103 no 51 pp 19219ndash19220 2006

[21] P Tangkraingkij C Lursinsap S Sanguansintukul and TDesudchit ldquoInsider and outsider person authentication withminimum number of brain wave signals by neural and homo-geneous identity filteringrdquo Neural Computing and Applicationsvol 22 no 1 pp 463ndash476 2013

[22] D S Bassett and E D Bullmore ldquoSmall-world brain networksrdquoNeuroscientist vol 12 no 6 pp 512ndash523 2006

[23] D J Watts and S H Strogatz ldquoCollective dynamics of ldquosmall-worldrdquo networksrdquoNature vol 393 no 6684 pp 440ndash442 1998

[24] L Douw M M Schoonheim D Landi et al ldquoCognitionis related to resting-state small-world network topology anmagnetoencephalographic studyrdquo Neuroscience vol 175 pp169ndash177 2011

[25] T Fekete F D C C Beacher J Cha D Rubin and L RMujica-Parodi ldquoSmall-world network properties in prefrontal cortexcorrelate with predictors of psychopathology risk in youngchildren aNIRS studyrdquoNeuroImage vol 85 part 1 pp 345ndash3532014

[26] N R Taylor ldquoSmall world network strategies for studyingprotein structures and bindingrdquo Computational and StructuralBiotechnology Journal vol 5 Article ID e201302006 2013

[27] D Simard L Nadeau andH Kroger ldquoFastest learning in small-world neural networksrdquo Physics Letters A General Atomic andSolid State Physics vol 336 no 1 pp 8ndash15 2005

[28] Z Biolek D Biolek and V Biolkova ldquoSPICEmodel of memris-tor with nonlinear dopant driftrdquo Radioengineering vol 18 no2 pp 210ndash214 2009

[29] H Kim M P Sah C Yang T Roska and L O ChualdquoMemristor bridge synapsesrdquo Proceedings of the IEEE vol 100no 6 pp 2061ndash2070 2012

[30] M Esfandyari M A Fanaei and H Zohreie ldquoAdaptive fuzzytuning of PID controllersrdquo Neural Computing and Applicationsvol 23 no 1 pp 19ndash28 2013

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DistributedSensor Networks

International Journal of

2 The Scientific World Journal

Voltage

Dopingfront Pt

PtL

hz

TiO2 (width is Dminus w)

TiO2minusx (width is w)

Figure 1 Schematic model of the HP memristor

it becomes a hot issue to design bionics neural networkwith randomness in architecture based on the background ofneurobiology

Notably Watts and Strogatz revealed a significant effectthat is in common among complex networks They pointedout that the real architecture of network is nearly a middlemodel between regular connection and random connectionand defined it as small-world network (WS model) in 1998[23] Over the past several years a large number of inves-tigations on complex networks have provided new insightinto biological neural networks Bassett concluded thathuman brain functional networks have small-world networktopology derived from a series of magneto encephalographyexperiments [22] Douw et al found that the cognition isrelated to the resting-state small-world network topology[24] In literature [25] the authors applied small-world prop-erties into prefrontal cortex that correlate with predictors ofpsychopathology risk which holds promise as a potentialneurodiagnostic for young children Taylor has studied theprotein structures and binding based on small-world networkstrategies and has made great progress [26] Simard built upa small-world neural network through rewiring the regularconnections and found that the small-world neural networkhas faster learning speed and smaller error than that ofthe regular network and random network with the samesize [27] In this paper we incorporate the memristor intothe multilayer feedforward small-world neural network tobuild up a new type of memristive neural network thatis easy of VLSI implementation and closer to biologicalnetworks Furthermore based on the proposed memristiveneural network a novel memristive intelligent PID controlleris put forward The nanoscale memristor is beneficial foreasily adjusting the PID control parameters and the hardwarerealization of modern intelligent microcontrol system

This paper is organized as follows In Section 2 we derivethe mathematical model of a nonlinear memristor whichtakes into account the nonlinear dopant drift effect nearby theterminals and the boundary conditions and give its Simulinkmodel correspondingly Following that the concepts anddesign algorithm of the memristive small-world neural net-work are described in detail in Section 3 Section 4 designs

a memristive PID controller by combining the proposedneural network with the standard PID control theory Inorder to guarantee the feasibility and effectiveness of theproposed scheme the computer simulations are performedin Section 5 Finally we give the conclusions in Section 6

2 The Nonlinear Memristor Model

21 The Mathematical Model of the Memristor A memristoror memristive device is essentially a two-terminal passiveelectronic element with memory capacity Its memristancestate depends on the amplitude polarity and duration ofthe external applied power The physical model of the HPmemristor from [28] shown in Figure 1 consists of a two-layer thin film (thickness 119863 asymp 10 nm) of TiO

2sandwiched

between two platinum electrodes One of the layers which isdescribed as TiO

2minus119909 is doped with oxygen vacancies (called

dopants) and thus it exhibits high conductivity The width 119908of the doped region is modulated depending on the amountof electric charge passing through the memristor The otherTiO2layer owning an insulating property has a perfect 2 1

oxygen-to-titanium ratio and this layer is referred to theundoped regionGenerally an external excitation V(119905) appliedacross the memristor may cause the charged dopants todrift and the boundary between the two regions would bemoved correspondingly with the total memristance changedeventually

The total resistance of the memristor119872 is a sum of theresistances of the doped and undoped regions

119872(119905) = 119877on (119908 (119905)

119863) + 119877off (1 minus

119908 (119905)

119863) (1)

where 119877on and 119877off are the limited values of the memristancefor 119908 = 119863 and 119908 = 0 respectively Setting the internal statevariable as 119909 = 119908119863 isin [0 1] (1) can be rewritten as

119872(119905) = 119877off + (119877on minus 119877off) 119909 (119905) (2)

When 119905 = 0 the initial memristance is

1198720= 119877off + (119877on minus 119877off) 1199090 (3)

The Scientific World Journal 3

0 05 10

02

04

06

08

1

f(x)

x

P = 1

P = 3

P = 5

P = 10

(a)

2 4 6 8

Linear model

0 2 40

1

2

2

15

1

05

0

times104

times104

times10minus6

times10minus6

P = 1

P = 3

P = 5

P = 10

M(Ω

)

Q (C)

(b)

Figure 2 The influence of different values of the integer 119875 on the memristor (a) Joglekar window function for 119875 = 1 119875 = 3 119875 = 5 and119875 = 10 (b) Relationship between memristance versus charge for the nonlinear memristor model As the integer 119875 increases the graphs tendto linearity

Themovement speed of the boundary between the dopedand undoped regions depends on the resistance of dopedarea the passing current and other factors according to thestate equation

119889119909

119889119905= 119896119894 (119905) 119891 (119909) 119896 =

120583V119877ON1198632

(4)

where 120583] asymp 10minus14m2 sminus1 Vminus1 is the average ionic mobil-

ity parameter As we all known small voltages can yieldenormous electric fields in nanoscale devices which cansecondarily produce significant nonlinearities in the ionictransport As for amemristive device these nonlinearities aremanifested particularly at the thin film edges especially at thetwo boundaries This phenomenon called nonlinear dopantdrift can be simulated by multiplying a proper windowfunction 119891(119909) on the right side of (4) Based on [28] thereare several kinds of classical window functions such asJoglekar window function and Biolek window function Thispaper chooses the Joglekar window function which can bedescribed by

119891 (119909) = 1 minus (2119909 minus 1)2119875 (5)

where 119875 is a positive integer called the control parameterFigure 2(a) exhibits the behavior of the Joglekar window

function for different values of 119875 Figure 2(b) shows thegraphs of the memristance versus charge of the memristorAs the value of119875 becomes smaller the nonlinearity increasesOn the other hand as the integer119875 increases themodel tendsto the linear model Based upon this as well as the literature

[3 28] we set the value of the integer 119875 = 1 in this windowfunction and obtain

119891 (119909) = 4119909 minus 41199092 (6)

Substituting (6) into (4)

int

119909(119905)

1199090

(1

119909 (120591)+

1

1 minus 119909 (120591)) 119889119909 (120591) = int

119905

0

4119896119894 (120591) 119889120591 (7)

where the internal state variable satisfies 119909(120591) isin [1199090 119909(119905)] and

the integration time is 0 le 120591 le 119905Assume 119902

0= 0 we can get

119909 (119905)

1 minus 119909 (119905)=

1199090

1 minus 1199090

times 1198904119896119902(119905)

(8)

The initial value of the state variable can be expressed as

1199090=119877off minus 1198770Δ119877

(9)

Then the expression of 119909(119905) can be calculated as

119909 (119905) = 1 minus1

1198601198904119896119902(119905) + 1 (10)

where119860 is a constant and its value is determined by 119877off 119877onand 119877

0

119860 =119877off minus 11987701198770minus 119877on

(11)

4 The Scientific World Journal

+ minus

+ minus

Ronw

DRoff(1 minus

w

D)

(a)

30

25

20

15

10

5

0

minus5minus02 minus01 0 01 02

Positive to negativeNegative to positive

Q (mC)

M(kΩ

)

(b)

Figure 3 The nonlinear memristor model and its characteristic curves (a) The equivalent circuit of the memristor and its 3D symbol (b)The relationship between memristance and the charge

Integrator

Math function

+

+

+

Constant

Math function

ProductProduct

Constant1

Add

1Constant

+

+

+

+

Divide

D ivide

D

U

100

4

Ron

Roff

M0

U2 divide

divide

divide

minus

minus

minus

times times

times

times

times

times

times

times

eu

A q(t) 1

sI

100RonM(t)

Q minus M

VminusM

Vminus I

V20k

10k

Figure 4 The Simulink model of the nonlinear memristor

Combining (2)ndash(11) the resistance of memristor can berewritten as

119872(119905) = 119877on + Δ1198771

1198601198904119896119902(119905) + 1 (12)

where Δ119877 = 119877off minus 119877onGiving a sine stimulus to the memristor we get the

simulation results using MATLAB software It is noteworthythat the memristor is a two-terminal element with polaritywhich is shown in Figure 3(a) When the current flows intothe memristive device from the positive pole to the negativepole one can get the relationship curve (the blue line)between memristance and charge through it as shown inFigure 3(b) On the contrary when the current flows into thememristor from the negative pole to the positive pole therelationship curve is denoted by the red dashed line When

the charge is close to or exceeds the charge threshold valuesthe resistance of the memristor reaches and stays at 119877on and119877off respectively Notably the threshold value denotes thequantity of electric charge required when the memristancereaches the limit resistance The parameters of the model are119877on = 100Ω 119877off = 20 kΩ 119872

0= 10 kΩ 119863 = 10 nm

and 120583] asymp 10minus14m2sminus1Vminus1 Moreover the simulation results

in Figure 3(b) are consistent with the results concluded byAdhikari et al in [8 29]

22 The Simulink Model of the Memristor For the sakeof analyzing the characteristics of the memristor modelcomprehensively a Simulinkmodel is built upon (2)ndash(12) andillustrated in Figure 4 The model mainly consists of inputand outputmodules internal operationmodules (multipliers

The Scientific World Journal 5

0 05 1 15

Time (s)

I(m

A)

minus06

minus04

minus02

0

02

04

06

(a)

0

05

minus05

0 05minus05

I(m

A)

(V)

(b)

30

25

20

15

10

5

0

minus5minus02 minus01 0 01 02

Q (mC)

M(kΩ

)

(c)

M(kΩ

)

105

10

95

9

85

8

75

7

65minus04 minus02 0 02 04 06

(V)

(d)

Figure 5 The results of the memristor Simulink model (a) The input current source (b) Relationship between the current 119894 and the voltageV (c) Relationship between the memristance119872 and the charge 119902 (d) Relationship between the memristance119872 and the voltage V

adders and modules) and parameter control modules Themodel parameters are the same as those in Figure 3 Thesignal stimulus applied into the memristor is a sinusoidalcurrent source with amplitude of 05mA and frequency of1Hz

The simulation results are exhibited in Figure 5 The cur-rent flowing through the memristor is shown in Figure 5(a)The typical hysteresis loop in Figure 5(b) shows its switchingcharacteristic that is the memristance can switch betweenhigh resistance and low resistance Figure 5(c) illustrates thatthe memristance is a nonlinear function of the flow of chargeas discussed previously Figure 5(d) shows the relationshipbetween the memristance 119872 and the charge 119902 Notably inthe part of the higher memristance state the change ratioof the memristance is low while in the part of the lowermemristance state the change ratio of the memristance ishigh

3 The Memristive Multilayer FeedforwardSmall-World Neural Network

31 The Multilayer Feedforward Small-World Neural NetworkGenerally small-world phenomenon indicates that a networkhas highly concentrated local connections and also includes afew random long connections In real world a large numberof networks have the small-world effect such as diseasetransmission network social network and the food chainnetwork [22] As is known to all in the classical multilayerfeedforward neural network such as BP network the 119894thneuron in the 119897th layer V119897

119894only connects its neighboring

neuron sets 119881119897minus1 and 119881119897+1 In addition all connections

are feedforward and no connections exist between neuronswithin the same layerThis kind of network can be consideredas a regular network Based on [27] and the constructionprocess of WS small world model we introduce Algorithm 1

6 The Scientific World Journal

Input Output

(a) (b)

Figure 6The construction progress of the multilayer feedforwardWS small-world neural network (a)The regular network (119875 = 0) (b)Themultilayer feedforward WS small-world neural network (0 lt 119875 lt 1)

Set the network has 119871 layersSet each layer has 119899

119897nodes

For ℎ = 1 To 119871 minus 21199011(ℎ) = 120572119890

minus120573(ℎ+2)

For 1198971= 1 To 119871 minus 2

For 119899119897= 1 To 119899

119897

For 1198992= 1 To 119899

119897

If rand lt 1199011199081198971(1198971+1)

11989911198992= 0

1198993= randint(1 119899

119897)

119905 = 1

While 119905 = 11199012= rand

For 1198972= 119871 minus 2 To 119897

1+ 2

If 1199012lt 1199011(1198972)

1198871198971 1198972

11989911198993= rand

119905 = 0

BreakFinish

Algorithm 1 Constructive procedure of the small-world neuralnetwork

which is used to construct multilayer feedforward neuralnetwork model according to the rewiring probability Thespecific construction process is given as follows

Step 1 Initialization assuming the number of the networklayers is 119871 each layer has 119899

119897neuron nodes and the rewiring

probability is 119875

Step 2 Generate the multilayer feedforward regular neuralnetwork as shown in Figure 6(a)

Step 3As shown in Algorithm 1 where 1199011(ℎ) is the probabil-

ity to select reconnection layer selection probability betweentwo neurons decreases exponentially 120572 and 120573 are the distancecoefficients rand and randint both are MATLAB functionsthe former is used to generate a number between (0 1)

randomly and the latter can be used to randomly generatean integer from 1 to 119899

119897 Since the connections of the (119871 minus

1)th layer cannot generate new long-connections if they aredisconnected the connections of the last two layers are notreconnected in the network

As shown in Figure 6(a) when the rewiring probability119875 = 0 the connection of the network maintains completelyregular mode Nonetheless when 119875 ranges from 0 to 1the long cross-layer connections are generated accordingto the rewiring probability 119875 and the probability of recon-nection layer selecting The resulting structure is betweencompletely regular and random connection mode as shownin Figure 6(b)

More specially we set the network connection matrix as119882 where119882119897denotes the connection submatrix between the119897th layer and the (119897 + 1)th layer 119908119897

119894119895isin 119877 is the connection

weight between the neuron 119894 of the 119897th layer and the neuron 119895of the (119897 + 1)th layer If there exists connection between thesetwo neurons then 119908119897

119894119895= 0 otherwise 119908119897

119894119895= 0 Therefore the

regular network connection matrix can be expressed as thefollowing equation

119882 =(((

(

0 1198821

0 0 0 0 0

0 0 1198822

0 0 0 0

0 0 0 1198823

0 0 0

0 0 0 0 1198824

0 0

0 0 0 0 0 1198825

0

0 0 0 0 0 0 1198826

0 0 0 0 0 0 0

)))

)

(13)

in which the number zero means no connection existsbetween the corresponding layers As for multilayer feedfor-ward small-world neural network because of the reconnec-tion performance the connection matrix changes into as

1198821015840=(((

(

0 11988211198613

11198614

11198615

11198616

11198617

1

0 0 11988221198614

21198615

21198616

21198617

2

0 0 0 11988231198615

31198616

31198617

3

0 0 0 0 11988241198616

41198617

4

0 0 0 0 0 11988251198617

5

0 0 0 0 0 0 1198826

0 0 0 0 0 0 0

)))

)

(14)

where119882119897 represents the reconnection submatrix between the119897th layer and the (119897 + 1)th layer and the 119861119897

1015840

119897is the submatrix

between two nonadjacent layers which 119897 isin 1 2 5 1198971015840 isin3 4 7

The Scientific World Journal 7

32 The Combination of the MFSNN and the Memristor

321 The Memristive Synapse The nanoscale memristor hashigh potential of information storage on account of the non-volatility with respect to long periods of power-down soit can be used as electric synapse in the artificial neuralnetworks and the primary reasons are manifold Firstly asa kind of analog component this device can realize weightupdating continuously Moreover the memristor possessesthe capacity of information storage due to the nonvolatilityThis feature is consistent with the memory ability of the neu-rons in humanrsquos brain Additionally the memristive neuralnetwork can be further integrated in crossbar array whichhas significant advantages in better information processingcapacity and huger storage

According to the nonlinear memristor model inSection 2 the memristive conductance can be calculatedfrom (12) as

119866 (119905) =1

119872 (119905)=

1

119877on + Δ119877 (1 (1198601198904119896119902(119905) + 1))

(15)

Differentiating (15) with respect to time 119905 we can beobtain

119889119866 (119905)

119889119905=

41198961198601198904119896119902(119905)

Δ119877

(119877on1198601198904119896119902(119905) + 119877off)

2times119889119902 (119905)

119889119905 (16)

where the current 119894(119905) = 119889119902(119905)119889119905 Notably when Δ119905 rarr 0119889119866(119905) asymp Δ119866 Hence the rate of the memristive conductanceΔ119866 can be described as the synapse weight update ruleThe relationship curve between the rate of the memristiveconductance change and the current is shown in Figure 7When the current is tiny the memristive conductance isalmost invariant While the current tends to plusmn4mA thememristive conductance changes suddenly So the currentthreshold value of thememristive synapse can be set as |119868th| =4mA

322 The Memristive Activation Function In the standardMFSNN the activation function for each neuron is usuallythe nonlinear Sigmoid function Particularly the activationfunction of the hidden layer adopts bipolar Sigmoid functionbut the output layer activation function is unipolar Sigmoidfunction

Based on the constitutive relationship of the memristor alot of nonlinear curves can be simulated and substituted [5 611 12] Based on the Simulink model of the nonlinear mem-ristor described in Section 2 we get its simplified Simulinkmodel accordingly as shown in Figure 8(a) Furthermore wedesign a package of the memristive device (in Figure 8(b))which can be considered as a system with single-input anddouble-output In this system the input variable is the current119868 and the output variable is thememristance119872(119905) and charge119902(119905) respectively

Then the behavior of the output curve can be adjustedefficiently by the parameter control module gain moduleand internal operationmodule which is crucial to implementthe activation function in the neural network Here we set

ΔG

8

6

4

2

0

minus2

minus4

minus6

minus8

times10minus3

minus4 minus2 0 2 4

I (mA)

Figure 7The relationship curve between the rate of the memristiveconductance change and the current

the activation functions for the hidden layer and outputlayer neurons of the memristive MFSNN as ℎ(119909) and 119910(119909)respectively

Figure 9(a) exhibits the constructing principle diagramof the memristive activation function in the hidden layerin which the red dotted line frame represents the parameteradjustment area 119870

1is the adjustable gain which is used for

controlling the shape of the activation function and1198702is the

fixed gain whose value is 1198702= 10minus4 The suitable parameters

of the memristor are chosen as 119877on = 100Ω 119877off = 20 kΩ1198720= 10 kΩ 119863 = 10 nm and 120583] asymp 10

minus14m2sminus1Vminus1 Theinput signal is a sinusoidal current with an amplitude of05mA and a frequency of 1Hz Notably the polarity of thevoltage applied into the memristor is opposite to the polarityof the memristor itself that is the current flows throughthe memristor from the negative polar to the positive polarFigure 9(b) shows the memristive activation function of thehidden layer and its shape varies with different values of119870

1

Similarly Figure 10(a) is the constructing principle dia-gram of the activation function of the output layer In theparameter adjustment part (the red dotted line frame) 119870

3

is an adjustable gain and 1198704is the fixed gain whose value

is 1198704= 2 times 10

minus4 The parameters are the same with thesimulation in Figure 9 Figure 10(b) shows the memristiveactivation function of the output layer Obviously as the valueof 1198703increases the graphs tend to flatten

4 The Memristive Intelligent PID Controller

So far the PID control has found widespread applicationsin the modern control field By adjusting the control actionof the proportion integration and differentiation we getan interactive nonlinear relationship among these controlvariables The neural network has the ability of expressing

8 The Scientific World Journal

Integrator Product ProductAdd

Divide

Constant Constant

Math function

Ron

q(t)

M(t)

+

+

+1

+

divide

times

times

times eu

A 100

ΔR4k

1

s

I

(a)

InOutOutI

q(t)

M(t)

(b)

Figure 8 The simplified memristor model (a) The simplified Simulink model of the memristor (b) The package of the memristor

InOut

OutI q(t)

M(t)

K1

K2

1

h(x)

minus

+

(a)

2

1

0

minus1

h(x)

minus4 minus2 0 2 4

K1 = 20000

K1 = 30000

K1 = 40000x

(b)

Figure 9Theprinciple diagramof thememristive activation function in the hidden layer (a)TheSimulinkmodel of thememristive activationfunction in the hidden layer (b) The curve of the memristive activation function ℎ(119909)

InOut

OutI q(t)

M(t)

K4

K3y(x)

(a)

minus4 minus2 0 2 4

K3 = 20000

K3 = 30000

K3 = 40000

x

2

15

1

05

0

minus05

y(x)

(b)

Figure 10 The principle diagram of the memristive activation function of the output layer (a) The Simulink model of the memristiveactivation function of the output layer (b) The curve of the memristive activation function 119910(119909)

The Scientific World Journal 9

the nonlinearity which can be used in the PID controlfor implementing the optimal nonlinear relationship amongcontrol variables In this work we build up a more intelligentPID controller with the parameters (119896

119901 119896119894 and 119896

119889) self-tune

based on the presented memristive multilayer feedforwardsmall-world neural network

According to the literature [30] the classical incrementaldigital PID control algorithm can be described as

119906 (119896) = 119906 (119896 minus 1) + 119896119901(119890 (119896) minus 119890 (119896 minus 1))

+ 119896119894119890 (119896) + 119896

119889(119890 (119896) minus 2119890 (119896 minus 1) + 119890 (119896 minus 2))

(17)

where the 119896119901 119896119894 and 119896

119889are the coefficient of the proportion

integration and differentiation respectivelyIn Figure 11 the ANN is the memristive multilayer feed-

forward small-world neural network Its learning algorithmconsisted of the backward error propagation and the forwardinput signal propagation Different from the traditional mul-tilayer feedforward neural network the state of the neurons ineach layer not only affects the state of the neurons in the nextlayer but also affects the state of the neurons in the cross-layer

Based on the novel neural network presented in Section 3we set the 119895 119894 and 119897 that represent the input layer hiddenlayer and output layer respectively The number of the inputlayer is 1 which is same with that of the output layer andthe number of the hidden layer is 119878 119909

119894represents the input

vector of the network then the set of the input samples is1198741

119895= [1199091 1199092 1199093 119909

119873] The number of the input vectors

is dependent on the complexity of the system Notably thesuperscript 1 represents the first layer in the whole neuralnetwork

The input and output vectors of the first hidden layer canbe expressed as

net1198941119894(119896) =

119873

sum

119895=0

1199081198941

1198941198951198741

119895

1198741198941

119894(119896) = ℎ (net1198941

119894(119896))

119894 = 1 2 3 119876

(18)

where the superscript 1198941denotes the first hidden layer of the

network and ℎ(119909) is the memristive bipolar sigmoid functionproposed in Section 3

By that analogy the input and output vectors of the 119878thhidden layer can be written as

net119894119904119894(119896) =

119873

sum

119895=0

119908119894119904

1198941198951198741

119895+

119904minus1

sum

119886=1

119904

sum

119887=1

119908119894119904

119894119886119894119904net119894119904119887(119896)

119874119894119904

119894(119896) = ℎ (net119894119904

119894(119896))

119894 = 1 2 3 119876

(19)

Finally the input and output vectors of the output layercan be obtained as

net119904+2119897(119896) =

119873

sum

119895=0

119908119904+2

1198941198971198741

119895+

119904

sum

119886=1

119904

sum

119887=1

119908119904+2

119894119886119897net119904+2119887(119896)

119874119904+2

119897(119896) = 119910 (net119904+2

119897(119896))

119897 = 1 2 3

(20)

where 119910(119909) denotes the memristive unipolar sigmoid func-tion The three nodes of the output layer are correspondingwith the nonnegative adjustable parameters 119896

119901 119896119894 and 119896

119889of

the PID controller respectivelyFrom [27] we conclude the weight update algorithm of

the memristive multilayer feedforward small-world neuralnetwork as below

Δ119908119904+2

119897119894(119896) = 120572Δ119908

119904+2

119897119894(119896 minus 1) + 120578120575

119904+2

119897119874119904+1

119894(119896)

119908119904+2

119897119894(119896 + 1) = 119908

119904+2

119897119894(119896) + Δ119908

119904+2

119897119894(119896)

(21)

where 120572 is the inertial coefficient whose scope ranges from 0to 1 and 120578 isin (0 1) is the learning rate

5 Computer Simulations and Results

In this section somenumerical simulations of thememristivemultilayer feedforward small-world neural network PIDcontroller have been executed on MATLAB software Themathematical model of the controlled plant is given as

119910out (119896) =119886 (119896) 119910out (119896 minus 1)

1 + 1199102out (119896 minus 1)+ 119906 (119896 minus 1) (22)

where the 119886(119896) is slow time-variant and its expression is119886(119896) = 12(1 minus 08119890

01119896)

The memristive neural network under investigation isconstituted by seven layers with four neurons in the inputlayer three neurons in the output layer and five in each of thefive hidden layers The learning rate of the network 120578 = 04and the inertial coefficient 120572 = 005 The initial weighs asrandom values fall in [minus05 05] and the value of the rewiringprobability is chosen as 119875 = 0 119875 = 008 119875 = 01 and 119875 = 02respectively The parameters are 119877on = 100Ω 119877off = 20 kΩ1198720= 10 kΩ 119863 = 10 nm and 120583] asymp 10

minus14m2sminus1Vminus1 1198701and

1198703are user-specified parameters whose value both are 20000

and the action time is 119905119904 = 0001 s When the system workssteadily the tracking results can be gotten as follows

Figure 12(a) shows the input signal (step response curve119903in(119896) = 10) and the output curves under a different rewiringprobability 119875 As can be seen from the figure when thetime 119905 = 05 s the whole system reaches the steady stateMaking a further analysis we can conclude that when therewiring probability 119875 = 0 the memristive neural networkkeeps regularly in architecture Its respond speed is slowerthan that of network when the rewiring probability 119875 =

008 and 119875 = 01 Moreover Figure 12(b) exhibits theerror curves between the input signal and the output signal

10 The Scientific World Journal

rin Error

aeat

ANN

PID Plant yout

kp ki kd

Figure 11 The structure diagram of the memristive neural network PID controller

0

05

1

15

20 05 1 15

Time (s)

Inpu

t ou

tput

Regularnetwork

InputP = 008

P = 01

P = 0

P = 02

(a)

0

08

06

04

02

1

20 05 1 15

Time (s)

Erro

rRegular network

P = 008

P = 01

P = 0

P = 02

(b)

001

002

003

0

005

01

0 2 4 6Time (s)

0 2 4 6Time (s)

0 2 4 60

2

4

Time (s)

P = 008

P = 008

P = 008

times10minus3

k pk i

k d

(c)

Figure 12The simulation results of the memristive neural network PID controller (119903in(119896) = 10) under a different rewiring probability 119875 (a)The step response curve (b) The error curves (c) The curves of the control parameters when the rewiring probability 119875 = 008

correspondingly When the rewiring probability 119875 = 008

and 119875 = 01 the network spends less time on approachingthe predefined approximation error than the regular network(when 119875 = 0) Figure 12(c) shows the output variables ofthe memristive multilayer feedforward small-world neural

network when 119875 = 008 which are the control parameters 119896119901

119896119894 and 119896

119889 correspondingly

In order to verify the superior performance of thememristive small-world neuronal networks andfigure out theoptimal structure we conducted a series of simulations to

The Scientific World Journal 11

4500

5000

5500

6000

6500

7000

0 02 04 06 08 1

Aver

age i

nter

actio

ns

Sampling points

P

(a)

35

37

39

41

43

45

47

0 02 04 06 08 1

Succ

essfu

l int

erac

tions

Sampling points

P

(b)

Figure 13 The convergence performance of the memristive neural network under different 119875 (a) The relationship between iteration andrewiring probability (b) The effective approximation number in 50 times simulations under varying rewiring probability

observe the convergence performance of the proposed net-work under different119875 Figure 13(a) shows the approximationspeed (iteration times) of different network structures thatis the smallest iteration number for reaching the predefinedapproximation error 120576 = 00001 Each drawn point is theaverage value of 50 times runs It can be observed thatthe small-world networks (0 lt 119875 lt 1) need much lessiteration times than the regular neural network (when 119875 =

