Hindawi Publishing CorporationJournal of OptimizationVolume 2013 Article ID 142390 11 pageshttpdxdoiorg1011552013142390

Research ArticleOptimization of Integer Order Integrators for DerivingImproved Models of Their Fractional Counterparts

Maneesha Gupta and Richa Yadav

Advanced Electronics Lab Department of ECE Netaji Subhas Institute of Technology Dwarka New Delhi 110078 India

Correspondence should be addressed to Richa Yadav yadavricha1gmailcom

Received 28 February 2013 Revised 21 May 2013 Accepted 30 May 2013

Academic Editor Manuel Lozano

Copyright copy 2013 M Gupta and R Yadav This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Second and third order digital integrators (DIs) have been optimized first using Particle SwarmOptimization (PSO)withminimizederror fitness function obtained by registering mean median and standard deviation values in different random iterations Laterindirect discretization using Continued Fraction Expansion (CFE) has been used to ascertain a better fitting of proposed integerorder optimized DIs into their corresponding fractional counterparts by utilizing their refined properties now restored in themdue to PSO algorithm Simulation results for the comparisons of the frequency responses of proposed 2nd and 3rd order optimizedDIs and proposed discretized mathematical models of half integrators based on them with their respective existing operators havebeen presented Proposed integer order PSO optimized integrators as well as fractional order integrators (FOIs) have been observedto outperform the existing recently published operators in their respective domains reasonably well in complete range of Nyquistfrequency

1 Introduction

In fractional calculus (FC) initially the development ofmathematical models of fractional order integrators (FOIs)and fractional order differentiators (FODs) [1 2] startedpresumably for searching different generalizing approachesfor switching from integer order to the fractional orderdomain The frequency response of ideal fractional orderdiffer-integrator (119904

plusmn120572) is

119867fractional (119895120596) = (119895120596)plusmn120572

(1)

where 119895 = radicminus1 and 120596 gives angular frequency in radiansVariable ldquo120572rdquo defines the order of fractional order operatorsand its value lies in between 0 and 1

Due to its exotic nature FC has strengthened its graspover different research areas such as statistical modelingmechanical system analysis [3] control [4] automated con-trol [5] instrumentation [6] signal processing [7] and radioengineering and image processing [8 9] At present timeFC is also extensively spreading its niche in different design

methods for improving fractional order controllers by frac-tional integral and derivative functions [10 11] An importantdesign method which has undoubtedly helped in sharpeningup the focus of an efficient formulation of new blocks offractional operators is ldquolinear interpolationrdquo method [12ndash16] This most vastly followed tri-step method involves thefollowing steps (1) First step deals with linear interpolationof two existing digital integrators (2) In second step a newfractional order integrator is formulated by discretization ofinteger order integrator developed in first step by anyone ofthe two existing discretization (direct and indirect) schemes(3) In the third step this expanded series is truncatedup to some arbitrary order (119873) which defines the orderof the new fractional operator (119904

plusmn120572) Direct discretization[17] works with the help of different expansion techniquessuch as Taylor Series Expansion (TSE) [18] Power SeriesExpansion (PSE) [19 20] and Continued Fraction Expansion(CFE) [21 22] whereas for indirect discretization [23ndash26]different rational approximations have been developed bymany mathematicians namely Carlson and Halijak [27]Steiglitz [28] Khovanskii [29] Roy [30] Oustaloup et al [31]

2 Journal of Optimization

andMaione [32] which proved like boon for the developmentof fractional integratorsdifferentiators

In spite of the efficient and systematic nature of dif-ferent previously mentioned approximations still thereexisted a dissent about performance of these approximationswhen these were later adapted by well-known discretiza-tion schemes for obtaining FOIs Moreover development ofnew approximations requiredmany exhaustivemathematicalapplications to make them perform in finite dimensionalform in proper range of frequencies during discretizationInstead of deriving new methods of rational approximationsauthors have suggested here an alternate solution in whichthe integrator operator (used as 119904-119911 transform) itself shouldbe pushed near boundaries of an optimal solution (near idealresponse) before converting it to fractional system Reasonbehind this is that every minute dispensation in propertiesof this integer order operator will directly propel into itsnoninteger order counterpart The main novelty proposed bythis paper is to put together unique combination of opti-mized integer order operators and an accurate approximationtechnique for configuring their efficient fractional order (fo)operators The current scenario of design methods for deriv-ing optimal operators [33ndash41] is cumulatively proceedingtowards ldquonot so distantrdquo phase of perfect optimization withnegligible errors So another motivating point behind theapproach proposed in this paper is to trespass all disadvan-tages of existing conventional design methods of FOIs byusing refined capabilities of an efficient optimization scheme

The frequency response of ideal integer order digitalintegrators (DIs) is

119867int (120596) = (1

119895120596) where |120596| le 120587 (2)

and the frequency response of ideal integer order digital dif-ferentiators (DDs) is

119867diff (120596) = 119895120596 where |120596| le 120587 (3)

where 119895 = radicminus1 120596 is the angular frequency in radians andldquo119879rdquo is the sampling period

In recent novel advancements different optimizationalgorithms namely Linear Programming (LP) [33 34]Genetic Algorithm (GA) [35ndash37] Simulated Annealing (SA)[38 39] and pole-zero (PZ) optimization [40 41] havebeen used for obtaining more refined integer order differ-integrators Jain et al [35] have recently derived optimalmodels of recursive DIs and DDs by GA algorithm whileUpadhyay [41] has developed these operators by analyzingpoles and zeros of existing recursive DDs by PZ optimizationtechnique for obtaining better results for frequency responsesover wideband of complete frequency spectrum Al-Alaoui[39] has done a credible work in improving design processof DIs and DDs based on Newton-Cotes integration rules byusing linear interpolation and SA

This paper attempts to find the optimal solutions ofDIs by using an efficient modified application of Parti-cle Swarm Optimization (PSO) technique [42ndash44] beforediscretizing them to their fractional mathematical models

Values of mean median and standard deviation in differentrandom iterations have been continuously registered usingPSO for the optimization of 2nd and 3rd order DIs withminimized error fitness function Later these optimized DIsare discretized by CFE of indirect discretization scheme [23]for finding half integrator models of different orders Thesimulation results of magnitude responses phase responsesand relative magnitude errors for the proposed optimizedDIs have been compared with all the recently publishedintegrators optimized by different optimization algorithmsnamely LP [34] GA [35] SA [39] and PZ [41] Frequencyresponses of the proposed FOIs based on 2nd and 3rd orderoptimized DIs have been also compared with their existingfractional models [2 17] in complete frequency range

The paper is organized as follows Section 2 deals withthe brief description of original and modified PSO algo-rithm for optimizing DIs along with its trade-off con-ditions and the resultant 2nd and 3rd order optimizedintegrators thus derived have been presented in this sec-tion Comparisons of proposed and existing DIs are alsogiven in the same section These optimized operators havebeen discretized by indirect discretization using CFE tech-nique of indirect discretization for deriving models of FOIsin Section 3 Section 4 presents the simulation results ofcomparisons of proposed half integrators with responsesof ideal and the existing models Section 5 concludes thepaper

2 PSO Algorithm for Optimizing 2nd and 3rdOrder Integer Order Integrators

21 Outline of Basic Functionality of Original PSO AlgorithmIn conventional PSO algorithm [42ndash44] first a popula-tion of random solutions has been initialized and differentgenerations are updated for searching optimal values Inthis algorithm the potential solutions called particles flythrough the problem space by following the current optimumparticles The position of each particle encodes a possiblesolution to the given problem The velocity of the particle isthe parameter to be added to the current position to find thenewposition of the next generationThe velocity and positionare updated for all particles in every generation unless anoptimal solution is found The particles keep track of itscoordinates in the problem space which are associated withthe best solution (fitness) it has achieved so far These valuesare called pbest and the best element among all the pbestvalues of all the particles is called gbest The velocities areinitialized to zero and after finding the two best values theparticle updates its velocity and positions in accordance withthe following update equations

V (119894) = V (119894 minus 1) + 11988811199031

(119901pbest minus 119901 (119894)) + 11988821199032

(119901gbest minus 119901 (119894))

119901 (119894 + 1) = 119901 (119894) + V (119894)

(4)

where V(119894) is the current particle velocity V(119894 minus 1) is previousparticle velocity 119901(119894) is current particle position 119901(119894 + 1) isthe position of particle in next generation 119901pbest is the best

Journal of Optimization 3

Define the solution space fitness function and population size

Initialise position velocity

For each particle

Evaluate fitness function

Next particle

orNext iteration

Solution is gbest

Update velocityand position

Yes

No

If fitness (x) better than

pbest and gbest

fitness (gbest)gbest = x

No iteration = max no iteration

fitness (gbest) is good enough

minus |H(ej120596)|)2d120596

If fitness (x) better thanfitness (pbest)

pbest = xEint = int

120587

0(1120596

Figure 1 Flowchart of modified PSO algorithm

position the particle has seen 119901gbests is the best position theswarm has seen 119888

1 1198882are constants set to 2 and 119903

1 1199032are the

random numbers between 0 and 1

22 Parameters Used for Optimization and Trade-Offs betweenMultiple Parameters during Convergence of Optimized DigitalIntegrators Amathematical model of transfer function (TF)of optimized digital integrator (DI) can be described ingeneralized form as

119867 (119911) = (119861119872

(119911)119887119872 + sdot sdot sdot + 119861

1(119911)1198871 + 119861

0(119911)1198870

119860119873

(119911)119886119873 + sdot sdot sdot + 119860

1(119911)1198861 + 119860

0(119911)1198860

) (5)

where119860119894and 119861

119895 are arbitrary constants whereas 119886

119894and 119887119895are

real numbers with (119894 = 0 1 119873) and (119895 = 0 1 119872)

Transfer functions (TFs) of 2nd and 3rd order DIs whichhave been optimized using PSO in this paper are given in (6)and (7) respectively

119867PSO 2nd opt (119911) = (11986121199112 + 119861

1119911 + 1198610

11986021199112 + 119860

1119911 + 119860

0

) (6)

119867PSO 3rd opt (119911) = (1198613(119911)3

+ 1198612(119911)2

+ 1198611119911 + 1198610

1198603(119911)3

+ 1198602(119911)2

+ 1198601119911 + 119860

0

) (7)

The coefficients of the previously mentioned TFs are gener-ated by random iterations of PSO algorithm as it takes realnumbers as particles

4 Journal of Optimization

Table 1 Transfer functions and fitness values of worst and average cases of 2nd order PSO optimized DIs

TFs of worst cases of 2nd order DIs along with their fitnessfunction values (FV)

TFs of average cases of 2nd order DIs along with their fitnessfunction values (FV)

Transfer function FV Transfer function FV

1198671(119911) = (

03733241199112 + 0427523119911 + 0397575

1199112 minus 0042561119911 minus 0999916) 0643052 119867

1(119911) = (

0228961199112 minus 0825829119911 minus 0502494

1199112 minus 0776284119911 minus 0181729) 0382107

1198672(119911) = (

minus09292511199112 + 0381076119911 minus 075834

1199112 minus 0725805119911 minus 0306558) 390016 119867

2(119911) = (

minus03364781199112 minus 082224119911 + 0101197

1199112 minus 0818946119911 minus 0147828) 0407093

1198673(119911) = (

05416991199112 + 0241831119911 + 0506633

1199112 minus 00552516119911 minus 1) 473234 119867

3(119911) = (

008498351199112 + 0752164119911 + 0756218

1199112 minus 0167219119911 minus 0882869) 050102

1198674(119911) = (

minus03545831199112 minus 0833923119911 minus 0813869

1199112 minus 00637125119911 minus 1) 589578 119867

4(119911) = (

minus0397291199112 minus 0795585119911 minus 0130415

1199112 minus 0722962119911 minus 0318656) 0618717

1198675(119911) = (

03538521199112 + 0203374119911 + 0466221

1199112 minus 00406162119911 minus 1) 18799 119867

5(119911) = (

02843821199112 + 0949776119911 + 0363676

1199112 minus 0401952119911 minus 0648204) 0737534

Magnitude response of the optimized DI is

119867mag = |119867 (119908)| =abs (numerator)abs (denominator)

(8)

Phase response of the optimized DI is

119867phase = arg |119867 (119908)| (9)

Relative magnitude error (dB) or RME value is given by

RME = 20 log10

100381610038161003816100381610038161003816100381610038161003816

119867ideal (120596) minus 119867approx (120596)

119867ideal (120596)

100381610038161003816100381610038161003816100381610038161003816

= 20 log10

1003816100381610038161003816100381610038161003816100381610038161003816

(1119895120596) minus 119867approx (120596)

(1119895120596)

1003816100381610038161003816100381610038161003816100381610038161003816

(10)

where119867ideal(120596) is themagnitude of ideal value of the operatorand 119867approx(120596) is the value of the digital integrator that isapproximated

In the recently published literature on digital and frac-tional operators most of the researchers have focused onformulating the fitness function for optimization techniquebased on only single parameter [38ndash41] that is magnitudeerror because practically if phase and magnitude are opti-mized for a multiparameter problem the solution is alwaysstuck in a trade-off due to their contradictory nature Thereason behind this is that when the optimal solution startsconverging towards the best results of one parameter (mag-nitude response) it simultaneously degrades optimality ofsecond parameter (phase responses) after certain number ofiterations Solution for such trade-offs is to choose domi-nating parameter in problem under consideration In thispaper the design strategy to find optimized DIs is kept moreconcerned for minimum magnitude response error whichis comparatively more significant characteristic The fitnessfunction used to approximate a digital integrator is taken asthe mean square error of the transfer function and it is givenas

119864int = int120587

0

(1

120596minus

10038161003816100381610038161003816119867 (119890119895120596

)10038161003816100381610038161003816)2

119889120596 (11)

1 15 2 25Frequency

Mag

nitu

de re

spon

se

Comparison of magnitude responses of proposed DIs

IdealAl-Alaoui 3-segment (SA)Al-Alaoui 4-segment (SA)Jain-Gupta-Jain (GA)

Upadhyay (pole-zero optim)Gupta-Jain-Kumar (LP)Proposed 2nd orderProposed 3rd order

1

09

08

07

06

05

04

03

02

with existing optimized DIs

Figure 2 Comparison ofmagnitude responses of proposed 2nd and3rd order optimized DIs with existing [26 34 39 41] optimizedmodels

Aim of PSO optimization is to search the best optimalsolution (TF of DI) with minimized mean square error

23 Modified PSO Algorithm for Derivation of OptimizedDIs In this paper a slight modification in the originalPSO algorithm has been introduced in velocity taking intoaccount the given problem of deriving the best feasibleoptimized DIs with very small magnitude response errorswithout worsening their corresponding phase responses Inmodified PSO used in this paper the maximum velocitywhich constraints the movement of swarms to the range is

Journal of Optimization 5

0 05 1 15 2 25 3

0

20

40

Frequency

Rel

ativ

e mag

nitu

de er

ror (

dB)

Comparison of RME values of proposed DIs

minus20

minus40

minus60

minus80

minus100

minus120

Al-Alaoui 3-segment (SA)Al-Alaoui 4-segment (SA)Jain-Gupta-Jain (GA)Upadhyay (pole-zero optim)

Gupta-Jain-Kumar (LP)Proposed 2nd orderProposed 3rd order

with existing optimized DIs

Figure 3 Comparison of RMEvalues of proposed 2nd and 3rd orderoptimized DIs with existing [26 34 39 41] optimized models

0

50

100

150

200

Frequency

Phas

e (de

g)

Comparison of phase responses of of proposed DIs

minus50

minus100

minus150

minus2000 05 1 15 2 25 3

with existing optimized DIs

IdealAl-Alaoui 3-segment (SA)Al-Alaoui 4-segment (SA)Jain-Gupta-Jain (GA)

Upadhyay (pole-zero optim)Gupta-Jain-Kumar (LP)Proposed 2nd orderProposed 3rd order

Figure 4 Comparison of phase responses of proposed 2nd and 3rdorder optimized DIs with existing [26 34 39 41] optimized models

divided by a factor of ten after a certain number of iterationsHere we have used dynamic velocity control variant ofPSO in which velocity is scaled down after every specificnumber of generationsThismodification results in improvedperformance as compared to the original algorithm In thispaper the parameters used for PSO algorithm for finding

Frequency

Mag

nitu

de re

spon

se

Magnitude responses of proposed half integrators

Ideal

Gupta-Madhu-JainKrishna half integrator

25

2

15

1

050 02 04 06 08 1 12 14 16 18 2

H2nd fo 2(z)H2nd fo 4(z)H2nd fo 6(z)

H3rd fo 3(z)H3rd fo 6(z)

based on PSO optimized DIs

Figure 5 Comparison of magnitude responses of proposed halfintegrators based on 2nd and 3rd order optimized DIs with existing[2 17] models

optimized solutions of 2nd and 3rd order DIs are as followsparticle range = (minus1 1) learning factors (119888

1and 119888

2) = 2

diversity factor = 1 maximum velocity factor = 05 swarmsize = 200 and number of generations = 200 The flowchartof modified PSO has been given in Figure 1

Intel Core 2 Duo CPU T6600 220GHz (InstalledMemory (RAM) of 4GB) has been used for simulationsTransfer functions of worst and average cases of 2nd and 3rdorderDIs have been given inTables 1 and 2 respectively alongwith their corresponding fitness function values (FV)

In this paper when the code was run in C++ for 100iterations the best optimized (with least mean square error)results for 2nd and 3rd order proposed DIs have beenobserved for 200 generations using 200 particles Transferfunctions (TFs) of proposed optimized integrators have beengiven next in pole zero form in (12) and (13) respectively

TF of proposed 2nd order optimized integrator is

119867PSO 2nd opt (119911) = ((08651 (119911 + 01042) (119911 + 06276))

((119911 minus 1) (119911 + 05547)))

(12)

TF of proposed 3rd order optimized integrator is

119867PSO 3rd opt (119911)

= ((08656 (119911 + 01057) (119911 + 02021) (119911 + 0503))

((119911 + 038) (119911 + 02522) (119911 minus 09997)))

(13)

Proposed 2nd and 3rd order DIs optimized here by PSOhave been compared with the recently published integratorsoptimized by different optimization algorithms namely LP

6 Journal of Optimization

0

20

40

Frequency

Rel

ativ

e mag

nitu

de er

ror (

dB)

Relative magnitude errors (dB) of proposed half integrators

minus20

minus40

minus60

minus80

minus100

minus1200 05 1 15 2 25 3

based on PSO optimized DIs

Gupta-Madhu-JainKrishna half integrator

H2nd fo 2(z)H2nd fo 4(z)H2nd fo 6(z)H3rd fo 3(z)

H3rd fo 6(z)

Figure 6 Comparison of RME values of proposed half integratorsbased on 2nd and 3rd order optimized DIs with existing [2 17]models

[34] GA [35] SA [39] and PZ [41] for validating efficientperformance of proposed PSO Comparisons of bode dia-grams (ie magnitude versus the frequency (119908) and phase(degrees) versus frequency (119908)) and RME responses (in dB)have been presented in Figures 2 3 and 4 respectively

Results for performance values (RME) for each opti-mization method have been compared in Table 3 It canbe observed from Figure 3 that the proposed PSO methodclearly outperforms existing SA [39] and LP [34] techniquesand gives comparable results for PZ [41] and GA [35] meth-ods The 2nd order DI namely 119867PSO 2nd opt(119911) given in (12)outperforms Al-Alaoui 3-segment and 4-segment optimizedDIs in complete spectrum 119867PSO 2nd opt(119911) excels Jain-Gupta-Jain DI [35] in frequency range over 0 le 120596 le 05120587 radiansand 258 le 120596 le 3120587 radians PSO optimized proposed 3rdorder DI namely 119867PSO 3rd opt(119911) clearly outperforms theexisting Al-Alaoui optimized 3-segment 4-segment [39] andUpadhyay [41] DIs in almost complete range with RME of le

minus41 dB 119867PSO 3rd opt(119911) given in (13) gives comparable resultswith GA optimized [35] operator Proposed optimized DIsare observed to satisfy stability criterion and show linearlydecreasing phase responses (see Figure 4)

Proposed DIs optimized by PSO clearly excel operatorsoptimized by SA [39] LP [34] and PZ [41] techniques butlie in close vicinity of the GA [35] responses So a statisticaltest has been done for comparing performances of PSOand GA for validating superiority of the proposed modifiedalgorithm The products of number of particles and numberof generations (119873

119901lowast 119873119892) have been varied from (50 lowast 50) to

(100 lowast 100) and then to (200 lowast 200) for different iterationsand values of mean median and standard deviation have

Pha

se (d

eg)

Phase responses of proposed half integrators

10

0

minus10

minus20

minus30

minus40

minus50

based on PSO optimized DIs

Frequency0 05 1 15 2 25 3

Ideal

Gupta-Madhu-JainKrishna half integrator

H2nd fo 2(z)H2nd fo 4(z)H2nd fo 6(z)

H3rd fo 3(z)H3rd fo 6(z)

Figure 7 Comparison of phase responses of proposed half integra-tors based on 2nd and 3rd order optimized DIs with existing [2 17]models

been registered for all the worst average and best results of2nd and 3rd order optimized DIs (see Table 4) From thistable it is clear that PSO technique performs better than GAmethod and also it was noticed that with the increase invalues of 119873

119901 119873119892 and number of iterations the response of

resultant DIs comes closer to the ideal response as meanmedian and standard deviation values go on decreasing withefficientminimization of error function In this paperwe haveconsidered values of root mean square error

3 Discretization ofProposed 2nd and 3rd Order OptimizedDIs for Half Integrator Models

Theoptimized integer order DIs derived in Section 2 are usedas 119904-to-119911 transformations for CFE based indirect discretiza-tion [2 23 24] scheme Formula given by Khovanskii in [29]has been used for different rational approximations of FOIsThat is

(119904)plusmn120572

=1

1minus

120572119904

1+(1+120572) 119904minus

(1 + 120572) 119904 (1 + 119904)

2+(3+120572) 119904minus

2 (2 + 120572) 119904 (1 + 119904)

3 + (5+120572) 119904minus

times3 (3 + 120572) 119904 (1 + 119904)

4 + (9 + 120572) 119904minus

4 (4 + 120572) 119904 (1 + 119904)

5 + (9 + 120572) 119904minus

times5 (5 + 120572) 119904 (1 + 119904)

6 + (11 + 120572) 119904minus

6 (6 + 120572) 119904 (1 + 119904)

7 + (13 + 120572) 119904minus

(14)

Infinite series of (14) can be terminated up to 2 4and 6 terms and CFE based indirect discretization used in

Journal of Optimization 7

Table2Transfe

rfun

ctions

andfitnessvalues

ofworstandaveragec

ases

of3rdorderP

SOop

timized

DIs

TFso

fworstcaseso

f3rd

orderD

Isalon

gwith

theirfi

tnessfun

ctionvalues

(FV)

TFso

faverage

caseso

f3rd

orderD

Isalon

gwith

theirfi

tnessfun

ctionvalues

(FV)

Transfe

rfun

ction

FV

Transfe

rfun

ction

FV

1198671(119911

)=

(01

423941199113

+04

285111199112

minus03

06611119911

+02

17758

1199113

minus09

031351199112

minus06

74295119911

+05

68774

)066

6364

1198671(119911

)=

(minus

06

56851199113

minus08

062451199112

minus08

20988119911

minus01

9093

1199113

+01

005341199112

minus04

73273119911

minus05

50301

)0548217

1198672(119911

)=

(minus

07

387211199113

minus04

255611199112

minus07

87252119911

+05

88187

1199113

minus00

8455311199112

minus07

14103119911

minus02

4424

)238383

1198672(119911

)=

(minus

01

711661199113

+07

950271199112

+03

39133119911

+04

81229

1199113

minus07

403291199112

minus00

821587119911

minus01

30453

)0651617

1198673(119911

)=

(05

248521199113

+09

588131199112

+05

85323119911

+07

79817

1199113

minus04

627791199112

minus03

80859119911

minus02

47247

)391385

1198673(119911

)=

(05

229891199113

+01

636031199112

minus00

262705119911

+04

83852

1199113

minus09

798611199112

minus01

40418119911

+00

831846

)0673745

1198674(119911

)=

(09

790991199113

minus08

529211199112

+09

60434119911

+08

75076

1199113

minus08

270921199112

minus02

46012119911

+01

35716

)406169

1198674(119911

)=

(02

258041199113

+07

800811199112

+07

58662119911

minus00

994224

1199113

minus02

373481199112

minus05

75985119911

minus01

3299

)0713815

1198675(119911

)=

(00

5946561199113

+07

000181199112

+03

46537119911

+03

14347

1199113

minus07

918671199112

minus05

73836119887119911

+04

10591

)468934

1198675(119911

)=

(01

238551199113

+04

917661199112

+07

27444119911

+05

68523

1199113

minus03

196421199112

minus02

87981119911

minus03

31056

)09760

67

8 Journal of Optimization

Table 3 Comparisons of RME values of the proposed PSO optimized DIs with other integer order existing integrators optimized by differentoptimization techniques

120596 (120587 rad)Al-Alaoui3-segment

(SA)

Al-Alaoui4-segment

(SA)

Jain-Gupta-Jain(GA)

Upadhyay(P-Z)

Gupta-Jain-Kumar(LP)

Proposed2nd orderDI (PSO)

Proposed3rd orderDI (PSO)

025 minus7203 minus6018 minus5446 minus4831 minus7106 minus7203 minus638205 minus4915 minus4121 minus5947 minus5165 minus5989 minus5968 minus7304075 minus3725 minus3402 minus8004 minus6171 minus6391 minus5327 minus714510 minus3029 minus3034 minus5728 minus5728 minus4994 minus4988 minus6463125 minus2272 minus3296 minus5384 minus4871 minus3933 minus4871 minus682175 minus1164 minus3505 minus5385 minus4791 minus3146 minus6262 minus54220 minus824 minus1911 minus6053 minus5289 minus3143 minus4998 minus4994225 minus692 minus1599 minus5384 minus615 minus345 minus4277 minus5366275 minus1819 minus1973 minus3754 minus3776 minus4588 minus5555 minus406130 minus417 minus1811 minus3552 minus3644 minus3014 minus4134 minus4501

Table 4 Statistical test for comparison of proposed PSO and GA techniques for approximations of 2nd and 3rd order digital integrators

Genetic algorithm Modified PSO algorithmApproximating 2nd order digital integrator for comparing PSO and GA

119873119901

lowast 119873119892

50 lowast 50 100 lowast 100 200 lowast 200 50 lowast 50 100 lowast 100 200 lowast 200

Mean 0915405 0230096 00874599 0614370 0201256 00936231Median 0482815 0194402 0075695 0491575 0114377 00488172Standard deviation 0737526 0131617 00400587 043579 0137471 0042671

Approximating 3rd order digital integrator for comparing PSO and GA119873119901

lowast 119873119892

50 lowast 50 100 lowast 100 200 lowast 200 50 lowast 50 100 lowast 100 200 lowast 200

Mean 0915406 0230096 00874599 0633554 0221398 00969881Median 01478565 0069402 002937085 01801123 01207528 00861992Standard deviation 0737526 0131617 00400587 0491451 0152073 00530145

[24] has been adopted for deriving half integrators fitted incontinuous-time domain (see Table 5) Also 2nd order opti-mized DI resulted in 2nd 4th and 6th order FOIs whereas3rd order optimized DI when substituted for different termsresulted in 3rd and 6th order half integrators In the TFs ofderived proposed half integrators given in Table 5 order ofoptimized DIs and order of FOIs are used as subscripts ofsymbol ldquo119867rdquo

4 Comparisons of Proposed FOIs Based on2nd and 3rd Order Optimized DIs with theExisting Models

Simulation results of comparison of magnitude responsesRME and phase responses of half integrators (see Table 5)based on different 2nd and 3rd order optimized DIs (given in(12) and (13)) with the existing half integrators given in [2 17]and ideal half integrator have been presented in Figures 5ndash7

Comparison of RME values of proposed and existingFOIs has been presented in Table 6 It is observed fromFigure 6 that the proposed 2nd and 3rd order FOIs namely1198672nd fo 2(119911) and 119867

3rd fo 3(119911) only lag in performance butstill these outperform the existing [2 17] models in higherfrequency spectrumThe6th order operator119867

2nd fo 6(119911) is the

best among the proposed FOIs and it clearly outperforms allthe existing models [2 17] in complete frequency spectrumwith RME of leminus40 dB Proposed 119867

2nd fo 2(119911) and 1198673rd fo 3(119911)

outperform Gupta-Madhu-Jain half integrator [17] in fre-quency range over 088 le 120596 le 3120587 radians 119867

2nd fo 4(119911)1198672nd fo 6(119911) and 119867

3rd fo 6(119911) half integrators excel Krishnahalf integrator [2] in almost complete range All the proposedFOIs are observed to be stable and follow linear phase curvein almost complete frequency spectrum (see Figure 7)

5 Conclusions

This paper attempts to focus on a simple yet unaddressedfact which till date has remained in the backdrop as theoptimization of an integer order operator before convertingit into its fractional the order (fo) counterparts ProposedFOIs validate that properties of an efficiently optimized DIpass directly into its fractional domain when it is used as 119904-to-119911 transformation during indirect discretization Here 2ndand 3rd order integer order integrators have been exploredby PSO algorithm for finding their optimal solutions for aminimized error fitness function The simulation results ofinteger order operators have revealed the effectiveness ofthe proposed modified PSO algorithm because of their low

Journal of Optimization 9

Table 5 Mathematical models of proposed half integrators based on 2nd and 3rd order PSO optimized DIs

Mathematical models of half integrators based on 2nd and 3rd order PSO optimized DIsName of PSO optimized DIs Transfer functions of FOIs derived by indirect discretization of CFE

119867PSO 2nd opt(119911)

1198672nd-fo-2(119911) = (09302 (119911 minus 01825) (119911 + 05869)

(119911 minus 07422) (119911 + 05604))

1198672nd-fo-4(119911) = (093011 (119911 minus 0003372) (119911 + 05633) (119911 + 06071) (119911 minus 06396)

(119911 minus 09034) (119911 minus 0293) (119911 + 05788) (119911 + 05566))

1198672nd-fo-6(119911) = (

(093011 (119911 minus 08073) (119911 minus 03427) (119911 + 0616)

(119911 + 05758) (119911 + 05587) (119911 + 004899))