0) which demonstrates its advantage in processing speedFurthermore when 119875 = 008 the network has the fastapproximation speed

Notably the mathematical function of this system has thelocal minimum for getting out of the local minimum wedefine the maximum allowable iteration times to be 10000as previously mentioned for each 119875 and we performed thesimulation for 50 times where the effective approximationtimes that is error lt 00001 within 10000 iterations arepresented in Figure 13(b) It can be found that the small-worldnetworks have higher accuracy rate than the regular network

6 Conclusions

A mathematical closed-form charge-governed memristormodel is recalled firstly and the corresponding Simulinkmodel is presented Using the change rule of memconduc-tance a memristive realization scheme for synaptic weightis proposed Moreover the activation functions in electricneurons are also implemented based on the single-inputand double-output package of the memristor Combiningthe proposed memristive synapse and activation functions amemristor-basedMFSNN is addressed It exhibits advantagesin computation speed and accuracy over the traditionalmultilayer neural networks by considering the small-worldeffect Meanwhile it has potential of hardware realizationof the neural network because of the nanoscale size of thememristive synapse These superior properties can furtherimprove the application of the neural networks such as inthe intelligent controller design Motivated by this we apply

the memristor-based MFSNN to classical PID control andthe proposed memristive PID controller may possess thefollowing superiorities (i) Its nanoscale physical implemen-tation could promote the development of themicrocontroller(ii) Because of the participation of the memristive neuralnetwork the proposed PID controller can realize the param-eters self-adjustment (iii) The control speed and accuracyare improved Eventually extensive numerical simulationsjustify the effectiveness and efficiency of the memristive PIDcontroller over the regular neural network PID controllerThis work may provide a theoretical reference to physicallyrealize the small-world neural networks and further promotethe development of modern intelligent control technology

Conflict of Interests

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

Acknowledgments

The work was supported by Program for New CenturyExcellent Talents in University (Grant nos [2013] 47)National Natural Science Foundation of China (Grant nos61372139 61101233 and 60972155) ldquoSpring Sunshine PlanrdquoResearch Project of Ministry of Education of China (Grantno z2011148) Technology Foundation for Selected OverseasChinese Scholars Ministry of Personnel in China (Grantno 2012-186) University Excellent Talents Supporting Foun-dations in of Chongqing (Grant no 2011-65) UniversityKey Teacher Supporting Foundations of Chongqing (Grantno 2011-65) Fundamental Research Funds for the CentralUniversities (Grant nos XDJK2014A009 XDJK2013B011)

References

[1] L O Chua ldquoMemristors the missing circuit elementrdquo IEEETransactions on Circuits and Systems vol 18 pp 507ndash519 1971

12 The Scientific World Journal

[2] J M Tour and H Tao ldquoElectronics the fourth elementrdquoNaturevol 453 no 7191 pp 42ndash43 2008

[3] D B Strukov G S Snider D R Stewart and R S WilliamsldquoThe missing memristor foundrdquo Nature vol 453 pp 80ndash832008

[4] R S Williams ldquoHow we found the missing Memristorrdquo IEEESpectrum vol 45 no 12 pp 28ndash35 2008

[5] S K Duan X F Hu L D Wang C D Li and P MazumderldquoMemristor-based RRAM with applicationsrdquo Science ChinaInformation Sciences vol 55 no 6 pp 1446ndash1460 2012

[6] X F Hu S K Duan L DWang and C D Li ldquoAnalog memorybased on pulse controlled memristorrdquo Journal of the Universityof Electronic Science and Technology of China vol 40 no 5 pp642ndash647 2011

[7] S Shin K Kim and S Kang ldquoResistive computingmemristors-enabled signal multiplicationrdquo IEEE Transactions on Circuitsand Systems I Regular Papers vol 60 no 5 pp 1241ndash1249 2013

[8] S P Adhikari C Yang H Kim and L O Chua ldquoMemristorbridge synapse-based neural network and its learningrdquo IEEETransactions on Neural Networks and Learning Systems vol 23no 9 pp 1426ndash1435 2012

[9] S H Jo T Chang I Ebong B B Bhadviya P Mazumder andW Lu ldquoNanoscale memristor device as synapse in neuromor-phic systemsrdquo Nano Letters vol 10 no 4 pp 1297ndash1301 2010

[10] Y Dai C Li and H Wang ldquoExpanded HP memristor modeland simulation in STDP learningrdquo Neural Computing andApplications vol 24 no 1 pp 51ndash57 2014

[11] S K Duan X F Hu L DWang S Y Gao andCD Li ldquoHybridmemristorRTD structure-based cellular neural networks withapplications in image processingrdquoNeural Computing and Appli-cations vol 25 no 2 pp 291ndash296 2014

[12] L Chen C D Li T W Huang and Y R Chen ldquoMemristorcrossbar-based unsupervised image learningrdquo Neural Comput-ing and Applications vol 25 no 2 pp 393ndash400 2013

[13] X F Hu S K Duan L D Wang and X F Liao ldquoMemristivecrossbar array with applications in image processingrdquo ScienceChina Information Sciences vol 41 pp 500ndash512 2011

[14] B Mouttet ldquoProposal for memristors in signal processingrdquo inNano-Net 3rd International ICST Conference NanoNet BostonMS USA September 14ndash16 2008 Revised Selected Papersvol 3 of Lecture Notes of the Institute for Computer SciencesSocial Informatics and Telecommunications Engineering pp 11ndash13 2009

[15] H Abdalla andM D Pickett ldquoSPICEmodeling of memristorsrdquoin Proceedings of the 2011 IEEE International Symposium ofCircuits and Systems (ISCAS rsquo11) pp 1832ndash1835 May 2011

[16] X B Wang Y R Chen H W Xi and D Dimitrov ldquoSpin-tronic memristor through spin-thorque-induced magnetiza-tion motionrdquo IEEE Electron Device Letters vol 30 no 3 pp294ndash297 2009

[17] D B Strukov and R S Williams ldquoExponential ionic drift fastswitching and low volatility of thin-film memristorsrdquo AppliedPhysics A Materials Science amp Processing vol 94 pp 515ndash5192009

[18] F Corinto A Ascoli and M Gilli ldquoMathematical models andcircuit implementations of memristive systemsrdquo in Proceedingsof the 13th International Workshop on Cellular Nanoscale Net-works and their Applications (CNNA rsquo12) pp 1ndash6 August 2012

[19] S Kvatinsky E G Friedman A Kolodny and U C WeiserldquoTEAMThrEshold adaptive memristor modelrdquo IEEE Transac-tions on Circuits and Systems I Regular Papers vol 60 no 1 pp211ndash221 2013

[20] O Sporns and C J Honey ldquoSmall worlds inside big brainsrdquoProceedings of the National Academy of Sciences of the UnitedStates of America vol 103 no 51 pp 19219ndash19220 2006

[21] P Tangkraingkij C Lursinsap S Sanguansintukul and TDesudchit ldquoInsider and outsider person authentication withminimum number of brain wave signals by neural and homo-geneous identity filteringrdquo Neural Computing and Applicationsvol 22 no 1 pp 463ndash476 2013

[22] D S Bassett and E D Bullmore ldquoSmall-world brain networksrdquoNeuroscientist vol 12 no 6 pp 512ndash523 2006

[23] D J Watts and S H Strogatz ldquoCollective dynamics of ldquosmall-worldrdquo networksrdquoNature vol 393 no 6684 pp 440ndash442 1998

[24] L Douw M M Schoonheim D Landi et al ldquoCognitionis related to resting-state small-world network topology anmagnetoencephalographic studyrdquo Neuroscience vol 175 pp169ndash177 2011

[25] T Fekete F D C C Beacher J Cha D Rubin and L RMujica-Parodi ldquoSmall-world network properties in prefrontal cortexcorrelate with predictors of psychopathology risk in youngchildren aNIRS studyrdquoNeuroImage vol 85 part 1 pp 345ndash3532014

[26] N R Taylor ldquoSmall world network strategies for studyingprotein structures and bindingrdquo Computational and StructuralBiotechnology Journal vol 5 Article ID e201302006 2013

[27] D Simard L Nadeau andH Kroger ldquoFastest learning in small-world neural networksrdquo Physics Letters A General Atomic andSolid State Physics vol 336 no 1 pp 8ndash15 2005

[28] Z Biolek D Biolek and V Biolkova ldquoSPICEmodel of memris-tor with nonlinear dopant driftrdquo Radioengineering vol 18 no2 pp 210ndash214 2009

[29] H Kim M P Sah C Yang T Roska and L O ChualdquoMemristor bridge synapsesrdquo Proceedings of the IEEE vol 100no 6 pp 2061ndash2070 2012

[30] M Esfandyari M A Fanaei and H Zohreie ldquoAdaptive fuzzytuning of PID controllersrdquo Neural Computing and Applicationsvol 23 no 1 pp 19ndash28 2013

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International Journal of

The Scientific World Journal 3

0 05 10

02

04

06

08

1

f(x)

x

P = 1

P = 3

P = 5

P = 10

(a)

2 4 6 8

Linear model

0 2 40

1

2

2

15

1

05

0

times104

times104

times10minus6

times10minus6

P = 1

P = 3

P = 5

P = 10

M(Ω

)

Q (C)

(b)

Figure 2 The influence of different values of the integer 119875 on the memristor (a) Joglekar window function for 119875 = 1 119875 = 3 119875 = 5 and119875 = 10 (b) Relationship between memristance versus charge for the nonlinear memristor model As the integer 119875 increases the graphs tendto linearity

Themovement speed of the boundary between the dopedand undoped regions depends on the resistance of dopedarea the passing current and other factors according to thestate equation

119889119909

119889119905= 119896119894 (119905) 119891 (119909) 119896 =

120583V119877ON1198632

(4)

where 120583] asymp 10minus14m2 sminus1 Vminus1 is the average ionic mobil-

ity parameter As we all known small voltages can yieldenormous electric fields in nanoscale devices which cansecondarily produce significant nonlinearities in the ionictransport As for amemristive device these nonlinearities aremanifested particularly at the thin film edges especially at thetwo boundaries This phenomenon called nonlinear dopantdrift can be simulated by multiplying a proper windowfunction 119891(119909) on the right side of (4) Based on [28] thereare several kinds of classical window functions such asJoglekar window function and Biolek window function Thispaper chooses the Joglekar window function which can bedescribed by

119891 (119909) = 1 minus (2119909 minus 1)2119875 (5)

where 119875 is a positive integer called the control parameterFigure 2(a) exhibits the behavior of the Joglekar window

function for different values of 119875 Figure 2(b) shows thegraphs of the memristance versus charge of the memristorAs the value of119875 becomes smaller the nonlinearity increasesOn the other hand as the integer119875 increases themodel tendsto the linear model Based upon this as well as the literature

[3 28] we set the value of the integer 119875 = 1 in this windowfunction and obtain

119891 (119909) = 4119909 minus 41199092 (6)

Substituting (6) into (4)

int

119909(119905)

1199090

(1

119909 (120591)+

1

1 minus 119909 (120591)) 119889119909 (120591) = int

119905

0

4119896119894 (120591) 119889120591 (7)

where the internal state variable satisfies 119909(120591) isin [1199090 119909(119905)] and

the integration time is 0 le 120591 le 119905Assume 119902

0= 0 we can get

119909 (119905)

1 minus 119909 (119905)=

1199090

1 minus 1199090

times 1198904119896119902(119905)

(8)

The initial value of the state variable can be expressed as

1199090=119877off minus 1198770Δ119877

(9)

Then the expression of 119909(119905) can be calculated as

119909 (119905) = 1 minus1

1198601198904119896119902(119905) + 1 (10)

where119860 is a constant and its value is determined by 119877off 119877onand 119877

0

119860 =119877off minus 11987701198770minus 119877on

(11)

4 The Scientific World Journal

+ minus

+ minus

Ronw

DRoff(1 minus

w

D)

(a)

30

25

20

15

10

5

0

minus5minus02 minus01 0 01 02

Positive to negativeNegative to positive

Q (mC)

M(kΩ

)

(b)

Figure 3 The nonlinear memristor model and its characteristic curves (a) The equivalent circuit of the memristor and its 3D symbol (b)The relationship between memristance and the charge

Integrator

Math function

+

+

+

Constant

Math function

ProductProduct

Constant1

Add

1Constant

+

+

+

+

Divide

D ivide

D

U

100

4

Ron

Roff

M0

U2 divide

divide

divide

minus

minus

minus

times times

times

times

times

times

times

times

eu

A q(t) 1

sI

100RonM(t)

Q minus M

VminusM

Vminus I

V20k

10k

Figure 4 The Simulink model of the nonlinear memristor

Combining (2)ndash(11) the resistance of memristor can berewritten as

119872(119905) = 119877on + Δ1198771

1198601198904119896119902(119905) + 1 (12)

where Δ119877 = 119877off minus 119877onGiving a sine stimulus to the memristor we get the

simulation results using MATLAB software It is noteworthythat the memristor is a two-terminal element with polaritywhich is shown in Figure 3(a) When the current flows intothe memristive device from the positive pole to the negativepole one can get the relationship curve (the blue line)between memristance and charge through it as shown inFigure 3(b) On the contrary when the current flows into thememristor from the negative pole to the positive pole therelationship curve is denoted by the red dashed line When

the charge is close to or exceeds the charge threshold valuesthe resistance of the memristor reaches and stays at 119877on and119877off respectively Notably the threshold value denotes thequantity of electric charge required when the memristancereaches the limit resistance The parameters of the model are119877on = 100Ω 119877off = 20 kΩ 119872

0= 10 kΩ 119863 = 10 nm

and 120583] asymp 10minus14m2sminus1Vminus1 Moreover the simulation results

in Figure 3(b) are consistent with the results concluded byAdhikari et al in [8 29]

22 The Simulink Model of the Memristor For the sakeof analyzing the characteristics of the memristor modelcomprehensively a Simulinkmodel is built upon (2)ndash(12) andillustrated in Figure 4 The model mainly consists of inputand outputmodules internal operationmodules (multipliers

The Scientific World Journal 5

0 05 1 15

Time (s)

I(m

A)

minus06

minus04

minus02

0

02

04

06

(a)

0

05

minus05

0 05minus05

I(m

A)

(V)

(b)

30

25

20

15

10

5

0

minus5minus02 minus01 0 01 02

Q (mC)

M(kΩ

)

(c)

M(kΩ

)

105

10

95

9

85

8

75

7

65minus04 minus02 0 02 04 06

(V)

(d)

Figure 5 The results of the memristor Simulink model (a) The input current source (b) Relationship between the current 119894 and the voltageV (c) Relationship between the memristance119872 and the charge 119902 (d) Relationship between the memristance119872 and the voltage V

adders and modules) and parameter control modules Themodel parameters are the same as those in Figure 3 Thesignal stimulus applied into the memristor is a sinusoidalcurrent source with amplitude of 05mA and frequency of1Hz

The simulation results are exhibited in Figure 5 The cur-rent flowing through the memristor is shown in Figure 5(a)The typical hysteresis loop in Figure 5(b) shows its switchingcharacteristic that is the memristance can switch betweenhigh resistance and low resistance Figure 5(c) illustrates thatthe memristance is a nonlinear function of the flow of chargeas discussed previously Figure 5(d) shows the relationshipbetween the memristance 119872 and the charge 119902 Notably inthe part of the higher memristance state the change ratioof the memristance is low while in the part of the lowermemristance state the change ratio of the memristance ishigh

3 The Memristive Multilayer FeedforwardSmall-World Neural Network

31 The Multilayer Feedforward Small-World Neural NetworkGenerally small-world phenomenon indicates that a networkhas highly concentrated local connections and also includes afew random long connections In real world a large numberof networks have the small-world effect such as diseasetransmission network social network and the food chainnetwork [22] As is known to all in the classical multilayerfeedforward neural network such as BP network the 119894thneuron in the 119897th layer V119897

119894only connects its neighboring

neuron sets 119881119897minus1 and 119881119897+1 In addition all connections

are feedforward and no connections exist between neuronswithin the same layerThis kind of network can be consideredas a regular network Based on [27] and the constructionprocess of WS small world model we introduce Algorithm 1

6 The Scientific World Journal

Input Output

(a) (b)

Figure 6The construction progress of the multilayer feedforwardWS small-world neural network (a)The regular network (119875 = 0) (b)Themultilayer feedforward WS small-world neural network (0 lt 119875 lt 1)

Set the network has 119871 layersSet each layer has 119899

119897nodes

For ℎ = 1 To 119871 minus 21199011(ℎ) = 120572119890

minus120573(ℎ+2)

For 1198971= 1 To 119871 minus 2

For 119899119897= 1 To 119899

119897

For 1198992= 1 To 119899

119897

If rand lt 1199011199081198971(1198971+1)

11989911198992= 0

1198993= randint(1 119899

119897)

119905 = 1

While 119905 = 11199012= rand

For 1198972= 119871 minus 2 To 119897

1+ 2

If 1199012lt 1199011(1198972)

1198871198971 1198972

11989911198993= rand

119905 = 0

BreakFinish

Algorithm 1 Constructive procedure of the small-world neuralnetwork

which is used to construct multilayer feedforward neuralnetwork model according to the rewiring probability Thespecific construction process is given as follows

Step 1 Initialization assuming the number of the networklayers is 119871 each layer has 119899

119897neuron nodes and the rewiring

probability is 119875

Step 2 Generate the multilayer feedforward regular neuralnetwork as shown in Figure 6(a)

Step 3As shown in Algorithm 1 where 1199011(ℎ) is the probabil-

ity to select reconnection layer selection probability betweentwo neurons decreases exponentially 120572 and 120573 are the distancecoefficients rand and randint both are MATLAB functionsthe former is used to generate a number between (0 1)

randomly and the latter can be used to randomly generatean integer from 1 to 119899

119897 Since the connections of the (119871 minus

1)th layer cannot generate new long-connections if they aredisconnected the connections of the last two layers are notreconnected in the network

As shown in Figure 6(a) when the rewiring probability119875 = 0 the connection of the network maintains completelyregular mode Nonetheless when 119875 ranges from 0 to 1the long cross-layer connections are generated accordingto the rewiring probability 119875 and the probability of recon-nection layer selecting The resulting structure is betweencompletely regular and random connection mode as shownin Figure 6(b)

More specially we set the network connection matrix as119882 where119882119897denotes the connection submatrix between the119897th layer and the (119897 + 1)th layer 119908119897

119894119895isin 119877 is the connection

weight between the neuron 119894 of the 119897th layer and the neuron 119895of the (119897 + 1)th layer If there exists connection between thesetwo neurons then 119908119897

119894119895= 0 otherwise 119908119897

119894119895= 0 Therefore the

regular network connection matrix can be expressed as thefollowing equation

119882 =(((

(

0 1198821

0 0 0 0 0

0 0 1198822

0 0 0 0

0 0 0 1198823

0 0 0

0 0 0 0 1198824

0 0

0 0 0 0 0 1198825

0

0 0 0 0 0 0 1198826

0 0 0 0 0 0 0

)))

)

(13)

in which the number zero means no connection existsbetween the corresponding layers As for multilayer feedfor-ward small-world neural network because of the reconnec-tion performance the connection matrix changes into as

1198821015840=(((

(

0 11988211198613

11198614

11198615

11198616

11198617

1

0 0 11988221198614

21198615

21198616

21198617

2

0 0 0 11988231198615

31198616

31198617

3

0 0 0 0 11988241198616

41198617

4

0 0 0 0 0 11988251198617

5

0 0 0 0 0 0 1198826

0 0 0 0 0 0 0

)))

)

(14)

where119882119897 represents the reconnection submatrix between the119897th layer and the (119897 + 1)th layer and the 119861119897

1015840

119897is the submatrix

between two nonadjacent layers which 119897 isin 1 2 5 1198971015840 isin3 4 7

The Scientific World Journal 7

32 The Combination of the MFSNN and the Memristor

321 The Memristive Synapse The nanoscale memristor hashigh potential of information storage on account of the non-volatility with respect to long periods of power-down soit can be used as electric synapse in the artificial neuralnetworks and the primary reasons are manifold Firstly asa kind of analog component this device can realize weightupdating continuously Moreover the memristor possessesthe capacity of information storage due to the nonvolatilityThis feature is consistent with the memory ability of the neu-rons in humanrsquos brain Additionally the memristive neuralnetwork can be further integrated in crossbar array whichhas significant advantages in better information processingcapacity and huger storage

According to the nonlinear memristor model inSection 2 the memristive conductance can be calculatedfrom (12) as

119866 (119905) =1

119872 (119905)=

1

119877on + Δ119877 (1 (1198601198904119896119902(119905) + 1))

(15)

Differentiating (15) with respect to time 119905 we can beobtain

119889119866 (119905)

119889119905=

41198961198601198904119896119902(119905)

Δ119877

(119877on1198601198904119896119902(119905) + 119877off)

2times119889119902 (119905)

119889119905 (16)

where the current 119894(119905) = 119889119902(119905)119889119905 Notably when Δ119905 rarr 0119889119866(119905) asymp Δ119866 Hence the rate of the memristive conductanceΔ119866 can be described as the synapse weight update ruleThe relationship curve between the rate of the memristiveconductance change and the current is shown in Figure 7When the current is tiny the memristive conductance isalmost invariant While the current tends to plusmn4mA thememristive conductance changes suddenly So the currentthreshold value of thememristive synapse can be set as |119868th| =4mA

322 The Memristive Activation Function In the standardMFSNN the activation function for each neuron is usuallythe nonlinear Sigmoid function Particularly the activationfunction of the hidden layer adopts bipolar Sigmoid functionbut the output layer activation function is unipolar Sigmoidfunction

Based on the constitutive relationship of the memristor alot of nonlinear curves can be simulated and substituted [5 611 12] Based on the Simulink model of the nonlinear mem-ristor described in Section 2 we get its simplified Simulinkmodel accordingly as shown in Figure 8(a) Furthermore wedesign a package of the memristive device (in Figure 8(b))which can be considered as a system with single-input anddouble-output In this system the input variable is the current119868 and the output variable is thememristance119872(119905) and charge119902(119905) respectively

Then the behavior of the output curve can be adjustedefficiently by the parameter control module gain moduleand internal operationmodule which is crucial to implementthe activation function in the neural network Here we set

ΔG

8

6

4

2

0

minus2

minus4

minus6

minus8

times10minus3

minus4 minus2 0 2 4

I (mA)

Figure 7The relationship curve between the rate of the memristiveconductance change and the current

the activation functions for the hidden layer and outputlayer neurons of the memristive MFSNN as ℎ(119909) and 119910(119909)respectively

Figure 9(a) exhibits the constructing principle diagramof the memristive activation function in the hidden layerin which the red dotted line frame represents the parameteradjustment area 119870

1is the adjustable gain which is used for

controlling the shape of the activation function and1198702is the

fixed gain whose value is 1198702= 10minus4 The suitable parameters

of the memristor are chosen as 119877on = 100Ω 119877off = 20 kΩ1198720= 10 kΩ 119863 = 10 nm and 120583] asymp 10

minus14m2sminus1Vminus1 Theinput signal is a sinusoidal current with an amplitude of05mA and a frequency of 1Hz Notably the polarity of thevoltage applied into the memristor is opposite to the polarityof the memristor itself that is the current flows throughthe memristor from the negative polar to the positive polarFigure 9(b) shows the memristive activation function of thehidden layer and its shape varies with different values of119870

1

Similarly Figure 10(a) is the constructing principle dia-gram of the activation function of the output layer In theparameter adjustment part (the red dotted line frame) 119870

3

is an adjustable gain and 1198704is the fixed gain whose value

is 1198704= 2 times 10

minus4 The parameters are the same with thesimulation in Figure 9 Figure 10(b) shows the memristiveactivation function of the output layer Obviously as the valueof 1198703increases the graphs tend to flatten

4 The Memristive Intelligent PID Controller

So far the PID control has found widespread applicationsin the modern control field By adjusting the control actionof the proportion integration and differentiation we getan interactive nonlinear relationship among these controlvariables The neural network has the ability of expressing

8 The Scientific World Journal

Integrator Product ProductAdd

Divide

Constant Constant

Math function

Ron

q(t)

M(t)

+

+

+1

+

divide

times

times

times eu

A 100

ΔR4k

1

s

I

(a)

InOutOutI

q(t)

M(t)

(b)

Figure 8 The simplified memristor model (a) The simplified Simulink model of the memristor (b) The package of the memristor

InOut

OutI q(t)

M(t)

K1

K2

1

h(x)

minus

+

(a)

2

1

0

minus1

h(x)

minus4 minus2 0 2 4

K1 = 20000

K1 = 30000

K1 = 40000x

(b)

Figure 9Theprinciple diagramof thememristive activation function in the hidden layer (a)TheSimulinkmodel of thememristive activationfunction in the hidden layer (b) The curve of the memristive activation function ℎ(119909)

InOut

OutI q(t)

M(t)

K4

K3y(x)

(a)

minus4 minus2 0 2 4

K3 = 20000

K3 = 30000

K3 = 40000

x

2

15

1

05

0

minus05

y(x)

(b)

Figure 10 The principle diagram of the memristive activation function of the output layer (a) The Simulink model of the memristiveactivation function of the output layer (b) The curve of the memristive activation function 119910(119909)

The Scientific World Journal 9

the nonlinearity which can be used in the PID controlfor implementing the optimal nonlinear relationship amongcontrol variables In this work we build up a more intelligentPID controller with the parameters (119896

119901 119896119894 and 119896

119889) self-tune

based on the presented memristive multilayer feedforwardsmall-world neural network

According to the literature [30] the classical incrementaldigital PID control algorithm can be described as

119906 (119896) = 119906 (119896 minus 1) + 119896119901(119890 (119896) minus 119890 (119896 minus 1))

+ 119896119894119890 (119896) + 119896

119889(119890 (119896) minus 2119890 (119896 minus 1) + 119890 (119896 minus 2))

(17)

where the 119896119901 119896119894 and 119896

119889are the coefficient of the proportion

integration and differentiation respectivelyIn Figure 11 the ANN is the memristive multilayer feed-

forward small-world neural network Its learning algorithmconsisted of the backward error propagation and the forwardinput signal propagation Different from the traditional mul-tilayer feedforward neural network the state of the neurons ineach layer not only affects the state of the neurons in the nextlayer but also affects the state of the neurons in the cross-layer

Based on the novel neural network presented in Section 3we set the 119895 119894 and 119897 that represent the input layer hiddenlayer and output layer respectively The number of the inputlayer is 1 which is same with that of the output layer andthe number of the hidden layer is 119878 119909

119894represents the input

vector of the network then the set of the input samples is1198741

119895= [1199091 1199092 1199093 119909

119873] The number of the input vectors

is dependent on the complexity of the system Notably thesuperscript 1 represents the first layer in the whole neuralnetwork

The input and output vectors of the first hidden layer canbe expressed as

net1198941119894(119896) =

119873

sum

119895=0

1199081198941

1198941198951198741

119895

1198741198941

119894(119896) = ℎ (net1198941

119894(119896))

119894 = 1 2 3 119876

(18)

where the superscript 1198941denotes the first hidden layer of the

network and ℎ(119909) is the memristive bipolar sigmoid functionproposed in Section 3

By that analogy the input and output vectors of the 119878thhidden layer can be written as

net119894119904119894(119896) =

119873

sum

119895=0

119908119894119904

1198941198951198741

119895+

119904minus1

sum

119886=1

119904

sum

119887=1

119908119894119904

119894119886119894119904net119894119904119887(119896)

119874119894119904

119894(119896) = ℎ (net119894119904

119894(119896))

119894 = 1 2 3 119876

(19)

Finally the input and output vectors of the output layercan be obtained as

net119904+2119897(119896) =

119873

sum

119895=0

119908119904+2

1198941198971198741

119895+

119904

sum

119886=1

119904

sum

119887=1

119908119904+2

119894119886119897net119904+2119887(119896)

119874119904+2

119897(119896) = 119910 (net119904+2

119897(119896))

119897 = 1 2 3

(20)

where 119910(119909) denotes the memristive unipolar sigmoid func-tion The three nodes of the output layer are correspondingwith the nonnegative adjustable parameters 119896

119901 119896119894 and 119896

119889of

the PID controller respectivelyFrom [27] we conclude the weight update algorithm of

the memristive multilayer feedforward small-world neuralnetwork as below

Δ119908119904+2

119897119894(119896) = 120572Δ119908

119904+2

119897119894(119896 minus 1) + 120578120575

119904+2

119897119874119904+1

119894(119896)

119908119904+2

119897119894(119896 + 1) = 119908

119904+2

119897119894(119896) + Δ119908

119904+2

119897119894(119896)

(21)

where 120572 is the inertial coefficient whose scope ranges from 0to 1 and 120578 isin (0 1) is the learning rate

5 Computer Simulations and Results

In this section somenumerical simulations of thememristivemultilayer feedforward small-world neural network PIDcontroller have been executed on MATLAB software Themathematical model of the controlled plant is given as

119910out (119896) =119886 (119896) 119910out (119896 minus 1)

1 + 1199102out (119896 minus 1)+ 119906 (119896 minus 1) (22)

where the 119886(119896) is slow time-variant and its expression is119886(119896) = 12(1 minus 08119890

01119896)

The memristive neural network under investigation isconstituted by seven layers with four neurons in the inputlayer three neurons in the output layer and five in each of thefive hidden layers The learning rate of the network 120578 = 04and the inertial coefficient 120572 = 005 The initial weighs asrandom values fall in [minus05 05] and the value of the rewiringprobability is chosen as 119875 = 0 119875 = 008 119875 = 01 and 119875 = 02respectively The parameters are 119877on = 100Ω 119877off = 20 kΩ1198720= 10 kΩ 119863 = 10 nm and 120583] asymp 10

minus14m2sminus1Vminus1 1198701and

1198703are user-specified parameters whose value both are 20000

and the action time is 119905119904 = 0001 s When the system workssteadily the tracking results can be gotten as follows

Figure 12(a) shows the input signal (step response curve119903in(119896) = 10) and the output curves under a different rewiringprobability 119875 As can be seen from the figure when thetime 119905 = 05 s the whole system reaches the steady stateMaking a further analysis we can conclude that when therewiring probability 119875 = 0 the memristive neural networkkeeps regularly in architecture Its respond speed is slowerthan that of network when the rewiring probability 119875 =

008 and 119875 = 01 Moreover Figure 12(b) exhibits theerror curves between the input signal and the output signal

10 The Scientific World Journal

rin Error

aeat

ANN

PID Plant yout

kp ki kd

Figure 11 The structure diagram of the memristive neural network PID controller

0

05

1

15

20 05 1 15

Time (s)

Inpu

t ou

tput

Regularnetwork

InputP = 008

P = 01

P = 0

P = 02

(a)

0

08

06

04

02

1

20 05 1 15

Time (s)

Erro

rRegular network

P = 008

P = 01

P = 0

P = 02

(b)

001

002

003

0

005

01

0 2 4 6Time (s)

0 2 4 6Time (s)

0 2 4 60

2

4

Time (s)

P = 008

P = 008

P = 008

times10minus3

k pk i

k d

(c)

Figure 12The simulation results of the memristive neural network PID controller (119903in(119896) = 10) under a different rewiring probability 119875 (a)The step response curve (b) The error curves (c) The curves of the control parameters when the rewiring probability 119875 = 008

correspondingly When the rewiring probability 119875 = 008

and 119875 = 01 the network spends less time on approachingthe predefined approximation error than the regular network(when 119875 = 0) Figure 12(c) shows the output variables ofthe memristive multilayer feedforward small-world neural

network when 119875 = 008 which are the control parameters 119896119901

119896119894 and 119896

119889 correspondingly

In order to verify the superior performance of thememristive small-world neuronal networks andfigure out theoptimal structure we conducted a series of simulations to

The Scientific World Journal 11

4500

5000

5500

6000

6500

7000

0 02 04 06 08 1

Aver

age i

nter

actio

ns

Sampling points

P

(a)

35

37

39

41

43

45

47

0 02 04 06 08 1

Succ

essfu

l int

erac

tions

Sampling points

P

(b)

Figure 13 The convergence performance of the memristive neural network under different 119875 (a) The relationship between iteration andrewiring probability (b) The effective approximation number in 50 times simulations under varying rewiring probability

observe the convergence performance of the proposed net-work under different119875 Figure 13(a) shows the approximationspeed (iteration times) of different network structures thatis the smallest iteration number for reaching the predefinedapproximation error 120576 = 00001 Each drawn point is theaverage value of 50 times runs It can be observed thatthe small-world networks (0 lt 119875 lt 1) need much lessiteration times than the regular neural network (when 119875 =

0) which demonstrates its advantage in processing speedFurthermore when 119875 = 008 the network has the fastapproximation speed

Notably the mathematical function of this system has thelocal minimum for getting out of the local minimum wedefine the maximum allowable iteration times to be 10000as previously mentioned for each 119875 and we performed thesimulation for 50 times where the effective approximationtimes that is error lt 00001 within 10000 iterations arepresented in Figure 13(b) It can be found that the small-worldnetworks have higher accuracy rate than the regular network

6 Conclusions

A mathematical closed-form charge-governed memristormodel is recalled firstly and the corresponding Simulinkmodel is presented Using the change rule of memconduc-tance a memristive realization scheme for synaptic weightis proposed Moreover the activation functions in electricneurons are also implemented based on the single-inputand double-output package of the memristor Combiningthe proposed memristive synapse and activation functions amemristor-basedMFSNN is addressed It exhibits advantagesin computation speed and accuracy over the traditionalmultilayer neural networks by considering the small-worldeffect Meanwhile it has potential of hardware realizationof the neural network because of the nanoscale size of thememristive synapse These superior properties can furtherimprove the application of the neural networks such as inthe intelligent controller design Motivated by this we apply

the memristor-based MFSNN to classical PID control andthe proposed memristive PID controller may possess thefollowing superiorities (i) Its nanoscale physical implemen-tation could promote the development of themicrocontroller(ii) Because of the participation of the memristive neuralnetwork the proposed PID controller can realize the param-eters self-adjustment (iii) The control speed and accuracyare improved Eventually extensive numerical simulationsjustify the effectiveness and efficiency of the memristive PIDcontroller over the regular neural network PID controllerThis work may provide a theoretical reference to physicallyrealize the small-world neural networks and further promotethe development of modern intelligent control technology

Conflict of Interests

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

Acknowledgments

The work was supported by Program for New CenturyExcellent Talents in University (Grant nos [2013] 47)National Natural Science Foundation of China (Grant nos61372139 61101233 and 60972155) ldquoSpring Sunshine PlanrdquoResearch Project of Ministry of Education of China (Grantno z2011148) Technology Foundation for Selected OverseasChinese Scholars Ministry of Personnel in China (Grantno 2012-186) University Excellent Talents Supporting Foun-dations in of Chongqing (Grant no 2011-65) UniversityKey Teacher Supporting Foundations of Chongqing (Grantno 2011-65) Fundamental Research Funds for the CentralUniversities (Grant nos XDJK2014A009 XDJK2013B011)

References

[1] L O Chua ldquoMemristors the missing circuit elementrdquo IEEETransactions on Circuits and Systems vol 18 pp 507ndash519 1971

12 The Scientific World Journal

[2] J M Tour and H Tao ldquoElectronics the fourth elementrdquoNaturevol 453 no 7191 pp 42ndash43 2008

[3] D B Strukov G S Snider D R Stewart and R S WilliamsldquoThe missing memristor foundrdquo Nature vol 453 pp 80ndash832008

[4] R S Williams ldquoHow we found the missing Memristorrdquo IEEESpectrum vol 45 no 12 pp 28ndash35 2008

[5] S K Duan X F Hu L D Wang C D Li and P MazumderldquoMemristor-based RRAM with applicationsrdquo Science ChinaInformation Sciences vol 55 no 6 pp 1446ndash1460 2012

[6] X F Hu S K Duan L DWang and C D Li ldquoAnalog memorybased on pulse controlled memristorrdquo Journal of the Universityof Electronic Science and Technology of China vol 40 no 5 pp642ndash647 2011

[7] S Shin K Kim and S Kang ldquoResistive computingmemristors-enabled signal multiplicationrdquo IEEE Transactions on Circuitsand Systems I Regular Papers vol 60 no 5 pp 1241ndash1249 2013

[8] S P Adhikari C Yang H Kim and L O Chua ldquoMemristorbridge synapse-based neural network and its learningrdquo IEEETransactions on Neural Networks and Learning Systems vol 23no 9 pp 1426ndash1435 2012

[9] S H Jo T Chang I Ebong B B Bhadviya P Mazumder andW Lu ldquoNanoscale memristor device as synapse in neuromor-phic systemsrdquo Nano Letters vol 10 no 4 pp 1297ndash1301 2010

[10] Y Dai C Li and H Wang ldquoExpanded HP memristor modeland simulation in STDP learningrdquo Neural Computing andApplications vol 24 no 1 pp 51ndash57 2014

[11] S K Duan X F Hu L DWang S Y Gao andCD Li ldquoHybridmemristorRTD structure-based cellular neural networks withapplications in image processingrdquoNeural Computing and Appli-cations vol 25 no 2 pp 291ndash296 2014

[12] L Chen C D Li T W Huang and Y R Chen ldquoMemristorcrossbar-based unsupervised image learningrdquo Neural Comput-ing and Applications vol 25 no 2 pp 393ndash400 2013

[13] X F Hu S K Duan L D Wang and X F Liao ldquoMemristivecrossbar array with applications in image processingrdquo ScienceChina Information Sciences vol 41 pp 500ndash512 2011

[14] B Mouttet ldquoProposal for memristors in signal processingrdquo inNano-Net 3rd International ICST Conference NanoNet BostonMS USA September 14ndash16 2008 Revised Selected Papersvol 3 of Lecture Notes of the Institute for Computer SciencesSocial Informatics and Telecommunications Engineering pp 11ndash13 2009

[15] H Abdalla andM D Pickett ldquoSPICEmodeling of memristorsrdquoin Proceedings of the 2011 IEEE International Symposium ofCircuits and Systems (ISCAS rsquo11) pp 1832ndash1835 May 2011

[16] X B Wang Y R Chen H W Xi and D Dimitrov ldquoSpin-tronic memristor through spin-thorque-induced magnetiza-tion motionrdquo IEEE Electron Device Letters vol 30 no 3 pp294ndash297 2009

[17] D B Strukov and R S Williams ldquoExponential ionic drift fastswitching and low volatility of thin-film memristorsrdquo AppliedPhysics A Materials Science amp Processing vol 94 pp 515ndash5192009

[18] F Corinto A Ascoli and M Gilli ldquoMathematical models andcircuit implementations of memristive systemsrdquo in Proceedingsof the 13th International Workshop on Cellular Nanoscale Net-works and their Applications (CNNA rsquo12) pp 1ndash6 August 2012

[19] S Kvatinsky E G Friedman A Kolodny and U C WeiserldquoTEAMThrEshold adaptive memristor modelrdquo IEEE Transac-tions on Circuits and Systems I Regular Papers vol 60 no 1 pp211ndash221 2013

[20] O Sporns and C J Honey ldquoSmall worlds inside big brainsrdquoProceedings of the National Academy of Sciences of the UnitedStates of America vol 103 no 51 pp 19219ndash19220 2006

[21] P Tangkraingkij C Lursinsap S Sanguansintukul and TDesudchit ldquoInsider and outsider person authentication withminimum number of brain wave signals by neural and homo-geneous identity filteringrdquo Neural Computing and Applicationsvol 22 no 1 pp 463ndash476 2013

[22] D S Bassett and E D Bullmore ldquoSmall-world brain networksrdquoNeuroscientist vol 12 no 6 pp 512ndash523 2006

[23] D J Watts and S H Strogatz ldquoCollective dynamics of ldquosmall-worldrdquo networksrdquoNature vol 393 no 6684 pp 440ndash442 1998

[24] L Douw M M Schoonheim D Landi et al ldquoCognitionis related to resting-state small-world network topology anmagnetoencephalographic studyrdquo Neuroscience vol 175 pp169ndash177 2011

[25] T Fekete F D C C Beacher J Cha D Rubin and L RMujica-Parodi ldquoSmall-world network properties in prefrontal cortexcorrelate with predictors of psychopathology risk in youngchildren aNIRS studyrdquoNeuroImage vol 85 part 1 pp 345ndash3532014

[26] N R Taylor ldquoSmall world network strategies for studyingprotein structures and bindingrdquo Computational and StructuralBiotechnology Journal vol 5 Article ID e201302006 2013

[27] D Simard L Nadeau andH Kroger ldquoFastest learning in small-world neural networksrdquo Physics Letters A General Atomic andSolid State Physics vol 336 no 1 pp 8ndash15 2005

[28] Z Biolek D Biolek and V Biolkova ldquoSPICEmodel of memris-tor with nonlinear dopant driftrdquo Radioengineering vol 18 no2 pp 210ndash214 2009

[29] H Kim M P Sah C Yang T Roska and L O ChualdquoMemristor bridge synapsesrdquo Proceedings of the IEEE vol 100no 6 pp 2061ndash2070 2012

[30] M Esfandyari M A Fanaei and H Zohreie ldquoAdaptive fuzzytuning of PID controllersrdquo Neural Computing and Applicationsvol 23 no 1 pp 19ndash28 2013

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4 The Scientific World Journal

+ minus

+ minus

Ronw

DRoff(1 minus

w

D)

(a)

30

25

20

15

10

5

0

minus5minus02 minus01 0 01 02

Positive to negativeNegative to positive

Q (mC)

M(kΩ

)

(b)

Figure 3 The nonlinear memristor model and its characteristic curves (a) The equivalent circuit of the memristor and its 3D symbol (b)The relationship between memristance and the charge

Integrator

Math function

+

+

+

Constant

Math function

ProductProduct

Constant1

Add

1Constant

+

+

+

+

Divide

D ivide

D

U

100

4

Ron

Roff

M0

U2 divide

divide

divide

minus

minus

minus

times times

times

times

times

times

times

times

eu

A q(t) 1

sI

100RonM(t)

Q minus M

VminusM

Vminus I

V20k

10k

Figure 4 The Simulink model of the nonlinear memristor

Combining (2)ndash(11) the resistance of memristor can berewritten as

119872(119905) = 119877on + Δ1198771

1198601198904119896119902(119905) + 1 (12)

where Δ119877 = 119877off minus 119877onGiving a sine stimulus to the memristor we get the

simulation results using MATLAB software It is noteworthythat the memristor is a two-terminal element with polaritywhich is shown in Figure 3(a) When the current flows intothe memristive device from the positive pole to the negativepole one can get the relationship curve (the blue line)between memristance and charge through it as shown inFigure 3(b) On the contrary when the current flows into thememristor from the negative pole to the positive pole therelationship curve is denoted by the red dashed line When

the charge is close to or exceeds the charge threshold valuesthe resistance of the memristor reaches and stays at 119877on and119877off respectively Notably the threshold value denotes thequantity of electric charge required when the memristancereaches the limit resistance The parameters of the model are119877on = 100Ω 119877off = 20 kΩ 119872

0= 10 kΩ 119863 = 10 nm

and 120583] asymp 10minus14m2sminus1Vminus1 Moreover the simulation results

in Figure 3(b) are consistent with the results concluded byAdhikari et al in [8 29]

22 The Simulink Model of the Memristor For the sakeof analyzing the characteristics of the memristor modelcomprehensively a Simulinkmodel is built upon (2)ndash(12) andillustrated in Figure 4 The model mainly consists of inputand outputmodules internal operationmodules (multipliers

The Scientific World Journal 5

0 05 1 15

Time (s)

I(m

A)

minus06

minus04

minus02

0

02

04

06

(a)

0

05

minus05

0 05minus05

I(m

A)

(V)

(b)

30

25

20

15

10

5

0

minus5minus02 minus01 0 01 02

Q (mC)

M(kΩ

)

(c)

M(kΩ

)

105

10

95

9

85

8

75

7

65minus04 minus02 0 02 04 06

(V)

(d)

Figure 5 The results of the memristor Simulink model (a) The input current source (b) Relationship between the current 119894 and the voltageV (c) Relationship between the memristance119872 and the charge 119902 (d) Relationship between the memristance119872 and the voltage V

adders and modules) and parameter control modules Themodel parameters are the same as those in Figure 3 Thesignal stimulus applied into the memristor is a sinusoidalcurrent source with amplitude of 05mA and frequency of1Hz

The simulation results are exhibited in Figure 5 The cur-rent flowing through the memristor is shown in Figure 5(a)The typical hysteresis loop in Figure 5(b) shows its switchingcharacteristic that is the memristance can switch betweenhigh resistance and low resistance Figure 5(c) illustrates thatthe memristance is a nonlinear function of the flow of chargeas discussed previously Figure 5(d) shows the relationshipbetween the memristance 119872 and the charge 119902 Notably inthe part of the higher memristance state the change ratioof the memristance is low while in the part of the lowermemristance state the change ratio of the memristance ishigh

3 The Memristive Multilayer FeedforwardSmall-World Neural Network

31 The Multilayer Feedforward Small-World Neural NetworkGenerally small-world phenomenon indicates that a networkhas highly concentrated local connections and also includes afew random long connections In real world a large numberof networks have the small-world effect such as diseasetransmission network social network and the food chainnetwork [22] As is known to all in the classical multilayerfeedforward neural network such as BP network the 119894thneuron in the 119897th layer V119897

119894only connects its neighboring

neuron sets 119881119897minus1 and 119881119897+1 In addition all connections

are feedforward and no connections exist between neuronswithin the same layerThis kind of network can be consideredas a regular network Based on [27] and the constructionprocess of WS small world model we introduce Algorithm 1

6 The Scientific World Journal

Input Output

(a) (b)

Figure 6The construction progress of the multilayer feedforwardWS small-world neural network (a)The regular network (119875 = 0) (b)Themultilayer feedforward WS small-world neural network (0 lt 119875 lt 1)

Set the network has 119871 layersSet each layer has 119899

119897nodes

For ℎ = 1 To 119871 minus 21199011(ℎ) = 120572119890

minus120573(ℎ+2)

For 1198971= 1 To 119871 minus 2

For 119899119897= 1 To 119899

119897

For 1198992= 1 To 119899

119897

If rand lt 1199011199081198971(1198971+1)

11989911198992= 0

1198993= randint(1 119899

119897)

119905 = 1

While 119905 = 11199012= rand

For 1198972= 119871 minus 2 To 119897

1+ 2

If 1199012lt 1199011(1198972)

1198871198971 1198972

11989911198993= rand

119905 = 0

BreakFinish

Algorithm 1 Constructive procedure of the small-world neuralnetwork

which is used to construct multilayer feedforward neuralnetwork model according to the rewiring probability Thespecific construction process is given as follows

Step 1 Initialization assuming the number of the networklayers is 119871 each layer has 119899

119897neuron nodes and the rewiring

probability is 119875

Step 2 Generate the multilayer feedforward regular neuralnetwork as shown in Figure 6(a)

Step 3As shown in Algorithm 1 where 1199011(ℎ) is the probabil-

ity to select reconnection layer selection probability betweentwo neurons decreases exponentially 120572 and 120573 are the distancecoefficients rand and randint both are MATLAB functionsthe former is used to generate a number between (0 1)

randomly and the latter can be used to randomly generatean integer from 1 to 119899

119897 Since the connections of the (119871 minus

1)th layer cannot generate new long-connections if they aredisconnected the connections of the last two layers are notreconnected in the network

As shown in Figure 6(a) when the rewiring probability119875 = 0 the connection of the network maintains completelyregular mode Nonetheless when 119875 ranges from 0 to 1the long cross-layer connections are generated accordingto the rewiring probability 119875 and the probability of recon-nection layer selecting The resulting structure is betweencompletely regular and random connection mode as shownin Figure 6(b)

More specially we set the network connection matrix as119882 where119882119897denotes the connection submatrix between the119897th layer and the (119897 + 1)th layer 119908119897

119894119895isin 119877 is the connection

weight between the neuron 119894 of the 119897th layer and the neuron 119895of the (119897 + 1)th layer If there exists connection between thesetwo neurons then 119908119897

119894119895= 0 otherwise 119908119897

119894119895= 0 Therefore the

regular network connection matrix can be expressed as thefollowing equation

119882 =(((

(

0 1198821

0 0 0 0 0

0 0 1198822

0 0 0 0

0 0 0 1198823

0 0 0

0 0 0 0 1198824

0 0

0 0 0 0 0 1198825

0

0 0 0 0 0 0 1198826

0 0 0 0 0 0 0

)))

)

(13)

in which the number zero means no connection existsbetween the corresponding layers As for multilayer feedfor-ward small-world neural network because of the reconnec-tion performance the connection matrix changes into as

1198821015840=(((

(

0 11988211198613

11198614

11198615

11198616

11198617

1

0 0 11988221198614

21198615

21198616

21198617

2

0 0 0 11988231198615

31198616

31198617

3

0 0 0 0 11988241198616

41198617

4

0 0 0 0 0 11988251198617

5

0 0 0 0 0 0 1198826

0 0 0 0 0 0 0

)))

)

(14)

where119882119897 represents the reconnection submatrix between the119897th layer and the (119897 + 1)th layer and the 119861119897

1015840

119897is the submatrix

between two nonadjacent layers which 119897 isin 1 2 5 1198971015840 isin3 4 7

The Scientific World Journal 7

32 The Combination of the MFSNN and the Memristor

321 The Memristive Synapse The nanoscale memristor hashigh potential of information storage on account of the non-volatility with respect to long periods of power-down soit can be used as electric synapse in the artificial neuralnetworks and the primary reasons are manifold Firstly asa kind of analog component this device can realize weightupdating continuously Moreover the memristor possessesthe capacity of information storage due to the nonvolatilityThis feature is consistent with the memory ability of the neu-rons in humanrsquos brain Additionally the memristive neuralnetwork can be further integrated in crossbar array whichhas significant advantages in better information processingcapacity and huger storage

According to the nonlinear memristor model inSection 2 the memristive conductance can be calculatedfrom (12) as

119866 (119905) =1

119872 (119905)=

1

119877on + Δ119877 (1 (1198601198904119896119902(119905) + 1))

(15)

Differentiating (15) with respect to time 119905 we can beobtain

119889119866 (119905)

119889119905=

41198961198601198904119896119902(119905)

Δ119877

(119877on1198601198904119896119902(119905) + 119877off)

2times119889119902 (119905)

119889119905 (16)

where the current 119894(119905) = 119889119902(119905)119889119905 Notably when Δ119905 rarr 0119889119866(119905) asymp Δ119866 Hence the rate of the memristive conductanceΔ119866 can be described as the synapse weight update ruleThe relationship curve between the rate of the memristiveconductance change and the current is shown in Figure 7When the current is tiny the memristive conductance isalmost invariant While the current tends to plusmn4mA thememristive conductance changes suddenly So the currentthreshold value of thememristive synapse can be set as |119868th| =4mA

322 The Memristive Activation Function In the standardMFSNN the activation function for each neuron is usuallythe nonlinear Sigmoid function Particularly the activationfunction of the hidden layer adopts bipolar Sigmoid functionbut the output layer activation function is unipolar Sigmoidfunction

Based on the constitutive relationship of the memristor alot of nonlinear curves can be simulated and substituted [5 611 12] Based on the Simulink model of the nonlinear mem-ristor described in Section 2 we get its simplified Simulinkmodel accordingly as shown in Figure 8(a) Furthermore wedesign a package of the memristive device (in Figure 8(b))which can be considered as a system with single-input anddouble-output In this system the input variable is the current119868 and the output variable is thememristance119872(119905) and charge119902(119905) respectively

Then the behavior of the output curve can be adjustedefficiently by the parameter control module gain moduleand internal operationmodule which is crucial to implementthe activation function in the neural network Here we set

ΔG

8

6

4

2

0

minus2

minus4

minus6

minus8

times10minus3

minus4 minus2 0 2 4

I (mA)

Figure 7The relationship curve between the rate of the memristiveconductance change and the current

the activation functions for the hidden layer and outputlayer neurons of the memristive MFSNN as ℎ(119909) and 119910(119909)respectively

Figure 9(a) exhibits the constructing principle diagramof the memristive activation function in the hidden layerin which the red dotted line frame represents the parameteradjustment area 119870

1is the adjustable gain which is used for

controlling the shape of the activation function and1198702is the

fixed gain whose value is 1198702= 10minus4 The suitable parameters

of the memristor are chosen as 119877on = 100Ω 119877off = 20 kΩ1198720= 10 kΩ 119863 = 10 nm and 120583] asymp 10

minus14m2sminus1Vminus1 Theinput signal is a sinusoidal current with an amplitude of05mA and a frequency of 1Hz Notably the polarity of thevoltage applied into the memristor is opposite to the polarityof the memristor itself that is the current flows throughthe memristor from the negative polar to the positive polarFigure 9(b) shows the memristive activation function of thehidden layer and its shape varies with different values of119870

1

Similarly Figure 10(a) is the constructing principle dia-gram of the activation function of the output layer In theparameter adjustment part (the red dotted line frame) 119870

3

is an adjustable gain and 1198704is the fixed gain whose value

is 1198704= 2 times 10

minus4 The parameters are the same with thesimulation in Figure 9 Figure 10(b) shows the memristiveactivation function of the output layer Obviously as the valueof 1198703increases the graphs tend to flatten

4 The Memristive Intelligent PID Controller

So far the PID control has found widespread applicationsin the modern control field By adjusting the control actionof the proportion integration and differentiation we getan interactive nonlinear relationship among these controlvariables The neural network has the ability of expressing

8 The Scientific World Journal

Integrator Product ProductAdd

Divide

Constant Constant

Math function

Ron

q(t)

M(t)

+

+

+1

+

divide

times

times

times eu

A 100

ΔR4k

1

s

I

(a)

InOutOutI

q(t)

M(t)

(b)

Figure 8 The simplified memristor model (a) The simplified Simulink model of the memristor (b) The package of the memristor

InOut

OutI q(t)

M(t)

K1

K2

1

h(x)

minus

+

(a)

2

1

0

minus1

h(x)

minus4 minus2 0 2 4

K1 = 20000

K1 = 30000

K1 = 40000x

(b)

Figure 9Theprinciple diagramof thememristive activation function in the hidden layer (a)TheSimulinkmodel of thememristive activationfunction in the hidden layer (b) The curve of the memristive activation function ℎ(119909)

InOut

OutI q(t)

M(t)

K4

K3y(x)

(a)

minus4 minus2 0 2 4

K3 = 20000

K3 = 30000

K3 = 40000

x

2

15

1

05

0

minus05

y(x)

(b)

Figure 10 The principle diagram of the memristive activation function of the output layer (a) The Simulink model of the memristiveactivation function of the output layer (b) The curve of the memristive activation function 119910(119909)

The Scientific World Journal 9

the nonlinearity which can be used in the PID controlfor implementing the optimal nonlinear relationship amongcontrol variables In this work we build up a more intelligentPID controller with the parameters (119896

119901 119896119894 and 119896

119889) self-tune

based on the presented memristive multilayer feedforwardsmall-world neural network

According to the literature [30] the classical incrementaldigital PID control algorithm can be described as

119906 (119896) = 119906 (119896 minus 1) + 119896119901(119890 (119896) minus 119890 (119896 minus 1))

+ 119896119894119890 (119896) + 119896

119889(119890 (119896) minus 2119890 (119896 minus 1) + 119890 (119896 minus 2))

(17)

where the 119896119901 119896119894 and 119896

119889are the coefficient of the proportion

integration and differentiation respectivelyIn Figure 11 the ANN is the memristive multilayer feed-

forward small-world neural network Its learning algorithmconsisted of the backward error propagation and the forwardinput signal propagation Different from the traditional mul-tilayer feedforward neural network the state of the neurons ineach layer not only affects the state of the neurons in the nextlayer but also affects the state of the neurons in the cross-layer

Based on the novel neural network presented in Section 3we set the 119895 119894 and 119897 that represent the input layer hiddenlayer and output layer respectively The number of the inputlayer is 1 which is same with that of the output layer andthe number of the hidden layer is 119878 119909

119894represents the input

vector of the network then the set of the input samples is1198741

119895= [1199091 1199092 1199093 119909

119873] The number of the input vectors

is dependent on the complexity of the system Notably thesuperscript 1 represents the first layer in the whole neuralnetwork

The input and output vectors of the first hidden layer canbe expressed as

net1198941119894(119896) =

119873

sum

119895=0

1199081198941

1198941198951198741

119895

1198741198941

119894(119896) = ℎ (net1198941

119894(119896))

119894 = 1 2 3 119876

(18)

where the superscript 1198941denotes the first hidden layer of the

network and ℎ(119909) is the memristive bipolar sigmoid functionproposed in Section 3

By that analogy the input and output vectors of the 119878thhidden layer can be written as

net119894119904119894(119896) =

119873

sum

119895=0

119908119894119904

1198941198951198741

119895+

119904minus1

sum

119886=1

119904

sum

119887=1

119908119894119904

119894119886119894119904net119894119904119887(119896)

119874119894119904

119894(119896) = ℎ (net119894119904

119894(119896))

119894 = 1 2 3 119876

(19)

Finally the input and output vectors of the output layercan be obtained as

net119904+2119897(119896) =

119873

sum

119895=0

119908119904+2

1198941198971198741

119895+

119904

sum

119886=1

119904

sum

119887=1

119908119904+2

119894119886119897net119904+2119887(119896)

119874119904+2

119897(119896) = 119910 (net119904+2

119897(119896))

119897 = 1 2 3

(20)

where 119910(119909) denotes the memristive unipolar sigmoid func-tion The three nodes of the output layer are correspondingwith the nonnegative adjustable parameters 119896

119901 119896119894 and 119896

119889of

the PID controller respectivelyFrom [27] we conclude the weight update algorithm of

the memristive multilayer feedforward small-world neuralnetwork as below

Δ119908119904+2

119897119894(119896) = 120572Δ119908

119904+2

119897119894(119896 minus 1) + 120578120575

119904+2

119897119874119904+1

119894(119896)

119908119904+2

119897119894(119896 + 1) = 119908

119904+2

119897119894(119896) + Δ119908

119904+2

119897119894(119896)

(21)

where 120572 is the inertial coefficient whose scope ranges from 0to 1 and 120578 isin (0 1) is the learning rate

5 Computer Simulations and Results

In this section somenumerical simulations of thememristivemultilayer feedforward small-world neural network PIDcontroller have been executed on MATLAB software Themathematical model of the controlled plant is given as

119910out (119896) =119886 (119896) 119910out (119896 minus 1)