((119911 + 05936) (119911 + 05649) (119911 + 05556)

(119911 minus 05924) (119911 minus 09502) (119911 minus 01107))

)

119867PSO 3rd opt(119911)

1198673rd-fo-3(119911) = (093046 (119911 minus 01852) (119911 + 02354) (119911 + 0433)

(119911 minus 0742) (119911 + 03892) (119911 + 02492))

1198673rd-fo-6(119911) = (

(093038 (119911 minus 06395) (119911 + 04678) (119911 + 03939)

(119911 + 02476) (119911 + 0223) (119911 minus 0008033))

((119911 minus 09031) (119911 minus 02946) (119911 + 02512)

(119911 + 02397) (119911 + 0383) (119911 + 04194))

)

Table 6 Comparisons of RME values of the proposed half integrators based on optimized DIs with the existing models

120596

(120587 rad)

Proposed half integrators based on 2nd orderPSO optimized DI

Proposed half integrators based on 3rdorder PSO optimized DI

Gupta-Madhu-Jain halfintegrator

Krishna halfintegrator

1198672nd-fo-2(119911) 1198672nd-fo-4(119911) 1198672nd-fo-6(119911) 1198673rd-fo-3(119911) 1198673rd-fo-6(119911)

025 minus1579 minus2455 minus4325 minus1579 minus2926 minus3553 minus174205 minus1752 minus3874 minus5585 minus1752 minus3897 minus5362 minus2831075 minus2582 minus4156 minus8514 minus2581 minus4261 minus5365 minus355610 minus583 minus583 minus5544 minus6401 minus7475 minus4526 minus4205125 minus3125 minus5734 minus5348 minus3113 minus5067 minus3893 minus3713175 minus2894 minus8115 minus7532 minus2982 minus5825 minus3359 minus405720 minus3262 minus7583 minus5554 minus3262 minus7325 minus3488 minus4646225 minus4143 minus5713 minus4899 minus4115 minus5893 minus4369 minus5528275 minus3299 minus596 minus596 minus3633 minus4694 minus326 minus29830 minus2956 minus5603 minus4775 minus2827 minus4674 minus3374 minus243

orders and very lessmagnitude errors as compared to existingones Proposed FOIs based on the optimizedDIs present veryless RME values of the order of minus40 dB and show linear phasecurves in almost full band of Nyquist frequency

References

[1] L Debnath ldquoRecent applications of fractional calculus toscience and engineeringrdquo International Journal of Mathematicsand Mathematical Sciences vol 2003 no 54 pp 3413ndash34422003

[2] B T Krishna ldquoStudies on fractional order differentiators andintegrators a surveyrdquo Signal Processing vol 91 no 3 pp 386ndash426 2011

[3] N M F Ferreira F B Duarte M F M Lima M G Marcosand J A T Machado ldquoApplication of fractional calculus in thedynamical analysis and control of mechanical manipulatorsrdquoFractional Calculus and Applied Analysis vol 11 pp 91ndash1132008

[4] S Manabe ldquoThe noninteger integral and its application tocontrol systemsrdquo English Translation Journal Japan vol 6 pp83ndash87 1961

[5] J A T Machado and A M S Galhano ldquoA new methodfor approximating fractional derivatives application in non-linear controlrdquo in Proceedings of the 6th EUROMECHNonlinearDynamics Conference (ENOC rsquo08) pp 4ndash8 Saint PetersburgRussia July 2008

[6] Y Ferdi J P Herbeuval A Charef and B Boucheham ldquoR wavedetection using fractional digital differentiationrdquo ITBM-RBMvol 24 no 5-6 pp 273ndash280 2003

[7] J Mocak I Janiga M Rievaj and D Bustin ldquoThe use of frac-tional differentiation or integration for signal improvementrdquoMeasurement Science Review vol 7 pp 39ndash42 2007

[8] B Mathieu P Melchior A Oustaloup and C Ceyral ldquoFrac-tional differentiation for edge detectionrdquo Signal Processing vol83 no 11 pp 2421ndash2432 2003

[9] Y Pu W Wang J Zhou Y Wang and H Jia ldquoFractionaldifferential approach to detecting textural features of digital

10 Journal of Optimization

image and its fractional differential filter implementationrdquoScience in China F vol 51 no 9 pp 1319ndash1339 2008

[10] S Das I Pan S Das andA Gupta ldquoImprovedmodel reductionand tuning of fractional-order PI120582D120583 controllers for analyticalrule extraction with genetic programmingrdquo ISA Transactionsvol 51 no 2 pp 237ndash261 2012

[11] S Das I Pan K Halder S Das and A Gupta ldquoImpact offractional order integral performance indices in LQR basedPID controller design via optimum selection of weightingmatricesrdquo in Proceedings of the International Conference onComputer Communication and Informatics (ICCCI rsquo12) pp 1ndash6Coimbatore India January 2012

[12] Y Chen and B M Vinagre ldquoA new IIR-type digital fractionalorder differentiatorrdquo Signal Processing vol 83 no 11 pp 2359ndash2365 2003

[13] Y Chen B M Vinagre and I Podlubny ldquoContinued frac-tion expansion approaches to discretizing fractional orderderivatives-an expository reviewrdquo Nonlinear Dynamics vol 38no 1ndash4 pp 155ndash170 2004

[14] Y Q Chen and K L Moore ldquoDiscretization schemes forfractional-order differentiators and integratorsrdquo IEEE Transac-tions on Circuits and Systems I vol 49 no 3 pp 363ndash367 2002

[15] M A Al-Alaoui ldquoNovel class of digital integrators and differ-entiatorsrdquo IEEE Transactions on Signal Processing p 1 2009

[16] M A Al-Alaoui ldquoAl-Alaoui operator and the new transfor-mation polynomials for discretization of analogue systemsrdquoElectrical Engineering vol 90 no 6 pp 455ndash467 2008

[17] M Gupta M Jain and N Jain ldquoA new fractional orderrecursive digital integrator using continued fraction expansionrdquoin Proceedings of the India International Conference on Powerelectronics (IICPE) pp 1ndash3 New Delhi India 2010

[18] M Gupta P Varshney and G S Visweswaran ldquoDigitalfractional-order differentiator and integrator models based onfirst-order and higher order operatorsrdquo International Journal ofCircuitTheory and Applications vol 39 no 5 pp 461ndash474 2011

[19] F Leulmi and Y Ferdi ldquoAn improvement of the rationalapproximation of the fractional operator 119904

120572rdquo in Proceedingsof the Saudi International Electronics Communications andPhotonics Conference (SIECP rsquo11) pp 1ndash6 Riyadh Saudi Arabia2011

[20] Y Ferdi ldquoComputation of fractional order derivative andintegral via power series expansion and signal modellingrdquoNonlinear Dynamics vol 46 no 1-2 pp 1ndash15 2006

[21] G S Visweswaran P Varshney and M Gupta ldquoNew approachto realize fractional power in z-domain at low frequencyrdquo IEEETransactions on Circuits and Systems II vol 58 no 3 pp 179ndash183 2011

[22] P Varshney M Gupta and G S Visweswaran ldquoSwitchedcapacitor realizations of fractional-order differentiators andintegrators based on an operator with improved performancerdquoRadioengineering vol 20 no 1 pp 340ndash348 2011

[23] B T Krishna and K V V Reddy ldquoDesign of fractional orderdigital differentiators and integrators using indirect discretiza-tionrdquo An International Journal for Theory and Applications vol11 pp 143ndash151 2008

[24] B T Krishna and K V V Reddy ldquoDesign of digital differentia-tors and integrators of order 12rdquo World Journal of Modellingand Simulation vol 4 pp 182ndash187 2008

[25] R Yadav and M Gupta ldquoDesign of fractional order differen-tiators and integrators using indirect discretization approachrdquoin Proceedings of the 2nd International Conference on Advances

in Recent Technologies in Communication and Computing (ART-Com rsquo10) pp 126ndash130 Kottayam India October 2010

[26] R Yadav and M Gupta ldquoDesign of fractional order differen-tiators and integrators using indirect discretization schemerdquoin Proceedings of the India International Conference on PowerElectronics (IICPE rsquo10) New Delhi India January 2011

[27] G E Carlson and C A Halijak ldquoSimulation of the fractionalderivative operator (1119904)

12 and the fractional integral operator(1119904)12rdquo Kansas State University Bulletin vol 45 pp 1ndash22 1961

[28] S K Steiglitz ldquoComputer-aided design of recursive digitalfiltersrdquo IEEE Transactions Audio and Electroacoustics vol AU-18 no 2 pp 123ndash129 1970

[29] A N Khovanskii The Application of Continued Fractions andtheir Generalizations to Problems in Approximation Theory PNoordhoff Ltd 1963 Translate by Peter Wynn

[30] S C Roy ldquoSome exact steady-state sinewave solutions ofnonuniform RC linesrdquo British Journal of Applied Physics vol 14pp 378ndash380 1963

[31] A Oustaloup F Levron B Mathieu and F M NanotldquoFrequency-band complex noninteger differentiator charac-terization and synthesisrdquo IEEE Transactions on Circuits andSystems I vol 47 no 1 pp 25ndash39 2000

[32] G Maione ldquoLaguerre approximation of fractional systemsrdquoElectronics Letters vol 38 no 20 pp 1234ndash1236 2002

[33] N Papamarkos and C Chamzas ldquoA new approach for thedesign of digital integratorsrdquo IEEE Transactions on Circuits andSystems I vol 43 no 9 pp 785ndash791 1996

[34] M Gupta M Jain and B Kumar ldquoRecursive wideband digitalintegrator and differentiatorrdquo International Journal of CircuitTheory and Applications vol 39 no 7 pp 775ndash782 2011

[35] N Jain M Gupta and M Jain ldquoLinear phase second orderrecursive digital integrators and differentiatorsrdquo Radioengineer-ing Journal vol 21 pp 712ndash717 2012

[36] C-C Hse W-Y Wang and C-Y Yu ldquoGenetic algorithm-derived digital integrators and their applications in discretiza-tion of continuous systemsrdquo in Proceedings of the CEC Congresson Evolutionary Computation pp 443ndash448 Honolulu HawaiiUSA 2002

[37] S Das B Majumder A Pakhira I Pan S Das and AGupta ldquoOptimizing continued fraction expansion based iirrealization of fractional order differ-integrators with geneticalgorithmrdquo in Proceedings of the International Conference onProcess Automation Control and Computing (PACC rsquo11) pp 1ndash6Coimbatore India July 2011

[38] G K Thakur A Gupta and D Upadhyay ldquoOptimizationalgorithmapproach for errorminimization of digital integratorsand differentiatorsrdquo in Proceedings of the International Con-ference on Advances in Recent Technologies in Communicationand Computing (ARTCom rsquo09) pp 118ndash122 Kottayam IndiaOctober 2009

[39] M A Al-Alaoui ldquoClass of digital integrators and differentia-torsrdquo IET Signal Processing vol 5 no 2 pp 251ndash260 2011

[40] D K Upadhyay and R K Singh ldquoRecursive wideband digitaldifferentiator and integratorrdquo Electronics Letters vol 47 no 11pp 647ndash648 2011

[41] D K Upadhyay ldquoClass of recursive wide band digital differen-tiators and integratorsrdquo Radioengineering Journal vol 21 no 3pp 904ndash910 2012

[42] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks pp 1942ndash1948 December 1995

Journal of Optimization 11

[43] Y-T Kao and E Zahara ldquoA hybrid genetic algorithm andparticle swarm optimization formultimodal functionsrdquoAppliedSoft Computing Journal vol 8 no 2 pp 849ndash857 2008

[44] Y Shi and R C Eberhart ldquoEmpirical study of particle swarmoptimizationrdquo in Proceedings of the IEEE international Con-ference Evolution Computer pp 1945ndash1950 Washington DCUSA 1999

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Journal of Optimization

andMaione [32] which proved like boon for the developmentof fractional integratorsdifferentiators

In spite of the efficient and systematic nature of dif-ferent previously mentioned approximations still thereexisted a dissent about performance of these approximationswhen these were later adapted by well-known discretiza-tion schemes for obtaining FOIs Moreover development ofnew approximations requiredmany exhaustivemathematicalapplications to make them perform in finite dimensionalform in proper range of frequencies during discretizationInstead of deriving new methods of rational approximationsauthors have suggested here an alternate solution in whichthe integrator operator (used as 119904-119911 transform) itself shouldbe pushed near boundaries of an optimal solution (near idealresponse) before converting it to fractional system Reasonbehind this is that every minute dispensation in propertiesof this integer order operator will directly propel into itsnoninteger order counterpart The main novelty proposed bythis paper is to put together unique combination of opti-mized integer order operators and an accurate approximationtechnique for configuring their efficient fractional order (fo)operators The current scenario of design methods for deriv-ing optimal operators [33ndash41] is cumulatively proceedingtowards ldquonot so distantrdquo phase of perfect optimization withnegligible errors So another motivating point behind theapproach proposed in this paper is to trespass all disadvan-tages of existing conventional design methods of FOIs byusing refined capabilities of an efficient optimization scheme

The frequency response of ideal integer order digitalintegrators (DIs) is

119867int (120596) = (1

119895120596) where |120596| le 120587 (2)

and the frequency response of ideal integer order digital dif-ferentiators (DDs) is

119867diff (120596) = 119895120596 where |120596| le 120587 (3)

where 119895 = radicminus1 120596 is the angular frequency in radians andldquo119879rdquo is the sampling period

In recent novel advancements different optimizationalgorithms namely Linear Programming (LP) [33 34]Genetic Algorithm (GA) [35ndash37] Simulated Annealing (SA)[38 39] and pole-zero (PZ) optimization [40 41] havebeen used for obtaining more refined integer order differ-integrators Jain et al [35] have recently derived optimalmodels of recursive DIs and DDs by GA algorithm whileUpadhyay [41] has developed these operators by analyzingpoles and zeros of existing recursive DDs by PZ optimizationtechnique for obtaining better results for frequency responsesover wideband of complete frequency spectrum Al-Alaoui[39] has done a credible work in improving design processof DIs and DDs based on Newton-Cotes integration rules byusing linear interpolation and SA

This paper attempts to find the optimal solutions ofDIs by using an efficient modified application of Parti-cle Swarm Optimization (PSO) technique [42ndash44] beforediscretizing them to their fractional mathematical models

Values of mean median and standard deviation in differentrandom iterations have been continuously registered usingPSO for the optimization of 2nd and 3rd order DIs withminimized error fitness function Later these optimized DIsare discretized by CFE of indirect discretization scheme [23]for finding half integrator models of different orders Thesimulation results of magnitude responses phase responsesand relative magnitude errors for the proposed optimizedDIs have been compared with all the recently publishedintegrators optimized by different optimization algorithmsnamely LP [34] GA [35] SA [39] and PZ [41] Frequencyresponses of the proposed FOIs based on 2nd and 3rd orderoptimized DIs have been also compared with their existingfractional models [2 17] in complete frequency range

The paper is organized as follows Section 2 deals withthe brief description of original and modified PSO algo-rithm for optimizing DIs along with its trade-off con-ditions and the resultant 2nd and 3rd order optimizedintegrators thus derived have been presented in this sec-tion Comparisons of proposed and existing DIs are alsogiven in the same section These optimized operators havebeen discretized by indirect discretization using CFE tech-nique of indirect discretization for deriving models of FOIsin Section 3 Section 4 presents the simulation results ofcomparisons of proposed half integrators with responsesof ideal and the existing models Section 5 concludes thepaper

2 PSO Algorithm for Optimizing 2nd and 3rdOrder Integer Order Integrators

21 Outline of Basic Functionality of Original PSO AlgorithmIn conventional PSO algorithm [42ndash44] first a popula-tion of random solutions has been initialized and differentgenerations are updated for searching optimal values Inthis algorithm the potential solutions called particles flythrough the problem space by following the current optimumparticles The position of each particle encodes a possiblesolution to the given problem The velocity of the particle isthe parameter to be added to the current position to find thenewposition of the next generationThe velocity and positionare updated for all particles in every generation unless anoptimal solution is found The particles keep track of itscoordinates in the problem space which are associated withthe best solution (fitness) it has achieved so far These valuesare called pbest and the best element among all the pbestvalues of all the particles is called gbest The velocities areinitialized to zero and after finding the two best values theparticle updates its velocity and positions in accordance withthe following update equations

V (119894) = V (119894 minus 1) + 11988811199031

(119901pbest minus 119901 (119894)) + 11988821199032

(119901gbest minus 119901 (119894))

119901 (119894 + 1) = 119901 (119894) + V (119894)

(4)

where V(119894) is the current particle velocity V(119894 minus 1) is previousparticle velocity 119901(119894) is current particle position 119901(119894 + 1) isthe position of particle in next generation 119901pbest is the best

Journal of Optimization 3

Define the solution space fitness function and population size

Initialise position velocity

For each particle

Evaluate fitness function

Next particle

orNext iteration

Solution is gbest

Update velocityand position

Yes

No

If fitness (x) better than

pbest and gbest

fitness (gbest)gbest = x

No iteration = max no iteration

fitness (gbest) is good enough

minus |H(ej120596)|)2d120596

If fitness (x) better thanfitness (pbest)

pbest = xEint = int

120587

0(1120596

Figure 1 Flowchart of modified PSO algorithm

position the particle has seen 119901gbests is the best position theswarm has seen 119888

1 1198882are constants set to 2 and 119903

1 1199032are the

random numbers between 0 and 1

22 Parameters Used for Optimization and Trade-Offs betweenMultiple Parameters during Convergence of Optimized DigitalIntegrators Amathematical model of transfer function (TF)of optimized digital integrator (DI) can be described ingeneralized form as

119867 (119911) = (119861119872

(119911)119887119872 + sdot sdot sdot + 119861

1(119911)1198871 + 119861

0(119911)1198870

119860119873

(119911)119886119873 + sdot sdot sdot + 119860

1(119911)1198861 + 119860

0(119911)1198860

) (5)

where119860119894and 119861

119895 are arbitrary constants whereas 119886

119894and 119887119895are

real numbers with (119894 = 0 1 119873) and (119895 = 0 1 119872)

Transfer functions (TFs) of 2nd and 3rd order DIs whichhave been optimized using PSO in this paper are given in (6)and (7) respectively

119867PSO 2nd opt (119911) = (11986121199112 + 119861

1119911 + 1198610

11986021199112 + 119860

1119911 + 119860

0

) (6)

119867PSO 3rd opt (119911) = (1198613(119911)3

+ 1198612(119911)2

+ 1198611119911 + 1198610

1198603(119911)3

+ 1198602(119911)2

+ 1198601119911 + 119860

0

) (7)

The coefficients of the previously mentioned TFs are gener-ated by random iterations of PSO algorithm as it takes realnumbers as particles

4 Journal of Optimization

Table 1 Transfer functions and fitness values of worst and average cases of 2nd order PSO optimized DIs

TFs of worst cases of 2nd order DIs along with their fitnessfunction values (FV)

TFs of average cases of 2nd order DIs along with their fitnessfunction values (FV)

Transfer function FV Transfer function FV

1198671(119911) = (

03733241199112 + 0427523119911 + 0397575

1199112 minus 0042561119911 minus 0999916) 0643052 119867

1(119911) = (

0228961199112 minus 0825829119911 minus 0502494

1199112 minus 0776284119911 minus 0181729) 0382107

1198672(119911) = (

minus09292511199112 + 0381076119911 minus 075834

1199112 minus 0725805119911 minus 0306558) 390016 119867

2(119911) = (

minus03364781199112 minus 082224119911 + 0101197

1199112 minus 0818946119911 minus 0147828) 0407093

1198673(119911) = (

05416991199112 + 0241831119911 + 0506633

1199112 minus 00552516119911 minus 1) 473234 119867

3(119911) = (

008498351199112 + 0752164119911 + 0756218

1199112 minus 0167219119911 minus 0882869) 050102

1198674(119911) = (

minus03545831199112 minus 0833923119911 minus 0813869

1199112 minus 00637125119911 minus 1) 589578 119867

4(119911) = (

minus0397291199112 minus 0795585119911 minus 0130415

1199112 minus 0722962119911 minus 0318656) 0618717

1198675(119911) = (

03538521199112 + 0203374119911 + 0466221

1199112 minus 00406162119911 minus 1) 18799 119867

5(119911) = (

02843821199112 + 0949776119911 + 0363676

1199112 minus 0401952119911 minus 0648204) 0737534

Magnitude response of the optimized DI is

119867mag = |119867 (119908)| =abs (numerator)abs (denominator)

(8)

Phase response of the optimized DI is

119867phase = arg |119867 (119908)| (9)

Relative magnitude error (dB) or RME value is given by

RME = 20 log10

100381610038161003816100381610038161003816100381610038161003816

119867ideal (120596) minus 119867approx (120596)

119867ideal (120596)

100381610038161003816100381610038161003816100381610038161003816

= 20 log10

1003816100381610038161003816100381610038161003816100381610038161003816

(1119895120596) minus 119867approx (120596)

(1119895120596)

1003816100381610038161003816100381610038161003816100381610038161003816

(10)

where119867ideal(120596) is themagnitude of ideal value of the operatorand 119867approx(120596) is the value of the digital integrator that isapproximated

In the recently published literature on digital and frac-tional operators most of the researchers have focused onformulating the fitness function for optimization techniquebased on only single parameter [38ndash41] that is magnitudeerror because practically if phase and magnitude are opti-mized for a multiparameter problem the solution is alwaysstuck in a trade-off due to their contradictory nature Thereason behind this is that when the optimal solution startsconverging towards the best results of one parameter (mag-nitude response) it simultaneously degrades optimality ofsecond parameter (phase responses) after certain number ofiterations Solution for such trade-offs is to choose domi-nating parameter in problem under consideration In thispaper the design strategy to find optimized DIs is kept moreconcerned for minimum magnitude response error whichis comparatively more significant characteristic The fitnessfunction used to approximate a digital integrator is taken asthe mean square error of the transfer function and it is givenas

119864int = int120587

0

(1

120596minus

10038161003816100381610038161003816119867 (119890119895120596

)10038161003816100381610038161003816)2

119889120596 (11)

1 15 2 25Frequency

Mag

nitu

de re

spon

se

Comparison of magnitude responses of proposed DIs

IdealAl-Alaoui 3-segment (SA)Al-Alaoui 4-segment (SA)Jain-Gupta-Jain (GA)

Upadhyay (pole-zero optim)Gupta-Jain-Kumar (LP)Proposed 2nd orderProposed 3rd order

1

09

08

07

06

05

04

03

02

with existing optimized DIs

Figure 2 Comparison ofmagnitude responses of proposed 2nd and3rd order optimized DIs with existing [26 34 39 41] optimizedmodels

Aim of PSO optimization is to search the best optimalsolution (TF of DI) with minimized mean square error

23 Modified PSO Algorithm for Derivation of OptimizedDIs In this paper a slight modification in the originalPSO algorithm has been introduced in velocity taking intoaccount the given problem of deriving the best feasibleoptimized DIs with very small magnitude response errorswithout worsening their corresponding phase responses Inmodified PSO used in this paper the maximum velocitywhich constraints the movement of swarms to the range is

Journal of Optimization 5

0 05 1 15 2 25 3

0

20

40

Frequency

Rel

ativ

e mag

nitu

de er

ror (

dB)

Comparison of RME values of proposed DIs

minus20

minus40

minus60

minus80

minus100

minus120

Al-Alaoui 3-segment (SA)Al-Alaoui 4-segment (SA)Jain-Gupta-Jain (GA)Upadhyay (pole-zero optim)

Gupta-Jain-Kumar (LP)Proposed 2nd orderProposed 3rd order

with existing optimized DIs

Figure 3 Comparison of RMEvalues of proposed 2nd and 3rd orderoptimized DIs with existing [26 34 39 41] optimized models

0

50

100

150

200

Frequency

Phas

e (de

g)

Comparison of phase responses of of proposed DIs

minus50

minus100

minus150

minus2000 05 1 15 2 25 3

with existing optimized DIs

IdealAl-Alaoui 3-segment (SA)Al-Alaoui 4-segment (SA)Jain-Gupta-Jain (GA)

Upadhyay (pole-zero optim)Gupta-Jain-Kumar (LP)Proposed 2nd orderProposed 3rd order

Figure 4 Comparison of phase responses of proposed 2nd and 3rdorder optimized DIs with existing [26 34 39 41] optimized models

divided by a factor of ten after a certain number of iterationsHere we have used dynamic velocity control variant ofPSO in which velocity is scaled down after every specificnumber of generationsThismodification results in improvedperformance as compared to the original algorithm In thispaper the parameters used for PSO algorithm for finding

Frequency

Mag

nitu

de re

spon

se

Magnitude responses of proposed half integrators

Ideal

Gupta-Madhu-JainKrishna half integrator

25

2

15

1

050 02 04 06 08 1 12 14 16 18 2

H2nd fo 2(z)H2nd fo 4(z)H2nd fo 6(z)

H3rd fo 3(z)H3rd fo 6(z)

based on PSO optimized DIs

Figure 5 Comparison of magnitude responses of proposed halfintegrators based on 2nd and 3rd order optimized DIs with existing[2 17] models

optimized solutions of 2nd and 3rd order DIs are as followsparticle range = (minus1 1) learning factors (119888

1and 119888

2) = 2

diversity factor = 1 maximum velocity factor = 05 swarmsize = 200 and number of generations = 200 The flowchartof modified PSO has been given in Figure 1

Intel Core 2 Duo CPU T6600 220GHz (InstalledMemory (RAM) of 4GB) has been used for simulationsTransfer functions of worst and average cases of 2nd and 3rdorderDIs have been given inTables 1 and 2 respectively alongwith their corresponding fitness function values (FV)

In this paper when the code was run in C++ for 100iterations the best optimized (with least mean square error)results for 2nd and 3rd order proposed DIs have beenobserved for 200 generations using 200 particles Transferfunctions (TFs) of proposed optimized integrators have beengiven next in pole zero form in (12) and (13) respectively

TF of proposed 2nd order optimized integrator is

119867PSO 2nd opt (119911) = ((08651 (119911 + 01042) (119911 + 06276))

((119911 minus 1) (119911 + 05547)))

(12)

TF of proposed 3rd order optimized integrator is

119867PSO 3rd opt (119911)

= ((08656 (119911 + 01057) (119911 + 02021) (119911 + 0503))

((119911 + 038) (119911 + 02522) (119911 minus 09997)))

(13)

Proposed 2nd and 3rd order DIs optimized here by PSOhave been compared with the recently published integratorsoptimized by different optimization algorithms namely LP

6 Journal of Optimization

0

20

40

Frequency

Rel

ativ

e mag

nitu

de er

ror (

dB)

Relative magnitude errors (dB) of proposed half integrators

minus20

minus40

minus60

minus80

minus100

minus1200 05 1 15 2 25 3

based on PSO optimized DIs

Gupta-Madhu-JainKrishna half integrator

H2nd fo 2(z)H2nd fo 4(z)H2nd fo 6(z)H3rd fo 3(z)

H3rd fo 6(z)

Figure 6 Comparison of RME values of proposed half integratorsbased on 2nd and 3rd order optimized DIs with existing [2 17]models

[34] GA [35] SA [39] and PZ [41] for validating efficientperformance of proposed PSO Comparisons of bode dia-grams (ie magnitude versus the frequency (119908) and phase(degrees) versus frequency (119908)) and RME responses (in dB)have been presented in Figures 2 3 and 4 respectively

Results for performance values (RME) for each opti-mization method have been compared in Table 3 It canbe observed from Figure 3 that the proposed PSO methodclearly outperforms existing SA [39] and LP [34] techniquesand gives comparable results for PZ [41] and GA [35] meth-ods The 2nd order DI namely 119867PSO 2nd opt(119911) given in (12)outperforms Al-Alaoui 3-segment and 4-segment optimizedDIs in complete spectrum 119867PSO 2nd opt(119911) excels Jain-Gupta-Jain DI [35] in frequency range over 0 le 120596 le 05120587 radiansand 258 le 120596 le 3120587 radians PSO optimized proposed 3rdorder DI namely 119867PSO 3rd opt(119911) clearly outperforms theexisting Al-Alaoui optimized 3-segment 4-segment [39] andUpadhyay [41] DIs in almost complete range with RME of le

minus41 dB 119867PSO 3rd opt(119911) given in (13) gives comparable resultswith GA optimized [35] operator Proposed optimized DIsare observed to satisfy stability criterion and show linearlydecreasing phase responses (see Figure 4)

Proposed DIs optimized by PSO clearly excel operatorsoptimized by SA [39] LP [34] and PZ [41] techniques butlie in close vicinity of the GA [35] responses So a statisticaltest has been done for comparing performances of PSOand GA for validating superiority of the proposed modifiedalgorithm The products of number of particles and numberof generations (119873

119901lowast 119873119892) have been varied from (50 lowast 50) to

(100 lowast 100) and then to (200 lowast 200) for different iterationsand values of mean median and standard deviation have

Pha

se (d

eg)

Phase responses of proposed half integrators

10

0

minus10

minus20

minus30

minus40

minus50

based on PSO optimized DIs

Frequency0 05 1 15 2 25 3

Ideal

Gupta-Madhu-JainKrishna half integrator

H2nd fo 2(z)H2nd fo 4(z)H2nd fo 6(z)

H3rd fo 3(z)H3rd fo 6(z)

Figure 7 Comparison of phase responses of proposed half integra-tors based on 2nd and 3rd order optimized DIs with existing [2 17]models

been registered for all the worst average and best results of2nd and 3rd order optimized DIs (see Table 4) From thistable it is clear that PSO technique performs better than GAmethod and also it was noticed that with the increase invalues of 119873

119901 119873119892 and number of iterations the response of

resultant DIs comes closer to the ideal response as meanmedian and standard deviation values go on decreasing withefficientminimization of error function In this paperwe haveconsidered values of root mean square error

3 Discretization ofProposed 2nd and 3rd Order OptimizedDIs for Half Integrator Models

Theoptimized integer order DIs derived in Section 2 are usedas 119904-to-119911 transformations for CFE based indirect discretiza-tion [2 23 24] scheme Formula given by Khovanskii in [29]has been used for different rational approximations of FOIsThat is

(119904)plusmn120572

=1

1minus

120572119904

1+(1+120572) 119904minus

(1 + 120572) 119904 (1 + 119904)