1 + 1199102out (119896 minus 1)+ 119906 (119896 minus 1) (22)

where the 119886(119896) is slow time-variant and its expression is119886(119896) = 12(1 minus 08119890

01119896)

The memristive neural network under investigation isconstituted by seven layers with four neurons in the inputlayer three neurons in the output layer and five in each of thefive hidden layers The learning rate of the network 120578 = 04and the inertial coefficient 120572 = 005 The initial weighs asrandom values fall in [minus05 05] and the value of the rewiringprobability is chosen as 119875 = 0 119875 = 008 119875 = 01 and 119875 = 02respectively The parameters are 119877on = 100Ω 119877off = 20 kΩ1198720= 10 kΩ 119863 = 10 nm and 120583] asymp 10

minus14m2sminus1Vminus1 1198701and

1198703are user-specified parameters whose value both are 20000

and the action time is 119905119904 = 0001 s When the system workssteadily the tracking results can be gotten as follows

Figure 12(a) shows the input signal (step response curve119903in(119896) = 10) and the output curves under a different rewiringprobability 119875 As can be seen from the figure when thetime 119905 = 05 s the whole system reaches the steady stateMaking a further analysis we can conclude that when therewiring probability 119875 = 0 the memristive neural networkkeeps regularly in architecture Its respond speed is slowerthan that of network when the rewiring probability 119875 =

008 and 119875 = 01 Moreover Figure 12(b) exhibits theerror curves between the input signal and the output signal

10 The Scientific World Journal

rin Error

aeat

ANN

PID Plant yout

kp ki kd

Figure 11 The structure diagram of the memristive neural network PID controller

0

05

1

15

20 05 1 15

Time (s)

Inpu

t ou

tput

Regularnetwork

InputP = 008

P = 01

P = 0

P = 02

(a)

0

08

06

04

02

1

20 05 1 15

Time (s)

Erro

rRegular network

P = 008

P = 01

P = 0

P = 02

(b)

001

002

003

0

005

01

0 2 4 6Time (s)

0 2 4 6Time (s)

0 2 4 60

2

4

Time (s)

P = 008

P = 008

P = 008

times10minus3

k pk i

k d

(c)

Figure 12The simulation results of the memristive neural network PID controller (119903in(119896) = 10) under a different rewiring probability 119875 (a)The step response curve (b) The error curves (c) The curves of the control parameters when the rewiring probability 119875 = 008

correspondingly When the rewiring probability 119875 = 008

and 119875 = 01 the network spends less time on approachingthe predefined approximation error than the regular network(when 119875 = 0) Figure 12(c) shows the output variables ofthe memristive multilayer feedforward small-world neural

network when 119875 = 008 which are the control parameters 119896119901

119896119894 and 119896

119889 correspondingly

In order to verify the superior performance of thememristive small-world neuronal networks andfigure out theoptimal structure we conducted a series of simulations to

The Scientific World Journal 11

4500

5000

5500

6000

6500

7000

0 02 04 06 08 1

Aver

age i

nter

actio

ns

Sampling points

P

(a)

35

37

39

41

43

45

47

0 02 04 06 08 1

Succ

essfu

l int

erac

tions

Sampling points

P

(b)

Figure 13 The convergence performance of the memristive neural network under different 119875 (a) The relationship between iteration andrewiring probability (b) The effective approximation number in 50 times simulations under varying rewiring probability

observe the convergence performance of the proposed net-work under different119875 Figure 13(a) shows the approximationspeed (iteration times) of different network structures thatis the smallest iteration number for reaching the predefinedapproximation error 120576 = 00001 Each drawn point is theaverage value of 50 times runs It can be observed thatthe small-world networks (0 lt 119875 lt 1) need much lessiteration times than the regular neural network (when 119875 =

0) which demonstrates its advantage in processing speedFurthermore when 119875 = 008 the network has the fastapproximation speed

Notably the mathematical function of this system has thelocal minimum for getting out of the local minimum wedefine the maximum allowable iteration times to be 10000as previously mentioned for each 119875 and we performed thesimulation for 50 times where the effective approximationtimes that is error lt 00001 within 10000 iterations arepresented in Figure 13(b) It can be found that the small-worldnetworks have higher accuracy rate than the regular network

6 Conclusions

A mathematical closed-form charge-governed memristormodel is recalled firstly and the corresponding Simulinkmodel is presented Using the change rule of memconduc-tance a memristive realization scheme for synaptic weightis proposed Moreover the activation functions in electricneurons are also implemented based on the single-inputand double-output package of the memristor Combiningthe proposed memristive synapse and activation functions amemristor-basedMFSNN is addressed It exhibits advantagesin computation speed and accuracy over the traditionalmultilayer neural networks by considering the small-worldeffect Meanwhile it has potential of hardware realizationof the neural network because of the nanoscale size of thememristive synapse These superior properties can furtherimprove the application of the neural networks such as inthe intelligent controller design Motivated by this we apply

the memristor-based MFSNN to classical PID control andthe proposed memristive PID controller may possess thefollowing superiorities (i) Its nanoscale physical implemen-tation could promote the development of themicrocontroller(ii) Because of the participation of the memristive neuralnetwork the proposed PID controller can realize the param-eters self-adjustment (iii) The control speed and accuracyare improved Eventually extensive numerical simulationsjustify the effectiveness and efficiency of the memristive PIDcontroller over the regular neural network PID controllerThis work may provide a theoretical reference to physicallyrealize the small-world neural networks and further promotethe development of modern intelligent control technology

Conflict of Interests

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

Acknowledgments

The work was supported by Program for New CenturyExcellent Talents in University (Grant nos [2013] 47)National Natural Science Foundation of China (Grant nos61372139 61101233 and 60972155) ldquoSpring Sunshine PlanrdquoResearch Project of Ministry of Education of China (Grantno z2011148) Technology Foundation for Selected OverseasChinese Scholars Ministry of Personnel in China (Grantno 2012-186) University Excellent Talents Supporting Foun-dations in of Chongqing (Grant no 2011-65) UniversityKey Teacher Supporting Foundations of Chongqing (Grantno 2011-65) Fundamental Research Funds for the CentralUniversities (Grant nos XDJK2014A009 XDJK2013B011)

References

[1] L O Chua ldquoMemristors the missing circuit elementrdquo IEEETransactions on Circuits and Systems vol 18 pp 507ndash519 1971

12 The Scientific World Journal

[2] J M Tour and H Tao ldquoElectronics the fourth elementrdquoNaturevol 453 no 7191 pp 42ndash43 2008

[3] D B Strukov G S Snider D R Stewart and R S WilliamsldquoThe missing memristor foundrdquo Nature vol 453 pp 80ndash832008

[4] R S Williams ldquoHow we found the missing Memristorrdquo IEEESpectrum vol 45 no 12 pp 28ndash35 2008

[5] S K Duan X F Hu L D Wang C D Li and P MazumderldquoMemristor-based RRAM with applicationsrdquo Science ChinaInformation Sciences vol 55 no 6 pp 1446ndash1460 2012

[6] X F Hu S K Duan L DWang and C D Li ldquoAnalog memorybased on pulse controlled memristorrdquo Journal of the Universityof Electronic Science and Technology of China vol 40 no 5 pp642ndash647 2011

[7] S Shin K Kim and S Kang ldquoResistive computingmemristors-enabled signal multiplicationrdquo IEEE Transactions on Circuitsand Systems I Regular Papers vol 60 no 5 pp 1241ndash1249 2013

[8] S P Adhikari C Yang H Kim and L O Chua ldquoMemristorbridge synapse-based neural network and its learningrdquo IEEETransactions on Neural Networks and Learning Systems vol 23no 9 pp 1426ndash1435 2012

[9] S H Jo T Chang I Ebong B B Bhadviya P Mazumder andW Lu ldquoNanoscale memristor device as synapse in neuromor-phic systemsrdquo Nano Letters vol 10 no 4 pp 1297ndash1301 2010

[10] Y Dai C Li and H Wang ldquoExpanded HP memristor modeland simulation in STDP learningrdquo Neural Computing andApplications vol 24 no 1 pp 51ndash57 2014

[11] S K Duan X F Hu L DWang S Y Gao andCD Li ldquoHybridmemristorRTD structure-based cellular neural networks withapplications in image processingrdquoNeural Computing and Appli-cations vol 25 no 2 pp 291ndash296 2014

[12] L Chen C D Li T W Huang and Y R Chen ldquoMemristorcrossbar-based unsupervised image learningrdquo Neural Comput-ing and Applications vol 25 no 2 pp 393ndash400 2013

[13] X F Hu S K Duan L D Wang and X F Liao ldquoMemristivecrossbar array with applications in image processingrdquo ScienceChina Information Sciences vol 41 pp 500ndash512 2011

[14] B Mouttet ldquoProposal for memristors in signal processingrdquo inNano-Net 3rd International ICST Conference NanoNet BostonMS USA September 14ndash16 2008 Revised Selected Papersvol 3 of Lecture Notes of the Institute for Computer SciencesSocial Informatics and Telecommunications Engineering pp 11ndash13 2009

[15] H Abdalla andM D Pickett ldquoSPICEmodeling of memristorsrdquoin Proceedings of the 2011 IEEE International Symposium ofCircuits and Systems (ISCAS rsquo11) pp 1832ndash1835 May 2011

[16] X B Wang Y R Chen H W Xi and D Dimitrov ldquoSpin-tronic memristor through spin-thorque-induced magnetiza-tion motionrdquo IEEE Electron Device Letters vol 30 no 3 pp294ndash297 2009

[17] D B Strukov and R S Williams ldquoExponential ionic drift fastswitching and low volatility of thin-film memristorsrdquo AppliedPhysics A Materials Science amp Processing vol 94 pp 515ndash5192009

[18] F Corinto A Ascoli and M Gilli ldquoMathematical models andcircuit implementations of memristive systemsrdquo in Proceedingsof the 13th International Workshop on Cellular Nanoscale Net-works and their Applications (CNNA rsquo12) pp 1ndash6 August 2012

[19] S Kvatinsky E G Friedman A Kolodny and U C WeiserldquoTEAMThrEshold adaptive memristor modelrdquo IEEE Transac-tions on Circuits and Systems I Regular Papers vol 60 no 1 pp211ndash221 2013

[20] O Sporns and C J Honey ldquoSmall worlds inside big brainsrdquoProceedings of the National Academy of Sciences of the UnitedStates of America vol 103 no 51 pp 19219ndash19220 2006

[21] P Tangkraingkij C Lursinsap S Sanguansintukul and TDesudchit ldquoInsider and outsider person authentication withminimum number of brain wave signals by neural and homo-geneous identity filteringrdquo Neural Computing and Applicationsvol 22 no 1 pp 463ndash476 2013

[22] D S Bassett and E D Bullmore ldquoSmall-world brain networksrdquoNeuroscientist vol 12 no 6 pp 512ndash523 2006

[23] D J Watts and S H Strogatz ldquoCollective dynamics of ldquosmall-worldrdquo networksrdquoNature vol 393 no 6684 pp 440ndash442 1998

[24] L Douw M M Schoonheim D Landi et al ldquoCognitionis related to resting-state small-world network topology anmagnetoencephalographic studyrdquo Neuroscience vol 175 pp169ndash177 2011

[25] T Fekete F D C C Beacher J Cha D Rubin and L RMujica-Parodi ldquoSmall-world network properties in prefrontal cortexcorrelate with predictors of psychopathology risk in youngchildren aNIRS studyrdquoNeuroImage vol 85 part 1 pp 345ndash3532014

[26] N R Taylor ldquoSmall world network strategies for studyingprotein structures and bindingrdquo Computational and StructuralBiotechnology Journal vol 5 Article ID e201302006 2013

[27] D Simard L Nadeau andH Kroger ldquoFastest learning in small-world neural networksrdquo Physics Letters A General Atomic andSolid State Physics vol 336 no 1 pp 8ndash15 2005

[28] Z Biolek D Biolek and V Biolkova ldquoSPICEmodel of memris-tor with nonlinear dopant driftrdquo Radioengineering vol 18 no2 pp 210ndash214 2009

[29] H Kim M P Sah C Yang T Roska and L O ChualdquoMemristor bridge synapsesrdquo Proceedings of the IEEE vol 100no 6 pp 2061ndash2070 2012

[30] M Esfandyari M A Fanaei and H Zohreie ldquoAdaptive fuzzytuning of PID controllersrdquo Neural Computing and Applicationsvol 23 no 1 pp 19ndash28 2013

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RotatingMachinery

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Submit your manuscripts athttpwwwhindawicom

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Shock and Vibration

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Electrical and Computer Engineering

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Advances inOptoElectronics

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Chemical EngineeringInternational Journal of Antennas and

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DistributedSensor Networks

International Journal of

The Scientific World Journal 5

0 05 1 15

Time (s)

I(m

A)

minus06

minus04

minus02

0

02

04

06

(a)

0

05

minus05

0 05minus05

I(m

A)

(V)

(b)

30

25

20

15

10

5

0

minus5minus02 minus01 0 01 02

Q (mC)

M(kΩ

)

(c)

M(kΩ

)

105

10

95

9

85

8

75

7

65minus04 minus02 0 02 04 06

(V)

(d)

Figure 5 The results of the memristor Simulink model (a) The input current source (b) Relationship between the current 119894 and the voltageV (c) Relationship between the memristance119872 and the charge 119902 (d) Relationship between the memristance119872 and the voltage V

adders and modules) and parameter control modules Themodel parameters are the same as those in Figure 3 Thesignal stimulus applied into the memristor is a sinusoidalcurrent source with amplitude of 05mA and frequency of1Hz

The simulation results are exhibited in Figure 5 The cur-rent flowing through the memristor is shown in Figure 5(a)The typical hysteresis loop in Figure 5(b) shows its switchingcharacteristic that is the memristance can switch betweenhigh resistance and low resistance Figure 5(c) illustrates thatthe memristance is a nonlinear function of the flow of chargeas discussed previously Figure 5(d) shows the relationshipbetween the memristance 119872 and the charge 119902 Notably inthe part of the higher memristance state the change ratioof the memristance is low while in the part of the lowermemristance state the change ratio of the memristance ishigh

3 The Memristive Multilayer FeedforwardSmall-World Neural Network

31 The Multilayer Feedforward Small-World Neural NetworkGenerally small-world phenomenon indicates that a networkhas highly concentrated local connections and also includes afew random long connections In real world a large numberof networks have the small-world effect such as diseasetransmission network social network and the food chainnetwork [22] As is known to all in the classical multilayerfeedforward neural network such as BP network the 119894thneuron in the 119897th layer V119897

119894only connects its neighboring

neuron sets 119881119897minus1 and 119881119897+1 In addition all connections

are feedforward and no connections exist between neuronswithin the same layerThis kind of network can be consideredas a regular network Based on [27] and the constructionprocess of WS small world model we introduce Algorithm 1

6 The Scientific World Journal

Input Output

(a) (b)

Figure 6The construction progress of the multilayer feedforwardWS small-world neural network (a)The regular network (119875 = 0) (b)Themultilayer feedforward WS small-world neural network (0 lt 119875 lt 1)

Set the network has 119871 layersSet each layer has 119899

119897nodes

For ℎ = 1 To 119871 minus 21199011(ℎ) = 120572119890

minus120573(ℎ+2)

For 1198971= 1 To 119871 minus 2

For 119899119897= 1 To 119899

119897

For 1198992= 1 To 119899

119897

If rand lt 1199011199081198971(1198971+1)

11989911198992= 0

1198993= randint(1 119899

119897)

119905 = 1

While 119905 = 11199012= rand

For 1198972= 119871 minus 2 To 119897

1+ 2

If 1199012lt 1199011(1198972)

1198871198971 1198972

11989911198993= rand

119905 = 0

BreakFinish

Algorithm 1 Constructive procedure of the small-world neuralnetwork

which is used to construct multilayer feedforward neuralnetwork model according to the rewiring probability Thespecific construction process is given as follows

Step 1 Initialization assuming the number of the networklayers is 119871 each layer has 119899

119897neuron nodes and the rewiring

probability is 119875

Step 2 Generate the multilayer feedforward regular neuralnetwork as shown in Figure 6(a)

Step 3As shown in Algorithm 1 where 1199011(ℎ) is the probabil-

ity to select reconnection layer selection probability betweentwo neurons decreases exponentially 120572 and 120573 are the distancecoefficients rand and randint both are MATLAB functionsthe former is used to generate a number between (0 1)

randomly and the latter can be used to randomly generatean integer from 1 to 119899

119897 Since the connections of the (119871 minus

1)th layer cannot generate new long-connections if they aredisconnected the connections of the last two layers are notreconnected in the network

As shown in Figure 6(a) when the rewiring probability119875 = 0 the connection of the network maintains completelyregular mode Nonetheless when 119875 ranges from 0 to 1the long cross-layer connections are generated accordingto the rewiring probability 119875 and the probability of recon-nection layer selecting The resulting structure is betweencompletely regular and random connection mode as shownin Figure 6(b)

More specially we set the network connection matrix as119882 where119882119897denotes the connection submatrix between the119897th layer and the (119897 + 1)th layer 119908119897

119894119895isin 119877 is the connection

weight between the neuron 119894 of the 119897th layer and the neuron 119895of the (119897 + 1)th layer If there exists connection between thesetwo neurons then 119908119897

119894119895= 0 otherwise 119908119897

119894119895= 0 Therefore the

regular network connection matrix can be expressed as thefollowing equation

119882 =(((

(

0 1198821

0 0 0 0 0

0 0 1198822

0 0 0 0

0 0 0 1198823

0 0 0

0 0 0 0 1198824

0 0

0 0 0 0 0 1198825

0

0 0 0 0 0 0 1198826

0 0 0 0 0 0 0

)))

)

(13)

in which the number zero means no connection existsbetween the corresponding layers As for multilayer feedfor-ward small-world neural network because of the reconnec-tion performance the connection matrix changes into as

1198821015840=(((

(

0 11988211198613

11198614

11198615

11198616

11198617

1

0 0 11988221198614

21198615

21198616

21198617

2

0 0 0 11988231198615

31198616

31198617

3

0 0 0 0 11988241198616

41198617

4

0 0 0 0 0 11988251198617

5

0 0 0 0 0 0 1198826

0 0 0 0 0 0 0

)))

)

(14)

where119882119897 represents the reconnection submatrix between the119897th layer and the (119897 + 1)th layer and the 119861119897

1015840

119897is the submatrix

between two nonadjacent layers which 119897 isin 1 2 5 1198971015840 isin3 4 7

The Scientific World Journal 7

32 The Combination of the MFSNN and the Memristor

321 The Memristive Synapse The nanoscale memristor hashigh potential of information storage on account of the non-volatility with respect to long periods of power-down soit can be used as electric synapse in the artificial neuralnetworks and the primary reasons are manifold Firstly asa kind of analog component this device can realize weightupdating continuously Moreover the memristor possessesthe capacity of information storage due to the nonvolatilityThis feature is consistent with the memory ability of the neu-rons in humanrsquos brain Additionally the memristive neuralnetwork can be further integrated in crossbar array whichhas significant advantages in better information processingcapacity and huger storage

According to the nonlinear memristor model inSection 2 the memristive conductance can be calculatedfrom (12) as

119866 (119905) =1

119872 (119905)=

1

119877on + Δ119877 (1 (1198601198904119896119902(119905) + 1))

(15)

Differentiating (15) with respect to time 119905 we can beobtain

119889119866 (119905)

119889119905=

41198961198601198904119896119902(119905)

Δ119877

(119877on1198601198904119896119902(119905) + 119877off)

2times119889119902 (119905)

119889119905 (16)

where the current 119894(119905) = 119889119902(119905)119889119905 Notably when Δ119905 rarr 0119889119866(119905) asymp Δ119866 Hence the rate of the memristive conductanceΔ119866 can be described as the synapse weight update ruleThe relationship curve between the rate of the memristiveconductance change and the current is shown in Figure 7When the current is tiny the memristive conductance isalmost invariant While the current tends to plusmn4mA thememristive conductance changes suddenly So the currentthreshold value of thememristive synapse can be set as |119868th| =4mA

322 The Memristive Activation Function In the standardMFSNN the activation function for each neuron is usuallythe nonlinear Sigmoid function Particularly the activationfunction of the hidden layer adopts bipolar Sigmoid functionbut the output layer activation function is unipolar Sigmoidfunction

Based on the constitutive relationship of the memristor alot of nonlinear curves can be simulated and substituted [5 611 12] Based on the Simulink model of the nonlinear mem-ristor described in Section 2 we get its simplified Simulinkmodel accordingly as shown in Figure 8(a) Furthermore wedesign a package of the memristive device (in Figure 8(b))which can be considered as a system with single-input anddouble-output In this system the input variable is the current119868 and the output variable is thememristance119872(119905) and charge119902(119905) respectively

Then the behavior of the output curve can be adjustedefficiently by the parameter control module gain moduleand internal operationmodule which is crucial to implementthe activation function in the neural network Here we set

ΔG

8

6

4

2

0

minus2

minus4

minus6

minus8

times10minus3

minus4 minus2 0 2 4

I (mA)

Figure 7The relationship curve between the rate of the memristiveconductance change and the current

the activation functions for the hidden layer and outputlayer neurons of the memristive MFSNN as ℎ(119909) and 119910(119909)respectively

Figure 9(a) exhibits the constructing principle diagramof the memristive activation function in the hidden layerin which the red dotted line frame represents the parameteradjustment area 119870

1is the adjustable gain which is used for

controlling the shape of the activation function and1198702is the

fixed gain whose value is 1198702= 10minus4 The suitable parameters

of the memristor are chosen as 119877on = 100Ω 119877off = 20 kΩ1198720= 10 kΩ 119863 = 10 nm and 120583] asymp 10

minus14m2sminus1Vminus1 Theinput signal is a sinusoidal current with an amplitude of05mA and a frequency of 1Hz Notably the polarity of thevoltage applied into the memristor is opposite to the polarityof the memristor itself that is the current flows throughthe memristor from the negative polar to the positive polarFigure 9(b) shows the memristive activation function of thehidden layer and its shape varies with different values of119870

1

Similarly Figure 10(a) is the constructing principle dia-gram of the activation function of the output layer In theparameter adjustment part (the red dotted line frame) 119870

3

is an adjustable gain and 1198704is the fixed gain whose value

is 1198704= 2 times 10

minus4 The parameters are the same with thesimulation in Figure 9 Figure 10(b) shows the memristiveactivation function of the output layer Obviously as the valueof 1198703increases the graphs tend to flatten

4 The Memristive Intelligent PID Controller

So far the PID control has found widespread applicationsin the modern control field By adjusting the control actionof the proportion integration and differentiation we getan interactive nonlinear relationship among these controlvariables The neural network has the ability of expressing

8 The Scientific World Journal

Integrator Product ProductAdd

Divide

Constant Constant

Math function

Ron

q(t)

M(t)

+

+

+1

+

divide

times

times

times eu

A 100

ΔR4k

1

s

I

(a)

InOutOutI

q(t)

M(t)

(b)

Figure 8 The simplified memristor model (a) The simplified Simulink model of the memristor (b) The package of the memristor

InOut

OutI q(t)

M(t)

K1

K2

1

h(x)

minus

+

(a)

2

1

0

minus1

h(x)

minus4 minus2 0 2 4

K1 = 20000

K1 = 30000

K1 = 40000x

(b)

Figure 9Theprinciple diagramof thememristive activation function in the hidden layer (a)TheSimulinkmodel of thememristive activationfunction in the hidden layer (b) The curve of the memristive activation function ℎ(119909)

InOut

OutI q(t)

M(t)

K4

K3y(x)

(a)

minus4 minus2 0 2 4

K3 = 20000

K3 = 30000

K3 = 40000

x

2

15

1

05

0

minus05

y(x)

(b)

Figure 10 The principle diagram of the memristive activation function of the output layer (a) The Simulink model of the memristiveactivation function of the output layer (b) The curve of the memristive activation function 119910(119909)

The Scientific World Journal 9

the nonlinearity which can be used in the PID controlfor implementing the optimal nonlinear relationship amongcontrol variables In this work we build up a more intelligentPID controller with the parameters (119896

119901 119896119894 and 119896

119889) self-tune

based on the presented memristive multilayer feedforwardsmall-world neural network

According to the literature [30] the classical incrementaldigital PID control algorithm can be described as

119906 (119896) = 119906 (119896 minus 1) + 119896119901(119890 (119896) minus 119890 (119896 minus 1))

+ 119896119894119890 (119896) + 119896

119889(119890 (119896) minus 2119890 (119896 minus 1) + 119890 (119896 minus 2))

(17)

where the 119896119901 119896119894 and 119896

119889are the coefficient of the proportion

integration and differentiation respectivelyIn Figure 11 the ANN is the memristive multilayer feed-

forward small-world neural network Its learning algorithmconsisted of the backward error propagation and the forwardinput signal propagation Different from the traditional mul-tilayer feedforward neural network the state of the neurons ineach layer not only affects the state of the neurons in the nextlayer but also affects the state of the neurons in the cross-layer

Based on the novel neural network presented in Section 3we set the 119895 119894 and 119897 that represent the input layer hiddenlayer and output layer respectively The number of the inputlayer is 1 which is same with that of the output layer andthe number of the hidden layer is 119878 119909

119894represents the input

vector of the network then the set of the input samples is1198741

119895= [1199091 1199092 1199093 119909

119873] The number of the input vectors

is dependent on the complexity of the system Notably thesuperscript 1 represents the first layer in the whole neuralnetwork

The input and output vectors of the first hidden layer canbe expressed as

net1198941119894(119896) =

119873

sum

119895=0

1199081198941

1198941198951198741

119895

1198741198941

119894(119896) = ℎ (net1198941

119894(119896))

119894 = 1 2 3 119876

(18)

where the superscript 1198941denotes the first hidden layer of the

network and ℎ(119909) is the memristive bipolar sigmoid functionproposed in Section 3

By that analogy the input and output vectors of the 119878thhidden layer can be written as

net119894119904119894(119896) =

119873

sum

119895=0

119908119894119904

1198941198951198741

119895+

119904minus1

sum

119886=1

119904

sum

119887=1

119908119894119904

119894119886119894119904net119894119904119887(119896)

119874119894119904

119894(119896) = ℎ (net119894119904

119894(119896))

119894 = 1 2 3 119876

(19)

Finally the input and output vectors of the output layercan be obtained as

net119904+2119897(119896) =

119873

sum

119895=0

119908119904+2

1198941198971198741

119895+

119904

sum

119886=1

119904

sum

119887=1

119908119904+2

119894119886119897net119904+2119887(119896)

119874119904+2

119897(119896) = 119910 (net119904+2

119897(119896))

119897 = 1 2 3

(20)

where 119910(119909) denotes the memristive unipolar sigmoid func-tion The three nodes of the output layer are correspondingwith the nonnegative adjustable parameters 119896

119901 119896119894 and 119896

119889of

the PID controller respectivelyFrom [27] we conclude the weight update algorithm of

the memristive multilayer feedforward small-world neuralnetwork as below

Δ119908119904+2

119897119894(119896) = 120572Δ119908

119904+2

119897119894(119896 minus 1) + 120578120575

119904+2

119897119874119904+1

119894(119896)

119908119904+2

119897119894(119896 + 1) = 119908

119904+2

119897119894(119896) + Δ119908

119904+2

119897119894(119896)

(21)

where 120572 is the inertial coefficient whose scope ranges from 0to 1 and 120578 isin (0 1) is the learning rate

5 Computer Simulations and Results

In this section somenumerical simulations of thememristivemultilayer feedforward small-world neural network PIDcontroller have been executed on MATLAB software Themathematical model of the controlled plant is given as

119910out (119896) =119886 (119896) 119910out (119896 minus 1)

1 + 1199102out (119896 minus 1)+ 119906 (119896 minus 1) (22)

where the 119886(119896) is slow time-variant and its expression is119886(119896) = 12(1 minus 08119890

01119896)

The memristive neural network under investigation isconstituted by seven layers with four neurons in the inputlayer three neurons in the output layer and five in each of thefive hidden layers The learning rate of the network 120578 = 04and the inertial coefficient 120572 = 005 The initial weighs asrandom values fall in [minus05 05] and the value of the rewiringprobability is chosen as 119875 = 0 119875 = 008 119875 = 01 and 119875 = 02respectively The parameters are 119877on = 100Ω 119877off = 20 kΩ1198720= 10 kΩ 119863 = 10 nm and 120583] asymp 10

minus14m2sminus1Vminus1 1198701and

1198703are user-specified parameters whose value both are 20000

and the action time is 119905119904 = 0001 s When the system workssteadily the tracking results can be gotten as follows

Figure 12(a) shows the input signal (step response curve119903in(119896) = 10) and the output curves under a different rewiringprobability 119875 As can be seen from the figure when thetime 119905 = 05 s the whole system reaches the steady stateMaking a further analysis we can conclude that when therewiring probability 119875 = 0 the memristive neural networkkeeps regularly in architecture Its respond speed is slowerthan that of network when the rewiring probability 119875 =

008 and 119875 = 01 Moreover Figure 12(b) exhibits theerror curves between the input signal and the output signal

10 The Scientific World Journal

rin Error

aeat

ANN

PID Plant yout

kp ki kd

Figure 11 The structure diagram of the memristive neural network PID controller

0

05

1

15

20 05 1 15

Time (s)

Inpu

t ou

tput

Regularnetwork

InputP = 008

P = 01

P = 0

P = 02

(a)

0

08

06

04

02

1

20 05 1 15

Time (s)

Erro

rRegular network

P = 008

P = 01

P = 0

P = 02

(b)

001

002

003

0

005

01

0 2 4 6Time (s)

0 2 4 6Time (s)

0 2 4 60

2

4

Time (s)

P = 008

P = 008

P = 008

times10minus3

k pk i

k d

(c)

Figure 12The simulation results of the memristive neural network PID controller (119903in(119896) = 10) under a different rewiring probability 119875 (a)The step response curve (b) The error curves (c) The curves of the control parameters when the rewiring probability 119875 = 008

correspondingly When the rewiring probability 119875 = 008

and 119875 = 01 the network spends less time on approachingthe predefined approximation error than the regular network(when 119875 = 0) Figure 12(c) shows the output variables ofthe memristive multilayer feedforward small-world neural

network when 119875 = 008 which are the control parameters 119896119901

119896119894 and 119896

119889 correspondingly

In order to verify the superior performance of thememristive small-world neuronal networks andfigure out theoptimal structure we conducted a series of simulations to

The Scientific World Journal 11

4500

5000

5500

6000

6500

7000

0 02 04 06 08 1

Aver

age i

nter

actio

ns

Sampling points

P

(a)

35

37

39

41

43

45

47

0 02 04 06 08 1

Succ

essfu

l int

erac

tions

Sampling points

P

(b)

Figure 13 The convergence performance of the memristive neural network under different 119875 (a) The relationship between iteration andrewiring probability (b) The effective approximation number in 50 times simulations under varying rewiring probability

observe the convergence performance of the proposed net-work under different119875 Figure 13(a) shows the approximationspeed (iteration times) of different network structures thatis the smallest iteration number for reaching the predefinedapproximation error 120576 = 00001 Each drawn point is theaverage value of 50 times runs It can be observed thatthe small-world networks (0 lt 119875 lt 1) need much lessiteration times than the regular neural network (when 119875 =

0) which demonstrates its advantage in processing speedFurthermore when 119875 = 008 the network has the fastapproximation speed

Notably the mathematical function of this system has thelocal minimum for getting out of the local minimum wedefine the maximum allowable iteration times to be 10000as previously mentioned for each 119875 and we performed thesimulation for 50 times where the effective approximationtimes that is error lt 00001 within 10000 iterations arepresented in Figure 13(b) It can be found that the small-worldnetworks have higher accuracy rate than the regular network

6 Conclusions

A mathematical closed-form charge-governed memristormodel is recalled firstly and the corresponding Simulinkmodel is presented Using the change rule of memconduc-tance a memristive realization scheme for synaptic weightis proposed Moreover the activation functions in electricneurons are also implemented based on the single-inputand double-output package of the memristor Combiningthe proposed memristive synapse and activation functions amemristor-basedMFSNN is addressed It exhibits advantagesin computation speed and accuracy over the traditionalmultilayer neural networks by considering the small-worldeffect Meanwhile it has potential of hardware realizationof the neural network because of the nanoscale size of thememristive synapse These superior properties can furtherimprove the application of the neural networks such as inthe intelligent controller design Motivated by this we apply

the memristor-based MFSNN to classical PID control andthe proposed memristive PID controller may possess thefollowing superiorities (i) Its nanoscale physical implemen-tation could promote the development of themicrocontroller(ii) Because of the participation of the memristive neuralnetwork the proposed PID controller can realize the param-eters self-adjustment (iii) The control speed and accuracyare improved Eventually extensive numerical simulationsjustify the effectiveness and efficiency of the memristive PIDcontroller over the regular neural network PID controllerThis work may provide a theoretical reference to physicallyrealize the small-world neural networks and further promotethe development of modern intelligent control technology

Conflict of Interests

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

Acknowledgments

The work was supported by Program for New CenturyExcellent Talents in University (Grant nos [2013] 47)National Natural Science Foundation of China (Grant nos61372139 61101233 and 60972155) ldquoSpring Sunshine PlanrdquoResearch Project of Ministry of Education of China (Grantno z2011148) Technology Foundation for Selected OverseasChinese Scholars Ministry of Personnel in China (Grantno 2012-186) University Excellent Talents Supporting Foun-dations in of Chongqing (Grant no 2011-65) UniversityKey Teacher Supporting Foundations of Chongqing (Grantno 2011-65) Fundamental Research Funds for the CentralUniversities (Grant nos XDJK2014A009 XDJK2013B011)

References

[1] L O Chua ldquoMemristors the missing circuit elementrdquo IEEETransactions on Circuits and Systems vol 18 pp 507ndash519 1971

12 The Scientific World Journal

[2] J M Tour and H Tao ldquoElectronics the fourth elementrdquoNaturevol 453 no 7191 pp 42ndash43 2008

[3] D B Strukov G S Snider D R Stewart and R S WilliamsldquoThe missing memristor foundrdquo Nature vol 453 pp 80ndash832008

[4] R S Williams ldquoHow we found the missing Memristorrdquo IEEESpectrum vol 45 no 12 pp 28ndash35 2008

[5] S K Duan X F Hu L D Wang C D Li and P MazumderldquoMemristor-based RRAM with applicationsrdquo Science ChinaInformation Sciences vol 55 no 6 pp 1446ndash1460 2012

[6] X F Hu S K Duan L DWang and C D Li ldquoAnalog memorybased on pulse controlled memristorrdquo Journal of the Universityof Electronic Science and Technology of China vol 40 no 5 pp642ndash647 2011

[7] S Shin K Kim and S Kang ldquoResistive computingmemristors-enabled signal multiplicationrdquo IEEE Transactions on Circuitsand Systems I Regular Papers vol 60 no 5 pp 1241ndash1249 2013

[8] S P Adhikari C Yang H Kim and L O Chua ldquoMemristorbridge synapse-based neural network and its learningrdquo IEEETransactions on Neural Networks and Learning Systems vol 23no 9 pp 1426ndash1435 2012

[9] S H Jo T Chang I Ebong B B Bhadviya P Mazumder andW Lu ldquoNanoscale memristor device as synapse in neuromor-phic systemsrdquo Nano Letters vol 10 no 4 pp 1297ndash1301 2010

[10] Y Dai C Li and H Wang ldquoExpanded HP memristor modeland simulation in STDP learningrdquo Neural Computing andApplications vol 24 no 1 pp 51ndash57 2014

[11] S K Duan X F Hu L DWang S Y Gao andCD Li ldquoHybridmemristorRTD structure-based cellular neural networks withapplications in image processingrdquoNeural Computing and Appli-cations vol 25 no 2 pp 291ndash296 2014

[12] L Chen C D Li T W Huang and Y R Chen ldquoMemristorcrossbar-based unsupervised image learningrdquo Neural Comput-ing and Applications vol 25 no 2 pp 393ndash400 2013

[13] X F Hu S K Duan L D Wang and X F Liao ldquoMemristivecrossbar array with applications in image processingrdquo ScienceChina Information Sciences vol 41 pp 500ndash512 2011

[14] B Mouttet ldquoProposal for memristors in signal processingrdquo inNano-Net 3rd International ICST Conference NanoNet BostonMS USA September 14ndash16 2008 Revised Selected Papersvol 3 of Lecture Notes of the Institute for Computer SciencesSocial Informatics and Telecommunications Engineering pp 11ndash13 2009

[15] H Abdalla andM D Pickett ldquoSPICEmodeling of memristorsrdquoin Proceedings of the 2011 IEEE International Symposium ofCircuits and Systems (ISCAS rsquo11) pp 1832ndash1835 May 2011

[16] X B Wang Y R Chen H W Xi and D Dimitrov ldquoSpin-tronic memristor through spin-thorque-induced magnetiza-tion motionrdquo IEEE Electron Device Letters vol 30 no 3 pp294ndash297 2009

[17] D B Strukov and R S Williams ldquoExponential ionic drift fastswitching and low volatility of thin-film memristorsrdquo AppliedPhysics A Materials Science amp Processing vol 94 pp 515ndash5192009

[18] F Corinto A Ascoli and M Gilli ldquoMathematical models andcircuit implementations of memristive systemsrdquo in Proceedingsof the 13th International Workshop on Cellular Nanoscale Net-works and their Applications (CNNA rsquo12) pp 1ndash6 August 2012

[19] S Kvatinsky E G Friedman A Kolodny and U C WeiserldquoTEAMThrEshold adaptive memristor modelrdquo IEEE Transac-tions on Circuits and Systems I Regular Papers vol 60 no 1 pp211ndash221 2013

[20] O Sporns and C J Honey ldquoSmall worlds inside big brainsrdquoProceedings of the National Academy of Sciences of the UnitedStates of America vol 103 no 51 pp 19219ndash19220 2006

[21] P Tangkraingkij C Lursinsap S Sanguansintukul and TDesudchit ldquoInsider and outsider person authentication withminimum number of brain wave signals by neural and homo-geneous identity filteringrdquo Neural Computing and Applicationsvol 22 no 1 pp 463ndash476 2013

[22] D S Bassett and E D Bullmore ldquoSmall-world brain networksrdquoNeuroscientist vol 12 no 6 pp 512ndash523 2006

[23] D J Watts and S H Strogatz ldquoCollective dynamics of ldquosmall-worldrdquo networksrdquoNature vol 393 no 6684 pp 440ndash442 1998

[24] L Douw M M Schoonheim D Landi et al ldquoCognitionis related to resting-state small-world network topology anmagnetoencephalographic studyrdquo Neuroscience vol 175 pp169ndash177 2011

[25] T Fekete F D C C Beacher J Cha D Rubin and L RMujica-Parodi ldquoSmall-world network properties in prefrontal cortexcorrelate with predictors of psychopathology risk in youngchildren aNIRS studyrdquoNeuroImage vol 85 part 1 pp 345ndash3532014

[26] N R Taylor ldquoSmall world network strategies for studyingprotein structures and bindingrdquo Computational and StructuralBiotechnology Journal vol 5 Article ID e201302006 2013

[27] D Simard L Nadeau andH Kroger ldquoFastest learning in small-world neural networksrdquo Physics Letters A General Atomic andSolid State Physics vol 336 no 1 pp 8ndash15 2005

[28] Z Biolek D Biolek and V Biolkova ldquoSPICEmodel of memris-tor with nonlinear dopant driftrdquo Radioengineering vol 18 no2 pp 210ndash214 2009

[29] H Kim M P Sah C Yang T Roska and L O ChualdquoMemristor bridge synapsesrdquo Proceedings of the IEEE vol 100no 6 pp 2061ndash2070 2012

[30] M Esfandyari M A Fanaei and H Zohreie ldquoAdaptive fuzzytuning of PID controllersrdquo Neural Computing and Applicationsvol 23 no 1 pp 19ndash28 2013

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Active and Passive Electronic Components

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Submit your manuscripts athttpwwwhindawicom

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Advances inOptoElectronics

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Chemical EngineeringInternational Journal of Antennas and

Propagation

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Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

6 The Scientific World Journal

Input Output

(a) (b)

Figure 6The construction progress of the multilayer feedforwardWS small-world neural network (a)The regular network (119875 = 0) (b)Themultilayer feedforward WS small-world neural network (0 lt 119875 lt 1)

Set the network has 119871 layersSet each layer has 119899

119897nodes

For ℎ = 1 To 119871 minus 21199011(ℎ) = 120572119890

minus120573(ℎ+2)

For 1198971= 1 To 119871 minus 2

For 119899119897= 1 To 119899

119897

For 1198992= 1 To 119899

119897

If rand lt 1199011199081198971(1198971+1)

11989911198992= 0

1198993= randint(1 119899

119897)

119905 = 1

While 119905 = 11199012= rand

For 1198972= 119871 minus 2 To 119897

1+ 2

If 1199012lt 1199011(1198972)

1198871198971 1198972

11989911198993= rand

119905 = 0

BreakFinish

Algorithm 1 Constructive procedure of the small-world neuralnetwork

which is used to construct multilayer feedforward neuralnetwork model according to the rewiring probability Thespecific construction process is given as follows

Step 1 Initialization assuming the number of the networklayers is 119871 each layer has 119899

119897neuron nodes and the rewiring

probability is 119875

Step 2 Generate the multilayer feedforward regular neuralnetwork as shown in Figure 6(a)

Step 3As shown in Algorithm 1 where 1199011(ℎ) is the probabil-

ity to select reconnection layer selection probability betweentwo neurons decreases exponentially 120572 and 120573 are the distancecoefficients rand and randint both are MATLAB functionsthe former is used to generate a number between (0 1)

randomly and the latter can be used to randomly generatean integer from 1 to 119899

119897 Since the connections of the (119871 minus

1)th layer cannot generate new long-connections if they aredisconnected the connections of the last two layers are notreconnected in the network

As shown in Figure 6(a) when the rewiring probability119875 = 0 the connection of the network maintains completelyregular mode Nonetheless when 119875 ranges from 0 to 1the long cross-layer connections are generated accordingto the rewiring probability 119875 and the probability of recon-nection layer selecting The resulting structure is betweencompletely regular and random connection mode as shownin Figure 6(b)

More specially we set the network connection matrix as119882 where119882119897denotes the connection submatrix between the119897th layer and the (119897 + 1)th layer 119908119897

119894119895isin 119877 is the connection

weight between the neuron 119894 of the 119897th layer and the neuron 119895of the (119897 + 1)th layer If there exists connection between thesetwo neurons then 119908119897

119894119895= 0 otherwise 119908119897

119894119895= 0 Therefore the

regular network connection matrix can be expressed as thefollowing equation

119882 =(((

(

0 1198821

0 0 0 0 0

0 0 1198822

0 0 0 0

0 0 0 1198823

0 0 0

0 0 0 0 1198824

0 0

0 0 0 0 0 1198825

0

0 0 0 0 0 0 1198826

0 0 0 0 0 0 0

)))

)

(13)

in which the number zero means no connection existsbetween the corresponding layers As for multilayer feedfor-ward small-world neural network because of the reconnec-tion performance the connection matrix changes into as

1198821015840=(((

(

0 11988211198613

11198614

11198615

11198616

11198617

1

0 0 11988221198614

21198615

21198616

21198617

2

0 0 0 11988231198615

31198616

31198617

3

0 0 0 0 11988241198616

41198617

4

0 0 0 0 0 11988251198617

5

0 0 0 0 0 0 1198826

0 0 0 0 0 0 0

)))

)

(14)

where119882119897 represents the reconnection submatrix between the119897th layer and the (119897 + 1)th layer and the 119861119897

1015840

119897is the submatrix

between two nonadjacent layers which 119897 isin 1 2 5 1198971015840 isin3 4 7

The Scientific World Journal 7

32 The Combination of the MFSNN and the Memristor

321 The Memristive Synapse The nanoscale memristor hashigh potential of information storage on account of the non-volatility with respect to long periods of power-down soit can be used as electric synapse in the artificial neuralnetworks and the primary reasons are manifold Firstly asa kind of analog component this device can realize weightupdating continuously Moreover the memristor possessesthe capacity of information storage due to the nonvolatilityThis feature is consistent with the memory ability of the neu-rons in humanrsquos brain Additionally the memristive neuralnetwork can be further integrated in crossbar array whichhas significant advantages in better information processingcapacity and huger storage

According to the nonlinear memristor model inSection 2 the memristive conductance can be calculatedfrom (12) as

119866 (119905) =1

119872 (119905)=

1

119877on + Δ119877 (1 (1198601198904119896119902(119905) + 1))

(15)

Differentiating (15) with respect to time 119905 we can beobtain

119889119866 (119905)

119889119905=

41198961198601198904119896119902(119905)

Δ119877

(119877on1198601198904119896119902(119905) + 119877off)

2times119889119902 (119905)

119889119905 (16)

where the current 119894(119905) = 119889119902(119905)119889119905 Notably when Δ119905 rarr 0119889119866(119905) asymp Δ119866 Hence the rate of the memristive conductanceΔ119866 can be described as the synapse weight update ruleThe relationship curve between the rate of the memristiveconductance change and the current is shown in Figure 7When the current is tiny the memristive conductance isalmost invariant While the current tends to plusmn4mA thememristive conductance changes suddenly So the currentthreshold value of thememristive synapse can be set as |119868th| =4mA

322 The Memristive Activation Function In the standardMFSNN the activation function for each neuron is usuallythe nonlinear Sigmoid function Particularly the activationfunction of the hidden layer adopts bipolar Sigmoid functionbut the output layer activation function is unipolar Sigmoidfunction

Based on the constitutive relationship of the memristor alot of nonlinear curves can be simulated and substituted [5 611 12] Based on the Simulink model of the nonlinear mem-ristor described in Section 2 we get its simplified Simulinkmodel accordingly as shown in Figure 8(a) Furthermore wedesign a package of the memristive device (in Figure 8(b))which can be considered as a system with single-input anddouble-output In this system the input variable is the current119868 and the output variable is thememristance119872(119905) and charge119902(119905) respectively

Then the behavior of the output curve can be adjustedefficiently by the parameter control module gain moduleand internal operationmodule which is crucial to implementthe activation function in the neural network Here we set

ΔG

8

6

4

2

0

minus2

minus4

minus6

minus8

times10minus3

minus4 minus2 0 2 4

I (mA)

Figure 7The relationship curve between the rate of the memristiveconductance change and the current

the activation functions for the hidden layer and outputlayer neurons of the memristive MFSNN as ℎ(119909) and 119910(119909)respectively

Figure 9(a) exhibits the constructing principle diagramof the memristive activation function in the hidden layerin which the red dotted line frame represents the parameteradjustment area 119870

1is the adjustable gain which is used for

controlling the shape of the activation function and1198702is the

fixed gain whose value is 1198702= 10minus4 The suitable parameters

of the memristor are chosen as 119877on = 100Ω 119877off = 20 kΩ1198720= 10 kΩ 119863 = 10 nm and 120583] asymp 10

minus14m2sminus1Vminus1 Theinput signal is a sinusoidal current with an amplitude of05mA and a frequency of 1Hz Notably the polarity of thevoltage applied into the memristor is opposite to the polarityof the memristor itself that is the current flows throughthe memristor from the negative polar to the positive polarFigure 9(b) shows the memristive activation function of thehidden layer and its shape varies with different values of119870

1

Similarly Figure 10(a) is the constructing principle dia-gram of the activation function of the output layer In theparameter adjustment part (the red dotted line frame) 119870

3

is an adjustable gain and 1198704is the fixed gain whose value

is 1198704= 2 times 10

minus4 The parameters are the same with thesimulation in Figure 9 Figure 10(b) shows the memristiveactivation function of the output layer Obviously as the valueof 1198703increases the graphs tend to flatten

4 The Memristive Intelligent PID Controller

So far the PID control has found widespread applicationsin the modern control field By adjusting the control actionof the proportion integration and differentiation we getan interactive nonlinear relationship among these controlvariables The neural network has the ability of expressing

8 The Scientific World Journal

Integrator Product ProductAdd

Divide

Constant Constant

Math function

Ron

q(t)

M(t)

+

+

+1

+

divide

times

times

times eu

A 100

ΔR4k

1

s

I

(a)

InOutOutI

q(t)

M(t)

(b)

Figure 8 The simplified memristor model (a) The simplified Simulink model of the memristor (b) The package of the memristor

InOut

OutI q(t)

M(t)

K1

K2

1

h(x)

minus

+

(a)

2

1

0

minus1

h(x)

minus4 minus2 0 2 4

K1 = 20000

K1 = 30000

K1 = 40000x

(b)

Figure 9Theprinciple diagramof thememristive activation function in the hidden layer (a)TheSimulinkmodel of thememristive activationfunction in the hidden layer (b) The curve of the memristive activation function ℎ(119909)

InOut

OutI q(t)

M(t)

K4

K3y(x)

(a)

minus4 minus2 0 2 4

K3 = 20000

K3 = 30000

K3 = 40000

x

2

15

1

05

0

minus05

y(x)

(b)

Figure 10 The principle diagram of the memristive activation function of the output layer (a) The Simulink model of the memristiveactivation function of the output layer (b) The curve of the memristive activation function 119910(119909)

The Scientific World Journal 9

the nonlinearity which can be used in the PID controlfor implementing the optimal nonlinear relationship amongcontrol variables In this work we build up a more intelligentPID controller with the parameters (119896

119901 119896119894 and 119896

119889) self-tune

based on the presented memristive multilayer feedforwardsmall-world neural network

According to the literature [30] the classical incrementaldigital PID control algorithm can be described as

119906 (119896) = 119906 (119896 minus 1) + 119896119901(119890 (119896) minus 119890 (119896 minus 1))

+ 119896119894119890 (119896) + 119896

119889(119890 (119896) minus 2119890 (119896 minus 1) + 119890 (119896 minus 2))

(17)

where the 119896119901 119896119894 and 119896

119889are the coefficient of the proportion

integration and differentiation respectivelyIn Figure 11 the ANN is the memristive multilayer feed-

forward small-world neural network Its learning algorithmconsisted of the backward error propagation and the forwardinput signal propagation Different from the traditional mul-tilayer feedforward neural network the state of the neurons ineach layer not only affects the state of the neurons in the nextlayer but also affects the state of the neurons in the cross-layer

Based on the novel neural network presented in Section 3we set the 119895 119894 and 119897 that represent the input layer hiddenlayer and output layer respectively The number of the inputlayer is 1 which is same with that of the output layer andthe number of the hidden layer is 119878 119909

119894represents the input

vector of the network then the set of the input samples is1198741

119895= [1199091 1199092 1199093 119909

119873] The number of the input vectors

is dependent on the complexity of the system Notably thesuperscript 1 represents the first layer in the whole neuralnetwork

The input and output vectors of the first hidden layer canbe expressed as

net1198941119894(119896) =

119873

sum

119895=0

1199081198941

1198941198951198741

119895

1198741198941

119894(119896) = ℎ (net1198941

119894(119896))

119894 = 1 2 3 119876

(18)

where the superscript 1198941denotes the first hidden layer of the

network and ℎ(119909) is the memristive bipolar sigmoid functionproposed in Section 3

By that analogy the input and output vectors of the 119878thhidden layer can be written as

net119894119904119894(119896) =

119873

sum

119895=0

119908119894119904

1198941198951198741

119895+

119904minus1

sum

119886=1

119904

sum

119887=1

119908119894119904

119894119886119894119904net119894119904119887(119896)

119874119894119904

119894(119896) = ℎ (net119894119904

119894(119896))

119894 = 1 2 3 119876

(19)

Finally the input and output vectors of the output layercan be obtained as

net119904+2119897(119896) =

119873

sum

119895=0

119908119904+2

1198941198971198741

119895+

119904

sum

119886=1

119904

sum

119887=1

119908119904+2

119894119886119897net119904+2119887(119896)

119874119904+2

119897(119896) = 119910 (net119904+2

119897(119896))

119897 = 1 2 3

(20)

where 119910(119909) denotes the memristive unipolar sigmoid func-tion The three nodes of the output layer are correspondingwith the nonnegative adjustable parameters 119896

119901 119896119894 and 119896

119889of

the PID controller respectivelyFrom [27] we conclude the weight update algorithm of

the memristive multilayer feedforward small-world neuralnetwork as below

Δ119908119904+2

119897119894(119896) = 120572Δ119908

119904+2

119897119894(119896 minus 1) + 120578120575

119904+2

119897119874119904+1

119894(119896)

119908119904+2

119897119894(119896 + 1) = 119908

119904+2

119897119894(119896) + Δ119908

119904+2

119897119894(119896)

(21)

where 120572 is the inertial coefficient whose scope ranges from 0to 1 and 120578 isin (0 1) is the learning rate

5 Computer Simulations and Results

In this section somenumerical simulations of thememristivemultilayer feedforward small-world neural network PIDcontroller have been executed on MATLAB software Themathematical model of the controlled plant is given as

119910out (119896) =119886 (119896) 119910out (119896 minus 1)

1 + 1199102out (119896 minus 1)+ 119906 (119896 minus 1) (22)

where the 119886(119896) is slow time-variant and its expression is119886(119896) = 12(1 minus 08119890

01119896)

The memristive neural network under investigation isconstituted by seven layers with four neurons in the inputlayer three neurons in the output layer and five in each of thefive hidden layers The learning rate of the network 120578 = 04and the inertial coefficient 120572 = 005 The initial weighs asrandom values fall in [minus05 05] and the value of the rewiringprobability is chosen as 119875 = 0 119875 = 008 119875 = 01 and 119875 = 02respectively The parameters are 119877on = 100Ω 119877off = 20 kΩ1198720= 10 kΩ 119863 = 10 nm and 120583] asymp 10

minus14m2sminus1Vminus1 1198701and

1198703are user-specified parameters whose value both are 20000

and the action time is 119905119904 = 0001 s When the system workssteadily the tracking results can be gotten as follows

Figure 12(a) shows the input signal (step response curve119903in(119896) = 10) and the output curves under a different rewiringprobability 119875 As can be seen from the figure when thetime 119905 = 05 s the whole system reaches the steady stateMaking a further analysis we can conclude that when therewiring probability 119875 = 0 the memristive neural networkkeeps regularly in architecture Its respond speed is slowerthan that of network when the rewiring probability 119875 =

008 and 119875 = 01 Moreover Figure 12(b) exhibits theerror curves between the input signal and the output signal

10 The Scientific World Journal

rin Error

aeat

ANN

PID Plant yout

kp ki kd

Figure 11 The structure diagram of the memristive neural network PID controller

0

05

1

15

20 05 1 15

Time (s)

Inpu

t ou

tput

Regularnetwork

InputP = 008

P = 01

P = 0

P = 02

(a)

0

08

06

04

02

1

20 05 1 15

Time (s)

Erro

rRegular network

P = 008

P = 01

P = 0

P = 02

(b)

001

002

003

0

005

01

0 2 4 6Time (s)

0 2 4 6Time (s)

0 2 4 60

2

4

Time (s)

P = 008

P = 008

P = 008

times10minus3

k pk i

k d

(c)

Figure 12The simulation results of the memristive neural network PID controller (119903in(119896) = 10) under a different rewiring probability 119875 (a)The step response curve (b) The error curves (c) The curves of the control parameters when the rewiring probability 119875 = 008

correspondingly When the rewiring probability 119875 = 008

and 119875 = 01 the network spends less time on approachingthe predefined approximation error than the regular network(when 119875 = 0) Figure 12(c) shows the output variables ofthe memristive multilayer feedforward small-world neural

network when 119875 = 008 which are the control parameters 119896119901

119896119894 and 119896

119889 correspondingly

In order to verify the superior performance of thememristive small-world neuronal networks andfigure out theoptimal structure we conducted a series of simulations to

The Scientific World Journal 11

4500

5000

5500

6000

6500

7000

0 02 04 06 08 1

Aver

age i

nter

actio

ns

Sampling points

P

(a)

35

37

39

41

43

45

47

0 02 04 06 08 1

Succ

essfu

l int

erac

tions

Sampling points

P

(b)

Figure 13 The convergence performance of the memristive neural network under different 119875 (a) The relationship between iteration andrewiring probability (b) The effective approximation number in 50 times simulations under varying rewiring probability

observe the convergence performance of the proposed net-work under different119875 Figure 13(a) shows the approximationspeed (iteration times) of different network structures thatis the smallest iteration number for reaching the predefinedapproximation error 120576 = 00001 Each drawn point is theaverage value of 50 times runs It can be observed thatthe small-world networks (0 lt 119875 lt 1) need much lessiteration times than the regular neural network (when 119875 =

0) which demonstrates its advantage in processing speedFurthermore when 119875 = 008 the network has the fastapproximation speed

Notably the mathematical function of this system has thelocal minimum for getting out of the local minimum wedefine the maximum allowable iteration times to be 10000as previously mentioned for each 119875 and we performed thesimulation for 50 times where the effective approximationtimes that is error lt 00001 within 10000 iterations arepresented in Figure 13(b) It can be found that the small-worldnetworks have higher accuracy rate than the regular network

6 Conclusions

A mathematical closed-form charge-governed memristormodel is recalled firstly and the corresponding Simulinkmodel is presented Using the change rule of memconduc-tance a memristive realization scheme for synaptic weightis proposed Moreover the activation functions in electricneurons are also implemented based on the single-inputand double-output package of the memristor Combiningthe proposed memristive synapse and activation functions amemristor-basedMFSNN is addressed It exhibits advantagesin computation speed and accuracy over the traditionalmultilayer neural networks by considering the small-worldeffect Meanwhile it has potential of hardware realizationof the neural network because of the nanoscale size of thememristive synapse These superior properties can furtherimprove the application of the neural networks such as inthe intelligent controller design Motivated by this we apply

the memristor-based MFSNN to classical PID control andthe proposed memristive PID controller may possess thefollowing superiorities (i) Its nanoscale physical implemen-tation could promote the development of themicrocontroller(ii) Because of the participation of the memristive neuralnetwork the proposed PID controller can realize the param-eters self-adjustment (iii) The control speed and accuracyare improved Eventually extensive numerical simulationsjustify the effectiveness and efficiency of the memristive PIDcontroller over the regular neural network PID controllerThis work may provide a theoretical reference to physicallyrealize the small-world neural networks and further promotethe development of modern intelligent control technology

Conflict of Interests

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

Acknowledgments

The work was supported by Program for New CenturyExcellent Talents in University (Grant nos [2013] 47)National Natural Science Foundation of China (Grant nos61372139 61101233 and 60972155) ldquoSpring Sunshine PlanrdquoResearch Project of Ministry of Education of China (Grantno z2011148) Technology Foundation for Selected OverseasChinese Scholars Ministry of Personnel in China (Grantno 2012-186) University Excellent Talents Supporting Foun-dations in of Chongqing (Grant no 2011-65) UniversityKey Teacher Supporting Foundations of Chongqing (Grantno 2011-65) Fundamental Research Funds for the CentralUniversities (Grant nos XDJK2014A009 XDJK2013B011)