2+(3+120572) 119904minus

2 (2 + 120572) 119904 (1 + 119904)

3 + (5+120572) 119904minus

times3 (3 + 120572) 119904 (1 + 119904)

4 + (9 + 120572) 119904minus

4 (4 + 120572) 119904 (1 + 119904)

5 + (9 + 120572) 119904minus

times5 (5 + 120572) 119904 (1 + 119904)

6 + (11 + 120572) 119904minus

6 (6 + 120572) 119904 (1 + 119904)

7 + (13 + 120572) 119904minus

(14)

Infinite series of (14) can be terminated up to 2 4and 6 terms and CFE based indirect discretization used in

Journal of Optimization 7

Table2Transfe

rfun

ctions

andfitnessvalues

ofworstandaveragec

ases

of3rdorderP

SOop

timized

DIs

TFso

fworstcaseso

f3rd

orderD

Isalon

gwith

theirfi

tnessfun

ctionvalues

(FV)

TFso

faverage

caseso

f3rd

orderD

Isalon

gwith

theirfi

tnessfun

ctionvalues

(FV)

Transfe

rfun

ction

FV

Transfe

rfun

ction

FV

1198671(119911

)=

(01

423941199113

+04

285111199112

minus03

06611119911

+02

17758

1199113

minus09

031351199112

minus06

74295119911

+05

68774

)066

6364

1198671(119911

)=

(minus

06

56851199113

minus08

062451199112

minus08

20988119911

minus01

9093

1199113

+01

005341199112

minus04

73273119911

minus05

50301

)0548217

1198672(119911

)=

(minus

07

387211199113

minus04

255611199112

minus07

87252119911

+05

88187

1199113

minus00

8455311199112

minus07

14103119911

minus02

4424

)238383

1198672(119911

)=

(minus

01

711661199113

+07

950271199112

+03

39133119911

+04

81229

1199113

minus07

403291199112

minus00

821587119911

minus01

30453

)0651617

1198673(119911

)=

(05

248521199113

+09

588131199112

+05

85323119911

+07

79817

1199113

minus04

627791199112

minus03

80859119911

minus02

47247

)391385

1198673(119911

)=

(05

229891199113

+01

636031199112

minus00

262705119911

+04

83852

1199113

minus09

798611199112

minus01

40418119911

+00

831846

)0673745

1198674(119911

)=

(09

790991199113

minus08

529211199112

+09

60434119911

+08

75076

1199113

minus08

270921199112

minus02

46012119911

+01

35716

)406169

1198674(119911

)=

(02

258041199113

+07

800811199112

+07

58662119911

minus00

994224

1199113

minus02

373481199112

minus05

75985119911

minus01

3299

)0713815

1198675(119911

)=

(00

5946561199113

+07

000181199112

+03

46537119911

+03

14347

1199113

minus07

918671199112

minus05

73836119887119911

+04

10591

)468934

1198675(119911

)=

(01

238551199113

+04

917661199112

+07

27444119911

+05

68523

1199113

minus03

196421199112

minus02

87981119911

minus03

31056

)09760

67

8 Journal of Optimization

Table 3 Comparisons of RME values of the proposed PSO optimized DIs with other integer order existing integrators optimized by differentoptimization techniques

120596 (120587 rad)Al-Alaoui3-segment

(SA)

Al-Alaoui4-segment

(SA)

Jain-Gupta-Jain(GA)

Upadhyay(P-Z)

Gupta-Jain-Kumar(LP)

Proposed2nd orderDI (PSO)

Proposed3rd orderDI (PSO)

025 minus7203 minus6018 minus5446 minus4831 minus7106 minus7203 minus638205 minus4915 minus4121 minus5947 minus5165 minus5989 minus5968 minus7304075 minus3725 minus3402 minus8004 minus6171 minus6391 minus5327 minus714510 minus3029 minus3034 minus5728 minus5728 minus4994 minus4988 minus6463125 minus2272 minus3296 minus5384 minus4871 minus3933 minus4871 minus682175 minus1164 minus3505 minus5385 minus4791 minus3146 minus6262 minus54220 minus824 minus1911 minus6053 minus5289 minus3143 minus4998 minus4994225 minus692 minus1599 minus5384 minus615 minus345 minus4277 minus5366275 minus1819 minus1973 minus3754 minus3776 minus4588 minus5555 minus406130 minus417 minus1811 minus3552 minus3644 minus3014 minus4134 minus4501

Table 4 Statistical test for comparison of proposed PSO and GA techniques for approximations of 2nd and 3rd order digital integrators

Genetic algorithm Modified PSO algorithmApproximating 2nd order digital integrator for comparing PSO and GA

119873119901

lowast 119873119892

50 lowast 50 100 lowast 100 200 lowast 200 50 lowast 50 100 lowast 100 200 lowast 200

Mean 0915405 0230096 00874599 0614370 0201256 00936231Median 0482815 0194402 0075695 0491575 0114377 00488172Standard deviation 0737526 0131617 00400587 043579 0137471 0042671

Approximating 3rd order digital integrator for comparing PSO and GA119873119901

lowast 119873119892

50 lowast 50 100 lowast 100 200 lowast 200 50 lowast 50 100 lowast 100 200 lowast 200

Mean 0915406 0230096 00874599 0633554 0221398 00969881Median 01478565 0069402 002937085 01801123 01207528 00861992Standard deviation 0737526 0131617 00400587 0491451 0152073 00530145

[24] has been adopted for deriving half integrators fitted incontinuous-time domain (see Table 5) Also 2nd order opti-mized DI resulted in 2nd 4th and 6th order FOIs whereas3rd order optimized DI when substituted for different termsresulted in 3rd and 6th order half integrators In the TFs ofderived proposed half integrators given in Table 5 order ofoptimized DIs and order of FOIs are used as subscripts ofsymbol ldquo119867rdquo

4 Comparisons of Proposed FOIs Based on2nd and 3rd Order Optimized DIs with theExisting Models

Simulation results of comparison of magnitude responsesRME and phase responses of half integrators (see Table 5)based on different 2nd and 3rd order optimized DIs (given in(12) and (13)) with the existing half integrators given in [2 17]and ideal half integrator have been presented in Figures 5ndash7

Comparison of RME values of proposed and existingFOIs has been presented in Table 6 It is observed fromFigure 6 that the proposed 2nd and 3rd order FOIs namely1198672nd fo 2(119911) and 119867

3rd fo 3(119911) only lag in performance butstill these outperform the existing [2 17] models in higherfrequency spectrumThe6th order operator119867

2nd fo 6(119911) is the

best among the proposed FOIs and it clearly outperforms allthe existing models [2 17] in complete frequency spectrumwith RME of leminus40 dB Proposed 119867

2nd fo 2(119911) and 1198673rd fo 3(119911)

outperform Gupta-Madhu-Jain half integrator [17] in fre-quency range over 088 le 120596 le 3120587 radians 119867

2nd fo 4(119911)1198672nd fo 6(119911) and 119867

3rd fo 6(119911) half integrators excel Krishnahalf integrator [2] in almost complete range All the proposedFOIs are observed to be stable and follow linear phase curvein almost complete frequency spectrum (see Figure 7)

5 Conclusions

This paper attempts to focus on a simple yet unaddressedfact which till date has remained in the backdrop as theoptimization of an integer order operator before convertingit into its fractional the order (fo) counterparts ProposedFOIs validate that properties of an efficiently optimized DIpass directly into its fractional domain when it is used as 119904-to-119911 transformation during indirect discretization Here 2ndand 3rd order integer order integrators have been exploredby PSO algorithm for finding their optimal solutions for aminimized error fitness function The simulation results ofinteger order operators have revealed the effectiveness ofthe proposed modified PSO algorithm because of their low

Journal of Optimization 9

Table 5 Mathematical models of proposed half integrators based on 2nd and 3rd order PSO optimized DIs

Mathematical models of half integrators based on 2nd and 3rd order PSO optimized DIsName of PSO optimized DIs Transfer functions of FOIs derived by indirect discretization of CFE

119867PSO 2nd opt(119911)

1198672nd-fo-2(119911) = (09302 (119911 minus 01825) (119911 + 05869)

(119911 minus 07422) (119911 + 05604))

1198672nd-fo-4(119911) = (093011 (119911 minus 0003372) (119911 + 05633) (119911 + 06071) (119911 minus 06396)

(119911 minus 09034) (119911 minus 0293) (119911 + 05788) (119911 + 05566))

1198672nd-fo-6(119911) = (

(093011 (119911 minus 08073) (119911 minus 03427) (119911 + 0616)

(119911 + 05758) (119911 + 05587) (119911 + 004899))

((119911 + 05936) (119911 + 05649) (119911 + 05556)

(119911 minus 05924) (119911 minus 09502) (119911 minus 01107))

)

119867PSO 3rd opt(119911)

1198673rd-fo-3(119911) = (093046 (119911 minus 01852) (119911 + 02354) (119911 + 0433)

(119911 minus 0742) (119911 + 03892) (119911 + 02492))

1198673rd-fo-6(119911) = (

(093038 (119911 minus 06395) (119911 + 04678) (119911 + 03939)

(119911 + 02476) (119911 + 0223) (119911 minus 0008033))

((119911 minus 09031) (119911 minus 02946) (119911 + 02512)

(119911 + 02397) (119911 + 0383) (119911 + 04194))

)

Table 6 Comparisons of RME values of the proposed half integrators based on optimized DIs with the existing models

120596

(120587 rad)

Proposed half integrators based on 2nd orderPSO optimized DI

Proposed half integrators based on 3rdorder PSO optimized DI

Gupta-Madhu-Jain halfintegrator

Krishna halfintegrator

1198672nd-fo-2(119911) 1198672nd-fo-4(119911) 1198672nd-fo-6(119911) 1198673rd-fo-3(119911) 1198673rd-fo-6(119911)

025 minus1579 minus2455 minus4325 minus1579 minus2926 minus3553 minus174205 minus1752 minus3874 minus5585 minus1752 minus3897 minus5362 minus2831075 minus2582 minus4156 minus8514 minus2581 minus4261 minus5365 minus355610 minus583 minus583 minus5544 minus6401 minus7475 minus4526 minus4205125 minus3125 minus5734 minus5348 minus3113 minus5067 minus3893 minus3713175 minus2894 minus8115 minus7532 minus2982 minus5825 minus3359 minus405720 minus3262 minus7583 minus5554 minus3262 minus7325 minus3488 minus4646225 minus4143 minus5713 minus4899 minus4115 minus5893 minus4369 minus5528275 minus3299 minus596 minus596 minus3633 minus4694 minus326 minus29830 minus2956 minus5603 minus4775 minus2827 minus4674 minus3374 minus243

orders and very lessmagnitude errors as compared to existingones Proposed FOIs based on the optimizedDIs present veryless RME values of the order of minus40 dB and show linear phasecurves in almost full band of Nyquist frequency

References

[1] L Debnath ldquoRecent applications of fractional calculus toscience and engineeringrdquo International Journal of Mathematicsand Mathematical Sciences vol 2003 no 54 pp 3413ndash34422003

[2] B T Krishna ldquoStudies on fractional order differentiators andintegrators a surveyrdquo Signal Processing vol 91 no 3 pp 386ndash426 2011

[3] N M F Ferreira F B Duarte M F M Lima M G Marcosand J A T Machado ldquoApplication of fractional calculus in thedynamical analysis and control of mechanical manipulatorsrdquoFractional Calculus and Applied Analysis vol 11 pp 91ndash1132008

[4] S Manabe ldquoThe noninteger integral and its application tocontrol systemsrdquo English Translation Journal Japan vol 6 pp83ndash87 1961

[5] J A T Machado and A M S Galhano ldquoA new methodfor approximating fractional derivatives application in non-linear controlrdquo in Proceedings of the 6th EUROMECHNonlinearDynamics Conference (ENOC rsquo08) pp 4ndash8 Saint PetersburgRussia July 2008

[6] Y Ferdi J P Herbeuval A Charef and B Boucheham ldquoR wavedetection using fractional digital differentiationrdquo ITBM-RBMvol 24 no 5-6 pp 273ndash280 2003

[7] J Mocak I Janiga M Rievaj and D Bustin ldquoThe use of frac-tional differentiation or integration for signal improvementrdquoMeasurement Science Review vol 7 pp 39ndash42 2007

[8] B Mathieu P Melchior A Oustaloup and C Ceyral ldquoFrac-tional differentiation for edge detectionrdquo Signal Processing vol83 no 11 pp 2421ndash2432 2003

[9] Y Pu W Wang J Zhou Y Wang and H Jia ldquoFractionaldifferential approach to detecting textural features of digital

10 Journal of Optimization

image and its fractional differential filter implementationrdquoScience in China F vol 51 no 9 pp 1319ndash1339 2008

[10] S Das I Pan S Das andA Gupta ldquoImprovedmodel reductionand tuning of fractional-order PI120582D120583 controllers for analyticalrule extraction with genetic programmingrdquo ISA Transactionsvol 51 no 2 pp 237ndash261 2012

[11] S Das I Pan K Halder S Das and A Gupta ldquoImpact offractional order integral performance indices in LQR basedPID controller design via optimum selection of weightingmatricesrdquo in Proceedings of the International Conference onComputer Communication and Informatics (ICCCI rsquo12) pp 1ndash6Coimbatore India January 2012

[12] Y Chen and B M Vinagre ldquoA new IIR-type digital fractionalorder differentiatorrdquo Signal Processing vol 83 no 11 pp 2359ndash2365 2003

[13] Y Chen B M Vinagre and I Podlubny ldquoContinued frac-tion expansion approaches to discretizing fractional orderderivatives-an expository reviewrdquo Nonlinear Dynamics vol 38no 1ndash4 pp 155ndash170 2004

[14] Y Q Chen and K L Moore ldquoDiscretization schemes forfractional-order differentiators and integratorsrdquo IEEE Transac-tions on Circuits and Systems I vol 49 no 3 pp 363ndash367 2002

[15] M A Al-Alaoui ldquoNovel class of digital integrators and differ-entiatorsrdquo IEEE Transactions on Signal Processing p 1 2009

[16] M A Al-Alaoui ldquoAl-Alaoui operator and the new transfor-mation polynomials for discretization of analogue systemsrdquoElectrical Engineering vol 90 no 6 pp 455ndash467 2008

[17] M Gupta M Jain and N Jain ldquoA new fractional orderrecursive digital integrator using continued fraction expansionrdquoin Proceedings of the India International Conference on Powerelectronics (IICPE) pp 1ndash3 New Delhi India 2010

[18] M Gupta P Varshney and G S Visweswaran ldquoDigitalfractional-order differentiator and integrator models based onfirst-order and higher order operatorsrdquo International Journal ofCircuitTheory and Applications vol 39 no 5 pp 461ndash474 2011

[19] F Leulmi and Y Ferdi ldquoAn improvement of the rationalapproximation of the fractional operator 119904

120572rdquo in Proceedingsof the Saudi International Electronics Communications andPhotonics Conference (SIECP rsquo11) pp 1ndash6 Riyadh Saudi Arabia2011

[20] Y Ferdi ldquoComputation of fractional order derivative andintegral via power series expansion and signal modellingrdquoNonlinear Dynamics vol 46 no 1-2 pp 1ndash15 2006

[21] G S Visweswaran P Varshney and M Gupta ldquoNew approachto realize fractional power in z-domain at low frequencyrdquo IEEETransactions on Circuits and Systems II vol 58 no 3 pp 179ndash183 2011

[22] P Varshney M Gupta and G S Visweswaran ldquoSwitchedcapacitor realizations of fractional-order differentiators andintegrators based on an operator with improved performancerdquoRadioengineering vol 20 no 1 pp 340ndash348 2011

[23] B T Krishna and K V V Reddy ldquoDesign of fractional orderdigital differentiators and integrators using indirect discretiza-tionrdquo An International Journal for Theory and Applications vol11 pp 143ndash151 2008

[24] B T Krishna and K V V Reddy ldquoDesign of digital differentia-tors and integrators of order 12rdquo World Journal of Modellingand Simulation vol 4 pp 182ndash187 2008

[25] R Yadav and M Gupta ldquoDesign of fractional order differen-tiators and integrators using indirect discretization approachrdquoin Proceedings of the 2nd International Conference on Advances

in Recent Technologies in Communication and Computing (ART-Com rsquo10) pp 126ndash130 Kottayam India October 2010

[26] R Yadav and M Gupta ldquoDesign of fractional order differen-tiators and integrators using indirect discretization schemerdquoin Proceedings of the India International Conference on PowerElectronics (IICPE rsquo10) New Delhi India January 2011

[27] G E Carlson and C A Halijak ldquoSimulation of the fractionalderivative operator (1119904)

12 and the fractional integral operator(1119904)12rdquo Kansas State University Bulletin vol 45 pp 1ndash22 1961

[28] S K Steiglitz ldquoComputer-aided design of recursive digitalfiltersrdquo IEEE Transactions Audio and Electroacoustics vol AU-18 no 2 pp 123ndash129 1970

[29] A N Khovanskii The Application of Continued Fractions andtheir Generalizations to Problems in Approximation Theory PNoordhoff Ltd 1963 Translate by Peter Wynn

[30] S C Roy ldquoSome exact steady-state sinewave solutions ofnonuniform RC linesrdquo British Journal of Applied Physics vol 14pp 378ndash380 1963

[31] A Oustaloup F Levron B Mathieu and F M NanotldquoFrequency-band complex noninteger differentiator charac-terization and synthesisrdquo IEEE Transactions on Circuits andSystems I vol 47 no 1 pp 25ndash39 2000

[32] G Maione ldquoLaguerre approximation of fractional systemsrdquoElectronics Letters vol 38 no 20 pp 1234ndash1236 2002

[33] N Papamarkos and C Chamzas ldquoA new approach for thedesign of digital integratorsrdquo IEEE Transactions on Circuits andSystems I vol 43 no 9 pp 785ndash791 1996

[34] M Gupta M Jain and B Kumar ldquoRecursive wideband digitalintegrator and differentiatorrdquo International Journal of CircuitTheory and Applications vol 39 no 7 pp 775ndash782 2011

[35] N Jain M Gupta and M Jain ldquoLinear phase second orderrecursive digital integrators and differentiatorsrdquo Radioengineer-ing Journal vol 21 pp 712ndash717 2012

[36] C-C Hse W-Y Wang and C-Y Yu ldquoGenetic algorithm-derived digital integrators and their applications in discretiza-tion of continuous systemsrdquo in Proceedings of the CEC Congresson Evolutionary Computation pp 443ndash448 Honolulu HawaiiUSA 2002

[37] S Das B Majumder A Pakhira I Pan S Das and AGupta ldquoOptimizing continued fraction expansion based iirrealization of fractional order differ-integrators with geneticalgorithmrdquo in Proceedings of the International Conference onProcess Automation Control and Computing (PACC rsquo11) pp 1ndash6Coimbatore India July 2011

[38] G K Thakur A Gupta and D Upadhyay ldquoOptimizationalgorithmapproach for errorminimization of digital integratorsand differentiatorsrdquo in Proceedings of the International Con-ference on Advances in Recent Technologies in Communicationand Computing (ARTCom rsquo09) pp 118ndash122 Kottayam IndiaOctober 2009

[39] M A Al-Alaoui ldquoClass of digital integrators and differentia-torsrdquo IET Signal Processing vol 5 no 2 pp 251ndash260 2011

[40] D K Upadhyay and R K Singh ldquoRecursive wideband digitaldifferentiator and integratorrdquo Electronics Letters vol 47 no 11pp 647ndash648 2011

[41] D K Upadhyay ldquoClass of recursive wide band digital differen-tiators and integratorsrdquo Radioengineering Journal vol 21 no 3pp 904ndash910 2012

[42] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks pp 1942ndash1948 December 1995

Journal of Optimization 11

[43] Y-T Kao and E Zahara ldquoA hybrid genetic algorithm andparticle swarm optimization formultimodal functionsrdquoAppliedSoft Computing Journal vol 8 no 2 pp 849ndash857 2008

[44] Y Shi and R C Eberhart ldquoEmpirical study of particle swarmoptimizationrdquo in Proceedings of the IEEE international Con-ference Evolution Computer pp 1945ndash1950 Washington DCUSA 1999

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Stochastic AnalysisInternational Journal of

Journal of Optimization 3

Define the solution space fitness function and population size

Initialise position velocity

For each particle

Evaluate fitness function

Next particle

orNext iteration

Solution is gbest

Update velocityand position

Yes

No

If fitness (x) better than

pbest and gbest

fitness (gbest)gbest = x

No iteration = max no iteration

fitness (gbest) is good enough

minus |H(ej120596)|)2d120596

If fitness (x) better thanfitness (pbest)

pbest = xEint = int

120587

0(1120596

Figure 1 Flowchart of modified PSO algorithm

position the particle has seen 119901gbests is the best position theswarm has seen 119888

1 1198882are constants set to 2 and 119903

1 1199032are the

random numbers between 0 and 1

22 Parameters Used for Optimization and Trade-Offs betweenMultiple Parameters during Convergence of Optimized DigitalIntegrators Amathematical model of transfer function (TF)of optimized digital integrator (DI) can be described ingeneralized form as

119867 (119911) = (119861119872

(119911)119887119872 + sdot sdot sdot + 119861

1(119911)1198871 + 119861

0(119911)1198870

119860119873

(119911)119886119873 + sdot sdot sdot + 119860

1(119911)1198861 + 119860

0(119911)1198860

) (5)

where119860119894and 119861

119895 are arbitrary constants whereas 119886

119894and 119887119895are

real numbers with (119894 = 0 1 119873) and (119895 = 0 1 119872)

Transfer functions (TFs) of 2nd and 3rd order DIs whichhave been optimized using PSO in this paper are given in (6)and (7) respectively

119867PSO 2nd opt (119911) = (11986121199112 + 119861

1119911 + 1198610

11986021199112 + 119860

1119911 + 119860

0

) (6)

119867PSO 3rd opt (119911) = (1198613(119911)3

+ 1198612(119911)2

+ 1198611119911 + 1198610

1198603(119911)3

+ 1198602(119911)2

+ 1198601119911 + 119860

0

) (7)

The coefficients of the previously mentioned TFs are gener-ated by random iterations of PSO algorithm as it takes realnumbers as particles

4 Journal of Optimization

Table 1 Transfer functions and fitness values of worst and average cases of 2nd order PSO optimized DIs

TFs of worst cases of 2nd order DIs along with their fitnessfunction values (FV)

TFs of average cases of 2nd order DIs along with their fitnessfunction values (FV)

Transfer function FV Transfer function FV

1198671(119911) = (

03733241199112 + 0427523119911 + 0397575

1199112 minus 0042561119911 minus 0999916) 0643052 119867

1(119911) = (

0228961199112 minus 0825829119911 minus 0502494

1199112 minus 0776284119911 minus 0181729) 0382107

1198672(119911) = (

minus09292511199112 + 0381076119911 minus 075834

1199112 minus 0725805119911 minus 0306558) 390016 119867

2(119911) = (

minus03364781199112 minus 082224119911 + 0101197

1199112 minus 0818946119911 minus 0147828) 0407093

1198673(119911) = (

05416991199112 + 0241831119911 + 0506633

1199112 minus 00552516119911 minus 1) 473234 119867

3(119911) = (

008498351199112 + 0752164119911 + 0756218

1199112 minus 0167219119911 minus 0882869) 050102

1198674(119911) = (

minus03545831199112 minus 0833923119911 minus 0813869

1199112 minus 00637125119911 minus 1) 589578 119867

4(119911) = (

minus0397291199112 minus 0795585119911 minus 0130415

1199112 minus 0722962119911 minus 0318656) 0618717

1198675(119911) = (

03538521199112 + 0203374119911 + 0466221

1199112 minus 00406162119911 minus 1) 18799 119867

5(119911) = (

02843821199112 + 0949776119911 + 0363676

1199112 minus 0401952119911 minus 0648204) 0737534

Magnitude response of the optimized DI is

119867mag = |119867 (119908)| =abs (numerator)abs (denominator)

(8)

Phase response of the optimized DI is

119867phase = arg |119867 (119908)| (9)

Relative magnitude error (dB) or RME value is given by

RME = 20 log10

100381610038161003816100381610038161003816100381610038161003816

119867ideal (120596) minus 119867approx (120596)

119867ideal (120596)

100381610038161003816100381610038161003816100381610038161003816

= 20 log10

1003816100381610038161003816100381610038161003816100381610038161003816

(1119895120596) minus 119867approx (120596)

(1119895120596)

1003816100381610038161003816100381610038161003816100381610038161003816

(10)

where119867ideal(120596) is themagnitude of ideal value of the operatorand 119867approx(120596) is the value of the digital integrator that isapproximated

In the recently published literature on digital and frac-tional operators most of the researchers have focused onformulating the fitness function for optimization techniquebased on only single parameter [38ndash41] that is magnitudeerror because practically if phase and magnitude are opti-mized for a multiparameter problem the solution is alwaysstuck in a trade-off due to their contradictory nature Thereason behind this is that when the optimal solution startsconverging towards the best results of one parameter (mag-nitude response) it simultaneously degrades optimality ofsecond parameter (phase responses) after certain number ofiterations Solution for such trade-offs is to choose domi-nating parameter in problem under consideration In thispaper the design strategy to find optimized DIs is kept moreconcerned for minimum magnitude response error whichis comparatively more significant characteristic The fitnessfunction used to approximate a digital integrator is taken asthe mean square error of the transfer function and it is givenas

119864int = int120587

0

(1

120596minus

10038161003816100381610038161003816119867 (119890119895120596

)10038161003816100381610038161003816)2

119889120596 (11)

1 15 2 25Frequency

Mag

nitu

de re

spon

se

Comparison of magnitude responses of proposed DIs

IdealAl-Alaoui 3-segment (SA)Al-Alaoui 4-segment (SA)Jain-Gupta-Jain (GA)

Upadhyay (pole-zero optim)Gupta-Jain-Kumar (LP)Proposed 2nd orderProposed 3rd order

1

09

08

07

06

05

04

03

02

with existing optimized DIs

Figure 2 Comparison ofmagnitude responses of proposed 2nd and3rd order optimized DIs with existing [26 34 39 41] optimizedmodels

Aim of PSO optimization is to search the best optimalsolution (TF of DI) with minimized mean square error

23 Modified PSO Algorithm for Derivation of OptimizedDIs In this paper a slight modification in the originalPSO algorithm has been introduced in velocity taking intoaccount the given problem of deriving the best feasibleoptimized DIs with very small magnitude response errorswithout worsening their corresponding phase responses Inmodified PSO used in this paper the maximum velocitywhich constraints the movement of swarms to the range is

Journal of Optimization 5

0 05 1 15 2 25 3

0

20

40

Frequency

Rel

ativ

e mag

nitu

de er

ror (

dB)

Comparison of RME values of proposed DIs

minus20

minus40

minus60

minus80

minus100

minus120

Al-Alaoui 3-segment (SA)Al-Alaoui 4-segment (SA)Jain-Gupta-Jain (GA)Upadhyay (pole-zero optim)

Gupta-Jain-Kumar (LP)Proposed 2nd orderProposed 3rd order

with existing optimized DIs

Figure 3 Comparison of RMEvalues of proposed 2nd and 3rd orderoptimized DIs with existing [26 34 39 41] optimized models

0

50

100

150

200

Frequency

Phas

e (de

g)

Comparison of phase responses of of proposed DIs

minus50

minus100

minus150

minus2000 05 1 15 2 25 3

with existing optimized DIs

IdealAl-Alaoui 3-segment (SA)Al-Alaoui 4-segment (SA)Jain-Gupta-Jain (GA)

Upadhyay (pole-zero optim)Gupta-Jain-Kumar (LP)Proposed 2nd orderProposed 3rd order

Figure 4 Comparison of phase responses of proposed 2nd and 3rdorder optimized DIs with existing [26 34 39 41] optimized models

divided by a factor of ten after a certain number of iterationsHere we have used dynamic velocity control variant ofPSO in which velocity is scaled down after every specificnumber of generationsThismodification results in improvedperformance as compared to the original algorithm In thispaper the parameters used for PSO algorithm for finding

Frequency

Mag

nitu

de re

spon

se

Magnitude responses of proposed half integrators

Ideal

Gupta-Madhu-JainKrishna half integrator

25

2

15

1

050 02 04 06 08 1 12 14 16 18 2

H2nd fo 2(z)H2nd fo 4(z)H2nd fo 6(z)

H3rd fo 3(z)H3rd fo 6(z)

based on PSO optimized DIs

Figure 5 Comparison of magnitude responses of proposed halfintegrators based on 2nd and 3rd order optimized DIs with existing[2 17] models

optimized solutions of 2nd and 3rd order DIs are as followsparticle range = (minus1 1) learning factors (119888

1and 119888

2) = 2

diversity factor = 1 maximum velocity factor = 05 swarmsize = 200 and number of generations = 200 The flowchartof modified PSO has been given in Figure 1

Intel Core 2 Duo CPU T6600 220GHz (InstalledMemory (RAM) of 4GB) has been used for simulationsTransfer functions of worst and average cases of 2nd and 3rdorderDIs have been given inTables 1 and 2 respectively alongwith their corresponding fitness function values (FV)

In this paper when the code was run in C++ for 100iterations the best optimized (with least mean square error)results for 2nd and 3rd order proposed DIs have beenobserved for 200 generations using 200 particles Transferfunctions (TFs) of proposed optimized integrators have beengiven next in pole zero form in (12) and (13) respectively

TF of proposed 2nd order optimized integrator is

119867PSO 2nd opt (119911) = ((08651 (119911 + 01042) (119911 + 06276))

((119911 minus 1) (119911 + 05547)))

(12)

TF of proposed 3rd order optimized integrator is

119867PSO 3rd opt (119911)

= ((08656 (119911 + 01057) (119911 + 02021) (119911 + 0503))

((119911 + 038) (119911 + 02522) (119911 minus 09997)))

(13)

Proposed 2nd and 3rd order DIs optimized here by PSOhave been compared with the recently published integratorsoptimized by different optimization algorithms namely LP

6 Journal of Optimization

0

20

40

Frequency

Rel

ativ

e mag

nitu

de er

ror (

dB)

Relative magnitude errors (dB) of proposed half integrators

minus20

minus40

minus60

minus80

minus100

minus1200 05 1 15 2 25 3

based on PSO optimized DIs

Gupta-Madhu-JainKrishna half integrator

H2nd fo 2(z)H2nd fo 4(z)H2nd fo 6(z)H3rd fo 3(z)