References

[1] L O Chua ldquoMemristors the missing circuit elementrdquo IEEETransactions on Circuits and Systems vol 18 pp 507ndash519 1971

12 The Scientific World Journal

[2] J M Tour and H Tao ldquoElectronics the fourth elementrdquoNaturevol 453 no 7191 pp 42ndash43 2008

[3] D B Strukov G S Snider D R Stewart and R S WilliamsldquoThe missing memristor foundrdquo Nature vol 453 pp 80ndash832008

[4] R S Williams ldquoHow we found the missing Memristorrdquo IEEESpectrum vol 45 no 12 pp 28ndash35 2008

[5] S K Duan X F Hu L D Wang C D Li and P MazumderldquoMemristor-based RRAM with applicationsrdquo Science ChinaInformation Sciences vol 55 no 6 pp 1446ndash1460 2012

[6] X F Hu S K Duan L DWang and C D Li ldquoAnalog memorybased on pulse controlled memristorrdquo Journal of the Universityof Electronic Science and Technology of China vol 40 no 5 pp642ndash647 2011

[7] S Shin K Kim and S Kang ldquoResistive computingmemristors-enabled signal multiplicationrdquo IEEE Transactions on Circuitsand Systems I Regular Papers vol 60 no 5 pp 1241ndash1249 2013

[8] S P Adhikari C Yang H Kim and L O Chua ldquoMemristorbridge synapse-based neural network and its learningrdquo IEEETransactions on Neural Networks and Learning Systems vol 23no 9 pp 1426ndash1435 2012

[9] S H Jo T Chang I Ebong B B Bhadviya P Mazumder andW Lu ldquoNanoscale memristor device as synapse in neuromor-phic systemsrdquo Nano Letters vol 10 no 4 pp 1297ndash1301 2010

[10] Y Dai C Li and H Wang ldquoExpanded HP memristor modeland simulation in STDP learningrdquo Neural Computing andApplications vol 24 no 1 pp 51ndash57 2014

[11] S K Duan X F Hu L DWang S Y Gao andCD Li ldquoHybridmemristorRTD structure-based cellular neural networks withapplications in image processingrdquoNeural Computing and Appli-cations vol 25 no 2 pp 291ndash296 2014

[12] L Chen C D Li T W Huang and Y R Chen ldquoMemristorcrossbar-based unsupervised image learningrdquo Neural Comput-ing and Applications vol 25 no 2 pp 393ndash400 2013

[13] X F Hu S K Duan L D Wang and X F Liao ldquoMemristivecrossbar array with applications in image processingrdquo ScienceChina Information Sciences vol 41 pp 500ndash512 2011

[14] B Mouttet ldquoProposal for memristors in signal processingrdquo inNano-Net 3rd International ICST Conference NanoNet BostonMS USA September 14ndash16 2008 Revised Selected Papersvol 3 of Lecture Notes of the Institute for Computer SciencesSocial Informatics and Telecommunications Engineering pp 11ndash13 2009

[15] H Abdalla andM D Pickett ldquoSPICEmodeling of memristorsrdquoin Proceedings of the 2011 IEEE International Symposium ofCircuits and Systems (ISCAS rsquo11) pp 1832ndash1835 May 2011

[16] X B Wang Y R Chen H W Xi and D Dimitrov ldquoSpin-tronic memristor through spin-thorque-induced magnetiza-tion motionrdquo IEEE Electron Device Letters vol 30 no 3 pp294ndash297 2009

[17] D B Strukov and R S Williams ldquoExponential ionic drift fastswitching and low volatility of thin-film memristorsrdquo AppliedPhysics A Materials Science amp Processing vol 94 pp 515ndash5192009

[18] F Corinto A Ascoli and M Gilli ldquoMathematical models andcircuit implementations of memristive systemsrdquo in Proceedingsof the 13th International Workshop on Cellular Nanoscale Net-works and their Applications (CNNA rsquo12) pp 1ndash6 August 2012

[19] S Kvatinsky E G Friedman A Kolodny and U C WeiserldquoTEAMThrEshold adaptive memristor modelrdquo IEEE Transac-tions on Circuits and Systems I Regular Papers vol 60 no 1 pp211ndash221 2013

[20] O Sporns and C J Honey ldquoSmall worlds inside big brainsrdquoProceedings of the National Academy of Sciences of the UnitedStates of America vol 103 no 51 pp 19219ndash19220 2006

[21] P Tangkraingkij C Lursinsap S Sanguansintukul and TDesudchit ldquoInsider and outsider person authentication withminimum number of brain wave signals by neural and homo-geneous identity filteringrdquo Neural Computing and Applicationsvol 22 no 1 pp 463ndash476 2013

[22] D S Bassett and E D Bullmore ldquoSmall-world brain networksrdquoNeuroscientist vol 12 no 6 pp 512ndash523 2006

[23] D J Watts and S H Strogatz ldquoCollective dynamics of ldquosmall-worldrdquo networksrdquoNature vol 393 no 6684 pp 440ndash442 1998

[24] L Douw M M Schoonheim D Landi et al ldquoCognitionis related to resting-state small-world network topology anmagnetoencephalographic studyrdquo Neuroscience vol 175 pp169ndash177 2011

[25] T Fekete F D C C Beacher J Cha D Rubin and L RMujica-Parodi ldquoSmall-world network properties in prefrontal cortexcorrelate with predictors of psychopathology risk in youngchildren aNIRS studyrdquoNeuroImage vol 85 part 1 pp 345ndash3532014

[26] N R Taylor ldquoSmall world network strategies for studyingprotein structures and bindingrdquo Computational and StructuralBiotechnology Journal vol 5 Article ID e201302006 2013

[27] D Simard L Nadeau andH Kroger ldquoFastest learning in small-world neural networksrdquo Physics Letters A General Atomic andSolid State Physics vol 336 no 1 pp 8ndash15 2005

[28] Z Biolek D Biolek and V Biolkova ldquoSPICEmodel of memris-tor with nonlinear dopant driftrdquo Radioengineering vol 18 no2 pp 210ndash214 2009

[29] H Kim M P Sah C Yang T Roska and L O ChualdquoMemristor bridge synapsesrdquo Proceedings of the IEEE vol 100no 6 pp 2061ndash2070 2012

[30] M Esfandyari M A Fanaei and H Zohreie ldquoAdaptive fuzzytuning of PID controllersrdquo Neural Computing and Applicationsvol 23 no 1 pp 19ndash28 2013

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Submit your manuscripts athttpwwwhindawicom

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Chemical EngineeringInternational Journal of Antennas and

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Navigation and Observation

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DistributedSensor Networks

International Journal of

The Scientific World Journal 7

32 The Combination of the MFSNN and the Memristor

321 The Memristive Synapse The nanoscale memristor hashigh potential of information storage on account of the non-volatility with respect to long periods of power-down soit can be used as electric synapse in the artificial neuralnetworks and the primary reasons are manifold Firstly asa kind of analog component this device can realize weightupdating continuously Moreover the memristor possessesthe capacity of information storage due to the nonvolatilityThis feature is consistent with the memory ability of the neu-rons in humanrsquos brain Additionally the memristive neuralnetwork can be further integrated in crossbar array whichhas significant advantages in better information processingcapacity and huger storage

According to the nonlinear memristor model inSection 2 the memristive conductance can be calculatedfrom (12) as

119866 (119905) =1

119872 (119905)=

1

119877on + Δ119877 (1 (1198601198904119896119902(119905) + 1))

(15)

Differentiating (15) with respect to time 119905 we can beobtain

119889119866 (119905)

119889119905=

41198961198601198904119896119902(119905)

Δ119877

(119877on1198601198904119896119902(119905) + 119877off)

2times119889119902 (119905)

119889119905 (16)

where the current 119894(119905) = 119889119902(119905)119889119905 Notably when Δ119905 rarr 0119889119866(119905) asymp Δ119866 Hence the rate of the memristive conductanceΔ119866 can be described as the synapse weight update ruleThe relationship curve between the rate of the memristiveconductance change and the current is shown in Figure 7When the current is tiny the memristive conductance isalmost invariant While the current tends to plusmn4mA thememristive conductance changes suddenly So the currentthreshold value of thememristive synapse can be set as |119868th| =4mA

322 The Memristive Activation Function In the standardMFSNN the activation function for each neuron is usuallythe nonlinear Sigmoid function Particularly the activationfunction of the hidden layer adopts bipolar Sigmoid functionbut the output layer activation function is unipolar Sigmoidfunction

Based on the constitutive relationship of the memristor alot of nonlinear curves can be simulated and substituted [5 611 12] Based on the Simulink model of the nonlinear mem-ristor described in Section 2 we get its simplified Simulinkmodel accordingly as shown in Figure 8(a) Furthermore wedesign a package of the memristive device (in Figure 8(b))which can be considered as a system with single-input anddouble-output In this system the input variable is the current119868 and the output variable is thememristance119872(119905) and charge119902(119905) respectively

Then the behavior of the output curve can be adjustedefficiently by the parameter control module gain moduleand internal operationmodule which is crucial to implementthe activation function in the neural network Here we set

ΔG

8

6

4

2

0

minus2

minus4

minus6

minus8

times10minus3

minus4 minus2 0 2 4

I (mA)

Figure 7The relationship curve between the rate of the memristiveconductance change and the current

the activation functions for the hidden layer and outputlayer neurons of the memristive MFSNN as ℎ(119909) and 119910(119909)respectively

Figure 9(a) exhibits the constructing principle diagramof the memristive activation function in the hidden layerin which the red dotted line frame represents the parameteradjustment area 119870

1is the adjustable gain which is used for

controlling the shape of the activation function and1198702is the

fixed gain whose value is 1198702= 10minus4 The suitable parameters

of the memristor are chosen as 119877on = 100Ω 119877off = 20 kΩ1198720= 10 kΩ 119863 = 10 nm and 120583] asymp 10

minus14m2sminus1Vminus1 Theinput signal is a sinusoidal current with an amplitude of05mA and a frequency of 1Hz Notably the polarity of thevoltage applied into the memristor is opposite to the polarityof the memristor itself that is the current flows throughthe memristor from the negative polar to the positive polarFigure 9(b) shows the memristive activation function of thehidden layer and its shape varies with different values of119870

1

Similarly Figure 10(a) is the constructing principle dia-gram of the activation function of the output layer In theparameter adjustment part (the red dotted line frame) 119870

3

is an adjustable gain and 1198704is the fixed gain whose value

is 1198704= 2 times 10

minus4 The parameters are the same with thesimulation in Figure 9 Figure 10(b) shows the memristiveactivation function of the output layer Obviously as the valueof 1198703increases the graphs tend to flatten

4 The Memristive Intelligent PID Controller

So far the PID control has found widespread applicationsin the modern control field By adjusting the control actionof the proportion integration and differentiation we getan interactive nonlinear relationship among these controlvariables The neural network has the ability of expressing

8 The Scientific World Journal

Integrator Product ProductAdd

Divide

Constant Constant

Math function

Ron

q(t)

M(t)

+

+

+1

+

divide

times

times

times eu

A 100

ΔR4k

1

s

I

(a)

InOutOutI

q(t)

M(t)

(b)

Figure 8 The simplified memristor model (a) The simplified Simulink model of the memristor (b) The package of the memristor

InOut

OutI q(t)

M(t)

K1

K2

1

h(x)

minus

+

(a)

2

1

0

minus1

h(x)

minus4 minus2 0 2 4

K1 = 20000

K1 = 30000

K1 = 40000x

(b)

Figure 9Theprinciple diagramof thememristive activation function in the hidden layer (a)TheSimulinkmodel of thememristive activationfunction in the hidden layer (b) The curve of the memristive activation function ℎ(119909)

InOut

OutI q(t)

M(t)

K4

K3y(x)

(a)

minus4 minus2 0 2 4

K3 = 20000

K3 = 30000

K3 = 40000

x

2

15

1

05

0

minus05

y(x)

(b)

Figure 10 The principle diagram of the memristive activation function of the output layer (a) The Simulink model of the memristiveactivation function of the output layer (b) The curve of the memristive activation function 119910(119909)

The Scientific World Journal 9

the nonlinearity which can be used in the PID controlfor implementing the optimal nonlinear relationship amongcontrol variables In this work we build up a more intelligentPID controller with the parameters (119896

119901 119896119894 and 119896

119889) self-tune

based on the presented memristive multilayer feedforwardsmall-world neural network

According to the literature [30] the classical incrementaldigital PID control algorithm can be described as

119906 (119896) = 119906 (119896 minus 1) + 119896119901(119890 (119896) minus 119890 (119896 minus 1))

+ 119896119894119890 (119896) + 119896

119889(119890 (119896) minus 2119890 (119896 minus 1) + 119890 (119896 minus 2))

(17)

where the 119896119901 119896119894 and 119896

119889are the coefficient of the proportion

integration and differentiation respectivelyIn Figure 11 the ANN is the memristive multilayer feed-

forward small-world neural network Its learning algorithmconsisted of the backward error propagation and the forwardinput signal propagation Different from the traditional mul-tilayer feedforward neural network the state of the neurons ineach layer not only affects the state of the neurons in the nextlayer but also affects the state of the neurons in the cross-layer

Based on the novel neural network presented in Section 3we set the 119895 119894 and 119897 that represent the input layer hiddenlayer and output layer respectively The number of the inputlayer is 1 which is same with that of the output layer andthe number of the hidden layer is 119878 119909

119894represents the input

vector of the network then the set of the input samples is1198741

119895= [1199091 1199092 1199093 119909

119873] The number of the input vectors

is dependent on the complexity of the system Notably thesuperscript 1 represents the first layer in the whole neuralnetwork

The input and output vectors of the first hidden layer canbe expressed as

net1198941119894(119896) =

119873

sum

119895=0

1199081198941

1198941198951198741

119895

1198741198941

119894(119896) = ℎ (net1198941

119894(119896))

119894 = 1 2 3 119876

(18)

where the superscript 1198941denotes the first hidden layer of the

network and ℎ(119909) is the memristive bipolar sigmoid functionproposed in Section 3

By that analogy the input and output vectors of the 119878thhidden layer can be written as

net119894119904119894(119896) =

119873

sum

119895=0

119908119894119904

1198941198951198741

119895+

119904minus1

sum

119886=1

119904

sum

119887=1

119908119894119904

119894119886119894119904net119894119904119887(119896)

119874119894119904

119894(119896) = ℎ (net119894119904

119894(119896))

119894 = 1 2 3 119876

(19)

Finally the input and output vectors of the output layercan be obtained as

net119904+2119897(119896) =

119873

sum

119895=0

119908119904+2

1198941198971198741

119895+

119904

sum

119886=1

119904

sum

119887=1

119908119904+2

119894119886119897net119904+2119887(119896)

119874119904+2

119897(119896) = 119910 (net119904+2

119897(119896))

119897 = 1 2 3

(20)

where 119910(119909) denotes the memristive unipolar sigmoid func-tion The three nodes of the output layer are correspondingwith the nonnegative adjustable parameters 119896

119901 119896119894 and 119896

119889of

the PID controller respectivelyFrom [27] we conclude the weight update algorithm of

the memristive multilayer feedforward small-world neuralnetwork as below

Δ119908119904+2

119897119894(119896) = 120572Δ119908

119904+2

119897119894(119896 minus 1) + 120578120575

119904+2

119897119874119904+1

119894(119896)

119908119904+2

119897119894(119896 + 1) = 119908

119904+2

119897119894(119896) + Δ119908

119904+2

119897119894(119896)

(21)

where 120572 is the inertial coefficient whose scope ranges from 0to 1 and 120578 isin (0 1) is the learning rate

5 Computer Simulations and Results

In this section somenumerical simulations of thememristivemultilayer feedforward small-world neural network PIDcontroller have been executed on MATLAB software Themathematical model of the controlled plant is given as

119910out (119896) =119886 (119896) 119910out (119896 minus 1)

1 + 1199102out (119896 minus 1)+ 119906 (119896 minus 1) (22)

where the 119886(119896) is slow time-variant and its expression is119886(119896) = 12(1 minus 08119890

01119896)

The memristive neural network under investigation isconstituted by seven layers with four neurons in the inputlayer three neurons in the output layer and five in each of thefive hidden layers The learning rate of the network 120578 = 04and the inertial coefficient 120572 = 005 The initial weighs asrandom values fall in [minus05 05] and the value of the rewiringprobability is chosen as 119875 = 0 119875 = 008 119875 = 01 and 119875 = 02respectively The parameters are 119877on = 100Ω 119877off = 20 kΩ1198720= 10 kΩ 119863 = 10 nm and 120583] asymp 10

minus14m2sminus1Vminus1 1198701and

1198703are user-specified parameters whose value both are 20000

and the action time is 119905119904 = 0001 s When the system workssteadily the tracking results can be gotten as follows

Figure 12(a) shows the input signal (step response curve119903in(119896) = 10) and the output curves under a different rewiringprobability 119875 As can be seen from the figure when thetime 119905 = 05 s the whole system reaches the steady stateMaking a further analysis we can conclude that when therewiring probability 119875 = 0 the memristive neural networkkeeps regularly in architecture Its respond speed is slowerthan that of network when the rewiring probability 119875 =

008 and 119875 = 01 Moreover Figure 12(b) exhibits theerror curves between the input signal and the output signal

10 The Scientific World Journal

rin Error

aeat

ANN

PID Plant yout

kp ki kd

Figure 11 The structure diagram of the memristive neural network PID controller

0

05

1

15

20 05 1 15

Time (s)

Inpu

t ou

tput

Regularnetwork

InputP = 008

P = 01

P = 0

P = 02

(a)

0

08

06

04

02

1

20 05 1 15

Time (s)

Erro

rRegular network

P = 008

P = 01

P = 0

P = 02

(b)

001

002

003

0

005

01

0 2 4 6Time (s)

0 2 4 6Time (s)

0 2 4 60

2

4

Time (s)

P = 008

P = 008

P = 008

times10minus3

k pk i

k d

(c)

Figure 12The simulation results of the memristive neural network PID controller (119903in(119896) = 10) under a different rewiring probability 119875 (a)The step response curve (b) The error curves (c) The curves of the control parameters when the rewiring probability 119875 = 008

correspondingly When the rewiring probability 119875 = 008

and 119875 = 01 the network spends less time on approachingthe predefined approximation error than the regular network(when 119875 = 0) Figure 12(c) shows the output variables ofthe memristive multilayer feedforward small-world neural

network when 119875 = 008 which are the control parameters 119896119901

119896119894 and 119896

119889 correspondingly

In order to verify the superior performance of thememristive small-world neuronal networks andfigure out theoptimal structure we conducted a series of simulations to

The Scientific World Journal 11

4500

5000

5500

6000

6500

7000

0 02 04 06 08 1

Aver

age i

nter

actio

ns

Sampling points

P

(a)

35

37

39

41

43

45

47

0 02 04 06 08 1

Succ

essfu

l int

erac

tions

Sampling points

P

(b)

Figure 13 The convergence performance of the memristive neural network under different 119875 (a) The relationship between iteration andrewiring probability (b) The effective approximation number in 50 times simulations under varying rewiring probability

observe the convergence performance of the proposed net-work under different119875 Figure 13(a) shows the approximationspeed (iteration times) of different network structures thatis the smallest iteration number for reaching the predefinedapproximation error 120576 = 00001 Each drawn point is theaverage value of 50 times runs It can be observed thatthe small-world networks (0 lt 119875 lt 1) need much lessiteration times than the regular neural network (when 119875 =

0) which demonstrates its advantage in processing speedFurthermore when 119875 = 008 the network has the fastapproximation speed

Notably the mathematical function of this system has thelocal minimum for getting out of the local minimum wedefine the maximum allowable iteration times to be 10000as previously mentioned for each 119875 and we performed thesimulation for 50 times where the effective approximationtimes that is error lt 00001 within 10000 iterations arepresented in Figure 13(b) It can be found that the small-worldnetworks have higher accuracy rate than the regular network

6 Conclusions

A mathematical closed-form charge-governed memristormodel is recalled firstly and the corresponding Simulinkmodel is presented Using the change rule of memconduc-tance a memristive realization scheme for synaptic weightis proposed Moreover the activation functions in electricneurons are also implemented based on the single-inputand double-output package of the memristor Combiningthe proposed memristive synapse and activation functions amemristor-basedMFSNN is addressed It exhibits advantagesin computation speed and accuracy over the traditionalmultilayer neural networks by considering the small-worldeffect Meanwhile it has potential of hardware realizationof the neural network because of the nanoscale size of thememristive synapse These superior properties can furtherimprove the application of the neural networks such as inthe intelligent controller design Motivated by this we apply

the memristor-based MFSNN to classical PID control andthe proposed memristive PID controller may possess thefollowing superiorities (i) Its nanoscale physical implemen-tation could promote the development of themicrocontroller(ii) Because of the participation of the memristive neuralnetwork the proposed PID controller can realize the param-eters self-adjustment (iii) The control speed and accuracyare improved Eventually extensive numerical simulationsjustify the effectiveness and efficiency of the memristive PIDcontroller over the regular neural network PID controllerThis work may provide a theoretical reference to physicallyrealize the small-world neural networks and further promotethe development of modern intelligent control technology

Conflict of Interests

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

Acknowledgments

The work was supported by Program for New CenturyExcellent Talents in University (Grant nos [2013] 47)National Natural Science Foundation of China (Grant nos61372139 61101233 and 60972155) ldquoSpring Sunshine PlanrdquoResearch Project of Ministry of Education of China (Grantno z2011148) Technology Foundation for Selected OverseasChinese Scholars Ministry of Personnel in China (Grantno 2012-186) University Excellent Talents Supporting Foun-dations in of Chongqing (Grant no 2011-65) UniversityKey Teacher Supporting Foundations of Chongqing (Grantno 2011-65) Fundamental Research Funds for the CentralUniversities (Grant nos XDJK2014A009 XDJK2013B011)

References

[1] L O Chua ldquoMemristors the missing circuit elementrdquo IEEETransactions on Circuits and Systems vol 18 pp 507ndash519 1971

12 The Scientific World Journal

[2] J M Tour and H Tao ldquoElectronics the fourth elementrdquoNaturevol 453 no 7191 pp 42ndash43 2008

[3] D B Strukov G S Snider D R Stewart and R S WilliamsldquoThe missing memristor foundrdquo Nature vol 453 pp 80ndash832008

[4] R S Williams ldquoHow we found the missing Memristorrdquo IEEESpectrum vol 45 no 12 pp 28ndash35 2008

[5] S K Duan X F Hu L D Wang C D Li and P MazumderldquoMemristor-based RRAM with applicationsrdquo Science ChinaInformation Sciences vol 55 no 6 pp 1446ndash1460 2012

[6] X F Hu S K Duan L DWang and C D Li ldquoAnalog memorybased on pulse controlled memristorrdquo Journal of the Universityof Electronic Science and Technology of China vol 40 no 5 pp642ndash647 2011

[7] S Shin K Kim and S Kang ldquoResistive computingmemristors-enabled signal multiplicationrdquo IEEE Transactions on Circuitsand Systems I Regular Papers vol 60 no 5 pp 1241ndash1249 2013

[8] S P Adhikari C Yang H Kim and L O Chua ldquoMemristorbridge synapse-based neural network and its learningrdquo IEEETransactions on Neural Networks and Learning Systems vol 23no 9 pp 1426ndash1435 2012

[9] S H Jo T Chang I Ebong B B Bhadviya P Mazumder andW Lu ldquoNanoscale memristor device as synapse in neuromor-phic systemsrdquo Nano Letters vol 10 no 4 pp 1297ndash1301 2010

[10] Y Dai C Li and H Wang ldquoExpanded HP memristor modeland simulation in STDP learningrdquo Neural Computing andApplications vol 24 no 1 pp 51ndash57 2014

[11] S K Duan X F Hu L DWang S Y Gao andCD Li ldquoHybridmemristorRTD structure-based cellular neural networks withapplications in image processingrdquoNeural Computing and Appli-cations vol 25 no 2 pp 291ndash296 2014

[12] L Chen C D Li T W Huang and Y R Chen ldquoMemristorcrossbar-based unsupervised image learningrdquo Neural Comput-ing and Applications vol 25 no 2 pp 393ndash400 2013

[13] X F Hu S K Duan L D Wang and X F Liao ldquoMemristivecrossbar array with applications in image processingrdquo ScienceChina Information Sciences vol 41 pp 500ndash512 2011

[14] B Mouttet ldquoProposal for memristors in signal processingrdquo inNano-Net 3rd International ICST Conference NanoNet BostonMS USA September 14ndash16 2008 Revised Selected Papersvol 3 of Lecture Notes of the Institute for Computer SciencesSocial Informatics and Telecommunications Engineering pp 11ndash13 2009

[15] H Abdalla andM D Pickett ldquoSPICEmodeling of memristorsrdquoin Proceedings of the 2011 IEEE International Symposium ofCircuits and Systems (ISCAS rsquo11) pp 1832ndash1835 May 2011

[16] X B Wang Y R Chen H W Xi and D Dimitrov ldquoSpin-tronic memristor through spin-thorque-induced magnetiza-tion motionrdquo IEEE Electron Device Letters vol 30 no 3 pp294ndash297 2009

[17] D B Strukov and R S Williams ldquoExponential ionic drift fastswitching and low volatility of thin-film memristorsrdquo AppliedPhysics A Materials Science amp Processing vol 94 pp 515ndash5192009

[18] F Corinto A Ascoli and M Gilli ldquoMathematical models andcircuit implementations of memristive systemsrdquo in Proceedingsof the 13th International Workshop on Cellular Nanoscale Net-works and their Applications (CNNA rsquo12) pp 1ndash6 August 2012

[19] S Kvatinsky E G Friedman A Kolodny and U C WeiserldquoTEAMThrEshold adaptive memristor modelrdquo IEEE Transac-tions on Circuits and Systems I Regular Papers vol 60 no 1 pp211ndash221 2013

[20] O Sporns and C J Honey ldquoSmall worlds inside big brainsrdquoProceedings of the National Academy of Sciences of the UnitedStates of America vol 103 no 51 pp 19219ndash19220 2006

[21] P Tangkraingkij C Lursinsap S Sanguansintukul and TDesudchit ldquoInsider and outsider person authentication withminimum number of brain wave signals by neural and homo-geneous identity filteringrdquo Neural Computing and Applicationsvol 22 no 1 pp 463ndash476 2013

[22] D S Bassett and E D Bullmore ldquoSmall-world brain networksrdquoNeuroscientist vol 12 no 6 pp 512ndash523 2006

[23] D J Watts and S H Strogatz ldquoCollective dynamics of ldquosmall-worldrdquo networksrdquoNature vol 393 no 6684 pp 440ndash442 1998

[24] L Douw M M Schoonheim D Landi et al ldquoCognitionis related to resting-state small-world network topology anmagnetoencephalographic studyrdquo Neuroscience vol 175 pp169ndash177 2011

[25] T Fekete F D C C Beacher J Cha D Rubin and L RMujica-Parodi ldquoSmall-world network properties in prefrontal cortexcorrelate with predictors of psychopathology risk in youngchildren aNIRS studyrdquoNeuroImage vol 85 part 1 pp 345ndash3532014

[26] N R Taylor ldquoSmall world network strategies for studyingprotein structures and bindingrdquo Computational and StructuralBiotechnology Journal vol 5 Article ID e201302006 2013

[27] D Simard L Nadeau andH Kroger ldquoFastest learning in small-world neural networksrdquo Physics Letters A General Atomic andSolid State Physics vol 336 no 1 pp 8ndash15 2005

[28] Z Biolek D Biolek and V Biolkova ldquoSPICEmodel of memris-tor with nonlinear dopant driftrdquo Radioengineering vol 18 no2 pp 210ndash214 2009

[29] H Kim M P Sah C Yang T Roska and L O ChualdquoMemristor bridge synapsesrdquo Proceedings of the IEEE vol 100no 6 pp 2061ndash2070 2012

[30] M Esfandyari M A Fanaei and H Zohreie ldquoAdaptive fuzzytuning of PID controllersrdquo Neural Computing and Applicationsvol 23 no 1 pp 19ndash28 2013

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

8 The Scientific World Journal

Integrator Product ProductAdd

Divide

Constant Constant

Math function

Ron

q(t)

M(t)

+

+

+1

+

divide

times

times

times eu

A 100

ΔR4k

1

s

I

(a)

InOutOutI

q(t)

M(t)

(b)

Figure 8 The simplified memristor model (a) The simplified Simulink model of the memristor (b) The package of the memristor

InOut

OutI q(t)

M(t)

K1

K2

1

h(x)

minus

+

(a)

2

1

0

minus1

h(x)

minus4 minus2 0 2 4

K1 = 20000

K1 = 30000

K1 = 40000x

(b)

Figure 9Theprinciple diagramof thememristive activation function in the hidden layer (a)TheSimulinkmodel of thememristive activationfunction in the hidden layer (b) The curve of the memristive activation function ℎ(119909)

InOut

OutI q(t)

M(t)

K4

K3y(x)

(a)

minus4 minus2 0 2 4

K3 = 20000

K3 = 30000

K3 = 40000

x

2

15

1

05

0

minus05

y(x)

(b)

Figure 10 The principle diagram of the memristive activation function of the output layer (a) The Simulink model of the memristiveactivation function of the output layer (b) The curve of the memristive activation function 119910(119909)