H3rd fo 6(z)

Figure 6 Comparison of RME values of proposed half integratorsbased on 2nd and 3rd order optimized DIs with existing [2 17]models

[34] GA [35] SA [39] and PZ [41] for validating efficientperformance of proposed PSO Comparisons of bode dia-grams (ie magnitude versus the frequency (119908) and phase(degrees) versus frequency (119908)) and RME responses (in dB)have been presented in Figures 2 3 and 4 respectively

Results for performance values (RME) for each opti-mization method have been compared in Table 3 It canbe observed from Figure 3 that the proposed PSO methodclearly outperforms existing SA [39] and LP [34] techniquesand gives comparable results for PZ [41] and GA [35] meth-ods The 2nd order DI namely 119867PSO 2nd opt(119911) given in (12)outperforms Al-Alaoui 3-segment and 4-segment optimizedDIs in complete spectrum 119867PSO 2nd opt(119911) excels Jain-Gupta-Jain DI [35] in frequency range over 0 le 120596 le 05120587 radiansand 258 le 120596 le 3120587 radians PSO optimized proposed 3rdorder DI namely 119867PSO 3rd opt(119911) clearly outperforms theexisting Al-Alaoui optimized 3-segment 4-segment [39] andUpadhyay [41] DIs in almost complete range with RME of le

minus41 dB 119867PSO 3rd opt(119911) given in (13) gives comparable resultswith GA optimized [35] operator Proposed optimized DIsare observed to satisfy stability criterion and show linearlydecreasing phase responses (see Figure 4)

Proposed DIs optimized by PSO clearly excel operatorsoptimized by SA [39] LP [34] and PZ [41] techniques butlie in close vicinity of the GA [35] responses So a statisticaltest has been done for comparing performances of PSOand GA for validating superiority of the proposed modifiedalgorithm The products of number of particles and numberof generations (119873

119901lowast 119873119892) have been varied from (50 lowast 50) to

(100 lowast 100) and then to (200 lowast 200) for different iterationsand values of mean median and standard deviation have

Pha

se (d

eg)

Phase responses of proposed half integrators

10

0

minus10

minus20

minus30

minus40

minus50

based on PSO optimized DIs

Frequency0 05 1 15 2 25 3

Ideal

Gupta-Madhu-JainKrishna half integrator

H2nd fo 2(z)H2nd fo 4(z)H2nd fo 6(z)

H3rd fo 3(z)H3rd fo 6(z)

Figure 7 Comparison of phase responses of proposed half integra-tors based on 2nd and 3rd order optimized DIs with existing [2 17]models

been registered for all the worst average and best results of2nd and 3rd order optimized DIs (see Table 4) From thistable it is clear that PSO technique performs better than GAmethod and also it was noticed that with the increase invalues of 119873

119901 119873119892 and number of iterations the response of

resultant DIs comes closer to the ideal response as meanmedian and standard deviation values go on decreasing withefficientminimization of error function In this paperwe haveconsidered values of root mean square error

3 Discretization ofProposed 2nd and 3rd Order OptimizedDIs for Half Integrator Models

Theoptimized integer order DIs derived in Section 2 are usedas 119904-to-119911 transformations for CFE based indirect discretiza-tion [2 23 24] scheme Formula given by Khovanskii in [29]has been used for different rational approximations of FOIsThat is

(119904)plusmn120572

=1

1minus

120572119904

1+(1+120572) 119904minus

(1 + 120572) 119904 (1 + 119904)

2+(3+120572) 119904minus

2 (2 + 120572) 119904 (1 + 119904)

3 + (5+120572) 119904minus

times3 (3 + 120572) 119904 (1 + 119904)

4 + (9 + 120572) 119904minus

4 (4 + 120572) 119904 (1 + 119904)

5 + (9 + 120572) 119904minus

times5 (5 + 120572) 119904 (1 + 119904)

6 + (11 + 120572) 119904minus

6 (6 + 120572) 119904 (1 + 119904)

7 + (13 + 120572) 119904minus

(14)

Infinite series of (14) can be terminated up to 2 4and 6 terms and CFE based indirect discretization used in

Journal of Optimization 7

Table2Transfe

rfun

ctions

andfitnessvalues

ofworstandaveragec

ases

of3rdorderP

SOop

timized

DIs

TFso

fworstcaseso

f3rd

orderD

Isalon

gwith

theirfi

tnessfun

ctionvalues

(FV)

TFso

faverage

caseso

f3rd

orderD

Isalon

gwith

theirfi

tnessfun

ctionvalues

(FV)

Transfe

rfun

ction

FV

Transfe

rfun

ction

FV

1198671(119911

)=

(01

423941199113

+04

285111199112

minus03

06611119911

+02

17758

1199113

minus09

031351199112

minus06

74295119911

+05

68774

)066

6364

1198671(119911

)=

(minus

06

56851199113

minus08

062451199112

minus08

20988119911

minus01

9093

1199113

+01

005341199112

minus04

73273119911

minus05

50301

)0548217

1198672(119911

)=

(minus

07

387211199113

minus04

255611199112

minus07

87252119911

+05

88187

1199113

minus00

8455311199112

minus07

14103119911

minus02

4424

)238383

1198672(119911

)=

(minus

01

711661199113

+07

950271199112

+03

39133119911

+04

81229

1199113

minus07

403291199112

minus00

821587119911

minus01

30453

)0651617

1198673(119911

)=

(05

248521199113

+09

588131199112

+05

85323119911

+07

79817

1199113

minus04

627791199112

minus03

80859119911

minus02

47247

)391385

1198673(119911

)=

(05

229891199113

+01

636031199112

minus00

262705119911

+04

83852

1199113

minus09

798611199112

minus01

40418119911

+00

831846

)0673745

1198674(119911

)=

(09

790991199113

minus08

529211199112

+09

60434119911

+08

75076

1199113

minus08

270921199112

minus02

46012119911

+01

35716

)406169

1198674(119911

)=

(02

258041199113

+07

800811199112

+07

58662119911

minus00

994224

1199113

minus02

373481199112

minus05

75985119911

minus01

3299

)0713815

1198675(119911

)=

(00

5946561199113

+07

000181199112

+03

46537119911

+03

14347

1199113

minus07

918671199112

minus05

73836119887119911

+04

10591

)468934

1198675(119911

)=

(01

238551199113

+04

917661199112

+07

27444119911

+05

68523

1199113

minus03

196421199112

minus02

87981119911

minus03

31056

)09760

67

8 Journal of Optimization

Table 3 Comparisons of RME values of the proposed PSO optimized DIs with other integer order existing integrators optimized by differentoptimization techniques

120596 (120587 rad)Al-Alaoui3-segment

(SA)

Al-Alaoui4-segment

(SA)

Jain-Gupta-Jain(GA)

Upadhyay(P-Z)

Gupta-Jain-Kumar(LP)

Proposed2nd orderDI (PSO)

Proposed3rd orderDI (PSO)

025 minus7203 minus6018 minus5446 minus4831 minus7106 minus7203 minus638205 minus4915 minus4121 minus5947 minus5165 minus5989 minus5968 minus7304075 minus3725 minus3402 minus8004 minus6171 minus6391 minus5327 minus714510 minus3029 minus3034 minus5728 minus5728 minus4994 minus4988 minus6463125 minus2272 minus3296 minus5384 minus4871 minus3933 minus4871 minus682175 minus1164 minus3505 minus5385 minus4791 minus3146 minus6262 minus54220 minus824 minus1911 minus6053 minus5289 minus3143 minus4998 minus4994225 minus692 minus1599 minus5384 minus615 minus345 minus4277 minus5366275 minus1819 minus1973 minus3754 minus3776 minus4588 minus5555 minus406130 minus417 minus1811 minus3552 minus3644 minus3014 minus4134 minus4501

Table 4 Statistical test for comparison of proposed PSO and GA techniques for approximations of 2nd and 3rd order digital integrators

Genetic algorithm Modified PSO algorithmApproximating 2nd order digital integrator for comparing PSO and GA

119873119901

lowast 119873119892

50 lowast 50 100 lowast 100 200 lowast 200 50 lowast 50 100 lowast 100 200 lowast 200

Mean 0915405 0230096 00874599 0614370 0201256 00936231Median 0482815 0194402 0075695 0491575 0114377 00488172Standard deviation 0737526 0131617 00400587 043579 0137471 0042671

Approximating 3rd order digital integrator for comparing PSO and GA119873119901

lowast 119873119892

50 lowast 50 100 lowast 100 200 lowast 200 50 lowast 50 100 lowast 100 200 lowast 200

Mean 0915406 0230096 00874599 0633554 0221398 00969881Median 01478565 0069402 002937085 01801123 01207528 00861992Standard deviation 0737526 0131617 00400587 0491451 0152073 00530145

[24] has been adopted for deriving half integrators fitted incontinuous-time domain (see Table 5) Also 2nd order opti-mized DI resulted in 2nd 4th and 6th order FOIs whereas3rd order optimized DI when substituted for different termsresulted in 3rd and 6th order half integrators In the TFs ofderived proposed half integrators given in Table 5 order ofoptimized DIs and order of FOIs are used as subscripts ofsymbol ldquo119867rdquo

4 Comparisons of Proposed FOIs Based on2nd and 3rd Order Optimized DIs with theExisting Models

Simulation results of comparison of magnitude responsesRME and phase responses of half integrators (see Table 5)based on different 2nd and 3rd order optimized DIs (given in(12) and (13)) with the existing half integrators given in [2 17]and ideal half integrator have been presented in Figures 5ndash7

Comparison of RME values of proposed and existingFOIs has been presented in Table 6 It is observed fromFigure 6 that the proposed 2nd and 3rd order FOIs namely1198672nd fo 2(119911) and 119867

3rd fo 3(119911) only lag in performance butstill these outperform the existing [2 17] models in higherfrequency spectrumThe6th order operator119867

2nd fo 6(119911) is the

best among the proposed FOIs and it clearly outperforms allthe existing models [2 17] in complete frequency spectrumwith RME of leminus40 dB Proposed 119867

2nd fo 2(119911) and 1198673rd fo 3(119911)

outperform Gupta-Madhu-Jain half integrator [17] in fre-quency range over 088 le 120596 le 3120587 radians 119867

2nd fo 4(119911)1198672nd fo 6(119911) and 119867

3rd fo 6(119911) half integrators excel Krishnahalf integrator [2] in almost complete range All the proposedFOIs are observed to be stable and follow linear phase curvein almost complete frequency spectrum (see Figure 7)

5 Conclusions

This paper attempts to focus on a simple yet unaddressedfact which till date has remained in the backdrop as theoptimization of an integer order operator before convertingit into its fractional the order (fo) counterparts ProposedFOIs validate that properties of an efficiently optimized DIpass directly into its fractional domain when it is used as 119904-to-119911 transformation during indirect discretization Here 2ndand 3rd order integer order integrators have been exploredby PSO algorithm for finding their optimal solutions for aminimized error fitness function The simulation results ofinteger order operators have revealed the effectiveness ofthe proposed modified PSO algorithm because of their low

Journal of Optimization 9

Table 5 Mathematical models of proposed half integrators based on 2nd and 3rd order PSO optimized DIs

Mathematical models of half integrators based on 2nd and 3rd order PSO optimized DIsName of PSO optimized DIs Transfer functions of FOIs derived by indirect discretization of CFE

119867PSO 2nd opt(119911)

1198672nd-fo-2(119911) = (09302 (119911 minus 01825) (119911 + 05869)

(119911 minus 07422) (119911 + 05604))

1198672nd-fo-4(119911) = (093011 (119911 minus 0003372) (119911 + 05633) (119911 + 06071) (119911 minus 06396)

(119911 minus 09034) (119911 minus 0293) (119911 + 05788) (119911 + 05566))

1198672nd-fo-6(119911) = (

(093011 (119911 minus 08073) (119911 minus 03427) (119911 + 0616)

(119911 + 05758) (119911 + 05587) (119911 + 004899))

((119911 + 05936) (119911 + 05649) (119911 + 05556)

(119911 minus 05924) (119911 minus 09502) (119911 minus 01107))

)

119867PSO 3rd opt(119911)

1198673rd-fo-3(119911) = (093046 (119911 minus 01852) (119911 + 02354) (119911 + 0433)

(119911 minus 0742) (119911 + 03892) (119911 + 02492))

1198673rd-fo-6(119911) = (

(093038 (119911 minus 06395) (119911 + 04678) (119911 + 03939)

(119911 + 02476) (119911 + 0223) (119911 minus 0008033))

((119911 minus 09031) (119911 minus 02946) (119911 + 02512)

(119911 + 02397) (119911 + 0383) (119911 + 04194))

)

Table 6 Comparisons of RME values of the proposed half integrators based on optimized DIs with the existing models

120596

(120587 rad)

Proposed half integrators based on 2nd orderPSO optimized DI

Proposed half integrators based on 3rdorder PSO optimized DI

Gupta-Madhu-Jain halfintegrator

Krishna halfintegrator

1198672nd-fo-2(119911) 1198672nd-fo-4(119911) 1198672nd-fo-6(119911) 1198673rd-fo-3(119911) 1198673rd-fo-6(119911)

025 minus1579 minus2455 minus4325 minus1579 minus2926 minus3553 minus174205 minus1752 minus3874 minus5585 minus1752 minus3897 minus5362 minus2831075 minus2582 minus4156 minus8514 minus2581 minus4261 minus5365 minus355610 minus583 minus583 minus5544 minus6401 minus7475 minus4526 minus4205125 minus3125 minus5734 minus5348 minus3113 minus5067 minus3893 minus3713175 minus2894 minus8115 minus7532 minus2982 minus5825 minus3359 minus405720 minus3262 minus7583 minus5554 minus3262 minus7325 minus3488 minus4646225 minus4143 minus5713 minus4899 minus4115 minus5893 minus4369 minus5528275 minus3299 minus596 minus596 minus3633 minus4694 minus326 minus29830 minus2956 minus5603 minus4775 minus2827 minus4674 minus3374 minus243

orders and very lessmagnitude errors as compared to existingones Proposed FOIs based on the optimizedDIs present veryless RME values of the order of minus40 dB and show linear phasecurves in almost full band of Nyquist frequency

References

[1] L Debnath ldquoRecent applications of fractional calculus toscience and engineeringrdquo International Journal of Mathematicsand Mathematical Sciences vol 2003 no 54 pp 3413ndash34422003

[2] B T Krishna ldquoStudies on fractional order differentiators andintegrators a surveyrdquo Signal Processing vol 91 no 3 pp 386ndash426 2011

[3] N M F Ferreira F B Duarte M F M Lima M G Marcosand J A T Machado ldquoApplication of fractional calculus in thedynamical analysis and control of mechanical manipulatorsrdquoFractional Calculus and Applied Analysis vol 11 pp 91ndash1132008

[4] S Manabe ldquoThe noninteger integral and its application tocontrol systemsrdquo English Translation Journal Japan vol 6 pp83ndash87 1961

[5] J A T Machado and A M S Galhano ldquoA new methodfor approximating fractional derivatives application in non-linear controlrdquo in Proceedings of the 6th EUROMECHNonlinearDynamics Conference (ENOC rsquo08) pp 4ndash8 Saint PetersburgRussia July 2008

[6] Y Ferdi J P Herbeuval A Charef and B Boucheham ldquoR wavedetection using fractional digital differentiationrdquo ITBM-RBMvol 24 no 5-6 pp 273ndash280 2003

[7] J Mocak I Janiga M Rievaj and D Bustin ldquoThe use of frac-tional differentiation or integration for signal improvementrdquoMeasurement Science Review vol 7 pp 39ndash42 2007

[8] B Mathieu P Melchior A Oustaloup and C Ceyral ldquoFrac-tional differentiation for edge detectionrdquo Signal Processing vol83 no 11 pp 2421ndash2432 2003

[9] Y Pu W Wang J Zhou Y Wang and H Jia ldquoFractionaldifferential approach to detecting textural features of digital

10 Journal of Optimization

image and its fractional differential filter implementationrdquoScience in China F vol 51 no 9 pp 1319ndash1339 2008

[10] S Das I Pan S Das andA Gupta ldquoImprovedmodel reductionand tuning of fractional-order PI120582D120583 controllers for analyticalrule extraction with genetic programmingrdquo ISA Transactionsvol 51 no 2 pp 237ndash261 2012

[11] S Das I Pan K Halder S Das and A Gupta ldquoImpact offractional order integral performance indices in LQR basedPID controller design via optimum selection of weightingmatricesrdquo in Proceedings of the International Conference onComputer Communication and Informatics (ICCCI rsquo12) pp 1ndash6Coimbatore India January 2012

[12] Y Chen and B M Vinagre ldquoA new IIR-type digital fractionalorder differentiatorrdquo Signal Processing vol 83 no 11 pp 2359ndash2365 2003

[13] Y Chen B M Vinagre and I Podlubny ldquoContinued frac-tion expansion approaches to discretizing fractional orderderivatives-an expository reviewrdquo Nonlinear Dynamics vol 38no 1ndash4 pp 155ndash170 2004

[14] Y Q Chen and K L Moore ldquoDiscretization schemes forfractional-order differentiators and integratorsrdquo IEEE Transac-tions on Circuits and Systems I vol 49 no 3 pp 363ndash367 2002

[15] M A Al-Alaoui ldquoNovel class of digital integrators and differ-entiatorsrdquo IEEE Transactions on Signal Processing p 1 2009

[16] M A Al-Alaoui ldquoAl-Alaoui operator and the new transfor-mation polynomials for discretization of analogue systemsrdquoElectrical Engineering vol 90 no 6 pp 455ndash467 2008

[17] M Gupta M Jain and N Jain ldquoA new fractional orderrecursive digital integrator using continued fraction expansionrdquoin Proceedings of the India International Conference on Powerelectronics (IICPE) pp 1ndash3 New Delhi India 2010

[18] M Gupta P Varshney and G S Visweswaran ldquoDigitalfractional-order differentiator and integrator models based onfirst-order and higher order operatorsrdquo International Journal ofCircuitTheory and Applications vol 39 no 5 pp 461ndash474 2011

[19] F Leulmi and Y Ferdi ldquoAn improvement of the rationalapproximation of the fractional operator 119904

120572rdquo in Proceedingsof the Saudi International Electronics Communications andPhotonics Conference (SIECP rsquo11) pp 1ndash6 Riyadh Saudi Arabia2011

[20] Y Ferdi ldquoComputation of fractional order derivative andintegral via power series expansion and signal modellingrdquoNonlinear Dynamics vol 46 no 1-2 pp 1ndash15 2006

[21] G S Visweswaran P Varshney and M Gupta ldquoNew approachto realize fractional power in z-domain at low frequencyrdquo IEEETransactions on Circuits and Systems II vol 58 no 3 pp 179ndash183 2011

[22] P Varshney M Gupta and G S Visweswaran ldquoSwitchedcapacitor realizations of fractional-order differentiators andintegrators based on an operator with improved performancerdquoRadioengineering vol 20 no 1 pp 340ndash348 2011

[23] B T Krishna and K V V Reddy ldquoDesign of fractional orderdigital differentiators and integrators using indirect discretiza-tionrdquo An International Journal for Theory and Applications vol11 pp 143ndash151 2008

[24] B T Krishna and K V V Reddy ldquoDesign of digital differentia-tors and integrators of order 12rdquo World Journal of Modellingand Simulation vol 4 pp 182ndash187 2008

[25] R Yadav and M Gupta ldquoDesign of fractional order differen-tiators and integrators using indirect discretization approachrdquoin Proceedings of the 2nd International Conference on Advances

in Recent Technologies in Communication and Computing (ART-Com rsquo10) pp 126ndash130 Kottayam India October 2010

[26] R Yadav and M Gupta ldquoDesign of fractional order differen-tiators and integrators using indirect discretization schemerdquoin Proceedings of the India International Conference on PowerElectronics (IICPE rsquo10) New Delhi India January 2011

[27] G E Carlson and C A Halijak ldquoSimulation of the fractionalderivative operator (1119904)

12 and the fractional integral operator(1119904)12rdquo Kansas State University Bulletin vol 45 pp 1ndash22 1961

[28] S K Steiglitz ldquoComputer-aided design of recursive digitalfiltersrdquo IEEE Transactions Audio and Electroacoustics vol AU-18 no 2 pp 123ndash129 1970

[29] A N Khovanskii The Application of Continued Fractions andtheir Generalizations to Problems in Approximation Theory PNoordhoff Ltd 1963 Translate by Peter Wynn

[30] S C Roy ldquoSome exact steady-state sinewave solutions ofnonuniform RC linesrdquo British Journal of Applied Physics vol 14pp 378ndash380 1963

[31] A Oustaloup F Levron B Mathieu and F M NanotldquoFrequency-band complex noninteger differentiator charac-terization and synthesisrdquo IEEE Transactions on Circuits andSystems I vol 47 no 1 pp 25ndash39 2000

[32] G Maione ldquoLaguerre approximation of fractional systemsrdquoElectronics Letters vol 38 no 20 pp 1234ndash1236 2002

[33] N Papamarkos and C Chamzas ldquoA new approach for thedesign of digital integratorsrdquo IEEE Transactions on Circuits andSystems I vol 43 no 9 pp 785ndash791 1996

[34] M Gupta M Jain and B Kumar ldquoRecursive wideband digitalintegrator and differentiatorrdquo International Journal of CircuitTheory and Applications vol 39 no 7 pp 775ndash782 2011

[35] N Jain M Gupta and M Jain ldquoLinear phase second orderrecursive digital integrators and differentiatorsrdquo Radioengineer-ing Journal vol 21 pp 712ndash717 2012

[36] C-C Hse W-Y Wang and C-Y Yu ldquoGenetic algorithm-derived digital integrators and their applications in discretiza-tion of continuous systemsrdquo in Proceedings of the CEC Congresson Evolutionary Computation pp 443ndash448 Honolulu HawaiiUSA 2002

[37] S Das B Majumder A Pakhira I Pan S Das and AGupta ldquoOptimizing continued fraction expansion based iirrealization of fractional order differ-integrators with geneticalgorithmrdquo in Proceedings of the International Conference onProcess Automation Control and Computing (PACC rsquo11) pp 1ndash6Coimbatore India July 2011

[38] G K Thakur A Gupta and D Upadhyay ldquoOptimizationalgorithmapproach for errorminimization of digital integratorsand differentiatorsrdquo in Proceedings of the International Con-ference on Advances in Recent Technologies in Communicationand Computing (ARTCom rsquo09) pp 118ndash122 Kottayam IndiaOctober 2009

[39] M A Al-Alaoui ldquoClass of digital integrators and differentia-torsrdquo IET Signal Processing vol 5 no 2 pp 251ndash260 2011

[40] D K Upadhyay and R K Singh ldquoRecursive wideband digitaldifferentiator and integratorrdquo Electronics Letters vol 47 no 11pp 647ndash648 2011

[41] D K Upadhyay ldquoClass of recursive wide band digital differen-tiators and integratorsrdquo Radioengineering Journal vol 21 no 3pp 904ndash910 2012

[42] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks pp 1942ndash1948 December 1995

Journal of Optimization 11

[43] Y-T Kao and E Zahara ldquoA hybrid genetic algorithm andparticle swarm optimization formultimodal functionsrdquoAppliedSoft Computing Journal vol 8 no 2 pp 849ndash857 2008

[44] Y Shi and R C Eberhart ldquoEmpirical study of particle swarmoptimizationrdquo in Proceedings of the IEEE international Con-ference Evolution Computer pp 1945ndash1950 Washington DCUSA 1999

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Journal of Optimization

Table 1 Transfer functions and fitness values of worst and average cases of 2nd order PSO optimized DIs

TFs of worst cases of 2nd order DIs along with their fitnessfunction values (FV)

TFs of average cases of 2nd order DIs along with their fitnessfunction values (FV)

Transfer function FV Transfer function FV

1198671(119911) = (

03733241199112 + 0427523119911 + 0397575

1199112 minus 0042561119911 minus 0999916) 0643052 119867

1(119911) = (

0228961199112 minus 0825829119911 minus 0502494

1199112 minus 0776284119911 minus 0181729) 0382107

1198672(119911) = (

minus09292511199112 + 0381076119911 minus 075834

1199112 minus 0725805119911 minus 0306558) 390016 119867

2(119911) = (

minus03364781199112 minus 082224119911 + 0101197

1199112 minus 0818946119911 minus 0147828) 0407093

1198673(119911) = (

05416991199112 + 0241831119911 + 0506633

1199112 minus 00552516119911 minus 1) 473234 119867

3(119911) = (

008498351199112 + 0752164119911 + 0756218

1199112 minus 0167219119911 minus 0882869) 050102

1198674(119911) = (

minus03545831199112 minus 0833923119911 minus 0813869

1199112 minus 00637125119911 minus 1) 589578 119867

4(119911) = (

minus0397291199112 minus 0795585119911 minus 0130415

1199112 minus 0722962119911 minus 0318656) 0618717

1198675(119911) = (

03538521199112 + 0203374119911 + 0466221

1199112 minus 00406162119911 minus 1) 18799 119867

5(119911) = (

02843821199112 + 0949776119911 + 0363676

1199112 minus 0401952119911 minus 0648204) 0737534

Magnitude response of the optimized DI is

119867mag = |119867 (119908)| =abs (numerator)abs (denominator)

(8)

Phase response of the optimized DI is

119867phase = arg |119867 (119908)| (9)

Relative magnitude error (dB) or RME value is given by

RME = 20 log10

100381610038161003816100381610038161003816100381610038161003816

119867ideal (120596) minus 119867approx (120596)

119867ideal (120596)

100381610038161003816100381610038161003816100381610038161003816

= 20 log10

1003816100381610038161003816100381610038161003816100381610038161003816

(1119895120596) minus 119867approx (120596)

(1119895120596)

1003816100381610038161003816100381610038161003816100381610038161003816

(10)

where119867ideal(120596) is themagnitude of ideal value of the operatorand 119867approx(120596) is the value of the digital integrator that isapproximated

In the recently published literature on digital and frac-tional operators most of the researchers have focused onformulating the fitness function for optimization techniquebased on only single parameter [38ndash41] that is magnitudeerror because practically if phase and magnitude are opti-mized for a multiparameter problem the solution is alwaysstuck in a trade-off due to their contradictory nature Thereason behind this is that when the optimal solution startsconverging towards the best results of one parameter (mag-nitude response) it simultaneously degrades optimality ofsecond parameter (phase responses) after certain number ofiterations Solution for such trade-offs is to choose domi-nating parameter in problem under consideration In thispaper the design strategy to find optimized DIs is kept moreconcerned for minimum magnitude response error whichis comparatively more significant characteristic The fitnessfunction used to approximate a digital integrator is taken asthe mean square error of the transfer function and it is givenas

119864int = int120587

0

(1

120596minus

10038161003816100381610038161003816119867 (119890119895120596

)10038161003816100381610038161003816)2

119889120596 (11)

1 15 2 25Frequency

Mag

nitu

de re

spon

se

Comparison of magnitude responses of proposed DIs

IdealAl-Alaoui 3-segment (SA)Al-Alaoui 4-segment (SA)Jain-Gupta-Jain (GA)

Upadhyay (pole-zero optim)Gupta-Jain-Kumar (LP)Proposed 2nd orderProposed 3rd order

1

09

08

07

06

05

04

03

02

with existing optimized DIs

Figure 2 Comparison ofmagnitude responses of proposed 2nd and3rd order optimized DIs with existing [26 34 39 41] optimizedmodels

Aim of PSO optimization is to search the best optimalsolution (TF of DI) with minimized mean square error

23 Modified PSO Algorithm for Derivation of OptimizedDIs In this paper a slight modification in the originalPSO algorithm has been introduced in velocity taking intoaccount the given problem of deriving the best feasibleoptimized DIs with very small magnitude response errorswithout worsening their corresponding phase responses Inmodified PSO used in this paper the maximum velocitywhich constraints the movement of swarms to the range is

Journal of Optimization 5

0 05 1 15 2 25 3

0

20

40

Frequency

Rel

ativ

e mag

nitu

de er

ror (

dB)

Comparison of RME values of proposed DIs

minus20

minus40

minus60

minus80

minus100

minus120

Al-Alaoui 3-segment (SA)Al-Alaoui 4-segment (SA)Jain-Gupta-Jain (GA)Upadhyay (pole-zero optim)

Gupta-Jain-Kumar (LP)Proposed 2nd orderProposed 3rd order

with existing optimized DIs

Figure 3 Comparison of RMEvalues of proposed 2nd and 3rd orderoptimized DIs with existing [26 34 39 41] optimized models

0

50

100

150

200

Frequency

Phas

e (de

g)

Comparison of phase responses of of proposed DIs

minus50

minus100

minus150

minus2000 05 1 15 2 25 3

with existing optimized DIs

IdealAl-Alaoui 3-segment (SA)Al-Alaoui 4-segment (SA)Jain-Gupta-Jain (GA)

Upadhyay (pole-zero optim)Gupta-Jain-Kumar (LP)Proposed 2nd orderProposed 3rd order

Figure 4 Comparison of phase responses of proposed 2nd and 3rdorder optimized DIs with existing [26 34 39 41] optimized models

divided by a factor of ten after a certain number of iterationsHere we have used dynamic velocity control variant ofPSO in which velocity is scaled down after every specificnumber of generationsThismodification results in improvedperformance as compared to the original algorithm In thispaper the parameters used for PSO algorithm for finding

Frequency

Mag

nitu

de re

spon

se

Magnitude responses of proposed half integrators

Ideal

Gupta-Madhu-JainKrishna half integrator

25

2

15

1

050 02 04 06 08 1 12 14 16 18 2

H2nd fo 2(z)H2nd fo 4(z)H2nd fo 6(z)

H3rd fo 3(z)H3rd fo 6(z)

based on PSO optimized DIs

Figure 5 Comparison of magnitude responses of proposed halfintegrators based on 2nd and 3rd order optimized DIs with existing[2 17] models

optimized solutions of 2nd and 3rd order DIs are as followsparticle range = (minus1 1) learning factors (119888

1and 119888

2) = 2

diversity factor = 1 maximum velocity factor = 05 swarmsize = 200 and number of generations = 200 The flowchartof modified PSO has been given in Figure 1

Intel Core 2 Duo CPU T6600 220GHz (InstalledMemory (RAM) of 4GB) has been used for simulationsTransfer functions of worst and average cases of 2nd and 3rdorderDIs have been given inTables 1 and 2 respectively alongwith their corresponding fitness function values (FV)

In this paper when the code was run in C++ for 100iterations the best optimized (with least mean square error)results for 2nd and 3rd order proposed DIs have beenobserved for 200 generations using 200 particles Transferfunctions (TFs) of proposed optimized integrators have beengiven next in pole zero form in (12) and (13) respectively

TF of proposed 2nd order optimized integrator is

119867PSO 2nd opt (119911) = ((08651 (119911 + 01042) (119911 + 06276))

((119911 minus 1) (119911 + 05547)))

(12)

TF of proposed 3rd order optimized integrator is

119867PSO 3rd opt (119911)

= ((08656 (119911 + 01057) (119911 + 02021) (119911 + 0503))

((119911 + 038) (119911 + 02522) (119911 minus 09997)))

(13)

Proposed 2nd and 3rd order DIs optimized here by PSOhave been compared with the recently published integratorsoptimized by different optimization algorithms namely LP

6 Journal of Optimization

0

20

40

Frequency

Rel

ativ

e mag

nitu

de er

ror (

dB)

Relative magnitude errors (dB) of proposed half integrators

minus20

minus40

minus60

minus80

minus100

minus1200 05 1 15 2 25 3

based on PSO optimized DIs

Gupta-Madhu-JainKrishna half integrator

H2nd fo 2(z)H2nd fo 4(z)H2nd fo 6(z)H3rd fo 3(z)

H3rd fo 6(z)

Figure 6 Comparison of RME values of proposed half integratorsbased on 2nd and 3rd order optimized DIs with existing [2 17]models

[34] GA [35] SA [39] and PZ [41] for validating efficientperformance of proposed PSO Comparisons of bode dia-grams (ie magnitude versus the frequency (119908) and phase(degrees) versus frequency (119908)) and RME responses (in dB)have been presented in Figures 2 3 and 4 respectively

Results for performance values (RME) for each opti-mization method have been compared in Table 3 It canbe observed from Figure 3 that the proposed PSO methodclearly outperforms existing SA [39] and LP [34] techniquesand gives comparable results for PZ [41] and GA [35] meth-ods The 2nd order DI namely 119867PSO 2nd opt(119911) given in (12)outperforms Al-Alaoui 3-segment and 4-segment optimizedDIs in complete spectrum 119867PSO 2nd opt(119911) excels Jain-Gupta-Jain DI [35] in frequency range over 0 le 120596 le 05120587 radiansand 258 le 120596 le 3120587 radians PSO optimized proposed 3rdorder DI namely 119867PSO 3rd opt(119911) clearly outperforms theexisting Al-Alaoui optimized 3-segment 4-segment [39] andUpadhyay [41] DIs in almost complete range with RME of le

minus41 dB 119867PSO 3rd opt(119911) given in (13) gives comparable resultswith GA optimized [35] operator Proposed optimized DIsare observed to satisfy stability criterion and show linearlydecreasing phase responses (see Figure 4)

Proposed DIs optimized by PSO clearly excel operatorsoptimized by SA [39] LP [34] and PZ [41] techniques butlie in close vicinity of the GA [35] responses So a statisticaltest has been done for comparing performances of PSOand GA for validating superiority of the proposed modifiedalgorithm The products of number of particles and numberof generations (119873

119901lowast 119873119892) have been varied from (50 lowast 50) to

(100 lowast 100) and then to (200 lowast 200) for different iterationsand values of mean median and standard deviation have

Pha

se (d

eg)

Phase responses of proposed half integrators

10

0

minus10

minus20

minus30

minus40

minus50

based on PSO optimized DIs

Frequency0 05 1 15 2 25 3

Ideal

Gupta-Madhu-JainKrishna half integrator

H2nd fo 2(z)H2nd fo 4(z)H2nd fo 6(z)

H3rd fo 3(z)H3rd fo 6(z)

Figure 7 Comparison of phase responses of proposed half integra-tors based on 2nd and 3rd order optimized DIs with existing [2 17]models

been registered for all the worst average and best results of2nd and 3rd order optimized DIs (see Table 4) From thistable it is clear that PSO technique performs better than GAmethod and also it was noticed that with the increase invalues of 119873

119901 119873119892 and number of iterations the response of

resultant DIs comes closer to the ideal response as meanmedian and standard deviation values go on decreasing withefficientminimization of error function In this paperwe haveconsidered values of root mean square error

3 Discretization ofProposed 2nd and 3rd Order OptimizedDIs for Half Integrator Models

Theoptimized integer order DIs derived in Section 2 are usedas 119904-to-119911 transformations for CFE based indirect discretiza-tion [2 23 24] scheme Formula given by Khovanskii in [29]has been used for different rational approximations of FOIsThat is

(119904)plusmn120572

=1

1minus

120572119904

1+(1+120572) 119904minus

(1 + 120572) 119904 (1 + 119904)

2+(3+120572) 119904minus

2 (2 + 120572) 119904 (1 + 119904)

3 + (5+120572) 119904minus

times3 (3 + 120572) 119904 (1 + 119904)

4 + (9 + 120572) 119904minus

4 (4 + 120572) 119904 (1 + 119904)

5 + (9 + 120572) 119904minus

times5 (5 + 120572) 119904 (1 + 119904)

6 + (11 + 120572) 119904minus

6 (6 + 120572) 119904 (1 + 119904)

7 + (13 + 120572) 119904minus

(14)

Infinite series of (14) can be terminated up to 2 4and 6 terms and CFE based indirect discretization used in

Journal of Optimization 7

Table2Transfe

rfun

ctions

andfitnessvalues

ofworstandaveragec

ases

of3rdorderP

SOop

timized

DIs

TFso

fworstcaseso

f3rd

orderD

Isalon

gwith

theirfi

tnessfun

ctionvalues

(FV)

TFso

faverage

caseso

f3rd

orderD

Isalon

gwith

theirfi

tnessfun

ctionvalues

(FV)

Transfe

rfun

ction

FV

Transfe

rfun

ction

FV

1198671(119911

)=

(01

423941199113

+04

285111199112

minus03

06611119911

+02

17758

1199113

minus09

031351199112

minus06

74295119911

+05

68774

)066

6364

1198671(119911

)=

(minus

06

56851199113

minus08

062451199112

minus08

20988119911

minus01

9093

1199113

+01

005341199112

minus04

73273119911

minus05

50301

)0548217

1198672(119911

)=

(minus

07

387211199113

minus04

255611199112

minus07

87252119911

+05

88187

1199113

minus00

8455311199112

minus07

14103119911

minus02

4424

)238383

1198672(119911

)=

(minus

01

711661199113

+07

950271199112

+03

39133119911

+04

81229

1199113

minus07

403291199112

minus00

821587119911

minus01

30453

)0651617

1198673(119911

)=

(05

248521199113

+09

588131199112

+05

85323119911

+07

79817

1199113

minus04

627791199112

minus03

80859119911

minus02

47247

)391385

1198673(119911

)=

(05

229891199113

+01

636031199112

minus00

262705119911

+04

83852

1199113

minus09

798611199112

minus01

40418119911

+00

831846

)0673745

1198674(119911

)=

(09

790991199113

minus08

529211199112

+09

60434119911

+08

75076

1199113

minus08

270921199112

minus02

46012119911

+01

35716

)406169

1198674(119911

)=

(02

258041199113

+07

800811199112

+07

58662119911

minus00

994224

1199113

minus02

373481199112

minus05

75985119911

minus01

3299

)0713815

1198675(119911

)=

(00

5946561199113

+07

000181199112

+03

46537119911

+03

14347

1199113

minus07

918671199112

minus05

73836119887119911

+04

10591

)468934

1198675(119911

)=

(01

238551199113

+04

917661199112

+07

27444119911

+05

68523

1199113

minus03

196421199112

minus02

87981119911

minus03

31056

)09760

67

8 Journal of Optimization

Table 3 Comparisons of RME values of the proposed PSO optimized DIs with other integer order existing integrators optimized by differentoptimization techniques

120596 (120587 rad)Al-Alaoui3-segment

(SA)

Al-Alaoui4-segment

(SA)

Jain-Gupta-Jain(GA)

Upadhyay(P-Z)

Gupta-Jain-Kumar(LP)

Proposed2nd orderDI (PSO)

Proposed3rd orderDI (PSO)

025 minus7203 minus6018 minus5446 minus4831 minus7106 minus7203 minus638205 minus4915 minus4121 minus5947 minus5165 minus5989 minus5968 minus7304075 minus3725 minus3402 minus8004 minus6171 minus6391 minus5327 minus714510 minus3029 minus3034 minus5728 minus5728 minus4994 minus4988 minus6463125 minus2272 minus3296 minus5384 minus4871 minus3933 minus4871 minus682175 minus1164 minus3505 minus5385 minus4791 minus3146 minus6262 minus54220 minus824 minus1911 minus6053 minus5289 minus3143 minus4998 minus4994225 minus692 minus1599 minus5384 minus615 minus345 minus4277 minus5366275 minus1819 minus1973 minus3754 minus3776 minus4588 minus5555 minus406130 minus417 minus1811 minus3552 minus3644 minus3014 minus4134 minus4501

Table 4 Statistical test for comparison of proposed PSO and GA techniques for approximations of 2nd and 3rd order digital integrators

Genetic algorithm Modified PSO algorithmApproximating 2nd order digital integrator for comparing PSO and GA

119873119901

lowast 119873119892

50 lowast 50 100 lowast 100 200 lowast 200 50 lowast 50 100 lowast 100 200 lowast 200

Mean 0915405 0230096 00874599 0614370 0201256 00936231Median 0482815 0194402 0075695 0491575 0114377 00488172Standard deviation 0737526 0131617 00400587 043579 0137471 0042671

Approximating 3rd order digital integrator for comparing PSO and GA119873119901

lowast 119873119892

50 lowast 50 100 lowast 100 200 lowast 200 50 lowast 50 100 lowast 100 200 lowast 200

Mean 0915406 0230096 00874599 0633554 0221398 00969881Median 01478565 0069402 002937085 01801123 01207528 00861992Standard deviation 0737526 0131617 00400587 0491451 0152073 00530145

[24] has been adopted for deriving half integrators fitted incontinuous-time domain (see Table 5) Also 2nd order opti-mized DI resulted in 2nd 4th and 6th order FOIs whereas3rd order optimized DI when substituted for different termsresulted in 3rd and 6th order half integrators In the TFs ofderived proposed half integrators given in Table 5 order ofoptimized DIs and order of FOIs are used as subscripts ofsymbol ldquo119867rdquo

4 Comparisons of Proposed FOIs Based on2nd and 3rd Order Optimized DIs with theExisting Models

Simulation results of comparison of magnitude responsesRME and phase responses of half integrators (see Table 5)based on different 2nd and 3rd order optimized DIs (given in(12) and (13)) with the existing half integrators given in [2 17]and ideal half integrator have been presented in Figures 5ndash7

Comparison of RME values of proposed and existingFOIs has been presented in Table 6 It is observed fromFigure 6 that the proposed 2nd and 3rd order FOIs namely1198672nd fo 2(119911) and 119867

3rd fo 3(119911) only lag in performance butstill these outperform the existing [2 17] models in higherfrequency spectrumThe6th order operator119867

2nd fo 6(119911) is the

best among the proposed FOIs and it clearly outperforms allthe existing models [2 17] in complete frequency spectrumwith RME of leminus40 dB Proposed 119867

2nd fo 2(119911) and 1198673rd fo 3(119911)

outperform Gupta-Madhu-Jain half integrator [17] in fre-quency range over 088 le 120596 le 3120587 radians 119867

2nd fo 4(119911)1198672nd fo 6(119911) and 119867

3rd fo 6(119911) half integrators excel Krishnahalf integrator [2] in almost complete range All the proposedFOIs are observed to be stable and follow linear phase curvein almost complete frequency spectrum (see Figure 7)

5 Conclusions

This paper attempts to focus on a simple yet unaddressedfact which till date has remained in the backdrop as theoptimization of an integer order operator before convertingit into its fractional the order (fo) counterparts ProposedFOIs validate that properties of an efficiently optimized DIpass directly into its fractional domain when it is used as 119904-to-119911 transformation during indirect discretization Here 2ndand 3rd order integer order integrators have been exploredby PSO algorithm for finding their optimal solutions for aminimized error fitness function The simulation results ofinteger order operators have revealed the effectiveness ofthe proposed modified PSO algorithm because of their low

Journal of Optimization 9

Table 5 Mathematical models of proposed half integrators based on 2nd and 3rd order PSO optimized DIs

Mathematical models of half integrators based on 2nd and 3rd order PSO optimized DIsName of PSO optimized DIs Transfer functions of FOIs derived by indirect discretization of CFE

119867PSO 2nd opt(119911)

1198672nd-fo-2(119911) = (09302 (119911 minus 01825) (119911 + 05869)

(119911 minus 07422) (119911 + 05604))

1198672nd-fo-4(119911) = (093011 (119911 minus 0003372) (119911 + 05633) (119911 + 06071) (119911 minus 06396)

(119911 minus 09034) (119911 minus 0293) (119911 + 05788) (119911 + 05566))

1198672nd-fo-6(119911) = (

(093011 (119911 minus 08073) (119911 minus 03427) (119911 + 0616)

(119911 + 05758) (119911 + 05587) (119911 + 004899))

((119911 + 05936) (119911 + 05649) (119911 + 05556)

(119911 minus 05924) (119911 minus 09502) (119911 minus 01107))

)

119867PSO 3rd opt(119911)

1198673rd-fo-3(119911) = (093046 (119911 minus 01852) (119911 + 02354) (119911 + 0433)

(119911 minus 0742) (119911 + 03892) (119911 + 02492))

1198673rd-fo-6(119911) = (

(093038 (119911 minus 06395) (119911 + 04678) (119911 + 03939)

(119911 + 02476) (119911 + 0223) (119911 minus 0008033))

((119911 minus 09031) (119911 minus 02946) (119911 + 02512)

(119911 + 02397) (119911 + 0383) (119911 + 04194))

)

Table 6 Comparisons of RME values of the proposed half integrators based on optimized DIs with the existing models

120596

(120587 rad)

Proposed half integrators based on 2nd orderPSO optimized DI

Proposed half integrators based on 3rdorder PSO optimized DI

Gupta-Madhu-Jain halfintegrator

Krishna halfintegrator

1198672nd-fo-2(119911) 1198672nd-fo-4(119911) 1198672nd-fo-6(119911) 1198673rd-fo-3(119911) 1198673rd-fo-6(119911)

025 minus1579 minus2455 minus4325 minus1579 minus2926 minus3553 minus174205 minus1752 minus3874 minus5585 minus1752 minus3897 minus5362 minus2831075 minus2582 minus4156 minus8514 minus2581 minus4261 minus5365 minus355610 minus583 minus583 minus5544 minus6401 minus7475 minus4526 minus4205125 minus3125 minus5734 minus5348 minus3113 minus5067 minus3893 minus3713175 minus2894 minus8115 minus7532 minus2982 minus5825 minus3359 minus405720 minus3262 minus7583 minus5554 minus3262 minus7325 minus3488 minus4646225 minus4143 minus5713 minus4899 minus4115 minus5893 minus4369 minus5528275 minus3299 minus596 minus596 minus3633 minus4694 minus326 minus29830 minus2956 minus5603 minus4775 minus2827 minus4674 minus3374 minus243

orders and very lessmagnitude errors as compared to existingones Proposed FOIs based on the optimizedDIs present veryless RME values of the order of minus40 dB and show linear phasecurves in almost full band of Nyquist frequency

References

[1] L Debnath ldquoRecent applications of fractional calculus toscience and engineeringrdquo International Journal of Mathematicsand Mathematical Sciences vol 2003 no 54 pp 3413ndash34422003

[2] B T Krishna ldquoStudies on fractional order differentiators andintegrators a surveyrdquo Signal Processing vol 91 no 3 pp 386ndash426 2011

[3] N M F Ferreira F B Duarte M F M Lima M G Marcosand J A T Machado ldquoApplication of fractional calculus in thedynamical analysis and control of mechanical manipulatorsrdquoFractional Calculus and Applied Analysis vol 11 pp 91ndash1132008

[4] S Manabe ldquoThe noninteger integral and its application tocontrol systemsrdquo English Translation Journal Japan vol 6 pp83ndash87 1961

[5] J A T Machado and A M S Galhano ldquoA new methodfor approximating fractional derivatives application in non-linear controlrdquo in Proceedings of the 6th EUROMECHNonlinearDynamics Conference (ENOC rsquo08) pp 4ndash8 Saint PetersburgRussia July 2008

[6] Y Ferdi J P Herbeuval A Charef and B Boucheham ldquoR wavedetection using fractional digital differentiationrdquo ITBM-RBMvol 24 no 5-6 pp 273ndash280 2003

[7] J Mocak I Janiga M Rievaj and D Bustin ldquoThe use of frac-tional differentiation or integration for signal improvementrdquoMeasurement Science Review vol 7 pp 39ndash42 2007

[8] B Mathieu P Melchior A Oustaloup and C Ceyral ldquoFrac-tional differentiation for edge detectionrdquo Signal Processing vol83 no 11 pp 2421ndash2432 2003

[9] Y Pu W Wang J Zhou Y Wang and H Jia ldquoFractionaldifferential approach to detecting textural features of digital

10 Journal of Optimization

image and its fractional differential filter implementationrdquoScience in China F vol 51 no 9 pp 1319ndash1339 2008

[10] S Das I Pan S Das andA Gupta ldquoImprovedmodel reductionand tuning of fractional-order PI120582D120583 controllers for analyticalrule extraction with genetic programmingrdquo ISA Transactionsvol 51 no 2 pp 237ndash261 2012

[11] S Das I Pan K Halder S Das and A Gupta ldquoImpact offractional order integral performance indices in LQR basedPID controller design via optimum selection of weightingmatricesrdquo in Proceedings of the International Conference onComputer Communication and Informatics (ICCCI rsquo12) pp 1ndash6Coimbatore India January 2012

[12] Y Chen and B M Vinagre ldquoA new IIR-type digital fractionalorder differentiatorrdquo Signal Processing vol 83 no 11 pp 2359ndash2365 2003

[13] Y Chen B M Vinagre and I Podlubny ldquoContinued frac-tion expansion approaches to discretizing fractional orderderivatives-an expository reviewrdquo Nonlinear Dynamics vol 38no 1ndash4 pp 155ndash170 2004

[14] Y Q Chen and K L Moore ldquoDiscretization schemes forfractional-order differentiators and integratorsrdquo IEEE Transac-tions on Circuits and Systems I vol 49 no 3 pp 363ndash367 2002

[15] M A Al-Alaoui ldquoNovel class of digital integrators and differ-entiatorsrdquo IEEE Transactions on Signal Processing p 1 2009

[16] M A Al-Alaoui ldquoAl-Alaoui operator and the new transfor-mation polynomials for discretization of analogue systemsrdquoElectrical Engineering vol 90 no 6 pp 455ndash467 2008

[17] M Gupta M Jain and N Jain ldquoA new fractional orderrecursive digital integrator using continued fraction expansionrdquoin Proceedings of the India International Conference on Powerelectronics (IICPE) pp 1ndash3 New Delhi India 2010

[18] M Gupta P Varshney and G S Visweswaran ldquoDigitalfractional-order differentiator and integrator models based onfirst-order and higher order operatorsrdquo International Journal ofCircuitTheory and Applications vol 39 no 5 pp 461ndash474 2011

[19] F Leulmi and Y Ferdi ldquoAn improvement of the rationalapproximation of the fractional operator 119904

120572rdquo in Proceedingsof the Saudi International Electronics Communications andPhotonics Conference (SIECP rsquo11) pp 1ndash6 Riyadh Saudi Arabia2011

[20] Y Ferdi ldquoComputation of fractional order derivative andintegral via power series expansion and signal modellingrdquoNonlinear Dynamics vol 46 no 1-2 pp 1ndash15 2006

[21] G S Visweswaran P Varshney and M Gupta ldquoNew approachto realize fractional power in z-domain at low frequencyrdquo IEEETransactions on Circuits and Systems II vol 58 no 3 pp 179ndash183 2011

[22] P Varshney M Gupta and G S Visweswaran ldquoSwitchedcapacitor realizations of fractional-order differentiators andintegrators based on an operator with improved performancerdquoRadioengineering vol 20 no 1 pp 340ndash348 2011

[23] B T Krishna and K V V Reddy ldquoDesign of fractional orderdigital differentiators and integrators using indirect discretiza-tionrdquo An International Journal for Theory and Applications vol11 pp 143ndash151 2008

[24] B T Krishna and K V V Reddy ldquoDesign of digital differentia-tors and integrators of order 12rdquo World Journal of Modellingand Simulation vol 4 pp 182ndash187 2008

[25] R Yadav and M Gupta ldquoDesign of fractional order differen-tiators and integrators using indirect discretization approachrdquoin Proceedings of the 2nd International Conference on Advances

in Recent Technologies in Communication and Computing (ART-Com rsquo10) pp 126ndash130 Kottayam India October 2010

[26] R Yadav and M Gupta ldquoDesign of fractional order differen-tiators and integrators using indirect discretization schemerdquoin Proceedings of the India International Conference on PowerElectronics (IICPE rsquo10) New Delhi India January 2011

[27] G E Carlson and C A Halijak ldquoSimulation of the fractionalderivative operator (1119904)

12 and the fractional integral operator(1119904)12rdquo Kansas State University Bulletin vol 45 pp 1ndash22 1961

[28] S K Steiglitz ldquoComputer-aided design of recursive digitalfiltersrdquo IEEE Transactions Audio and Electroacoustics vol AU-18 no 2 pp 123ndash129 1970

[29] A N Khovanskii The Application of Continued Fractions andtheir Generalizations to Problems in Approximation Theory PNoordhoff Ltd 1963 Translate by Peter Wynn

[30] S C Roy ldquoSome exact steady-state sinewave solutions ofnonuniform RC linesrdquo British Journal of Applied Physics vol 14pp 378ndash380 1963

[31] A Oustaloup F Levron B Mathieu and F M NanotldquoFrequency-band complex noninteger differentiator charac-terization and synthesisrdquo IEEE Transactions on Circuits andSystems I vol 47 no 1 pp 25ndash39 2000

[32] G Maione ldquoLaguerre approximation of fractional systemsrdquoElectronics Letters vol 38 no 20 pp 1234ndash1236 2002

[33] N Papamarkos and C Chamzas ldquoA new approach for thedesign of digital integratorsrdquo IEEE Transactions on Circuits andSystems I vol 43 no 9 pp 785ndash791 1996

[34] M Gupta M Jain and B Kumar ldquoRecursive wideband digitalintegrator and differentiatorrdquo International Journal of CircuitTheory and Applications vol 39 no 7 pp 775ndash782 2011

[35] N Jain M Gupta and M Jain ldquoLinear phase second orderrecursive digital integrators and differentiatorsrdquo Radioengineer-ing Journal vol 21 pp 712ndash717 2012

[36] C-C Hse W-Y Wang and C-Y Yu ldquoGenetic algorithm-derived digital integrators and their applications in discretiza-tion of continuous systemsrdquo in Proceedings of the CEC Congresson Evolutionary Computation pp 443ndash448 Honolulu HawaiiUSA 2002

[37] S Das B Majumder A Pakhira I Pan S Das and AGupta ldquoOptimizing continued fraction expansion based iirrealization of fractional order differ-integrators with geneticalgorithmrdquo in Proceedings of the International Conference onProcess Automation Control and Computing (PACC rsquo11) pp 1ndash6Coimbatore India July 2011

[38] G K Thakur A Gupta and D Upadhyay ldquoOptimizationalgorithmapproach for errorminimization of digital integratorsand differentiatorsrdquo in Proceedings of the International Con-ference on Advances in Recent Technologies in Communicationand Computing (ARTCom rsquo09) pp 118ndash122 Kottayam IndiaOctober 2009

[39] M A Al-Alaoui ldquoClass of digital integrators and differentia-torsrdquo IET Signal Processing vol 5 no 2 pp 251ndash260 2011

[40] D K Upadhyay and R K Singh ldquoRecursive wideband digitaldifferentiator and integratorrdquo Electronics Letters vol 47 no 11pp 647ndash648 2011

[41] D K Upadhyay ldquoClass of recursive wide band digital differen-tiators and integratorsrdquo Radioengineering Journal vol 21 no 3pp 904ndash910 2012

[42] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks pp 1942ndash1948 December 1995

Journal of Optimization 11

[43] Y-T Kao and E Zahara ldquoA hybrid genetic algorithm andparticle swarm optimization formultimodal functionsrdquoAppliedSoft Computing Journal vol 8 no 2 pp 849ndash857 2008

[44] Y Shi and R C Eberhart ldquoEmpirical study of particle swarmoptimizationrdquo in Proceedings of the IEEE international Con-ference Evolution Computer pp 1945ndash1950 Washington DCUSA 1999

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Optimization 5

0 05 1 15 2 25 3

0

20

40

Frequency

Rel

ativ

e mag

nitu

de er

ror (

dB)

Comparison of RME values of proposed DIs

minus20

minus40

minus60

minus80

minus100

minus120

Al-Alaoui 3-segment (SA)Al-Alaoui 4-segment (SA)Jain-Gupta-Jain (GA)Upadhyay (pole-zero optim)

Gupta-Jain-Kumar (LP)Proposed 2nd orderProposed 3rd order

with existing optimized DIs

Figure 3 Comparison of RMEvalues of proposed 2nd and 3rd orderoptimized DIs with existing [26 34 39 41] optimized models

0

50

100

150

200

Frequency

Phas

e (de

g)

Comparison of phase responses of of proposed DIs

minus50

minus100

minus150

minus2000 05 1 15 2 25 3

with existing optimized DIs

IdealAl-Alaoui 3-segment (SA)Al-Alaoui 4-segment (SA)Jain-Gupta-Jain (GA)

Upadhyay (pole-zero optim)Gupta-Jain-Kumar (LP)Proposed 2nd orderProposed 3rd order

Figure 4 Comparison of phase responses of proposed 2nd and 3rdorder optimized DIs with existing [26 34 39 41] optimized models

divided by a factor of ten after a certain number of iterationsHere we have used dynamic velocity control variant ofPSO in which velocity is scaled down after every specificnumber of generationsThismodification results in improvedperformance as compared to the original algorithm In thispaper the parameters used for PSO algorithm for finding

Frequency

Mag

nitu

de re

spon

se

Magnitude responses of proposed half integrators

Ideal

Gupta-Madhu-JainKrishna half integrator

25

2

15

1

050 02 04 06 08 1 12 14 16 18 2

H2nd fo 2(z)H2nd fo 4(z)H2nd fo 6(z)

H3rd fo 3(z)H3rd fo 6(z)

based on PSO optimized DIs

Figure 5 Comparison of magnitude responses of proposed halfintegrators based on 2nd and 3rd order optimized DIs with existing[2 17] models

optimized solutions of 2nd and 3rd order DIs are as followsparticle range = (minus1 1) learning factors (119888

1and 119888

2) = 2

diversity factor = 1 maximum velocity factor = 05 swarmsize = 200 and number of generations = 200 The flowchartof modified PSO has been given in Figure 1

Intel Core 2 Duo CPU T6600 220GHz (InstalledMemory (RAM) of 4GB) has been used for simulationsTransfer functions of worst and average cases of 2nd and 3rdorderDIs have been given inTables 1 and 2 respectively alongwith their corresponding fitness function values (FV)

In this paper when the code was run in C++ for 100iterations the best optimized (with least mean square error)results for 2nd and 3rd order proposed DIs have beenobserved for 200 generations using 200 particles Transferfunctions (TFs) of proposed optimized integrators have beengiven next in pole zero form in (12) and (13) respectively

TF of proposed 2nd order optimized integrator is

119867PSO 2nd opt (119911) = ((08651 (119911 + 01042) (119911 + 06276))

((119911 minus 1) (119911 + 05547)))

(12)

TF of proposed 3rd order optimized integrator is

119867PSO 3rd opt (119911)

= ((08656 (119911 + 01057) (119911 + 02021) (119911 + 0503))

((119911 + 038) (119911 + 02522) (119911 minus 09997)))

(13)

Proposed 2nd and 3rd order DIs optimized here by PSOhave been compared with the recently published integratorsoptimized by different optimization algorithms namely LP

6 Journal of Optimization

0

20

40

Frequency

Rel

ativ

e mag

nitu

de er

ror (

dB)

Relative magnitude errors (dB) of proposed half integrators

minus20

minus40

minus60

minus80

minus100

minus1200 05 1 15 2 25 3

based on PSO optimized DIs

Gupta-Madhu-JainKrishna half integrator

H2nd fo 2(z)H2nd fo 4(z)H2nd fo 6(z)H3rd fo 3(z)

H3rd fo 6(z)

Figure 6 Comparison of RME values of proposed half integratorsbased on 2nd and 3rd order optimized DIs with existing [2 17]models

[34] GA [35] SA [39] and PZ [41] for validating efficientperformance of proposed PSO Comparisons of bode dia-grams (ie magnitude versus the frequency (119908) and phase(degrees) versus frequency (119908)) and RME responses (in dB)have been presented in Figures 2 3 and 4 respectively

Results for performance values (RME) for each opti-mization method have been compared in Table 3 It canbe observed from Figure 3 that the proposed PSO methodclearly outperforms existing SA [39] and LP [34] techniquesand gives comparable results for PZ [41] and GA [35] meth-ods The 2nd order DI namely 119867PSO 2nd opt(119911) given in (12)outperforms Al-Alaoui 3-segment and 4-segment optimizedDIs in complete spectrum 119867PSO 2nd opt(119911) excels Jain-Gupta-Jain DI [35] in frequency range over 0 le 120596 le 05120587 radiansand 258 le 120596 le 3120587 radians PSO optimized proposed 3rdorder DI namely 119867PSO 3rd opt(119911) clearly outperforms theexisting Al-Alaoui optimized 3-segment 4-segment [39] andUpadhyay [41] DIs in almost complete range with RME of le

minus41 dB 119867PSO 3rd opt(119911) given in (13) gives comparable resultswith GA optimized [35] operator Proposed optimized DIsare observed to satisfy stability criterion and show linearlydecreasing phase responses (see Figure 4)