The Scientific World Journal 9

the nonlinearity which can be used in the PID controlfor implementing the optimal nonlinear relationship amongcontrol variables In this work we build up a more intelligentPID controller with the parameters (119896

119901 119896119894 and 119896

119889) self-tune

based on the presented memristive multilayer feedforwardsmall-world neural network

According to the literature [30] the classical incrementaldigital PID control algorithm can be described as

119906 (119896) = 119906 (119896 minus 1) + 119896119901(119890 (119896) minus 119890 (119896 minus 1))

+ 119896119894119890 (119896) + 119896

119889(119890 (119896) minus 2119890 (119896 minus 1) + 119890 (119896 minus 2))

(17)

where the 119896119901 119896119894 and 119896

119889are the coefficient of the proportion

integration and differentiation respectivelyIn Figure 11 the ANN is the memristive multilayer feed-

forward small-world neural network Its learning algorithmconsisted of the backward error propagation and the forwardinput signal propagation Different from the traditional mul-tilayer feedforward neural network the state of the neurons ineach layer not only affects the state of the neurons in the nextlayer but also affects the state of the neurons in the cross-layer

Based on the novel neural network presented in Section 3we set the 119895 119894 and 119897 that represent the input layer hiddenlayer and output layer respectively The number of the inputlayer is 1 which is same with that of the output layer andthe number of the hidden layer is 119878 119909

119894represents the input

vector of the network then the set of the input samples is1198741

119895= [1199091 1199092 1199093 119909

119873] The number of the input vectors

is dependent on the complexity of the system Notably thesuperscript 1 represents the first layer in the whole neuralnetwork

The input and output vectors of the first hidden layer canbe expressed as

net1198941119894(119896) =

119873

sum

119895=0

1199081198941

1198941198951198741

119895

1198741198941

119894(119896) = ℎ (net1198941

119894(119896))

119894 = 1 2 3 119876

(18)

where the superscript 1198941denotes the first hidden layer of the

network and ℎ(119909) is the memristive bipolar sigmoid functionproposed in Section 3

By that analogy the input and output vectors of the 119878thhidden layer can be written as

net119894119904119894(119896) =

119873

sum

119895=0

119908119894119904

1198941198951198741

119895+

119904minus1

sum

119886=1

119904

sum

119887=1

119908119894119904

119894119886119894119904net119894119904119887(119896)

119874119894119904

119894(119896) = ℎ (net119894119904

119894(119896))

119894 = 1 2 3 119876

(19)

Finally the input and output vectors of the output layercan be obtained as

net119904+2119897(119896) =

119873

sum

119895=0

119908119904+2

1198941198971198741

119895+

119904

sum

119886=1

119904

sum

119887=1

119908119904+2

119894119886119897net119904+2119887(119896)

119874119904+2

119897(119896) = 119910 (net119904+2

119897(119896))

119897 = 1 2 3

(20)

where 119910(119909) denotes the memristive unipolar sigmoid func-tion The three nodes of the output layer are correspondingwith the nonnegative adjustable parameters 119896

119901 119896119894 and 119896

119889of

the PID controller respectivelyFrom [27] we conclude the weight update algorithm of

the memristive multilayer feedforward small-world neuralnetwork as below

Δ119908119904+2

119897119894(119896) = 120572Δ119908

119904+2

119897119894(119896 minus 1) + 120578120575

119904+2

119897119874119904+1

119894(119896)

119908119904+2

119897119894(119896 + 1) = 119908

119904+2

119897119894(119896) + Δ119908

119904+2

119897119894(119896)

(21)

where 120572 is the inertial coefficient whose scope ranges from 0to 1 and 120578 isin (0 1) is the learning rate

5 Computer Simulations and Results

In this section somenumerical simulations of thememristivemultilayer feedforward small-world neural network PIDcontroller have been executed on MATLAB software Themathematical model of the controlled plant is given as

119910out (119896) =119886 (119896) 119910out (119896 minus 1)

1 + 1199102out (119896 minus 1)+ 119906 (119896 minus 1) (22)

where the 119886(119896) is slow time-variant and its expression is119886(119896) = 12(1 minus 08119890

01119896)

The memristive neural network under investigation isconstituted by seven layers with four neurons in the inputlayer three neurons in the output layer and five in each of thefive hidden layers The learning rate of the network 120578 = 04and the inertial coefficient 120572 = 005 The initial weighs asrandom values fall in [minus05 05] and the value of the rewiringprobability is chosen as 119875 = 0 119875 = 008 119875 = 01 and 119875 = 02respectively The parameters are 119877on = 100Ω 119877off = 20 kΩ1198720= 10 kΩ 119863 = 10 nm and 120583] asymp 10

minus14m2sminus1Vminus1 1198701and

1198703are user-specified parameters whose value both are 20000

and the action time is 119905119904 = 0001 s When the system workssteadily the tracking results can be gotten as follows

Figure 12(a) shows the input signal (step response curve119903in(119896) = 10) and the output curves under a different rewiringprobability 119875 As can be seen from the figure when thetime 119905 = 05 s the whole system reaches the steady stateMaking a further analysis we can conclude that when therewiring probability 119875 = 0 the memristive neural networkkeeps regularly in architecture Its respond speed is slowerthan that of network when the rewiring probability 119875 =

008 and 119875 = 01 Moreover Figure 12(b) exhibits theerror curves between the input signal and the output signal

10 The Scientific World Journal

rin Error

aeat

ANN

PID Plant yout

kp ki kd

Figure 11 The structure diagram of the memristive neural network PID controller

0

05

1

15

20 05 1 15

Time (s)

Inpu

t ou

tput

Regularnetwork

InputP = 008

P = 01

P = 0

P = 02

(a)

0

08

06

04

02

1

20 05 1 15

Time (s)

Erro

rRegular network

P = 008

P = 01

P = 0

P = 02

(b)

001

002

003

0

005

01

0 2 4 6Time (s)

0 2 4 6Time (s)

0 2 4 60

2

4

Time (s)

P = 008

P = 008

P = 008

times10minus3

k pk i

k d

(c)

Figure 12The simulation results of the memristive neural network PID controller (119903in(119896) = 10) under a different rewiring probability 119875 (a)The step response curve (b) The error curves (c) The curves of the control parameters when the rewiring probability 119875 = 008

correspondingly When the rewiring probability 119875 = 008

and 119875 = 01 the network spends less time on approachingthe predefined approximation error than the regular network(when 119875 = 0) Figure 12(c) shows the output variables ofthe memristive multilayer feedforward small-world neural

network when 119875 = 008 which are the control parameters 119896119901

119896119894 and 119896

119889 correspondingly

In order to verify the superior performance of thememristive small-world neuronal networks andfigure out theoptimal structure we conducted a series of simulations to

The Scientific World Journal 11

4500

5000

5500

6000

6500

7000

0 02 04 06 08 1

Aver

age i

nter

actio

ns

Sampling points

P

(a)

35

37

39

41

43

45

47

0 02 04 06 08 1

Succ

essfu

l int

erac

tions

Sampling points

P

(b)

Figure 13 The convergence performance of the memristive neural network under different 119875 (a) The relationship between iteration andrewiring probability (b) The effective approximation number in 50 times simulations under varying rewiring probability

observe the convergence performance of the proposed net-work under different119875 Figure 13(a) shows the approximationspeed (iteration times) of different network structures thatis the smallest iteration number for reaching the predefinedapproximation error 120576 = 00001 Each drawn point is theaverage value of 50 times runs It can be observed thatthe small-world networks (0 lt 119875 lt 1) need much lessiteration times than the regular neural network (when 119875 =

0) which demonstrates its advantage in processing speedFurthermore when 119875 = 008 the network has the fastapproximation speed

Notably the mathematical function of this system has thelocal minimum for getting out of the local minimum wedefine the maximum allowable iteration times to be 10000as previously mentioned for each 119875 and we performed thesimulation for 50 times where the effective approximationtimes that is error lt 00001 within 10000 iterations arepresented in Figure 13(b) It can be found that the small-worldnetworks have higher accuracy rate than the regular network

6 Conclusions

A mathematical closed-form charge-governed memristormodel is recalled firstly and the corresponding Simulinkmodel is presented Using the change rule of memconduc-tance a memristive realization scheme for synaptic weightis proposed Moreover the activation functions in electricneurons are also implemented based on the single-inputand double-output package of the memristor Combiningthe proposed memristive synapse and activation functions amemristor-basedMFSNN is addressed It exhibits advantagesin computation speed and accuracy over the traditionalmultilayer neural networks by considering the small-worldeffect Meanwhile it has potential of hardware realizationof the neural network because of the nanoscale size of thememristive synapse These superior properties can furtherimprove the application of the neural networks such as inthe intelligent controller design Motivated by this we apply

the memristor-based MFSNN to classical PID control andthe proposed memristive PID controller may possess thefollowing superiorities (i) Its nanoscale physical implemen-tation could promote the development of themicrocontroller(ii) Because of the participation of the memristive neuralnetwork the proposed PID controller can realize the param-eters self-adjustment (iii) The control speed and accuracyare improved Eventually extensive numerical simulationsjustify the effectiveness and efficiency of the memristive PIDcontroller over the regular neural network PID controllerThis work may provide a theoretical reference to physicallyrealize the small-world neural networks and further promotethe development of modern intelligent control technology

Conflict of Interests

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

Acknowledgments

The work was supported by Program for New CenturyExcellent Talents in University (Grant nos [2013] 47)National Natural Science Foundation of China (Grant nos61372139 61101233 and 60972155) ldquoSpring Sunshine PlanrdquoResearch Project of Ministry of Education of China (Grantno z2011148) Technology Foundation for Selected OverseasChinese Scholars Ministry of Personnel in China (Grantno 2012-186) University Excellent Talents Supporting Foun-dations in of Chongqing (Grant no 2011-65) UniversityKey Teacher Supporting Foundations of Chongqing (Grantno 2011-65) Fundamental Research Funds for the CentralUniversities (Grant nos XDJK2014A009 XDJK2013B011)

References

[1] L O Chua ldquoMemristors the missing circuit elementrdquo IEEETransactions on Circuits and Systems vol 18 pp 507ndash519 1971

12 The Scientific World Journal

[2] J M Tour and H Tao ldquoElectronics the fourth elementrdquoNaturevol 453 no 7191 pp 42ndash43 2008

[3] D B Strukov G S Snider D R Stewart and R S WilliamsldquoThe missing memristor foundrdquo Nature vol 453 pp 80ndash832008

[4] R S Williams ldquoHow we found the missing Memristorrdquo IEEESpectrum vol 45 no 12 pp 28ndash35 2008

[5] S K Duan X F Hu L D Wang C D Li and P MazumderldquoMemristor-based RRAM with applicationsrdquo Science ChinaInformation Sciences vol 55 no 6 pp 1446ndash1460 2012

[6] X F Hu S K Duan L DWang and C D Li ldquoAnalog memorybased on pulse controlled memristorrdquo Journal of the Universityof Electronic Science and Technology of China vol 40 no 5 pp642ndash647 2011

[7] S Shin K Kim and S Kang ldquoResistive computingmemristors-enabled signal multiplicationrdquo IEEE Transactions on Circuitsand Systems I Regular Papers vol 60 no 5 pp 1241ndash1249 2013

[8] S P Adhikari C Yang H Kim and L O Chua ldquoMemristorbridge synapse-based neural network and its learningrdquo IEEETransactions on Neural Networks and Learning Systems vol 23no 9 pp 1426ndash1435 2012

[9] S H Jo T Chang I Ebong B B Bhadviya P Mazumder andW Lu ldquoNanoscale memristor device as synapse in neuromor-phic systemsrdquo Nano Letters vol 10 no 4 pp 1297ndash1301 2010

[10] Y Dai C Li and H Wang ldquoExpanded HP memristor modeland simulation in STDP learningrdquo Neural Computing andApplications vol 24 no 1 pp 51ndash57 2014

[11] S K Duan X F Hu L DWang S Y Gao andCD Li ldquoHybridmemristorRTD structure-based cellular neural networks withapplications in image processingrdquoNeural Computing and Appli-cations vol 25 no 2 pp 291ndash296 2014

[12] L Chen C D Li T W Huang and Y R Chen ldquoMemristorcrossbar-based unsupervised image learningrdquo Neural Comput-ing and Applications vol 25 no 2 pp 393ndash400 2013

[13] X F Hu S K Duan L D Wang and X F Liao ldquoMemristivecrossbar array with applications in image processingrdquo ScienceChina Information Sciences vol 41 pp 500ndash512 2011

[14] B Mouttet ldquoProposal for memristors in signal processingrdquo inNano-Net 3rd International ICST Conference NanoNet BostonMS USA September 14ndash16 2008 Revised Selected Papersvol 3 of Lecture Notes of the Institute for Computer SciencesSocial Informatics and Telecommunications Engineering pp 11ndash13 2009

[15] H Abdalla andM D Pickett ldquoSPICEmodeling of memristorsrdquoin Proceedings of the 2011 IEEE International Symposium ofCircuits and Systems (ISCAS rsquo11) pp 1832ndash1835 May 2011

[16] X B Wang Y R Chen H W Xi and D Dimitrov ldquoSpin-tronic memristor through spin-thorque-induced magnetiza-tion motionrdquo IEEE Electron Device Letters vol 30 no 3 pp294ndash297 2009

[17] D B Strukov and R S Williams ldquoExponential ionic drift fastswitching and low volatility of thin-film memristorsrdquo AppliedPhysics A Materials Science amp Processing vol 94 pp 515ndash5192009

[18] F Corinto A Ascoli and M Gilli ldquoMathematical models andcircuit implementations of memristive systemsrdquo in Proceedingsof the 13th International Workshop on Cellular Nanoscale Net-works and their Applications (CNNA rsquo12) pp 1ndash6 August 2012

[19] S Kvatinsky E G Friedman A Kolodny and U C WeiserldquoTEAMThrEshold adaptive memristor modelrdquo IEEE Transac-tions on Circuits and Systems I Regular Papers vol 60 no 1 pp211ndash221 2013

[20] O Sporns and C J Honey ldquoSmall worlds inside big brainsrdquoProceedings of the National Academy of Sciences of the UnitedStates of America vol 103 no 51 pp 19219ndash19220 2006

[21] P Tangkraingkij C Lursinsap S Sanguansintukul and TDesudchit ldquoInsider and outsider person authentication withminimum number of brain wave signals by neural and homo-geneous identity filteringrdquo Neural Computing and Applicationsvol 22 no 1 pp 463ndash476 2013

[22] D S Bassett and E D Bullmore ldquoSmall-world brain networksrdquoNeuroscientist vol 12 no 6 pp 512ndash523 2006

[23] D J Watts and S H Strogatz ldquoCollective dynamics of ldquosmall-worldrdquo networksrdquoNature vol 393 no 6684 pp 440ndash442 1998

[24] L Douw M M Schoonheim D Landi et al ldquoCognitionis related to resting-state small-world network topology anmagnetoencephalographic studyrdquo Neuroscience vol 175 pp169ndash177 2011

[25] T Fekete F D C C Beacher J Cha D Rubin and L RMujica-Parodi ldquoSmall-world network properties in prefrontal cortexcorrelate with predictors of psychopathology risk in youngchildren aNIRS studyrdquoNeuroImage vol 85 part 1 pp 345ndash3532014

[26] N R Taylor ldquoSmall world network strategies for studyingprotein structures and bindingrdquo Computational and StructuralBiotechnology Journal vol 5 Article ID e201302006 2013

[27] D Simard L Nadeau andH Kroger ldquoFastest learning in small-world neural networksrdquo Physics Letters A General Atomic andSolid State Physics vol 336 no 1 pp 8ndash15 2005

[28] Z Biolek D Biolek and V Biolkova ldquoSPICEmodel of memris-tor with nonlinear dopant driftrdquo Radioengineering vol 18 no2 pp 210ndash214 2009

[29] H Kim M P Sah C Yang T Roska and L O ChualdquoMemristor bridge synapsesrdquo Proceedings of the IEEE vol 100no 6 pp 2061ndash2070 2012

[30] M Esfandyari M A Fanaei and H Zohreie ldquoAdaptive fuzzytuning of PID controllersrdquo Neural Computing and Applicationsvol 23 no 1 pp 19ndash28 2013

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

The Scientific World Journal 9

the nonlinearity which can be used in the PID controlfor implementing the optimal nonlinear relationship amongcontrol variables In this work we build up a more intelligentPID controller with the parameters (119896

119901 119896119894 and 119896

119889) self-tune

based on the presented memristive multilayer feedforwardsmall-world neural network

According to the literature [30] the classical incrementaldigital PID control algorithm can be described as

119906 (119896) = 119906 (119896 minus 1) + 119896119901(119890 (119896) minus 119890 (119896 minus 1))

+ 119896119894119890 (119896) + 119896

119889(119890 (119896) minus 2119890 (119896 minus 1) + 119890 (119896 minus 2))

(17)

where the 119896119901 119896119894 and 119896

119889are the coefficient of the proportion

integration and differentiation respectivelyIn Figure 11 the ANN is the memristive multilayer feed-

forward small-world neural network Its learning algorithmconsisted of the backward error propagation and the forwardinput signal propagation Different from the traditional mul-tilayer feedforward neural network the state of the neurons ineach layer not only affects the state of the neurons in the nextlayer but also affects the state of the neurons in the cross-layer

Based on the novel neural network presented in Section 3we set the 119895 119894 and 119897 that represent the input layer hiddenlayer and output layer respectively The number of the inputlayer is 1 which is same with that of the output layer andthe number of the hidden layer is 119878 119909

119894represents the input

vector of the network then the set of the input samples is1198741

119895= [1199091 1199092 1199093 119909

119873] The number of the input vectors

is dependent on the complexity of the system Notably thesuperscript 1 represents the first layer in the whole neuralnetwork

The input and output vectors of the first hidden layer canbe expressed as

net1198941119894(119896) =

119873

sum

119895=0

1199081198941

1198941198951198741

119895

1198741198941

119894(119896) = ℎ (net1198941

119894(119896))

119894 = 1 2 3 119876

(18)

where the superscript 1198941denotes the first hidden layer of the

network and ℎ(119909) is the memristive bipolar sigmoid functionproposed in Section 3

By that analogy the input and output vectors of the 119878thhidden layer can be written as

net119894119904119894(119896) =

119873

sum

119895=0

119908119894119904

1198941198951198741

119895+

119904minus1

sum

119886=1

119904

sum

119887=1

119908119894119904

119894119886119894119904net119894119904119887(119896)

119874119894119904

119894(119896) = ℎ (net119894119904

119894(119896))

119894 = 1 2 3 119876

(19)

Finally the input and output vectors of the output layercan be obtained as

net119904+2119897(119896) =

119873

sum

119895=0

119908119904+2

1198941198971198741

119895+

119904

sum

119886=1

119904

sum

119887=1

119908119904+2

119894119886119897net119904+2119887(119896)

119874119904+2

119897(119896) = 119910 (net119904+2

119897(119896))

119897 = 1 2 3

(20)

where 119910(119909) denotes the memristive unipolar sigmoid func-tion The three nodes of the output layer are correspondingwith the nonnegative adjustable parameters 119896

119901 119896119894 and 119896

119889of

the PID controller respectivelyFrom [27] we conclude the weight update algorithm of

the memristive multilayer feedforward small-world neuralnetwork as below

Δ119908119904+2

119897119894(119896) = 120572Δ119908

119904+2

119897119894(119896 minus 1) + 120578120575

119904+2

119897119874119904+1

119894(119896)

119908119904+2

119897119894(119896 + 1) = 119908

119904+2

119897119894(119896) + Δ119908

119904+2

119897119894(119896)

(21)

where 120572 is the inertial coefficient whose scope ranges from 0to 1 and 120578 isin (0 1) is the learning rate

5 Computer Simulations and Results

In this section somenumerical simulations of thememristivemultilayer feedforward small-world neural network PIDcontroller have been executed on MATLAB software Themathematical model of the controlled plant is given as

119910out (119896) =119886 (119896) 119910out (119896 minus 1)

1 + 1199102out (119896 minus 1)+ 119906 (119896 minus 1) (22)

where the 119886(119896) is slow time-variant and its expression is119886(119896) = 12(1 minus 08119890

01119896)

The memristive neural network under investigation isconstituted by seven layers with four neurons in the inputlayer three neurons in the output layer and five in each of thefive hidden layers The learning rate of the network 120578 = 04and the inertial coefficient 120572 = 005 The initial weighs asrandom values fall in [minus05 05] and the value of the rewiringprobability is chosen as 119875 = 0 119875 = 008 119875 = 01 and 119875 = 02respectively The parameters are 119877on = 100Ω 119877off = 20 kΩ1198720= 10 kΩ 119863 = 10 nm and 120583] asymp 10

minus14m2sminus1Vminus1 1198701and

1198703are user-specified parameters whose value both are 20000

and the action time is 119905119904 = 0001 s When the system workssteadily the tracking results can be gotten as follows

Figure 12(a) shows the input signal (step response curve119903in(119896) = 10) and the output curves under a different rewiringprobability 119875 As can be seen from the figure when thetime 119905 = 05 s the whole system reaches the steady stateMaking a further analysis we can conclude that when therewiring probability 119875 = 0 the memristive neural networkkeeps regularly in architecture Its respond speed is slowerthan that of network when the rewiring probability 119875 =

008 and 119875 = 01 Moreover Figure 12(b) exhibits theerror curves between the input signal and the output signal

10 The Scientific World Journal

rin Error

aeat

ANN

PID Plant yout

kp ki kd

Figure 11 The structure diagram of the memristive neural network PID controller

0

05

1

15

20 05 1 15

Time (s)

Inpu

t ou

tput

Regularnetwork

InputP = 008

P = 01

P = 0

P = 02

(a)

0

08

06

04

02

1

20 05 1 15

Time (s)

Erro

rRegular network

P = 008

P = 01

P = 0

P = 02

(b)

001

002

003

0

005

01

0 2 4 6Time (s)

0 2 4 6Time (s)

0 2 4 60

2

4

Time (s)

P = 008

P = 008

P = 008

times10minus3

k pk i

k d

(c)

Figure 12The simulation results of the memristive neural network PID controller (119903in(119896) = 10) under a different rewiring probability 119875 (a)The step response curve (b) The error curves (c) The curves of the control parameters when the rewiring probability 119875 = 008

correspondingly When the rewiring probability 119875 = 008

and 119875 = 01 the network spends less time on approachingthe predefined approximation error than the regular network(when 119875 = 0) Figure 12(c) shows the output variables ofthe memristive multilayer feedforward small-world neural

network when 119875 = 008 which are the control parameters 119896119901

119896119894 and 119896

119889 correspondingly

In order to verify the superior performance of thememristive small-world neuronal networks andfigure out theoptimal structure we conducted a series of simulations to

The Scientific World Journal 11

4500

5000

5500

6000

6500

7000

0 02 04 06 08 1

Aver

age i

nter

actio

ns

Sampling points

P

(a)

35

37

39

41

43

45

47

0 02 04 06 08 1

Succ

essfu

l int

erac

tions

Sampling points

P

(b)

Figure 13 The convergence performance of the memristive neural network under different 119875 (a) The relationship between iteration andrewiring probability (b) The effective approximation number in 50 times simulations under varying rewiring probability

observe the convergence performance of the proposed net-work under different119875 Figure 13(a) shows the approximationspeed (iteration times) of different network structures thatis the smallest iteration number for reaching the predefinedapproximation error 120576 = 00001 Each drawn point is theaverage value of 50 times runs It can be observed thatthe small-world networks (0 lt 119875 lt 1) need much lessiteration times than the regular neural network (when 119875 =

0) which demonstrates its advantage in processing speedFurthermore when 119875 = 008 the network has the fastapproximation speed

Notably the mathematical function of this system has thelocal minimum for getting out of the local minimum wedefine the maximum allowable iteration times to be 10000as previously mentioned for each 119875 and we performed thesimulation for 50 times where the effective approximationtimes that is error lt 00001 within 10000 iterations arepresented in Figure 13(b) It can be found that the small-worldnetworks have higher accuracy rate than the regular network

6 Conclusions

A mathematical closed-form charge-governed memristormodel is recalled firstly and the corresponding Simulinkmodel is presented Using the change rule of memconduc-tance a memristive realization scheme for synaptic weightis proposed Moreover the activation functions in electricneurons are also implemented based on the single-inputand double-output package of the memristor Combiningthe proposed memristive synapse and activation functions amemristor-basedMFSNN is addressed It exhibits advantagesin computation speed and accuracy over the traditionalmultilayer neural networks by considering the small-worldeffect Meanwhile it has potential of hardware realizationof the neural network because of the nanoscale size of thememristive synapse These superior properties can furtherimprove the application of the neural networks such as inthe intelligent controller design Motivated by this we apply

the memristor-based MFSNN to classical PID control andthe proposed memristive PID controller may possess thefollowing superiorities (i) Its nanoscale physical implemen-tation could promote the development of themicrocontroller(ii) Because of the participation of the memristive neuralnetwork the proposed PID controller can realize the param-eters self-adjustment (iii) The control speed and accuracyare improved Eventually extensive numerical simulationsjustify the effectiveness and efficiency of the memristive PIDcontroller over the regular neural network PID controllerThis work may provide a theoretical reference to physicallyrealize the small-world neural networks and further promotethe development of modern intelligent control technology

Conflict of Interests

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

Acknowledgments

The work was supported by Program for New CenturyExcellent Talents in University (Grant nos [2013] 47)National Natural Science Foundation of China (Grant nos61372139 61101233 and 60972155) ldquoSpring Sunshine PlanrdquoResearch Project of Ministry of Education of China (Grantno z2011148) Technology Foundation for Selected OverseasChinese Scholars Ministry of Personnel in China (Grantno 2012-186) University Excellent Talents Supporting Foun-dations in of Chongqing (Grant no 2011-65) UniversityKey Teacher Supporting Foundations of Chongqing (Grantno 2011-65) Fundamental Research Funds for the CentralUniversities (Grant nos XDJK2014A009 XDJK2013B011)

References

[1] L O Chua ldquoMemristors the missing circuit elementrdquo IEEETransactions on Circuits and Systems vol 18 pp 507ndash519 1971

12 The Scientific World Journal

[2] J M Tour and H Tao ldquoElectronics the fourth elementrdquoNaturevol 453 no 7191 pp 42ndash43 2008

[3] D B Strukov G S Snider D R Stewart and R S WilliamsldquoThe missing memristor foundrdquo Nature vol 453 pp 80ndash832008

[4] R S Williams ldquoHow we found the missing Memristorrdquo IEEESpectrum vol 45 no 12 pp 28ndash35 2008

[5] S K Duan X F Hu L D Wang C D Li and P MazumderldquoMemristor-based RRAM with applicationsrdquo Science ChinaInformation Sciences vol 55 no 6 pp 1446ndash1460 2012

[6] X F Hu S K Duan L DWang and C D Li ldquoAnalog memorybased on pulse controlled memristorrdquo Journal of the Universityof Electronic Science and Technology of China vol 40 no 5 pp642ndash647 2011

[7] S Shin K Kim and S Kang ldquoResistive computingmemristors-enabled signal multiplicationrdquo IEEE Transactions on Circuitsand Systems I Regular Papers vol 60 no 5 pp 1241ndash1249 2013

[8] S P Adhikari C Yang H Kim and L O Chua ldquoMemristorbridge synapse-based neural network and its learningrdquo IEEETransactions on Neural Networks and Learning Systems vol 23no 9 pp 1426ndash1435 2012

[9] S H Jo T Chang I Ebong B B Bhadviya P Mazumder andW Lu ldquoNanoscale memristor device as synapse in neuromor-phic systemsrdquo Nano Letters vol 10 no 4 pp 1297ndash1301 2010

[10] Y Dai C Li and H Wang ldquoExpanded HP memristor modeland simulation in STDP learningrdquo Neural Computing andApplications vol 24 no 1 pp 51ndash57 2014

[11] S K Duan X F Hu L DWang S Y Gao andCD Li ldquoHybridmemristorRTD structure-based cellular neural networks withapplications in image processingrdquoNeural Computing and Appli-cations vol 25 no 2 pp 291ndash296 2014

[12] L Chen C D Li T W Huang and Y R Chen ldquoMemristorcrossbar-based unsupervised image learningrdquo Neural Comput-ing and Applications vol 25 no 2 pp 393ndash400 2013

[13] X F Hu S K Duan L D Wang and X F Liao ldquoMemristivecrossbar array with applications in image processingrdquo ScienceChina Information Sciences vol 41 pp 500ndash512 2011

[14] B Mouttet ldquoProposal for memristors in signal processingrdquo inNano-Net 3rd International ICST Conference NanoNet BostonMS USA September 14ndash16 2008 Revised Selected Papersvol 3 of Lecture Notes of the Institute for Computer SciencesSocial Informatics and Telecommunications Engineering pp 11ndash13 2009

[15] H Abdalla andM D Pickett ldquoSPICEmodeling of memristorsrdquoin Proceedings of the 2011 IEEE International Symposium ofCircuits and Systems (ISCAS rsquo11) pp 1832ndash1835 May 2011

[16] X B Wang Y R Chen H W Xi and D Dimitrov ldquoSpin-tronic memristor through spin-thorque-induced magnetiza-tion motionrdquo IEEE Electron Device Letters vol 30 no 3 pp294ndash297 2009

[17] D B Strukov and R S Williams ldquoExponential ionic drift fastswitching and low volatility of thin-film memristorsrdquo AppliedPhysics A Materials Science amp Processing vol 94 pp 515ndash5192009