Proposed DIs optimized by PSO clearly excel operatorsoptimized by SA [39] LP [34] and PZ [41] techniques butlie in close vicinity of the GA [35] responses So a statisticaltest has been done for comparing performances of PSOand GA for validating superiority of the proposed modifiedalgorithm The products of number of particles and numberof generations (119873

119901lowast 119873119892) have been varied from (50 lowast 50) to

(100 lowast 100) and then to (200 lowast 200) for different iterationsand values of mean median and standard deviation have

Pha

se (d

eg)

Phase responses of proposed half integrators

10

0

minus10

minus20

minus30

minus40

minus50

based on PSO optimized DIs

Frequency0 05 1 15 2 25 3

Ideal

Gupta-Madhu-JainKrishna half integrator

H2nd fo 2(z)H2nd fo 4(z)H2nd fo 6(z)

H3rd fo 3(z)H3rd fo 6(z)

Figure 7 Comparison of phase responses of proposed half integra-tors based on 2nd and 3rd order optimized DIs with existing [2 17]models

been registered for all the worst average and best results of2nd and 3rd order optimized DIs (see Table 4) From thistable it is clear that PSO technique performs better than GAmethod and also it was noticed that with the increase invalues of 119873

119901 119873119892 and number of iterations the response of

resultant DIs comes closer to the ideal response as meanmedian and standard deviation values go on decreasing withefficientminimization of error function In this paperwe haveconsidered values of root mean square error

3 Discretization ofProposed 2nd and 3rd Order OptimizedDIs for Half Integrator Models

Theoptimized integer order DIs derived in Section 2 are usedas 119904-to-119911 transformations for CFE based indirect discretiza-tion [2 23 24] scheme Formula given by Khovanskii in [29]has been used for different rational approximations of FOIsThat is

(119904)plusmn120572

=1

1minus

120572119904

1+(1+120572) 119904minus

(1 + 120572) 119904 (1 + 119904)

2+(3+120572) 119904minus

2 (2 + 120572) 119904 (1 + 119904)

3 + (5+120572) 119904minus

times3 (3 + 120572) 119904 (1 + 119904)

4 + (9 + 120572) 119904minus

4 (4 + 120572) 119904 (1 + 119904)

5 + (9 + 120572) 119904minus

times5 (5 + 120572) 119904 (1 + 119904)

6 + (11 + 120572) 119904minus

6 (6 + 120572) 119904 (1 + 119904)

7 + (13 + 120572) 119904minus

(14)

Infinite series of (14) can be terminated up to 2 4and 6 terms and CFE based indirect discretization used in

Journal of Optimization 7

Table2Transfe

rfun

ctions

andfitnessvalues

ofworstandaveragec

ases

of3rdorderP

SOop

timized

DIs

TFso

fworstcaseso

f3rd

orderD

Isalon

gwith

theirfi

tnessfun

ctionvalues

(FV)

TFso

faverage

caseso

f3rd

orderD

Isalon

gwith

theirfi

tnessfun

ctionvalues

(FV)

Transfe

rfun

ction

FV

Transfe

rfun

ction

FV

1198671(119911

)=

(01

423941199113

+04

285111199112

minus03

06611119911

+02

17758

1199113

minus09

031351199112

minus06

74295119911

+05

68774

)066

6364

1198671(119911

)=

(minus

06

56851199113

minus08

062451199112

minus08

20988119911

minus01

9093

1199113

+01

005341199112

minus04

73273119911

minus05

50301

)0548217

1198672(119911

)=

(minus

07

387211199113

minus04

255611199112

minus07

87252119911

+05

88187

1199113

minus00

8455311199112

minus07

14103119911

minus02

4424

)238383

1198672(119911

)=

(minus

01

711661199113

+07

950271199112

+03

39133119911

+04

81229

1199113

minus07

403291199112

minus00

821587119911

minus01

30453

)0651617

1198673(119911

)=

(05

248521199113

+09

588131199112

+05

85323119911

+07

79817

1199113

minus04

627791199112

minus03

80859119911

minus02

47247

)391385

1198673(119911

)=

(05

229891199113

+01

636031199112

minus00

262705119911

+04

83852

1199113

minus09

798611199112

minus01

40418119911

+00

831846

)0673745

1198674(119911

)=

(09

790991199113

minus08

529211199112

+09

60434119911

+08

75076

1199113

minus08

270921199112

minus02

46012119911

+01

35716

)406169

1198674(119911

)=

(02

258041199113

+07

800811199112

+07

58662119911

minus00

994224

1199113

minus02

373481199112

minus05

75985119911

minus01

3299

)0713815

1198675(119911

)=

(00

5946561199113

+07

000181199112

+03

46537119911

+03

14347

1199113

minus07

918671199112

minus05

73836119887119911

+04

10591

)468934

1198675(119911

)=

(01

238551199113

+04

917661199112

+07

27444119911

+05

68523

1199113

minus03

196421199112

minus02

87981119911

minus03

31056

)09760

67

8 Journal of Optimization

Table 3 Comparisons of RME values of the proposed PSO optimized DIs with other integer order existing integrators optimized by differentoptimization techniques

120596 (120587 rad)Al-Alaoui3-segment

(SA)

Al-Alaoui4-segment

(SA)

Jain-Gupta-Jain(GA)

Upadhyay(P-Z)

Gupta-Jain-Kumar(LP)

Proposed2nd orderDI (PSO)

Proposed3rd orderDI (PSO)

025 minus7203 minus6018 minus5446 minus4831 minus7106 minus7203 minus638205 minus4915 minus4121 minus5947 minus5165 minus5989 minus5968 minus7304075 minus3725 minus3402 minus8004 minus6171 minus6391 minus5327 minus714510 minus3029 minus3034 minus5728 minus5728 minus4994 minus4988 minus6463125 minus2272 minus3296 minus5384 minus4871 minus3933 minus4871 minus682175 minus1164 minus3505 minus5385 minus4791 minus3146 minus6262 minus54220 minus824 minus1911 minus6053 minus5289 minus3143 minus4998 minus4994225 minus692 minus1599 minus5384 minus615 minus345 minus4277 minus5366275 minus1819 minus1973 minus3754 minus3776 minus4588 minus5555 minus406130 minus417 minus1811 minus3552 minus3644 minus3014 minus4134 minus4501

Table 4 Statistical test for comparison of proposed PSO and GA techniques for approximations of 2nd and 3rd order digital integrators

Genetic algorithm Modified PSO algorithmApproximating 2nd order digital integrator for comparing PSO and GA

119873119901

lowast 119873119892

50 lowast 50 100 lowast 100 200 lowast 200 50 lowast 50 100 lowast 100 200 lowast 200

Mean 0915405 0230096 00874599 0614370 0201256 00936231Median 0482815 0194402 0075695 0491575 0114377 00488172Standard deviation 0737526 0131617 00400587 043579 0137471 0042671

Approximating 3rd order digital integrator for comparing PSO and GA119873119901

lowast 119873119892

50 lowast 50 100 lowast 100 200 lowast 200 50 lowast 50 100 lowast 100 200 lowast 200

Mean 0915406 0230096 00874599 0633554 0221398 00969881Median 01478565 0069402 002937085 01801123 01207528 00861992Standard deviation 0737526 0131617 00400587 0491451 0152073 00530145

[24] has been adopted for deriving half integrators fitted incontinuous-time domain (see Table 5) Also 2nd order opti-mized DI resulted in 2nd 4th and 6th order FOIs whereas3rd order optimized DI when substituted for different termsresulted in 3rd and 6th order half integrators In the TFs ofderived proposed half integrators given in Table 5 order ofoptimized DIs and order of FOIs are used as subscripts ofsymbol ldquo119867rdquo

4 Comparisons of Proposed FOIs Based on2nd and 3rd Order Optimized DIs with theExisting Models

Simulation results of comparison of magnitude responsesRME and phase responses of half integrators (see Table 5)based on different 2nd and 3rd order optimized DIs (given in(12) and (13)) with the existing half integrators given in [2 17]and ideal half integrator have been presented in Figures 5ndash7

Comparison of RME values of proposed and existingFOIs has been presented in Table 6 It is observed fromFigure 6 that the proposed 2nd and 3rd order FOIs namely1198672nd fo 2(119911) and 119867

3rd fo 3(119911) only lag in performance butstill these outperform the existing [2 17] models in higherfrequency spectrumThe6th order operator119867

2nd fo 6(119911) is the

best among the proposed FOIs and it clearly outperforms allthe existing models [2 17] in complete frequency spectrumwith RME of leminus40 dB Proposed 119867

2nd fo 2(119911) and 1198673rd fo 3(119911)

outperform Gupta-Madhu-Jain half integrator [17] in fre-quency range over 088 le 120596 le 3120587 radians 119867

2nd fo 4(119911)1198672nd fo 6(119911) and 119867

3rd fo 6(119911) half integrators excel Krishnahalf integrator [2] in almost complete range All the proposedFOIs are observed to be stable and follow linear phase curvein almost complete frequency spectrum (see Figure 7)

5 Conclusions

This paper attempts to focus on a simple yet unaddressedfact which till date has remained in the backdrop as theoptimization of an integer order operator before convertingit into its fractional the order (fo) counterparts ProposedFOIs validate that properties of an efficiently optimized DIpass directly into its fractional domain when it is used as 119904-to-119911 transformation during indirect discretization Here 2ndand 3rd order integer order integrators have been exploredby PSO algorithm for finding their optimal solutions for aminimized error fitness function The simulation results ofinteger order operators have revealed the effectiveness ofthe proposed modified PSO algorithm because of their low

Journal of Optimization 9

Table 5 Mathematical models of proposed half integrators based on 2nd and 3rd order PSO optimized DIs

Mathematical models of half integrators based on 2nd and 3rd order PSO optimized DIsName of PSO optimized DIs Transfer functions of FOIs derived by indirect discretization of CFE

119867PSO 2nd opt(119911)

1198672nd-fo-2(119911) = (09302 (119911 minus 01825) (119911 + 05869)

(119911 minus 07422) (119911 + 05604))

1198672nd-fo-4(119911) = (093011 (119911 minus 0003372) (119911 + 05633) (119911 + 06071) (119911 minus 06396)

(119911 minus 09034) (119911 minus 0293) (119911 + 05788) (119911 + 05566))

1198672nd-fo-6(119911) = (

(093011 (119911 minus 08073) (119911 minus 03427) (119911 + 0616)

(119911 + 05758) (119911 + 05587) (119911 + 004899))

((119911 + 05936) (119911 + 05649) (119911 + 05556)

(119911 minus 05924) (119911 minus 09502) (119911 minus 01107))

)

119867PSO 3rd opt(119911)

1198673rd-fo-3(119911) = (093046 (119911 minus 01852) (119911 + 02354) (119911 + 0433)

(119911 minus 0742) (119911 + 03892) (119911 + 02492))

1198673rd-fo-6(119911) = (

(093038 (119911 minus 06395) (119911 + 04678) (119911 + 03939)

(119911 + 02476) (119911 + 0223) (119911 minus 0008033))

((119911 minus 09031) (119911 minus 02946) (119911 + 02512)

(119911 + 02397) (119911 + 0383) (119911 + 04194))

)

Table 6 Comparisons of RME values of the proposed half integrators based on optimized DIs with the existing models

120596

(120587 rad)

Proposed half integrators based on 2nd orderPSO optimized DI

Proposed half integrators based on 3rdorder PSO optimized DI

Gupta-Madhu-Jain halfintegrator

Krishna halfintegrator

1198672nd-fo-2(119911) 1198672nd-fo-4(119911) 1198672nd-fo-6(119911) 1198673rd-fo-3(119911) 1198673rd-fo-6(119911)

025 minus1579 minus2455 minus4325 minus1579 minus2926 minus3553 minus174205 minus1752 minus3874 minus5585 minus1752 minus3897 minus5362 minus2831075 minus2582 minus4156 minus8514 minus2581 minus4261 minus5365 minus355610 minus583 minus583 minus5544 minus6401 minus7475 minus4526 minus4205125 minus3125 minus5734 minus5348 minus3113 minus5067 minus3893 minus3713175 minus2894 minus8115 minus7532 minus2982 minus5825 minus3359 minus405720 minus3262 minus7583 minus5554 minus3262 minus7325 minus3488 minus4646225 minus4143 minus5713 minus4899 minus4115 minus5893 minus4369 minus5528275 minus3299 minus596 minus596 minus3633 minus4694 minus326 minus29830 minus2956 minus5603 minus4775 minus2827 minus4674 minus3374 minus243

orders and very lessmagnitude errors as compared to existingones Proposed FOIs based on the optimizedDIs present veryless RME values of the order of minus40 dB and show linear phasecurves in almost full band of Nyquist frequency

References

[1] L Debnath ldquoRecent applications of fractional calculus toscience and engineeringrdquo International Journal of Mathematicsand Mathematical Sciences vol 2003 no 54 pp 3413ndash34422003

[2] B T Krishna ldquoStudies on fractional order differentiators andintegrators a surveyrdquo Signal Processing vol 91 no 3 pp 386ndash426 2011

[3] N M F Ferreira F B Duarte M F M Lima M G Marcosand J A T Machado ldquoApplication of fractional calculus in thedynamical analysis and control of mechanical manipulatorsrdquoFractional Calculus and Applied Analysis vol 11 pp 91ndash1132008

[4] S Manabe ldquoThe noninteger integral and its application tocontrol systemsrdquo English Translation Journal Japan vol 6 pp83ndash87 1961

[5] J A T Machado and A M S Galhano ldquoA new methodfor approximating fractional derivatives application in non-linear controlrdquo in Proceedings of the 6th EUROMECHNonlinearDynamics Conference (ENOC rsquo08) pp 4ndash8 Saint PetersburgRussia July 2008

[6] Y Ferdi J P Herbeuval A Charef and B Boucheham ldquoR wavedetection using fractional digital differentiationrdquo ITBM-RBMvol 24 no 5-6 pp 273ndash280 2003

[7] J Mocak I Janiga M Rievaj and D Bustin ldquoThe use of frac-tional differentiation or integration for signal improvementrdquoMeasurement Science Review vol 7 pp 39ndash42 2007

[8] B Mathieu P Melchior A Oustaloup and C Ceyral ldquoFrac-tional differentiation for edge detectionrdquo Signal Processing vol83 no 11 pp 2421ndash2432 2003

[9] Y Pu W Wang J Zhou Y Wang and H Jia ldquoFractionaldifferential approach to detecting textural features of digital

10 Journal of Optimization

image and its fractional differential filter implementationrdquoScience in China F vol 51 no 9 pp 1319ndash1339 2008

[10] S Das I Pan S Das andA Gupta ldquoImprovedmodel reductionand tuning of fractional-order PI120582D120583 controllers for analyticalrule extraction with genetic programmingrdquo ISA Transactionsvol 51 no 2 pp 237ndash261 2012

[11] S Das I Pan K Halder S Das and A Gupta ldquoImpact offractional order integral performance indices in LQR basedPID controller design via optimum selection of weightingmatricesrdquo in Proceedings of the International Conference onComputer Communication and Informatics (ICCCI rsquo12) pp 1ndash6Coimbatore India January 2012

[12] Y Chen and B M Vinagre ldquoA new IIR-type digital fractionalorder differentiatorrdquo Signal Processing vol 83 no 11 pp 2359ndash2365 2003

[13] Y Chen B M Vinagre and I Podlubny ldquoContinued frac-tion expansion approaches to discretizing fractional orderderivatives-an expository reviewrdquo Nonlinear Dynamics vol 38no 1ndash4 pp 155ndash170 2004

[14] Y Q Chen and K L Moore ldquoDiscretization schemes forfractional-order differentiators and integratorsrdquo IEEE Transac-tions on Circuits and Systems I vol 49 no 3 pp 363ndash367 2002

[15] M A Al-Alaoui ldquoNovel class of digital integrators and differ-entiatorsrdquo IEEE Transactions on Signal Processing p 1 2009

[16] M A Al-Alaoui ldquoAl-Alaoui operator and the new transfor-mation polynomials for discretization of analogue systemsrdquoElectrical Engineering vol 90 no 6 pp 455ndash467 2008

[17] M Gupta M Jain and N Jain ldquoA new fractional orderrecursive digital integrator using continued fraction expansionrdquoin Proceedings of the India International Conference on Powerelectronics (IICPE) pp 1ndash3 New Delhi India 2010

[18] M Gupta P Varshney and G S Visweswaran ldquoDigitalfractional-order differentiator and integrator models based onfirst-order and higher order operatorsrdquo International Journal ofCircuitTheory and Applications vol 39 no 5 pp 461ndash474 2011

[19] F Leulmi and Y Ferdi ldquoAn improvement of the rationalapproximation of the fractional operator 119904

120572rdquo in Proceedingsof the Saudi International Electronics Communications andPhotonics Conference (SIECP rsquo11) pp 1ndash6 Riyadh Saudi Arabia2011

[20] Y Ferdi ldquoComputation of fractional order derivative andintegral via power series expansion and signal modellingrdquoNonlinear Dynamics vol 46 no 1-2 pp 1ndash15 2006

[21] G S Visweswaran P Varshney and M Gupta ldquoNew approachto realize fractional power in z-domain at low frequencyrdquo IEEETransactions on Circuits and Systems II vol 58 no 3 pp 179ndash183 2011

[22] P Varshney M Gupta and G S Visweswaran ldquoSwitchedcapacitor realizations of fractional-order differentiators andintegrators based on an operator with improved performancerdquoRadioengineering vol 20 no 1 pp 340ndash348 2011

[23] B T Krishna and K V V Reddy ldquoDesign of fractional orderdigital differentiators and integrators using indirect discretiza-tionrdquo An International Journal for Theory and Applications vol11 pp 143ndash151 2008

[24] B T Krishna and K V V Reddy ldquoDesign of digital differentia-tors and integrators of order 12rdquo World Journal of Modellingand Simulation vol 4 pp 182ndash187 2008

[25] R Yadav and M Gupta ldquoDesign of fractional order differen-tiators and integrators using indirect discretization approachrdquoin Proceedings of the 2nd International Conference on Advances

in Recent Technologies in Communication and Computing (ART-Com rsquo10) pp 126ndash130 Kottayam India October 2010

[26] R Yadav and M Gupta ldquoDesign of fractional order differen-tiators and integrators using indirect discretization schemerdquoin Proceedings of the India International Conference on PowerElectronics (IICPE rsquo10) New Delhi India January 2011

[27] G E Carlson and C A Halijak ldquoSimulation of the fractionalderivative operator (1119904)

12 and the fractional integral operator(1119904)12rdquo Kansas State University Bulletin vol 45 pp 1ndash22 1961

[28] S K Steiglitz ldquoComputer-aided design of recursive digitalfiltersrdquo IEEE Transactions Audio and Electroacoustics vol AU-18 no 2 pp 123ndash129 1970

[29] A N Khovanskii The Application of Continued Fractions andtheir Generalizations to Problems in Approximation Theory PNoordhoff Ltd 1963 Translate by Peter Wynn

[30] S C Roy ldquoSome exact steady-state sinewave solutions ofnonuniform RC linesrdquo British Journal of Applied Physics vol 14pp 378ndash380 1963

[31] A Oustaloup F Levron B Mathieu and F M NanotldquoFrequency-band complex noninteger differentiator charac-terization and synthesisrdquo IEEE Transactions on Circuits andSystems I vol 47 no 1 pp 25ndash39 2000

[32] G Maione ldquoLaguerre approximation of fractional systemsrdquoElectronics Letters vol 38 no 20 pp 1234ndash1236 2002

[33] N Papamarkos and C Chamzas ldquoA new approach for thedesign of digital integratorsrdquo IEEE Transactions on Circuits andSystems I vol 43 no 9 pp 785ndash791 1996

[34] M Gupta M Jain and B Kumar ldquoRecursive wideband digitalintegrator and differentiatorrdquo International Journal of CircuitTheory and Applications vol 39 no 7 pp 775ndash782 2011

[35] N Jain M Gupta and M Jain ldquoLinear phase second orderrecursive digital integrators and differentiatorsrdquo Radioengineer-ing Journal vol 21 pp 712ndash717 2012

[36] C-C Hse W-Y Wang and C-Y Yu ldquoGenetic algorithm-derived digital integrators and their applications in discretiza-tion of continuous systemsrdquo in Proceedings of the CEC Congresson Evolutionary Computation pp 443ndash448 Honolulu HawaiiUSA 2002

[37] S Das B Majumder A Pakhira I Pan S Das and AGupta ldquoOptimizing continued fraction expansion based iirrealization of fractional order differ-integrators with geneticalgorithmrdquo in Proceedings of the International Conference onProcess Automation Control and Computing (PACC rsquo11) pp 1ndash6Coimbatore India July 2011

[38] G K Thakur A Gupta and D Upadhyay ldquoOptimizationalgorithmapproach for errorminimization of digital integratorsand differentiatorsrdquo in Proceedings of the International Con-ference on Advances in Recent Technologies in Communicationand Computing (ARTCom rsquo09) pp 118ndash122 Kottayam IndiaOctober 2009

[39] M A Al-Alaoui ldquoClass of digital integrators and differentia-torsrdquo IET Signal Processing vol 5 no 2 pp 251ndash260 2011

[40] D K Upadhyay and R K Singh ldquoRecursive wideband digitaldifferentiator and integratorrdquo Electronics Letters vol 47 no 11pp 647ndash648 2011

[41] D K Upadhyay ldquoClass of recursive wide band digital differen-tiators and integratorsrdquo Radioengineering Journal vol 21 no 3pp 904ndash910 2012

[42] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks pp 1942ndash1948 December 1995

Journal of Optimization 11

[43] Y-T Kao and E Zahara ldquoA hybrid genetic algorithm andparticle swarm optimization formultimodal functionsrdquoAppliedSoft Computing Journal vol 8 no 2 pp 849ndash857 2008

[44] Y Shi and R C Eberhart ldquoEmpirical study of particle swarmoptimizationrdquo in Proceedings of the IEEE international Con-ference Evolution Computer pp 1945ndash1950 Washington DCUSA 1999

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Journal of Optimization

0

20

40

Frequency

Rel

ativ

e mag

nitu

de er

ror (

dB)

Relative magnitude errors (dB) of proposed half integrators

minus20

minus40

minus60

minus80

minus100

minus1200 05 1 15 2 25 3

based on PSO optimized DIs

Gupta-Madhu-JainKrishna half integrator

H2nd fo 2(z)H2nd fo 4(z)H2nd fo 6(z)H3rd fo 3(z)

H3rd fo 6(z)

Figure 6 Comparison of RME values of proposed half integratorsbased on 2nd and 3rd order optimized DIs with existing [2 17]models

[34] GA [35] SA [39] and PZ [41] for validating efficientperformance of proposed PSO Comparisons of bode dia-grams (ie magnitude versus the frequency (119908) and phase(degrees) versus frequency (119908)) and RME responses (in dB)have been presented in Figures 2 3 and 4 respectively

Results for performance values (RME) for each opti-mization method have been compared in Table 3 It canbe observed from Figure 3 that the proposed PSO methodclearly outperforms existing SA [39] and LP [34] techniquesand gives comparable results for PZ [41] and GA [35] meth-ods The 2nd order DI namely 119867PSO 2nd opt(119911) given in (12)outperforms Al-Alaoui 3-segment and 4-segment optimizedDIs in complete spectrum 119867PSO 2nd opt(119911) excels Jain-Gupta-Jain DI [35] in frequency range over 0 le 120596 le 05120587 radiansand 258 le 120596 le 3120587 radians PSO optimized proposed 3rdorder DI namely 119867PSO 3rd opt(119911) clearly outperforms theexisting Al-Alaoui optimized 3-segment 4-segment [39] andUpadhyay [41] DIs in almost complete range with RME of le

minus41 dB 119867PSO 3rd opt(119911) given in (13) gives comparable resultswith GA optimized [35] operator Proposed optimized DIsare observed to satisfy stability criterion and show linearlydecreasing phase responses (see Figure 4)

Proposed DIs optimized by PSO clearly excel operatorsoptimized by SA [39] LP [34] and PZ [41] techniques butlie in close vicinity of the GA [35] responses So a statisticaltest has been done for comparing performances of PSOand GA for validating superiority of the proposed modifiedalgorithm The products of number of particles and numberof generations (119873

119901lowast 119873119892) have been varied from (50 lowast 50) to

(100 lowast 100) and then to (200 lowast 200) for different iterationsand values of mean median and standard deviation have

Pha

se (d

eg)

Phase responses of proposed half integrators

10

0

minus10

minus20

minus30

minus40

minus50

based on PSO optimized DIs

Frequency0 05 1 15 2 25 3

Ideal

Gupta-Madhu-JainKrishna half integrator

H2nd fo 2(z)H2nd fo 4(z)H2nd fo 6(z)

H3rd fo 3(z)H3rd fo 6(z)

Figure 7 Comparison of phase responses of proposed half integra-tors based on 2nd and 3rd order optimized DIs with existing [2 17]models

been registered for all the worst average and best results of2nd and 3rd order optimized DIs (see Table 4) From thistable it is clear that PSO technique performs better than GAmethod and also it was noticed that with the increase invalues of 119873

119901 119873119892 and number of iterations the response of

resultant DIs comes closer to the ideal response as meanmedian and standard deviation values go on decreasing withefficientminimization of error function In this paperwe haveconsidered values of root mean square error

3 Discretization ofProposed 2nd and 3rd Order OptimizedDIs for Half Integrator Models

Theoptimized integer order DIs derived in Section 2 are usedas 119904-to-119911 transformations for CFE based indirect discretiza-tion [2 23 24] scheme Formula given by Khovanskii in [29]has been used for different rational approximations of FOIsThat is

(119904)plusmn120572

=1

1minus

120572119904

1+(1+120572) 119904minus

(1 + 120572) 119904 (1 + 119904)

2+(3+120572) 119904minus

2 (2 + 120572) 119904 (1 + 119904)

3 + (5+120572) 119904minus

times3 (3 + 120572) 119904 (1 + 119904)

4 + (9 + 120572) 119904minus

4 (4 + 120572) 119904 (1 + 119904)

5 + (9 + 120572) 119904minus

times5 (5 + 120572) 119904 (1 + 119904)

6 + (11 + 120572) 119904minus

6 (6 + 120572) 119904 (1 + 119904)

7 + (13 + 120572) 119904minus

(14)

Infinite series of (14) can be terminated up to 2 4and 6 terms and CFE based indirect discretization used in

Journal of Optimization 7

Table2Transfe

rfun

ctions

andfitnessvalues

ofworstandaveragec

ases

of3rdorderP

SOop

timized

DIs

TFso

fworstcaseso

f3rd

orderD

Isalon

gwith

theirfi

tnessfun

ctionvalues

(FV)

TFso

faverage

caseso

f3rd

orderD

Isalon

gwith

theirfi

tnessfun

ctionvalues

(FV)

Transfe

rfun

ction

FV

Transfe

rfun

ction

FV

1198671(119911

)=

(01

423941199113

+04

285111199112

minus03

06611119911

+02

17758

1199113

minus09

031351199112

minus06

74295119911

+05

68774

)066

6364

1198671(119911

)=

(minus

06

56851199113

minus08

062451199112

minus08

20988119911

minus01

9093

1199113

+01

005341199112

minus04

73273119911

minus05

50301

)0548217

1198672(119911

)=

(minus

07

387211199113

minus04

255611199112

minus07

87252119911

+05

88187

1199113

minus00

8455311199112

minus07

14103119911

minus02

4424

)238383

1198672(119911

)=

(minus

01

711661199113

+07

950271199112

+03

39133119911

+04

81229

1199113

minus07

403291199112

minus00

821587119911

minus01

30453

)0651617

1198673(119911

)=

(05

248521199113

+09

588131199112

+05

85323119911

+07

79817

1199113

minus04

627791199112

minus03

80859119911

minus02

47247

)391385

1198673(119911

)=

(05

229891199113

+01

636031199112

minus00

262705119911

+04

83852

1199113

minus09

798611199112

minus01

40418119911

+00

831846

)0673745

1198674(119911

)=

(09

790991199113

minus08

529211199112

+09

60434119911

+08

75076

1199113

minus08

270921199112

minus02

46012119911

+01

35716

)406169

1198674(119911

)=

(02

258041199113

+07

800811199112

+07

58662119911

minus00

994224

1199113

minus02

373481199112

minus05

75985119911

minus01

3299

)0713815

1198675(119911

)=

(00

5946561199113

+07

000181199112

+03

46537119911

+03

14347

1199113

minus07

918671199112

minus05

73836119887119911

+04

10591

)468934

1198675(119911

)=

(01

238551199113

+04

917661199112

+07

27444119911

+05

68523

1199113

minus03

196421199112

minus02

87981119911

minus03

31056

)09760

67

8 Journal of Optimization

Table 3 Comparisons of RME values of the proposed PSO optimized DIs with other integer order existing integrators optimized by differentoptimization techniques

120596 (120587 rad)Al-Alaoui3-segment

(SA)

Al-Alaoui4-segment

(SA)

Jain-Gupta-Jain(GA)

Upadhyay(P-Z)

Gupta-Jain-Kumar(LP)

Proposed2nd orderDI (PSO)

Proposed3rd orderDI (PSO)

025 minus7203 minus6018 minus5446 minus4831 minus7106 minus7203 minus638205 minus4915 minus4121 minus5947 minus5165 minus5989 minus5968 minus7304075 minus3725 minus3402 minus8004 minus6171 minus6391 minus5327 minus714510 minus3029 minus3034 minus5728 minus5728 minus4994 minus4988 minus6463125 minus2272 minus3296 minus5384 minus4871 minus3933 minus4871 minus682175 minus1164 minus3505 minus5385 minus4791 minus3146 minus6262 minus54220 minus824 minus1911 minus6053 minus5289 minus3143 minus4998 minus4994225 minus692 minus1599 minus5384 minus615 minus345 minus4277 minus5366275 minus1819 minus1973 minus3754 minus3776 minus4588 minus5555 minus406130 minus417 minus1811 minus3552 minus3644 minus3014 minus4134 minus4501

Table 4 Statistical test for comparison of proposed PSO and GA techniques for approximations of 2nd and 3rd order digital integrators

Genetic algorithm Modified PSO algorithmApproximating 2nd order digital integrator for comparing PSO and GA

119873119901

lowast 119873119892

50 lowast 50 100 lowast 100 200 lowast 200 50 lowast 50 100 lowast 100 200 lowast 200

Mean 0915405 0230096 00874599 0614370 0201256 00936231Median 0482815 0194402 0075695 0491575 0114377 00488172Standard deviation 0737526 0131617 00400587 043579 0137471 0042671

Approximating 3rd order digital integrator for comparing PSO and GA119873119901

lowast 119873119892

50 lowast 50 100 lowast 100 200 lowast 200 50 lowast 50 100 lowast 100 200 lowast 200

Mean 0915406 0230096 00874599 0633554 0221398 00969881Median 01478565 0069402 002937085 01801123 01207528 00861992Standard deviation 0737526 0131617 00400587 0491451 0152073 00530145

[24] has been adopted for deriving half integrators fitted incontinuous-time domain (see Table 5) Also 2nd order opti-mized DI resulted in 2nd 4th and 6th order FOIs whereas3rd order optimized DI when substituted for different termsresulted in 3rd and 6th order half integrators In the TFs ofderived proposed half integrators given in Table 5 order ofoptimized DIs and order of FOIs are used as subscripts ofsymbol ldquo119867rdquo

4 Comparisons of Proposed FOIs Based on2nd and 3rd Order Optimized DIs with theExisting Models

Simulation results of comparison of magnitude responsesRME and phase responses of half integrators (see Table 5)based on different 2nd and 3rd order optimized DIs (given in(12) and (13)) with the existing half integrators given in [2 17]and ideal half integrator have been presented in Figures 5ndash7

Comparison of RME values of proposed and existingFOIs has been presented in Table 6 It is observed fromFigure 6 that the proposed 2nd and 3rd order FOIs namely1198672nd fo 2(119911) and 119867

3rd fo 3(119911) only lag in performance butstill these outperform the existing [2 17] models in higherfrequency spectrumThe6th order operator119867

2nd fo 6(119911) is the

best among the proposed FOIs and it clearly outperforms allthe existing models [2 17] in complete frequency spectrumwith RME of leminus40 dB Proposed 119867

2nd fo 2(119911) and 1198673rd fo 3(119911)

outperform Gupta-Madhu-Jain half integrator [17] in fre-quency range over 088 le 120596 le 3120587 radians 119867

2nd fo 4(119911)1198672nd fo 6(119911) and 119867

3rd fo 6(119911) half integrators excel Krishnahalf integrator [2] in almost complete range All the proposedFOIs are observed to be stable and follow linear phase curvein almost complete frequency spectrum (see Figure 7)

5 Conclusions

This paper attempts to focus on a simple yet unaddressedfact which till date has remained in the backdrop as theoptimization of an integer order operator before convertingit into its fractional the order (fo) counterparts ProposedFOIs validate that properties of an efficiently optimized DIpass directly into its fractional domain when it is used as 119904-to-119911 transformation during indirect discretization Here 2ndand 3rd order integer order integrators have been exploredby PSO algorithm for finding their optimal solutions for aminimized error fitness function The simulation results ofinteger order operators have revealed the effectiveness ofthe proposed modified PSO algorithm because of their low

Journal of Optimization 9

Table 5 Mathematical models of proposed half integrators based on 2nd and 3rd order PSO optimized DIs

Mathematical models of half integrators based on 2nd and 3rd order PSO optimized DIsName of PSO optimized DIs Transfer functions of FOIs derived by indirect discretization of CFE

119867PSO 2nd opt(119911)

1198672nd-fo-2(119911) = (09302 (119911 minus 01825) (119911 + 05869)

(119911 minus 07422) (119911 + 05604))

1198672nd-fo-4(119911) = (093011 (119911 minus 0003372) (119911 + 05633) (119911 + 06071) (119911 minus 06396)

(119911 minus 09034) (119911 minus 0293) (119911 + 05788) (119911 + 05566))

1198672nd-fo-6(119911) = (

(093011 (119911 minus 08073) (119911 minus 03427) (119911 + 0616)

(119911 + 05758) (119911 + 05587) (119911 + 004899))

((119911 + 05936) (119911 + 05649) (119911 + 05556)

(119911 minus 05924) (119911 minus 09502) (119911 minus 01107))

)

119867PSO 3rd opt(119911)

1198673rd-fo-3(119911) = (093046 (119911 minus 01852) (119911 + 02354) (119911 + 0433)

(119911 minus 0742) (119911 + 03892) (119911 + 02492))

1198673rd-fo-6(119911) = (

(093038 (119911 minus 06395) (119911 + 04678) (119911 + 03939)

(119911 + 02476) (119911 + 0223) (119911 minus 0008033))

((119911 minus 09031) (119911 minus 02946) (119911 + 02512)

(119911 + 02397) (119911 + 0383) (119911 + 04194))

)

Table 6 Comparisons of RME values of the proposed half integrators based on optimized DIs with the existing models

120596

(120587 rad)

Proposed half integrators based on 2nd orderPSO optimized DI

Proposed half integrators based on 3rdorder PSO optimized DI

Gupta-Madhu-Jain halfintegrator

Krishna halfintegrator

1198672nd-fo-2(119911) 1198672nd-fo-4(119911) 1198672nd-fo-6(119911) 1198673rd-fo-3(119911) 1198673rd-fo-6(119911)

025 minus1579 minus2455 minus4325 minus1579 minus2926 minus3553 minus174205 minus1752 minus3874 minus5585 minus1752 minus3897 minus5362 minus2831075 minus2582 minus4156 minus8514 minus2581 minus4261 minus5365 minus355610 minus583 minus583 minus5544 minus6401 minus7475 minus4526 minus4205125 minus3125 minus5734 minus5348 minus3113 minus5067 minus3893 minus3713175 minus2894 minus8115 minus7532 minus2982 minus5825 minus3359 minus405720 minus3262 minus7583 minus5554 minus3262 minus7325 minus3488 minus4646225 minus4143 minus5713 minus4899 minus4115 minus5893 minus4369 minus5528275 minus3299 minus596 minus596 minus3633 minus4694 minus326 minus29830 minus2956 minus5603 minus4775 minus2827 minus4674 minus3374 minus243

orders and very lessmagnitude errors as compared to existingones Proposed FOIs based on the optimizedDIs present veryless RME values of the order of minus40 dB and show linear phasecurves in almost full band of Nyquist frequency

References

[1] L Debnath ldquoRecent applications of fractional calculus toscience and engineeringrdquo International Journal of Mathematicsand Mathematical Sciences vol 2003 no 54 pp 3413ndash34422003

[2] B T Krishna ldquoStudies on fractional order differentiators andintegrators a surveyrdquo Signal Processing vol 91 no 3 pp 386ndash426 2011

[3] N M F Ferreira F B Duarte M F M Lima M G Marcosand J A T Machado ldquoApplication of fractional calculus in thedynamical analysis and control of mechanical manipulatorsrdquoFractional Calculus and Applied Analysis vol 11 pp 91ndash1132008

[4] S Manabe ldquoThe noninteger integral and its application tocontrol systemsrdquo English Translation Journal Japan vol 6 pp83ndash87 1961

[5] J A T Machado and A M S Galhano ldquoA new methodfor approximating fractional derivatives application in non-linear controlrdquo in Proceedings of the 6th EUROMECHNonlinearDynamics Conference (ENOC rsquo08) pp 4ndash8 Saint PetersburgRussia July 2008

[6] Y Ferdi J P Herbeuval A Charef and B Boucheham ldquoR wavedetection using fractional digital differentiationrdquo ITBM-RBMvol 24 no 5-6 pp 273ndash280 2003

[7] J Mocak I Janiga M Rievaj and D Bustin ldquoThe use of frac-tional differentiation or integration for signal improvementrdquoMeasurement Science Review vol 7 pp 39ndash42 2007

[8] B Mathieu P Melchior A Oustaloup and C Ceyral ldquoFrac-tional differentiation for edge detectionrdquo Signal Processing vol83 no 11 pp 2421ndash2432 2003

[9] Y Pu W Wang J Zhou Y Wang and H Jia ldquoFractionaldifferential approach to detecting textural features of digital

10 Journal of Optimization

image and its fractional differential filter implementationrdquoScience in China F vol 51 no 9 pp 1319ndash1339 2008

[10] S Das I Pan S Das andA Gupta ldquoImprovedmodel reductionand tuning of fractional-order PI120582D120583 controllers for analyticalrule extraction with genetic programmingrdquo ISA Transactionsvol 51 no 2 pp 237ndash261 2012

[11] S Das I Pan K Halder S Das and A Gupta ldquoImpact offractional order integral performance indices in LQR basedPID controller design via optimum selection of weightingmatricesrdquo in Proceedings of the International Conference onComputer Communication and Informatics (ICCCI rsquo12) pp 1ndash6Coimbatore India January 2012

[12] Y Chen and B M Vinagre ldquoA new IIR-type digital fractionalorder differentiatorrdquo Signal Processing vol 83 no 11 pp 2359ndash2365 2003

[13] Y Chen B M Vinagre and I Podlubny ldquoContinued frac-tion expansion approaches to discretizing fractional orderderivatives-an expository reviewrdquo Nonlinear Dynamics vol 38no 1ndash4 pp 155ndash170 2004

[14] Y Q Chen and K L Moore ldquoDiscretization schemes forfractional-order differentiators and integratorsrdquo IEEE Transac-tions on Circuits and Systems I vol 49 no 3 pp 363ndash367 2002

[15] M A Al-Alaoui ldquoNovel class of digital integrators and differ-entiatorsrdquo IEEE Transactions on Signal Processing p 1 2009

[16] M A Al-Alaoui ldquoAl-Alaoui operator and the new transfor-mation polynomials for discretization of analogue systemsrdquoElectrical Engineering vol 90 no 6 pp 455ndash467 2008

[17] M Gupta M Jain and N Jain ldquoA new fractional orderrecursive digital integrator using continued fraction expansionrdquoin Proceedings of the India International Conference on Powerelectronics (IICPE) pp 1ndash3 New Delhi India 2010

[18] M Gupta P Varshney and G S Visweswaran ldquoDigitalfractional-order differentiator and integrator models based onfirst-order and higher order operatorsrdquo International Journal ofCircuitTheory and Applications vol 39 no 5 pp 461ndash474 2011

[19] F Leulmi and Y Ferdi ldquoAn improvement of the rationalapproximation of the fractional operator 119904

120572rdquo in Proceedingsof the Saudi International Electronics Communications andPhotonics Conference (SIECP rsquo11) pp 1ndash6 Riyadh Saudi Arabia2011

[20] Y Ferdi ldquoComputation of fractional order derivative andintegral via power series expansion and signal modellingrdquoNonlinear Dynamics vol 46 no 1-2 pp 1ndash15 2006

[21] G S Visweswaran P Varshney and M Gupta ldquoNew approachto realize fractional power in z-domain at low frequencyrdquo IEEETransactions on Circuits and Systems II vol 58 no 3 pp 179ndash183 2011

[22] P Varshney M Gupta and G S Visweswaran ldquoSwitchedcapacitor realizations of fractional-order differentiators andintegrators based on an operator with improved performancerdquoRadioengineering vol 20 no 1 pp 340ndash348 2011

[23] B T Krishna and K V V Reddy ldquoDesign of fractional orderdigital differentiators and integrators using indirect discretiza-tionrdquo An International Journal for Theory and Applications vol11 pp 143ndash151 2008

[24] B T Krishna and K V V Reddy ldquoDesign of digital differentia-tors and integrators of order 12rdquo World Journal of Modellingand Simulation vol 4 pp 182ndash187 2008

[25] R Yadav and M Gupta ldquoDesign of fractional order differen-tiators and integrators using indirect discretization approachrdquoin Proceedings of the 2nd International Conference on Advances

in Recent Technologies in Communication and Computing (ART-Com rsquo10) pp 126ndash130 Kottayam India October 2010

[26] R Yadav and M Gupta ldquoDesign of fractional order differen-tiators and integrators using indirect discretization schemerdquoin Proceedings of the India International Conference on PowerElectronics (IICPE rsquo10) New Delhi India January 2011

[27] G E Carlson and C A Halijak ldquoSimulation of the fractionalderivative operator (1119904)

12 and the fractional integral operator(1119904)12rdquo Kansas State University Bulletin vol 45 pp 1ndash22 1961

[28] S K Steiglitz ldquoComputer-aided design of recursive digitalfiltersrdquo IEEE Transactions Audio and Electroacoustics vol AU-18 no 2 pp 123ndash129 1970

[29] A N Khovanskii The Application of Continued Fractions andtheir Generalizations to Problems in Approximation Theory PNoordhoff Ltd 1963 Translate by Peter Wynn

[30] S C Roy ldquoSome exact steady-state sinewave solutions ofnonuniform RC linesrdquo British Journal of Applied Physics vol 14pp 378ndash380 1963

[31] A Oustaloup F Levron B Mathieu and F M NanotldquoFrequency-band complex noninteger differentiator charac-terization and synthesisrdquo IEEE Transactions on Circuits andSystems I vol 47 no 1 pp 25ndash39 2000

[32] G Maione ldquoLaguerre approximation of fractional systemsrdquoElectronics Letters vol 38 no 20 pp 1234ndash1236 2002

[33] N Papamarkos and C Chamzas ldquoA new approach for thedesign of digital integratorsrdquo IEEE Transactions on Circuits andSystems I vol 43 no 9 pp 785ndash791 1996

[34] M Gupta M Jain and B Kumar ldquoRecursive wideband digitalintegrator and differentiatorrdquo International Journal of CircuitTheory and Applications vol 39 no 7 pp 775ndash782 2011

[35] N Jain M Gupta and M Jain ldquoLinear phase second orderrecursive digital integrators and differentiatorsrdquo Radioengineer-ing Journal vol 21 pp 712ndash717 2012

[36] C-C Hse W-Y Wang and C-Y Yu ldquoGenetic algorithm-derived digital integrators and their applications in discretiza-tion of continuous systemsrdquo in Proceedings of the CEC Congresson Evolutionary Computation pp 443ndash448 Honolulu HawaiiUSA 2002

[37] S Das B Majumder A Pakhira I Pan S Das and AGupta ldquoOptimizing continued fraction expansion based iirrealization of fractional order differ-integrators with geneticalgorithmrdquo in Proceedings of the International Conference onProcess Automation Control and Computing (PACC rsquo11) pp 1ndash6Coimbatore India July 2011

[38] G K Thakur A Gupta and D Upadhyay ldquoOptimizationalgorithmapproach for errorminimization of digital integratorsand differentiatorsrdquo in Proceedings of the International Con-ference on Advances in Recent Technologies in Communicationand Computing (ARTCom rsquo09) pp 118ndash122 Kottayam IndiaOctober 2009

[39] M A Al-Alaoui ldquoClass of digital integrators and differentia-torsrdquo IET Signal Processing vol 5 no 2 pp 251ndash260 2011

[40] D K Upadhyay and R K Singh ldquoRecursive wideband digitaldifferentiator and integratorrdquo Electronics Letters vol 47 no 11pp 647ndash648 2011

[41] D K Upadhyay ldquoClass of recursive wide band digital differen-tiators and integratorsrdquo Radioengineering Journal vol 21 no 3pp 904ndash910 2012

[42] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks pp 1942ndash1948 December 1995

Journal of Optimization 11

[43] Y-T Kao and E Zahara ldquoA hybrid genetic algorithm andparticle swarm optimization formultimodal functionsrdquoAppliedSoft Computing Journal vol 8 no 2 pp 849ndash857 2008

[44] Y Shi and R C Eberhart ldquoEmpirical study of particle swarmoptimizationrdquo in Proceedings of the IEEE international Con-ference Evolution Computer pp 1945ndash1950 Washington DCUSA 1999

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Optimization 7

Table2Transfe

rfun

ctions

andfitnessvalues

ofworstandaveragec

ases

of3rdorderP

SOop

timized

DIs

TFso

fworstcaseso

f3rd

orderD

Isalon

gwith

theirfi

tnessfun

ctionvalues

(FV)

TFso

faverage

caseso

f3rd

orderD

Isalon

gwith

theirfi

tnessfun

ctionvalues

(FV)

Transfe

rfun

ction

FV

Transfe

rfun

ction

FV

1198671(119911

)=

(01

423941199113

+04

285111199112

minus03

06611119911

+02

17758

1199113

minus09

031351199112

minus06

74295119911

+05

68774

)066

6364

1198671(119911

)=

(minus

06

56851199113

minus08

062451199112

minus08

20988119911

minus01

9093

1199113

+01

005341199112

minus04

73273119911

minus05

50301

)0548217

1198672(119911

)=

(minus

07

387211199113

minus04

255611199112

minus07

87252119911

+05

88187

1199113

minus00

8455311199112

minus07

14103119911

minus02

4424

)238383

1198672(119911

)=

(minus

01

711661199113

+07

950271199112

+03

39133119911

+04

81229

1199113

minus07

403291199112

minus00

821587119911

minus01

30453

)0651617

1198673(119911

)=

(05

248521199113

+09

588131199112

+05

85323119911

+07

79817

1199113

minus04

627791199112

minus03

80859119911

minus02

47247

)391385

1198673(119911

)=

(05

229891199113

+01

636031199112

minus00

262705119911

+04

83852

1199113

minus09

798611199112

minus01

40418119911

+00

831846

)0673745

1198674(119911

)=

(09

790991199113

minus08

529211199112

+09

60434119911

+08

75076

1199113

minus08

270921199112

minus02

46012119911

+01

35716

)406169

1198674(119911

)=

(02

258041199113

+07

800811199112

+07

58662119911

minus00

994224

1199113

minus02

373481199112

minus05

75985119911

minus01

3299

)0713815

1198675(119911

)=

(00

5946561199113

+07

000181199112

+03

46537119911

+03

14347

1199113

minus07

918671199112

minus05

73836119887119911

+04

10591

)468934

1198675(119911

)=

(01

238551199113

+04

917661199112

+07

27444119911

+05

68523

1199113

minus03

196421199112

minus02

87981119911

minus03

31056

)09760

67

8 Journal of Optimization

Table 3 Comparisons of RME values of the proposed PSO optimized DIs with other integer order existing integrators optimized by differentoptimization techniques

120596 (120587 rad)Al-Alaoui3-segment

(SA)

Al-Alaoui4-segment

(SA)

Jain-Gupta-Jain(GA)

Upadhyay(P-Z)

Gupta-Jain-Kumar(LP)

Proposed2nd orderDI (PSO)

Proposed3rd orderDI (PSO)

025 minus7203 minus6018 minus5446 minus4831 minus7106 minus7203 minus638205 minus4915 minus4121 minus5947 minus5165 minus5989 minus5968 minus7304075 minus3725 minus3402 minus8004 minus6171 minus6391 minus5327 minus714510 minus3029 minus3034 minus5728 minus5728 minus4994 minus4988 minus6463125 minus2272 minus3296 minus5384 minus4871 minus3933 minus4871 minus682175 minus1164 minus3505 minus5385 minus4791 minus3146 minus6262 minus54220 minus824 minus1911 minus6053 minus5289 minus3143 minus4998 minus4994225 minus692 minus1599 minus5384 minus615 minus345 minus4277 minus5366275 minus1819 minus1973 minus3754 minus3776 minus4588 minus5555 minus406130 minus417 minus1811 minus3552 minus3644 minus3014 minus4134 minus4501

Table 4 Statistical test for comparison of proposed PSO and GA techniques for approximations of 2nd and 3rd order digital integrators

Genetic algorithm Modified PSO algorithmApproximating 2nd order digital integrator for comparing PSO and GA

119873119901

lowast 119873119892

50 lowast 50 100 lowast 100 200 lowast 200 50 lowast 50 100 lowast 100 200 lowast 200

Mean 0915405 0230096 00874599 0614370 0201256 00936231Median 0482815 0194402 0075695 0491575 0114377 00488172Standard deviation 0737526 0131617 00400587 043579 0137471 0042671

Approximating 3rd order digital integrator for comparing PSO and GA119873119901

lowast 119873119892

50 lowast 50 100 lowast 100 200 lowast 200 50 lowast 50 100 lowast 100 200 lowast 200

Mean 0915406 0230096 00874599 0633554 0221398 00969881Median 01478565 0069402 002937085 01801123 01207528 00861992Standard deviation 0737526 0131617 00400587 0491451 0152073 00530145

[24] has been adopted for deriving half integrators fitted incontinuous-time domain (see Table 5) Also 2nd order opti-mized DI resulted in 2nd 4th and 6th order FOIs whereas3rd order optimized DI when substituted for different termsresulted in 3rd and 6th order half integrators In the TFs ofderived proposed half integrators given in Table 5 order ofoptimized DIs and order of FOIs are used as subscripts ofsymbol ldquo119867rdquo

4 Comparisons of Proposed FOIs Based on2nd and 3rd Order Optimized DIs with theExisting Models

Simulation results of comparison of magnitude responsesRME and phase responses of half integrators (see Table 5)based on different 2nd and 3rd order optimized DIs (given in(12) and (13)) with the existing half integrators given in [2 17]and ideal half integrator have been presented in Figures 5ndash7

Comparison of RME values of proposed and existingFOIs has been presented in Table 6 It is observed fromFigure 6 that the proposed 2nd and 3rd order FOIs namely1198672nd fo 2(119911) and 119867

3rd fo 3(119911) only lag in performance butstill these outperform the existing [2 17] models in higherfrequency spectrumThe6th order operator119867

2nd fo 6(119911) is the

best among the proposed FOIs and it clearly outperforms allthe existing models [2 17] in complete frequency spectrumwith RME of leminus40 dB Proposed 119867

2nd fo 2(119911) and 1198673rd fo 3(119911)

outperform Gupta-Madhu-Jain half integrator [17] in fre-quency range over 088 le 120596 le 3120587 radians 119867

2nd fo 4(119911)1198672nd fo 6(119911) and 119867

3rd fo 6(119911) half integrators excel Krishnahalf integrator [2] in almost complete range All the proposedFOIs are observed to be stable and follow linear phase curvein almost complete frequency spectrum (see Figure 7)

5 Conclusions

This paper attempts to focus on a simple yet unaddressedfact which till date has remained in the backdrop as theoptimization of an integer order operator before convertingit into its fractional the order (fo) counterparts ProposedFOIs validate that properties of an efficiently optimized DIpass directly into its fractional domain when it is used as 119904-to-119911 transformation during indirect discretization Here 2ndand 3rd order integer order integrators have been exploredby PSO algorithm for finding their optimal solutions for aminimized error fitness function The simulation results ofinteger order operators have revealed the effectiveness ofthe proposed modified PSO algorithm because of their low

Journal of Optimization 9

Table 5 Mathematical models of proposed half integrators based on 2nd and 3rd order PSO optimized DIs

Mathematical models of half integrators based on 2nd and 3rd order PSO optimized DIsName of PSO optimized DIs Transfer functions of FOIs derived by indirect discretization of CFE

119867PSO 2nd opt(119911)

1198672nd-fo-2(119911) = (09302 (119911 minus 01825) (119911 + 05869)

(119911 minus 07422) (119911 + 05604))

1198672nd-fo-4(119911) = (093011 (119911 minus 0003372) (119911 + 05633) (119911 + 06071) (119911 minus 06396)

(119911 minus 09034) (119911 minus 0293) (119911 + 05788) (119911 + 05566))

1198672nd-fo-6(119911) = (

(093011 (119911 minus 08073) (119911 minus 03427) (119911 + 0616)

(119911 + 05758) (119911 + 05587) (119911 + 004899))

((119911 + 05936) (119911 + 05649) (119911 + 05556)

(119911 minus 05924) (119911 minus 09502) (119911 minus 01107))

)

119867PSO 3rd opt(119911)

1198673rd-fo-3(119911) = (093046 (119911 minus 01852) (119911 + 02354) (119911 + 0433)

(119911 minus 0742) (119911 + 03892) (119911 + 02492))

1198673rd-fo-6(119911) = (

(093038 (119911 minus 06395) (119911 + 04678) (119911 + 03939)

(119911 + 02476) (119911 + 0223) (119911 minus 0008033))

((119911 minus 09031) (119911 minus 02946) (119911 + 02512)

(119911 + 02397) (119911 + 0383) (119911 + 04194))

)

Table 6 Comparisons of RME values of the proposed half integrators based on optimized DIs with the existing models

120596

(120587 rad)

Proposed half integrators based on 2nd orderPSO optimized DI

Proposed half integrators based on 3rdorder PSO optimized DI

Gupta-Madhu-Jain halfintegrator

Krishna halfintegrator

1198672nd-fo-2(119911) 1198672nd-fo-4(119911) 1198672nd-fo-6(119911) 1198673rd-fo-3(119911) 1198673rd-fo-6(119911)

025 minus1579 minus2455 minus4325 minus1579 minus2926 minus3553 minus174205 minus1752 minus3874 minus5585 minus1752 minus3897 minus5362 minus2831075 minus2582 minus4156 minus8514 minus2581 minus4261 minus5365 minus355610 minus583 minus583 minus5544 minus6401 minus7475 minus4526 minus4205125 minus3125 minus5734 minus5348 minus3113 minus5067 minus3893 minus3713175 minus2894 minus8115 minus7532 minus2982 minus5825 minus3359 minus405720 minus3262 minus7583 minus5554 minus3262 minus7325 minus3488 minus4646225 minus4143 minus5713 minus4899 minus4115 minus5893 minus4369 minus5528275 minus3299 minus596 minus596 minus3633 minus4694 minus326 minus29830 minus2956 minus5603 minus4775 minus2827 minus4674 minus3374 minus243

orders and very lessmagnitude errors as compared to existingones Proposed FOIs based on the optimizedDIs present veryless RME values of the order of minus40 dB and show linear phasecurves in almost full band of Nyquist frequency

References

[1] L Debnath ldquoRecent applications of fractional calculus toscience and engineeringrdquo International Journal of Mathematicsand Mathematical Sciences vol 2003 no 54 pp 3413ndash34422003

[2] B T Krishna ldquoStudies on fractional order differentiators andintegrators a surveyrdquo Signal Processing vol 91 no 3 pp 386ndash426 2011

[3] N M F Ferreira F B Duarte M F M Lima M G Marcosand J A T Machado ldquoApplication of fractional calculus in thedynamical analysis and control of mechanical manipulatorsrdquoFractional Calculus and Applied Analysis vol 11 pp 91ndash1132008

[4] S Manabe ldquoThe noninteger integral and its application tocontrol systemsrdquo English Translation Journal Japan vol 6 pp83ndash87 1961

[5] J A T Machado and A M S Galhano ldquoA new methodfor approximating fractional derivatives application in non-linear controlrdquo in Proceedings of the 6th EUROMECHNonlinearDynamics Conference (ENOC rsquo08) pp 4ndash8 Saint PetersburgRussia July 2008

[6] Y Ferdi J P Herbeuval A Charef and B Boucheham ldquoR wavedetection using fractional digital differentiationrdquo ITBM-RBMvol 24 no 5-6 pp 273ndash280 2003

[7] J Mocak I Janiga M Rievaj and D Bustin ldquoThe use of frac-tional differentiation or integration for signal improvementrdquoMeasurement Science Review vol 7 pp 39ndash42 2007

[8] B Mathieu P Melchior A Oustaloup and C Ceyral ldquoFrac-tional differentiation for edge detectionrdquo Signal Processing vol83 no 11 pp 2421ndash2432 2003

[9] Y Pu W Wang J Zhou Y Wang and H Jia ldquoFractionaldifferential approach to detecting textural features of digital

10 Journal of Optimization

image and its fractional differential filter implementationrdquoScience in China F vol 51 no 9 pp 1319ndash1339 2008

[10] S Das I Pan S Das andA Gupta ldquoImprovedmodel reductionand tuning of fractional-order PI120582D120583 controllers for analyticalrule extraction with genetic programmingrdquo ISA Transactionsvol 51 no 2 pp 237ndash261 2012

[11] S Das I Pan K Halder S Das and A Gupta ldquoImpact offractional order integral performance indices in LQR basedPID controller design via optimum selection of weightingmatricesrdquo in Proceedings of the International Conference onComputer Communication and Informatics (ICCCI rsquo12) pp 1ndash6Coimbatore India January 2012

[12] Y Chen and B M Vinagre ldquoA new IIR-type digital fractionalorder differentiatorrdquo Signal Processing vol 83 no 11 pp 2359ndash2365 2003

[13] Y Chen B M Vinagre and I Podlubny ldquoContinued frac-tion expansion approaches to discretizing fractional orderderivatives-an expository reviewrdquo Nonlinear Dynamics vol 38no 1ndash4 pp 155ndash170 2004

[14] Y Q Chen and K L Moore ldquoDiscretization schemes forfractional-order differentiators and integratorsrdquo IEEE Transac-tions on Circuits and Systems I vol 49 no 3 pp 363ndash367 2002

[15] M A Al-Alaoui ldquoNovel class of digital integrators and differ-entiatorsrdquo IEEE Transactions on Signal Processing p 1 2009

[16] M A Al-Alaoui ldquoAl-Alaoui operator and the new transfor-mation polynomials for discretization of analogue systemsrdquoElectrical Engineering vol 90 no 6 pp 455ndash467 2008

[17] M Gupta M Jain and N Jain ldquoA new fractional orderrecursive digital integrator using continued fraction expansionrdquoin Proceedings of the India International Conference on Powerelectronics (IICPE) pp 1ndash3 New Delhi India 2010

[18] M Gupta P Varshney and G S Visweswaran ldquoDigitalfractional-order differentiator and integrator models based onfirst-order and higher order operatorsrdquo International Journal ofCircuitTheory and Applications vol 39 no 5 pp 461ndash474 2011

[19] F Leulmi and Y Ferdi ldquoAn improvement of the rationalapproximation of the fractional operator 119904

120572rdquo in Proceedingsof the Saudi International Electronics Communications andPhotonics Conference (SIECP rsquo11) pp 1ndash6 Riyadh Saudi Arabia2011

[20] Y Ferdi ldquoComputation of fractional order derivative andintegral via power series expansion and signal modellingrdquoNonlinear Dynamics vol 46 no 1-2 pp 1ndash15 2006

[21] G S Visweswaran P Varshney and M Gupta ldquoNew approachto realize fractional power in z-domain at low frequencyrdquo IEEETransactions on Circuits and Systems II vol 58 no 3 pp 179ndash183 2011