[18] F Corinto A Ascoli and M Gilli ldquoMathematical models andcircuit implementations of memristive systemsrdquo in Proceedingsof the 13th International Workshop on Cellular Nanoscale Net-works and their Applications (CNNA rsquo12) pp 1ndash6 August 2012

[19] S Kvatinsky E G Friedman A Kolodny and U C WeiserldquoTEAMThrEshold adaptive memristor modelrdquo IEEE Transac-tions on Circuits and Systems I Regular Papers vol 60 no 1 pp211ndash221 2013

[20] O Sporns and C J Honey ldquoSmall worlds inside big brainsrdquoProceedings of the National Academy of Sciences of the UnitedStates of America vol 103 no 51 pp 19219ndash19220 2006

[21] P Tangkraingkij C Lursinsap S Sanguansintukul and TDesudchit ldquoInsider and outsider person authentication withminimum number of brain wave signals by neural and homo-geneous identity filteringrdquo Neural Computing and Applicationsvol 22 no 1 pp 463ndash476 2013

[22] D S Bassett and E D Bullmore ldquoSmall-world brain networksrdquoNeuroscientist vol 12 no 6 pp 512ndash523 2006

[23] D J Watts and S H Strogatz ldquoCollective dynamics of ldquosmall-worldrdquo networksrdquoNature vol 393 no 6684 pp 440ndash442 1998

[24] L Douw M M Schoonheim D Landi et al ldquoCognitionis related to resting-state small-world network topology anmagnetoencephalographic studyrdquo Neuroscience vol 175 pp169ndash177 2011

[25] T Fekete F D C C Beacher J Cha D Rubin and L RMujica-Parodi ldquoSmall-world network properties in prefrontal cortexcorrelate with predictors of psychopathology risk in youngchildren aNIRS studyrdquoNeuroImage vol 85 part 1 pp 345ndash3532014

[26] N R Taylor ldquoSmall world network strategies for studyingprotein structures and bindingrdquo Computational and StructuralBiotechnology Journal vol 5 Article ID e201302006 2013

[27] D Simard L Nadeau andH Kroger ldquoFastest learning in small-world neural networksrdquo Physics Letters A General Atomic andSolid State Physics vol 336 no 1 pp 8ndash15 2005

[28] Z Biolek D Biolek and V Biolkova ldquoSPICEmodel of memris-tor with nonlinear dopant driftrdquo Radioengineering vol 18 no2 pp 210ndash214 2009

[29] H Kim M P Sah C Yang T Roska and L O ChualdquoMemristor bridge synapsesrdquo Proceedings of the IEEE vol 100no 6 pp 2061ndash2070 2012

[30] M Esfandyari M A Fanaei and H Zohreie ldquoAdaptive fuzzytuning of PID controllersrdquo Neural Computing and Applicationsvol 23 no 1 pp 19ndash28 2013

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

10 The Scientific World Journal

rin Error

aeat

ANN

PID Plant yout

kp ki kd

Figure 11 The structure diagram of the memristive neural network PID controller

0

05

1

15

20 05 1 15

Time (s)

Inpu

t ou

tput

Regularnetwork

InputP = 008

P = 01

P = 0

P = 02

(a)

0

08

06

04

02

1

20 05 1 15

Time (s)

Erro

rRegular network

P = 008

P = 01

P = 0

P = 02

(b)

001

002

003

0

005

01

0 2 4 6Time (s)

0 2 4 6Time (s)

0 2 4 60

2

4

Time (s)

P = 008

P = 008

P = 008

times10minus3

k pk i

k d

(c)

Figure 12The simulation results of the memristive neural network PID controller (119903in(119896) = 10) under a different rewiring probability 119875 (a)The step response curve (b) The error curves (c) The curves of the control parameters when the rewiring probability 119875 = 008

correspondingly When the rewiring probability 119875 = 008

and 119875 = 01 the network spends less time on approachingthe predefined approximation error than the regular network(when 119875 = 0) Figure 12(c) shows the output variables ofthe memristive multilayer feedforward small-world neural

network when 119875 = 008 which are the control parameters 119896119901

119896119894 and 119896

119889 correspondingly

In order to verify the superior performance of thememristive small-world neuronal networks andfigure out theoptimal structure we conducted a series of simulations to

The Scientific World Journal 11

4500

5000

5500

6000

6500

7000

0 02 04 06 08 1

Aver

age i

nter

actio

ns

Sampling points

P

(a)

35

37

39

41

43

45

47

0 02 04 06 08 1

Succ

essfu

l int

erac

tions

Sampling points

P

(b)

Figure 13 The convergence performance of the memristive neural network under different 119875 (a) The relationship between iteration andrewiring probability (b) The effective approximation number in 50 times simulations under varying rewiring probability

observe the convergence performance of the proposed net-work under different119875 Figure 13(a) shows the approximationspeed (iteration times) of different network structures thatis the smallest iteration number for reaching the predefinedapproximation error 120576 = 00001 Each drawn point is theaverage value of 50 times runs It can be observed thatthe small-world networks (0 lt 119875 lt 1) need much lessiteration times than the regular neural network (when 119875 =

0) which demonstrates its advantage in processing speedFurthermore when 119875 = 008 the network has the fastapproximation speed

Notably the mathematical function of this system has thelocal minimum for getting out of the local minimum wedefine the maximum allowable iteration times to be 10000as previously mentioned for each 119875 and we performed thesimulation for 50 times where the effective approximationtimes that is error lt 00001 within 10000 iterations arepresented in Figure 13(b) It can be found that the small-worldnetworks have higher accuracy rate than the regular network

6 Conclusions

A mathematical closed-form charge-governed memristormodel is recalled firstly and the corresponding Simulinkmodel is presented Using the change rule of memconduc-tance a memristive realization scheme for synaptic weightis proposed Moreover the activation functions in electricneurons are also implemented based on the single-inputand double-output package of the memristor Combiningthe proposed memristive synapse and activation functions amemristor-basedMFSNN is addressed It exhibits advantagesin computation speed and accuracy over the traditionalmultilayer neural networks by considering the small-worldeffect Meanwhile it has potential of hardware realizationof the neural network because of the nanoscale size of thememristive synapse These superior properties can furtherimprove the application of the neural networks such as inthe intelligent controller design Motivated by this we apply

the memristor-based MFSNN to classical PID control andthe proposed memristive PID controller may possess thefollowing superiorities (i) Its nanoscale physical implemen-tation could promote the development of themicrocontroller(ii) Because of the participation of the memristive neuralnetwork the proposed PID controller can realize the param-eters self-adjustment (iii) The control speed and accuracyare improved Eventually extensive numerical simulationsjustify the effectiveness and efficiency of the memristive PIDcontroller over the regular neural network PID controllerThis work may provide a theoretical reference to physicallyrealize the small-world neural networks and further promotethe development of modern intelligent control technology

Conflict of Interests

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

Acknowledgments

The work was supported by Program for New CenturyExcellent Talents in University (Grant nos [2013] 47)National Natural Science Foundation of China (Grant nos61372139 61101233 and 60972155) ldquoSpring Sunshine PlanrdquoResearch Project of Ministry of Education of China (Grantno z2011148) Technology Foundation for Selected OverseasChinese Scholars Ministry of Personnel in China (Grantno 2012-186) University Excellent Talents Supporting Foun-dations in of Chongqing (Grant no 2011-65) UniversityKey Teacher Supporting Foundations of Chongqing (Grantno 2011-65) Fundamental Research Funds for the CentralUniversities (Grant nos XDJK2014A009 XDJK2013B011)

References

[1] L O Chua ldquoMemristors the missing circuit elementrdquo IEEETransactions on Circuits and Systems vol 18 pp 507ndash519 1971

12 The Scientific World Journal

[2] J M Tour and H Tao ldquoElectronics the fourth elementrdquoNaturevol 453 no 7191 pp 42ndash43 2008

[3] D B Strukov G S Snider D R Stewart and R S WilliamsldquoThe missing memristor foundrdquo Nature vol 453 pp 80ndash832008

[4] R S Williams ldquoHow we found the missing Memristorrdquo IEEESpectrum vol 45 no 12 pp 28ndash35 2008

[5] S K Duan X F Hu L D Wang C D Li and P MazumderldquoMemristor-based RRAM with applicationsrdquo Science ChinaInformation Sciences vol 55 no 6 pp 1446ndash1460 2012

[6] X F Hu S K Duan L DWang and C D Li ldquoAnalog memorybased on pulse controlled memristorrdquo Journal of the Universityof Electronic Science and Technology of China vol 40 no 5 pp642ndash647 2011

[7] S Shin K Kim and S Kang ldquoResistive computingmemristors-enabled signal multiplicationrdquo IEEE Transactions on Circuitsand Systems I Regular Papers vol 60 no 5 pp 1241ndash1249 2013

[8] S P Adhikari C Yang H Kim and L O Chua ldquoMemristorbridge synapse-based neural network and its learningrdquo IEEETransactions on Neural Networks and Learning Systems vol 23no 9 pp 1426ndash1435 2012

[9] S H Jo T Chang I Ebong B B Bhadviya P Mazumder andW Lu ldquoNanoscale memristor device as synapse in neuromor-phic systemsrdquo Nano Letters vol 10 no 4 pp 1297ndash1301 2010

[10] Y Dai C Li and H Wang ldquoExpanded HP memristor modeland simulation in STDP learningrdquo Neural Computing andApplications vol 24 no 1 pp 51ndash57 2014

[11] S K Duan X F Hu L DWang S Y Gao andCD Li ldquoHybridmemristorRTD structure-based cellular neural networks withapplications in image processingrdquoNeural Computing and Appli-cations vol 25 no 2 pp 291ndash296 2014

[12] L Chen C D Li T W Huang and Y R Chen ldquoMemristorcrossbar-based unsupervised image learningrdquo Neural Comput-ing and Applications vol 25 no 2 pp 393ndash400 2013

[13] X F Hu S K Duan L D Wang and X F Liao ldquoMemristivecrossbar array with applications in image processingrdquo ScienceChina Information Sciences vol 41 pp 500ndash512 2011

[14] B Mouttet ldquoProposal for memristors in signal processingrdquo inNano-Net 3rd International ICST Conference NanoNet BostonMS USA September 14ndash16 2008 Revised Selected Papersvol 3 of Lecture Notes of the Institute for Computer SciencesSocial Informatics and Telecommunications Engineering pp 11ndash13 2009

[15] H Abdalla andM D Pickett ldquoSPICEmodeling of memristorsrdquoin Proceedings of the 2011 IEEE International Symposium ofCircuits and Systems (ISCAS rsquo11) pp 1832ndash1835 May 2011

[16] X B Wang Y R Chen H W Xi and D Dimitrov ldquoSpin-tronic memristor through spin-thorque-induced magnetiza-tion motionrdquo IEEE Electron Device Letters vol 30 no 3 pp294ndash297 2009

[17] D B Strukov and R S Williams ldquoExponential ionic drift fastswitching and low volatility of thin-film memristorsrdquo AppliedPhysics A Materials Science amp Processing vol 94 pp 515ndash5192009

[18] F Corinto A Ascoli and M Gilli ldquoMathematical models andcircuit implementations of memristive systemsrdquo in Proceedingsof the 13th International Workshop on Cellular Nanoscale Net-works and their Applications (CNNA rsquo12) pp 1ndash6 August 2012

[19] S Kvatinsky E G Friedman A Kolodny and U C WeiserldquoTEAMThrEshold adaptive memristor modelrdquo IEEE Transac-tions on Circuits and Systems I Regular Papers vol 60 no 1 pp211ndash221 2013

[20] O Sporns and C J Honey ldquoSmall worlds inside big brainsrdquoProceedings of the National Academy of Sciences of the UnitedStates of America vol 103 no 51 pp 19219ndash19220 2006

[21] P Tangkraingkij C Lursinsap S Sanguansintukul and TDesudchit ldquoInsider and outsider person authentication withminimum number of brain wave signals by neural and homo-geneous identity filteringrdquo Neural Computing and Applicationsvol 22 no 1 pp 463ndash476 2013

[22] D S Bassett and E D Bullmore ldquoSmall-world brain networksrdquoNeuroscientist vol 12 no 6 pp 512ndash523 2006

[23] D J Watts and S H Strogatz ldquoCollective dynamics of ldquosmall-worldrdquo networksrdquoNature vol 393 no 6684 pp 440ndash442 1998

[24] L Douw M M Schoonheim D Landi et al ldquoCognitionis related to resting-state small-world network topology anmagnetoencephalographic studyrdquo Neuroscience vol 175 pp169ndash177 2011

[25] T Fekete F D C C Beacher J Cha D Rubin and L RMujica-Parodi ldquoSmall-world network properties in prefrontal cortexcorrelate with predictors of psychopathology risk in youngchildren aNIRS studyrdquoNeuroImage vol 85 part 1 pp 345ndash3532014

[26] N R Taylor ldquoSmall world network strategies for studyingprotein structures and bindingrdquo Computational and StructuralBiotechnology Journal vol 5 Article ID e201302006 2013

[27] D Simard L Nadeau andH Kroger ldquoFastest learning in small-world neural networksrdquo Physics Letters A General Atomic andSolid State Physics vol 336 no 1 pp 8ndash15 2005

[28] Z Biolek D Biolek and V Biolkova ldquoSPICEmodel of memris-tor with nonlinear dopant driftrdquo Radioengineering vol 18 no2 pp 210ndash214 2009

[29] H Kim M P Sah C Yang T Roska and L O ChualdquoMemristor bridge synapsesrdquo Proceedings of the IEEE vol 100no 6 pp 2061ndash2070 2012

[30] M Esfandyari M A Fanaei and H Zohreie ldquoAdaptive fuzzytuning of PID controllersrdquo Neural Computing and Applicationsvol 23 no 1 pp 19ndash28 2013

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

The Scientific World Journal 11

4500

5000

5500

6000

6500

7000

0 02 04 06 08 1

Aver

age i

nter

actio

ns

Sampling points

P

(a)

35

37

39

41

43

45

47

0 02 04 06 08 1

Succ

essfu

l int

erac

tions

Sampling points

P

(b)

Figure 13 The convergence performance of the memristive neural network under different 119875 (a) The relationship between iteration andrewiring probability (b) The effective approximation number in 50 times simulations under varying rewiring probability

observe the convergence performance of the proposed net-work under different119875 Figure 13(a) shows the approximationspeed (iteration times) of different network structures thatis the smallest iteration number for reaching the predefinedapproximation error 120576 = 00001 Each drawn point is theaverage value of 50 times runs It can be observed thatthe small-world networks (0 lt 119875 lt 1) need much lessiteration times than the regular neural network (when 119875 =

0) which demonstrates its advantage in processing speedFurthermore when 119875 = 008 the network has the fastapproximation speed

Notably the mathematical function of this system has thelocal minimum for getting out of the local minimum wedefine the maximum allowable iteration times to be 10000as previously mentioned for each 119875 and we performed thesimulation for 50 times where the effective approximationtimes that is error lt 00001 within 10000 iterations arepresented in Figure 13(b) It can be found that the small-worldnetworks have higher accuracy rate than the regular network

6 Conclusions

A mathematical closed-form charge-governed memristormodel is recalled firstly and the corresponding Simulinkmodel is presented Using the change rule of memconduc-tance a memristive realization scheme for synaptic weightis proposed Moreover the activation functions in electricneurons are also implemented based on the single-inputand double-output package of the memristor Combiningthe proposed memristive synapse and activation functions amemristor-basedMFSNN is addressed It exhibits advantagesin computation speed and accuracy over the traditionalmultilayer neural networks by considering the small-worldeffect Meanwhile it has potential of hardware realizationof the neural network because of the nanoscale size of thememristive synapse These superior properties can furtherimprove the application of the neural networks such as inthe intelligent controller design Motivated by this we apply

the memristor-based MFSNN to classical PID control andthe proposed memristive PID controller may possess thefollowing superiorities (i) Its nanoscale physical implemen-tation could promote the development of themicrocontroller(ii) Because of the participation of the memristive neuralnetwork the proposed PID controller can realize the param-eters self-adjustment (iii) The control speed and accuracyare improved Eventually extensive numerical simulationsjustify the effectiveness and efficiency of the memristive PIDcontroller over the regular neural network PID controllerThis work may provide a theoretical reference to physicallyrealize the small-world neural networks and further promotethe development of modern intelligent control technology

Conflict of Interests

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

Acknowledgments

The work was supported by Program for New CenturyExcellent Talents in University (Grant nos [2013] 47)National Natural Science Foundation of China (Grant nos61372139 61101233 and 60972155) ldquoSpring Sunshine PlanrdquoResearch Project of Ministry of Education of China (Grantno z2011148) Technology Foundation for Selected OverseasChinese Scholars Ministry of Personnel in China (Grantno 2012-186) University Excellent Talents Supporting Foun-dations in of Chongqing (Grant no 2011-65) UniversityKey Teacher Supporting Foundations of Chongqing (Grantno 2011-65) Fundamental Research Funds for the CentralUniversities (Grant nos XDJK2014A009 XDJK2013B011)

References

[1] L O Chua ldquoMemristors the missing circuit elementrdquo IEEETransactions on Circuits and Systems vol 18 pp 507ndash519 1971

12 The Scientific World Journal

[2] J M Tour and H Tao ldquoElectronics the fourth elementrdquoNaturevol 453 no 7191 pp 42ndash43 2008

[3] D B Strukov G S Snider D R Stewart and R S WilliamsldquoThe missing memristor foundrdquo Nature vol 453 pp 80ndash832008

[4] R S Williams ldquoHow we found the missing Memristorrdquo IEEESpectrum vol 45 no 12 pp 28ndash35 2008

[5] S K Duan X F Hu L D Wang C D Li and P MazumderldquoMemristor-based RRAM with applicationsrdquo Science ChinaInformation Sciences vol 55 no 6 pp 1446ndash1460 2012

[6] X F Hu S K Duan L DWang and C D Li ldquoAnalog memorybased on pulse controlled memristorrdquo Journal of the Universityof Electronic Science and Technology of China vol 40 no 5 pp642ndash647 2011

[7] S Shin K Kim and S Kang ldquoResistive computingmemristors-enabled signal multiplicationrdquo IEEE Transactions on Circuitsand Systems I Regular Papers vol 60 no 5 pp 1241ndash1249 2013

[8] S P Adhikari C Yang H Kim and L O Chua ldquoMemristorbridge synapse-based neural network and its learningrdquo IEEETransactions on Neural Networks and Learning Systems vol 23no 9 pp 1426ndash1435 2012

[9] S H Jo T Chang I Ebong B B Bhadviya P Mazumder andW Lu ldquoNanoscale memristor device as synapse in neuromor-phic systemsrdquo Nano Letters vol 10 no 4 pp 1297ndash1301 2010

[10] Y Dai C Li and H Wang ldquoExpanded HP memristor modeland simulation in STDP learningrdquo Neural Computing andApplications vol 24 no 1 pp 51ndash57 2014

[11] S K Duan X F Hu L DWang S Y Gao andCD Li ldquoHybridmemristorRTD structure-based cellular neural networks withapplications in image processingrdquoNeural Computing and Appli-cations vol 25 no 2 pp 291ndash296 2014

[12] L Chen C D Li T W Huang and Y R Chen ldquoMemristorcrossbar-based unsupervised image learningrdquo Neural Comput-ing and Applications vol 25 no 2 pp 393ndash400 2013

[13] X F Hu S K Duan L D Wang and X F Liao ldquoMemristivecrossbar array with applications in image processingrdquo ScienceChina Information Sciences vol 41 pp 500ndash512 2011

[14] B Mouttet ldquoProposal for memristors in signal processingrdquo inNano-Net 3rd International ICST Conference NanoNet BostonMS USA September 14ndash16 2008 Revised Selected Papersvol 3 of Lecture Notes of the Institute for Computer SciencesSocial Informatics and Telecommunications Engineering pp 11ndash13 2009

[15] H Abdalla andM D Pickett ldquoSPICEmodeling of memristorsrdquoin Proceedings of the 2011 IEEE International Symposium ofCircuits and Systems (ISCAS rsquo11) pp 1832ndash1835 May 2011

[16] X B Wang Y R Chen H W Xi and D Dimitrov ldquoSpin-tronic memristor through spin-thorque-induced magnetiza-tion motionrdquo IEEE Electron Device Letters vol 30 no 3 pp294ndash297 2009

[17] D B Strukov and R S Williams ldquoExponential ionic drift fastswitching and low volatility of thin-film memristorsrdquo AppliedPhysics A Materials Science amp Processing vol 94 pp 515ndash5192009

[18] F Corinto A Ascoli and M Gilli ldquoMathematical models andcircuit implementations of memristive systemsrdquo in Proceedingsof the 13th International Workshop on Cellular Nanoscale Net-works and their Applications (CNNA rsquo12) pp 1ndash6 August 2012

[19] S Kvatinsky E G Friedman A Kolodny and U C WeiserldquoTEAMThrEshold adaptive memristor modelrdquo IEEE Transac-tions on Circuits and Systems I Regular Papers vol 60 no 1 pp211ndash221 2013

[20] O Sporns and C J Honey ldquoSmall worlds inside big brainsrdquoProceedings of the National Academy of Sciences of the UnitedStates of America vol 103 no 51 pp 19219ndash19220 2006

[21] P Tangkraingkij C Lursinsap S Sanguansintukul and TDesudchit ldquoInsider and outsider person authentication withminimum number of brain wave signals by neural and homo-geneous identity filteringrdquo Neural Computing and Applicationsvol 22 no 1 pp 463ndash476 2013

[22] D S Bassett and E D Bullmore ldquoSmall-world brain networksrdquoNeuroscientist vol 12 no 6 pp 512ndash523 2006

[23] D J Watts and S H Strogatz ldquoCollective dynamics of ldquosmall-worldrdquo networksrdquoNature vol 393 no 6684 pp 440ndash442 1998

[24] L Douw M M Schoonheim D Landi et al ldquoCognitionis related to resting-state small-world network topology anmagnetoencephalographic studyrdquo Neuroscience vol 175 pp169ndash177 2011

[25] T Fekete F D C C Beacher J Cha D Rubin and L RMujica-Parodi ldquoSmall-world network properties in prefrontal cortexcorrelate with predictors of psychopathology risk in youngchildren aNIRS studyrdquoNeuroImage vol 85 part 1 pp 345ndash3532014

[26] N R Taylor ldquoSmall world network strategies for studyingprotein structures and bindingrdquo Computational and StructuralBiotechnology Journal vol 5 Article ID e201302006 2013

[27] D Simard L Nadeau andH Kroger ldquoFastest learning in small-world neural networksrdquo Physics Letters A General Atomic andSolid State Physics vol 336 no 1 pp 8ndash15 2005

[28] Z Biolek D Biolek and V Biolkova ldquoSPICEmodel of memris-tor with nonlinear dopant driftrdquo Radioengineering vol 18 no2 pp 210ndash214 2009

[29] H Kim M P Sah C Yang T Roska and L O ChualdquoMemristor bridge synapsesrdquo Proceedings of the IEEE vol 100no 6 pp 2061ndash2070 2012

[30] M Esfandyari M A Fanaei and H Zohreie ldquoAdaptive fuzzytuning of PID controllersrdquo Neural Computing and Applicationsvol 23 no 1 pp 19ndash28 2013

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

12 The Scientific World Journal

[2] J M Tour and H Tao ldquoElectronics the fourth elementrdquoNaturevol 453 no 7191 pp 42ndash43 2008

[3] D B Strukov G S Snider D R Stewart and R S WilliamsldquoThe missing memristor foundrdquo Nature vol 453 pp 80ndash832008

[4] R S Williams ldquoHow we found the missing Memristorrdquo IEEESpectrum vol 45 no 12 pp 28ndash35 2008

[5] S K Duan X F Hu L D Wang C D Li and P MazumderldquoMemristor-based RRAM with applicationsrdquo Science ChinaInformation Sciences vol 55 no 6 pp 1446ndash1460 2012

[6] X F Hu S K Duan L DWang and C D Li ldquoAnalog memorybased on pulse controlled memristorrdquo Journal of the Universityof Electronic Science and Technology of China vol 40 no 5 pp642ndash647 2011

[7] S Shin K Kim and S Kang ldquoResistive computingmemristors-enabled signal multiplicationrdquo IEEE Transactions on Circuitsand Systems I Regular Papers vol 60 no 5 pp 1241ndash1249 2013

[8] S P Adhikari C Yang H Kim and L O Chua ldquoMemristorbridge synapse-based neural network and its learningrdquo IEEETransactions on Neural Networks and Learning Systems vol 23no 9 pp 1426ndash1435 2012

[9] S H Jo T Chang I Ebong B B Bhadviya P Mazumder andW Lu ldquoNanoscale memristor device as synapse in neuromor-phic systemsrdquo Nano Letters vol 10 no 4 pp 1297ndash1301 2010

[10] Y Dai C Li and H Wang ldquoExpanded HP memristor modeland simulation in STDP learningrdquo Neural Computing andApplications vol 24 no 1 pp 51ndash57 2014

[11] S K Duan X F Hu L DWang S Y Gao andCD Li ldquoHybridmemristorRTD structure-based cellular neural networks withapplications in image processingrdquoNeural Computing and Appli-cations vol 25 no 2 pp 291ndash296 2014

[12] L Chen C D Li T W Huang and Y R Chen ldquoMemristorcrossbar-based unsupervised image learningrdquo Neural Comput-ing and Applications vol 25 no 2 pp 393ndash400 2013

[13] X F Hu S K Duan L D Wang and X F Liao ldquoMemristivecrossbar array with applications in image processingrdquo ScienceChina Information Sciences vol 41 pp 500ndash512 2011

[14] B Mouttet ldquoProposal for memristors in signal processingrdquo inNano-Net 3rd International ICST Conference NanoNet BostonMS USA September 14ndash16 2008 Revised Selected Papersvol 3 of Lecture Notes of the Institute for Computer SciencesSocial Informatics and Telecommunications Engineering pp 11ndash13 2009

[15] H Abdalla andM D Pickett ldquoSPICEmodeling of memristorsrdquoin Proceedings of the 2011 IEEE International Symposium ofCircuits and Systems (ISCAS rsquo11) pp 1832ndash1835 May 2011

[16] X B Wang Y R Chen H W Xi and D Dimitrov ldquoSpin-tronic memristor through spin-thorque-induced magnetiza-tion motionrdquo IEEE Electron Device Letters vol 30 no 3 pp294ndash297 2009

[17] D B Strukov and R S Williams ldquoExponential ionic drift fastswitching and low volatility of thin-film memristorsrdquo AppliedPhysics A Materials Science amp Processing vol 94 pp 515ndash5192009

[18] F Corinto A Ascoli and M Gilli ldquoMathematical models andcircuit implementations of memristive systemsrdquo in Proceedingsof the 13th International Workshop on Cellular Nanoscale Net-works and their Applications (CNNA rsquo12) pp 1ndash6 August 2012

[19] S Kvatinsky E G Friedman A Kolodny and U C WeiserldquoTEAMThrEshold adaptive memristor modelrdquo IEEE Transac-tions on Circuits and Systems I Regular Papers vol 60 no 1 pp211ndash221 2013

[20] O Sporns and C J Honey ldquoSmall worlds inside big brainsrdquoProceedings of the National Academy of Sciences of the UnitedStates of America vol 103 no 51 pp 19219ndash19220 2006

[21] P Tangkraingkij C Lursinsap S Sanguansintukul and TDesudchit ldquoInsider and outsider person authentication withminimum number of brain wave signals by neural and homo-geneous identity filteringrdquo Neural Computing and Applicationsvol 22 no 1 pp 463ndash476 2013

[22] D S Bassett and E D Bullmore ldquoSmall-world brain networksrdquoNeuroscientist vol 12 no 6 pp 512ndash523 2006

[23] D J Watts and S H Strogatz ldquoCollective dynamics of ldquosmall-worldrdquo networksrdquoNature vol 393 no 6684 pp 440ndash442 1998

[24] L Douw M M Schoonheim D Landi et al ldquoCognitionis related to resting-state small-world network topology anmagnetoencephalographic studyrdquo Neuroscience vol 175 pp169ndash177 2011

[25] T Fekete F D C C Beacher J Cha D Rubin and L RMujica-Parodi ldquoSmall-world network properties in prefrontal cortexcorrelate with predictors of psychopathology risk in youngchildren aNIRS studyrdquoNeuroImage vol 85 part 1 pp 345ndash3532014

[26] N R Taylor ldquoSmall world network strategies for studyingprotein structures and bindingrdquo Computational and StructuralBiotechnology Journal vol 5 Article ID e201302006 2013

[27] D Simard L Nadeau andH Kroger ldquoFastest learning in small-world neural networksrdquo Physics Letters A General Atomic andSolid State Physics vol 336 no 1 pp 8ndash15 2005

[28] Z Biolek D Biolek and V Biolkova ldquoSPICEmodel of memris-tor with nonlinear dopant driftrdquo Radioengineering vol 18 no2 pp 210ndash214 2009

[29] H Kim M P Sah C Yang T Roska and L O ChualdquoMemristor bridge synapsesrdquo Proceedings of the IEEE vol 100no 6 pp 2061ndash2070 2012

[30] M Esfandyari M A Fanaei and H Zohreie ldquoAdaptive fuzzytuning of PID controllersrdquo Neural Computing and Applicationsvol 23 no 1 pp 19ndash28 2013

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of