[22] P Varshney M Gupta and G S Visweswaran ldquoSwitchedcapacitor realizations of fractional-order differentiators andintegrators based on an operator with improved performancerdquoRadioengineering vol 20 no 1 pp 340ndash348 2011

[23] B T Krishna and K V V Reddy ldquoDesign of fractional orderdigital differentiators and integrators using indirect discretiza-tionrdquo An International Journal for Theory and Applications vol11 pp 143ndash151 2008

[24] B T Krishna and K V V Reddy ldquoDesign of digital differentia-tors and integrators of order 12rdquo World Journal of Modellingand Simulation vol 4 pp 182ndash187 2008

[25] R Yadav and M Gupta ldquoDesign of fractional order differen-tiators and integrators using indirect discretization approachrdquoin Proceedings of the 2nd International Conference on Advances

in Recent Technologies in Communication and Computing (ART-Com rsquo10) pp 126ndash130 Kottayam India October 2010

[26] R Yadav and M Gupta ldquoDesign of fractional order differen-tiators and integrators using indirect discretization schemerdquoin Proceedings of the India International Conference on PowerElectronics (IICPE rsquo10) New Delhi India January 2011

[27] G E Carlson and C A Halijak ldquoSimulation of the fractionalderivative operator (1119904)

12 and the fractional integral operator(1119904)12rdquo Kansas State University Bulletin vol 45 pp 1ndash22 1961

[28] S K Steiglitz ldquoComputer-aided design of recursive digitalfiltersrdquo IEEE Transactions Audio and Electroacoustics vol AU-18 no 2 pp 123ndash129 1970

[29] A N Khovanskii The Application of Continued Fractions andtheir Generalizations to Problems in Approximation Theory PNoordhoff Ltd 1963 Translate by Peter Wynn

[30] S C Roy ldquoSome exact steady-state sinewave solutions ofnonuniform RC linesrdquo British Journal of Applied Physics vol 14pp 378ndash380 1963

[31] A Oustaloup F Levron B Mathieu and F M NanotldquoFrequency-band complex noninteger differentiator charac-terization and synthesisrdquo IEEE Transactions on Circuits andSystems I vol 47 no 1 pp 25ndash39 2000

[32] G Maione ldquoLaguerre approximation of fractional systemsrdquoElectronics Letters vol 38 no 20 pp 1234ndash1236 2002

[33] N Papamarkos and C Chamzas ldquoA new approach for thedesign of digital integratorsrdquo IEEE Transactions on Circuits andSystems I vol 43 no 9 pp 785ndash791 1996

[34] M Gupta M Jain and B Kumar ldquoRecursive wideband digitalintegrator and differentiatorrdquo International Journal of CircuitTheory and Applications vol 39 no 7 pp 775ndash782 2011

[35] N Jain M Gupta and M Jain ldquoLinear phase second orderrecursive digital integrators and differentiatorsrdquo Radioengineer-ing Journal vol 21 pp 712ndash717 2012

[36] C-C Hse W-Y Wang and C-Y Yu ldquoGenetic algorithm-derived digital integrators and their applications in discretiza-tion of continuous systemsrdquo in Proceedings of the CEC Congresson Evolutionary Computation pp 443ndash448 Honolulu HawaiiUSA 2002

[37] S Das B Majumder A Pakhira I Pan S Das and AGupta ldquoOptimizing continued fraction expansion based iirrealization of fractional order differ-integrators with geneticalgorithmrdquo in Proceedings of the International Conference onProcess Automation Control and Computing (PACC rsquo11) pp 1ndash6Coimbatore India July 2011

[38] G K Thakur A Gupta and D Upadhyay ldquoOptimizationalgorithmapproach for errorminimization of digital integratorsand differentiatorsrdquo in Proceedings of the International Con-ference on Advances in Recent Technologies in Communicationand Computing (ARTCom rsquo09) pp 118ndash122 Kottayam IndiaOctober 2009

[39] M A Al-Alaoui ldquoClass of digital integrators and differentia-torsrdquo IET Signal Processing vol 5 no 2 pp 251ndash260 2011

[40] D K Upadhyay and R K Singh ldquoRecursive wideband digitaldifferentiator and integratorrdquo Electronics Letters vol 47 no 11pp 647ndash648 2011

[41] D K Upadhyay ldquoClass of recursive wide band digital differen-tiators and integratorsrdquo Radioengineering Journal vol 21 no 3pp 904ndash910 2012

[42] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks pp 1942ndash1948 December 1995

Journal of Optimization 11

[43] Y-T Kao and E Zahara ldquoA hybrid genetic algorithm andparticle swarm optimization formultimodal functionsrdquoAppliedSoft Computing Journal vol 8 no 2 pp 849ndash857 2008

[44] Y Shi and R C Eberhart ldquoEmpirical study of particle swarmoptimizationrdquo in Proceedings of the IEEE international Con-ference Evolution Computer pp 1945ndash1950 Washington DCUSA 1999

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Journal of Optimization

Table 3 Comparisons of RME values of the proposed PSO optimized DIs with other integer order existing integrators optimized by differentoptimization techniques

120596 (120587 rad)Al-Alaoui3-segment

(SA)

Al-Alaoui4-segment

(SA)

Jain-Gupta-Jain(GA)

Upadhyay(P-Z)

Gupta-Jain-Kumar(LP)

Proposed2nd orderDI (PSO)

Proposed3rd orderDI (PSO)

025 minus7203 minus6018 minus5446 minus4831 minus7106 minus7203 minus638205 minus4915 minus4121 minus5947 minus5165 minus5989 minus5968 minus7304075 minus3725 minus3402 minus8004 minus6171 minus6391 minus5327 minus714510 minus3029 minus3034 minus5728 minus5728 minus4994 minus4988 minus6463125 minus2272 minus3296 minus5384 minus4871 minus3933 minus4871 minus682175 minus1164 minus3505 minus5385 minus4791 minus3146 minus6262 minus54220 minus824 minus1911 minus6053 minus5289 minus3143 minus4998 minus4994225 minus692 minus1599 minus5384 minus615 minus345 minus4277 minus5366275 minus1819 minus1973 minus3754 minus3776 minus4588 minus5555 minus406130 minus417 minus1811 minus3552 minus3644 minus3014 minus4134 minus4501

Table 4 Statistical test for comparison of proposed PSO and GA techniques for approximations of 2nd and 3rd order digital integrators

Genetic algorithm Modified PSO algorithmApproximating 2nd order digital integrator for comparing PSO and GA

119873119901

lowast 119873119892

50 lowast 50 100 lowast 100 200 lowast 200 50 lowast 50 100 lowast 100 200 lowast 200

Mean 0915405 0230096 00874599 0614370 0201256 00936231Median 0482815 0194402 0075695 0491575 0114377 00488172Standard deviation 0737526 0131617 00400587 043579 0137471 0042671

Approximating 3rd order digital integrator for comparing PSO and GA119873119901

lowast 119873119892

50 lowast 50 100 lowast 100 200 lowast 200 50 lowast 50 100 lowast 100 200 lowast 200

Mean 0915406 0230096 00874599 0633554 0221398 00969881Median 01478565 0069402 002937085 01801123 01207528 00861992Standard deviation 0737526 0131617 00400587 0491451 0152073 00530145

[24] has been adopted for deriving half integrators fitted incontinuous-time domain (see Table 5) Also 2nd order opti-mized DI resulted in 2nd 4th and 6th order FOIs whereas3rd order optimized DI when substituted for different termsresulted in 3rd and 6th order half integrators In the TFs ofderived proposed half integrators given in Table 5 order ofoptimized DIs and order of FOIs are used as subscripts ofsymbol ldquo119867rdquo

4 Comparisons of Proposed FOIs Based on2nd and 3rd Order Optimized DIs with theExisting Models

Simulation results of comparison of magnitude responsesRME and phase responses of half integrators (see Table 5)based on different 2nd and 3rd order optimized DIs (given in(12) and (13)) with the existing half integrators given in [2 17]and ideal half integrator have been presented in Figures 5ndash7

Comparison of RME values of proposed and existingFOIs has been presented in Table 6 It is observed fromFigure 6 that the proposed 2nd and 3rd order FOIs namely1198672nd fo 2(119911) and 119867

3rd fo 3(119911) only lag in performance butstill these outperform the existing [2 17] models in higherfrequency spectrumThe6th order operator119867

2nd fo 6(119911) is the

best among the proposed FOIs and it clearly outperforms allthe existing models [2 17] in complete frequency spectrumwith RME of leminus40 dB Proposed 119867

2nd fo 2(119911) and 1198673rd fo 3(119911)

outperform Gupta-Madhu-Jain half integrator [17] in fre-quency range over 088 le 120596 le 3120587 radians 119867

2nd fo 4(119911)1198672nd fo 6(119911) and 119867

3rd fo 6(119911) half integrators excel Krishnahalf integrator [2] in almost complete range All the proposedFOIs are observed to be stable and follow linear phase curvein almost complete frequency spectrum (see Figure 7)

5 Conclusions

This paper attempts to focus on a simple yet unaddressedfact which till date has remained in the backdrop as theoptimization of an integer order operator before convertingit into its fractional the order (fo) counterparts ProposedFOIs validate that properties of an efficiently optimized DIpass directly into its fractional domain when it is used as 119904-to-119911 transformation during indirect discretization Here 2ndand 3rd order integer order integrators have been exploredby PSO algorithm for finding their optimal solutions for aminimized error fitness function The simulation results ofinteger order operators have revealed the effectiveness ofthe proposed modified PSO algorithm because of their low

Journal of Optimization 9

Table 5 Mathematical models of proposed half integrators based on 2nd and 3rd order PSO optimized DIs

Mathematical models of half integrators based on 2nd and 3rd order PSO optimized DIsName of PSO optimized DIs Transfer functions of FOIs derived by indirect discretization of CFE

119867PSO 2nd opt(119911)

1198672nd-fo-2(119911) = (09302 (119911 minus 01825) (119911 + 05869)

(119911 minus 07422) (119911 + 05604))

1198672nd-fo-4(119911) = (093011 (119911 minus 0003372) (119911 + 05633) (119911 + 06071) (119911 minus 06396)

(119911 minus 09034) (119911 minus 0293) (119911 + 05788) (119911 + 05566))

1198672nd-fo-6(119911) = (

(093011 (119911 minus 08073) (119911 minus 03427) (119911 + 0616)

(119911 + 05758) (119911 + 05587) (119911 + 004899))

((119911 + 05936) (119911 + 05649) (119911 + 05556)

(119911 minus 05924) (119911 minus 09502) (119911 minus 01107))

)

119867PSO 3rd opt(119911)

1198673rd-fo-3(119911) = (093046 (119911 minus 01852) (119911 + 02354) (119911 + 0433)

(119911 minus 0742) (119911 + 03892) (119911 + 02492))

1198673rd-fo-6(119911) = (

(093038 (119911 minus 06395) (119911 + 04678) (119911 + 03939)

(119911 + 02476) (119911 + 0223) (119911 minus 0008033))

((119911 minus 09031) (119911 minus 02946) (119911 + 02512)

(119911 + 02397) (119911 + 0383) (119911 + 04194))

)

Table 6 Comparisons of RME values of the proposed half integrators based on optimized DIs with the existing models

120596

(120587 rad)

Proposed half integrators based on 2nd orderPSO optimized DI

Proposed half integrators based on 3rdorder PSO optimized DI

Gupta-Madhu-Jain halfintegrator

Krishna halfintegrator

1198672nd-fo-2(119911) 1198672nd-fo-4(119911) 1198672nd-fo-6(119911) 1198673rd-fo-3(119911) 1198673rd-fo-6(119911)

025 minus1579 minus2455 minus4325 minus1579 minus2926 minus3553 minus174205 minus1752 minus3874 minus5585 minus1752 minus3897 minus5362 minus2831075 minus2582 minus4156 minus8514 minus2581 minus4261 minus5365 minus355610 minus583 minus583 minus5544 minus6401 minus7475 minus4526 minus4205125 minus3125 minus5734 minus5348 minus3113 minus5067 minus3893 minus3713175 minus2894 minus8115 minus7532 minus2982 minus5825 minus3359 minus405720 minus3262 minus7583 minus5554 minus3262 minus7325 minus3488 minus4646225 minus4143 minus5713 minus4899 minus4115 minus5893 minus4369 minus5528275 minus3299 minus596 minus596 minus3633 minus4694 minus326 minus29830 minus2956 minus5603 minus4775 minus2827 minus4674 minus3374 minus243

orders and very lessmagnitude errors as compared to existingones Proposed FOIs based on the optimizedDIs present veryless RME values of the order of minus40 dB and show linear phasecurves in almost full band of Nyquist frequency

References

[1] L Debnath ldquoRecent applications of fractional calculus toscience and engineeringrdquo International Journal of Mathematicsand Mathematical Sciences vol 2003 no 54 pp 3413ndash34422003

[2] B T Krishna ldquoStudies on fractional order differentiators andintegrators a surveyrdquo Signal Processing vol 91 no 3 pp 386ndash426 2011

[3] N M F Ferreira F B Duarte M F M Lima M G Marcosand J A T Machado ldquoApplication of fractional calculus in thedynamical analysis and control of mechanical manipulatorsrdquoFractional Calculus and Applied Analysis vol 11 pp 91ndash1132008

[4] S Manabe ldquoThe noninteger integral and its application tocontrol systemsrdquo English Translation Journal Japan vol 6 pp83ndash87 1961

[5] J A T Machado and A M S Galhano ldquoA new methodfor approximating fractional derivatives application in non-linear controlrdquo in Proceedings of the 6th EUROMECHNonlinearDynamics Conference (ENOC rsquo08) pp 4ndash8 Saint PetersburgRussia July 2008

[6] Y Ferdi J P Herbeuval A Charef and B Boucheham ldquoR wavedetection using fractional digital differentiationrdquo ITBM-RBMvol 24 no 5-6 pp 273ndash280 2003

[7] J Mocak I Janiga M Rievaj and D Bustin ldquoThe use of frac-tional differentiation or integration for signal improvementrdquoMeasurement Science Review vol 7 pp 39ndash42 2007

[8] B Mathieu P Melchior A Oustaloup and C Ceyral ldquoFrac-tional differentiation for edge detectionrdquo Signal Processing vol83 no 11 pp 2421ndash2432 2003

[9] Y Pu W Wang J Zhou Y Wang and H Jia ldquoFractionaldifferential approach to detecting textural features of digital

10 Journal of Optimization

image and its fractional differential filter implementationrdquoScience in China F vol 51 no 9 pp 1319ndash1339 2008

[10] S Das I Pan S Das andA Gupta ldquoImprovedmodel reductionand tuning of fractional-order PI120582D120583 controllers for analyticalrule extraction with genetic programmingrdquo ISA Transactionsvol 51 no 2 pp 237ndash261 2012

[11] S Das I Pan K Halder S Das and A Gupta ldquoImpact offractional order integral performance indices in LQR basedPID controller design via optimum selection of weightingmatricesrdquo in Proceedings of the International Conference onComputer Communication and Informatics (ICCCI rsquo12) pp 1ndash6Coimbatore India January 2012

[12] Y Chen and B M Vinagre ldquoA new IIR-type digital fractionalorder differentiatorrdquo Signal Processing vol 83 no 11 pp 2359ndash2365 2003

[13] Y Chen B M Vinagre and I Podlubny ldquoContinued frac-tion expansion approaches to discretizing fractional orderderivatives-an expository reviewrdquo Nonlinear Dynamics vol 38no 1ndash4 pp 155ndash170 2004

[14] Y Q Chen and K L Moore ldquoDiscretization schemes forfractional-order differentiators and integratorsrdquo IEEE Transac-tions on Circuits and Systems I vol 49 no 3 pp 363ndash367 2002

[15] M A Al-Alaoui ldquoNovel class of digital integrators and differ-entiatorsrdquo IEEE Transactions on Signal Processing p 1 2009

[16] M A Al-Alaoui ldquoAl-Alaoui operator and the new transfor-mation polynomials for discretization of analogue systemsrdquoElectrical Engineering vol 90 no 6 pp 455ndash467 2008

[17] M Gupta M Jain and N Jain ldquoA new fractional orderrecursive digital integrator using continued fraction expansionrdquoin Proceedings of the India International Conference on Powerelectronics (IICPE) pp 1ndash3 New Delhi India 2010

[18] M Gupta P Varshney and G S Visweswaran ldquoDigitalfractional-order differentiator and integrator models based onfirst-order and higher order operatorsrdquo International Journal ofCircuitTheory and Applications vol 39 no 5 pp 461ndash474 2011

[19] F Leulmi and Y Ferdi ldquoAn improvement of the rationalapproximation of the fractional operator 119904

120572rdquo in Proceedingsof the Saudi International Electronics Communications andPhotonics Conference (SIECP rsquo11) pp 1ndash6 Riyadh Saudi Arabia2011

[20] Y Ferdi ldquoComputation of fractional order derivative andintegral via power series expansion and signal modellingrdquoNonlinear Dynamics vol 46 no 1-2 pp 1ndash15 2006

[21] G S Visweswaran P Varshney and M Gupta ldquoNew approachto realize fractional power in z-domain at low frequencyrdquo IEEETransactions on Circuits and Systems II vol 58 no 3 pp 179ndash183 2011

[22] P Varshney M Gupta and G S Visweswaran ldquoSwitchedcapacitor realizations of fractional-order differentiators andintegrators based on an operator with improved performancerdquoRadioengineering vol 20 no 1 pp 340ndash348 2011

[23] B T Krishna and K V V Reddy ldquoDesign of fractional orderdigital differentiators and integrators using indirect discretiza-tionrdquo An International Journal for Theory and Applications vol11 pp 143ndash151 2008

[24] B T Krishna and K V V Reddy ldquoDesign of digital differentia-tors and integrators of order 12rdquo World Journal of Modellingand Simulation vol 4 pp 182ndash187 2008

[25] R Yadav and M Gupta ldquoDesign of fractional order differen-tiators and integrators using indirect discretization approachrdquoin Proceedings of the 2nd International Conference on Advances

in Recent Technologies in Communication and Computing (ART-Com rsquo10) pp 126ndash130 Kottayam India October 2010

[26] R Yadav and M Gupta ldquoDesign of fractional order differen-tiators and integrators using indirect discretization schemerdquoin Proceedings of the India International Conference on PowerElectronics (IICPE rsquo10) New Delhi India January 2011

[27] G E Carlson and C A Halijak ldquoSimulation of the fractionalderivative operator (1119904)

12 and the fractional integral operator(1119904)12rdquo Kansas State University Bulletin vol 45 pp 1ndash22 1961

[28] S K Steiglitz ldquoComputer-aided design of recursive digitalfiltersrdquo IEEE Transactions Audio and Electroacoustics vol AU-18 no 2 pp 123ndash129 1970

[29] A N Khovanskii The Application of Continued Fractions andtheir Generalizations to Problems in Approximation Theory PNoordhoff Ltd 1963 Translate by Peter Wynn

[30] S C Roy ldquoSome exact steady-state sinewave solutions ofnonuniform RC linesrdquo British Journal of Applied Physics vol 14pp 378ndash380 1963

[31] A Oustaloup F Levron B Mathieu and F M NanotldquoFrequency-band complex noninteger differentiator charac-terization and synthesisrdquo IEEE Transactions on Circuits andSystems I vol 47 no 1 pp 25ndash39 2000

[32] G Maione ldquoLaguerre approximation of fractional systemsrdquoElectronics Letters vol 38 no 20 pp 1234ndash1236 2002

[33] N Papamarkos and C Chamzas ldquoA new approach for thedesign of digital integratorsrdquo IEEE Transactions on Circuits andSystems I vol 43 no 9 pp 785ndash791 1996

[34] M Gupta M Jain and B Kumar ldquoRecursive wideband digitalintegrator and differentiatorrdquo International Journal of CircuitTheory and Applications vol 39 no 7 pp 775ndash782 2011

[35] N Jain M Gupta and M Jain ldquoLinear phase second orderrecursive digital integrators and differentiatorsrdquo Radioengineer-ing Journal vol 21 pp 712ndash717 2012

[36] C-C Hse W-Y Wang and C-Y Yu ldquoGenetic algorithm-derived digital integrators and their applications in discretiza-tion of continuous systemsrdquo in Proceedings of the CEC Congresson Evolutionary Computation pp 443ndash448 Honolulu HawaiiUSA 2002

[37] S Das B Majumder A Pakhira I Pan S Das and AGupta ldquoOptimizing continued fraction expansion based iirrealization of fractional order differ-integrators with geneticalgorithmrdquo in Proceedings of the International Conference onProcess Automation Control and Computing (PACC rsquo11) pp 1ndash6Coimbatore India July 2011

[38] G K Thakur A Gupta and D Upadhyay ldquoOptimizationalgorithmapproach for errorminimization of digital integratorsand differentiatorsrdquo in Proceedings of the International Con-ference on Advances in Recent Technologies in Communicationand Computing (ARTCom rsquo09) pp 118ndash122 Kottayam IndiaOctober 2009

[39] M A Al-Alaoui ldquoClass of digital integrators and differentia-torsrdquo IET Signal Processing vol 5 no 2 pp 251ndash260 2011

[40] D K Upadhyay and R K Singh ldquoRecursive wideband digitaldifferentiator and integratorrdquo Electronics Letters vol 47 no 11pp 647ndash648 2011

[41] D K Upadhyay ldquoClass of recursive wide band digital differen-tiators and integratorsrdquo Radioengineering Journal vol 21 no 3pp 904ndash910 2012

[42] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks pp 1942ndash1948 December 1995

Journal of Optimization 11

[43] Y-T Kao and E Zahara ldquoA hybrid genetic algorithm andparticle swarm optimization formultimodal functionsrdquoAppliedSoft Computing Journal vol 8 no 2 pp 849ndash857 2008

[44] Y Shi and R C Eberhart ldquoEmpirical study of particle swarmoptimizationrdquo in Proceedings of the IEEE international Con-ference Evolution Computer pp 1945ndash1950 Washington DCUSA 1999

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Optimization 9

Table 5 Mathematical models of proposed half integrators based on 2nd and 3rd order PSO optimized DIs

Mathematical models of half integrators based on 2nd and 3rd order PSO optimized DIsName of PSO optimized DIs Transfer functions of FOIs derived by indirect discretization of CFE

119867PSO 2nd opt(119911)

1198672nd-fo-2(119911) = (09302 (119911 minus 01825) (119911 + 05869)

(119911 minus 07422) (119911 + 05604))

1198672nd-fo-4(119911) = (093011 (119911 minus 0003372) (119911 + 05633) (119911 + 06071) (119911 minus 06396)

(119911 minus 09034) (119911 minus 0293) (119911 + 05788) (119911 + 05566))

1198672nd-fo-6(119911) = (

(093011 (119911 minus 08073) (119911 minus 03427) (119911 + 0616)

(119911 + 05758) (119911 + 05587) (119911 + 004899))

((119911 + 05936) (119911 + 05649) (119911 + 05556)

(119911 minus 05924) (119911 minus 09502) (119911 minus 01107))

)

119867PSO 3rd opt(119911)

1198673rd-fo-3(119911) = (093046 (119911 minus 01852) (119911 + 02354) (119911 + 0433)

(119911 minus 0742) (119911 + 03892) (119911 + 02492))

1198673rd-fo-6(119911) = (

(093038 (119911 minus 06395) (119911 + 04678) (119911 + 03939)

(119911 + 02476) (119911 + 0223) (119911 minus 0008033))

((119911 minus 09031) (119911 minus 02946) (119911 + 02512)

(119911 + 02397) (119911 + 0383) (119911 + 04194))

)

Table 6 Comparisons of RME values of the proposed half integrators based on optimized DIs with the existing models

120596

(120587 rad)

Proposed half integrators based on 2nd orderPSO optimized DI

Proposed half integrators based on 3rdorder PSO optimized DI

Gupta-Madhu-Jain halfintegrator

Krishna halfintegrator

1198672nd-fo-2(119911) 1198672nd-fo-4(119911) 1198672nd-fo-6(119911) 1198673rd-fo-3(119911) 1198673rd-fo-6(119911)

025 minus1579 minus2455 minus4325 minus1579 minus2926 minus3553 minus174205 minus1752 minus3874 minus5585 minus1752 minus3897 minus5362 minus2831075 minus2582 minus4156 minus8514 minus2581 minus4261 minus5365 minus355610 minus583 minus583 minus5544 minus6401 minus7475 minus4526 minus4205125 minus3125 minus5734 minus5348 minus3113 minus5067 minus3893 minus3713175 minus2894 minus8115 minus7532 minus2982 minus5825 minus3359 minus405720 minus3262 minus7583 minus5554 minus3262 minus7325 minus3488 minus4646225 minus4143 minus5713 minus4899 minus4115 minus5893 minus4369 minus5528275 minus3299 minus596 minus596 minus3633 minus4694 minus326 minus29830 minus2956 minus5603 minus4775 minus2827 minus4674 minus3374 minus243

orders and very lessmagnitude errors as compared to existingones Proposed FOIs based on the optimizedDIs present veryless RME values of the order of minus40 dB and show linear phasecurves in almost full band of Nyquist frequency

References

[1] L Debnath ldquoRecent applications of fractional calculus toscience and engineeringrdquo International Journal of Mathematicsand Mathematical Sciences vol 2003 no 54 pp 3413ndash34422003

[2] B T Krishna ldquoStudies on fractional order differentiators andintegrators a surveyrdquo Signal Processing vol 91 no 3 pp 386ndash426 2011

[3] N M F Ferreira F B Duarte M F M Lima M G Marcosand J A T Machado ldquoApplication of fractional calculus in thedynamical analysis and control of mechanical manipulatorsrdquoFractional Calculus and Applied Analysis vol 11 pp 91ndash1132008

[4] S Manabe ldquoThe noninteger integral and its application tocontrol systemsrdquo English Translation Journal Japan vol 6 pp83ndash87 1961

[5] J A T Machado and A M S Galhano ldquoA new methodfor approximating fractional derivatives application in non-linear controlrdquo in Proceedings of the 6th EUROMECHNonlinearDynamics Conference (ENOC rsquo08) pp 4ndash8 Saint PetersburgRussia July 2008

[6] Y Ferdi J P Herbeuval A Charef and B Boucheham ldquoR wavedetection using fractional digital differentiationrdquo ITBM-RBMvol 24 no 5-6 pp 273ndash280 2003

[7] J Mocak I Janiga M Rievaj and D Bustin ldquoThe use of frac-tional differentiation or integration for signal improvementrdquoMeasurement Science Review vol 7 pp 39ndash42 2007

[8] B Mathieu P Melchior A Oustaloup and C Ceyral ldquoFrac-tional differentiation for edge detectionrdquo Signal Processing vol83 no 11 pp 2421ndash2432 2003

[9] Y Pu W Wang J Zhou Y Wang and H Jia ldquoFractionaldifferential approach to detecting textural features of digital

10 Journal of Optimization

image and its fractional differential filter implementationrdquoScience in China F vol 51 no 9 pp 1319ndash1339 2008

[10] S Das I Pan S Das andA Gupta ldquoImprovedmodel reductionand tuning of fractional-order PI120582D120583 controllers for analyticalrule extraction with genetic programmingrdquo ISA Transactionsvol 51 no 2 pp 237ndash261 2012

[11] S Das I Pan K Halder S Das and A Gupta ldquoImpact offractional order integral performance indices in LQR basedPID controller design via optimum selection of weightingmatricesrdquo in Proceedings of the International Conference onComputer Communication and Informatics (ICCCI rsquo12) pp 1ndash6Coimbatore India January 2012

[12] Y Chen and B M Vinagre ldquoA new IIR-type digital fractionalorder differentiatorrdquo Signal Processing vol 83 no 11 pp 2359ndash2365 2003

[13] Y Chen B M Vinagre and I Podlubny ldquoContinued frac-tion expansion approaches to discretizing fractional orderderivatives-an expository reviewrdquo Nonlinear Dynamics vol 38no 1ndash4 pp 155ndash170 2004

[14] Y Q Chen and K L Moore ldquoDiscretization schemes forfractional-order differentiators and integratorsrdquo IEEE Transac-tions on Circuits and Systems I vol 49 no 3 pp 363ndash367 2002

[15] M A Al-Alaoui ldquoNovel class of digital integrators and differ-entiatorsrdquo IEEE Transactions on Signal Processing p 1 2009

[16] M A Al-Alaoui ldquoAl-Alaoui operator and the new transfor-mation polynomials for discretization of analogue systemsrdquoElectrical Engineering vol 90 no 6 pp 455ndash467 2008

[17] M Gupta M Jain and N Jain ldquoA new fractional orderrecursive digital integrator using continued fraction expansionrdquoin Proceedings of the India International Conference on Powerelectronics (IICPE) pp 1ndash3 New Delhi India 2010

[18] M Gupta P Varshney and G S Visweswaran ldquoDigitalfractional-order differentiator and integrator models based onfirst-order and higher order operatorsrdquo International Journal ofCircuitTheory and Applications vol 39 no 5 pp 461ndash474 2011

[19] F Leulmi and Y Ferdi ldquoAn improvement of the rationalapproximation of the fractional operator 119904

120572rdquo in Proceedingsof the Saudi International Electronics Communications andPhotonics Conference (SIECP rsquo11) pp 1ndash6 Riyadh Saudi Arabia2011

[20] Y Ferdi ldquoComputation of fractional order derivative andintegral via power series expansion and signal modellingrdquoNonlinear Dynamics vol 46 no 1-2 pp 1ndash15 2006

[21] G S Visweswaran P Varshney and M Gupta ldquoNew approachto realize fractional power in z-domain at low frequencyrdquo IEEETransactions on Circuits and Systems II vol 58 no 3 pp 179ndash183 2011

[22] P Varshney M Gupta and G S Visweswaran ldquoSwitchedcapacitor realizations of fractional-order differentiators andintegrators based on an operator with improved performancerdquoRadioengineering vol 20 no 1 pp 340ndash348 2011

[23] B T Krishna and K V V Reddy ldquoDesign of fractional orderdigital differentiators and integrators using indirect discretiza-tionrdquo An International Journal for Theory and Applications vol11 pp 143ndash151 2008

[24] B T Krishna and K V V Reddy ldquoDesign of digital differentia-tors and integrators of order 12rdquo World Journal of Modellingand Simulation vol 4 pp 182ndash187 2008

[25] R Yadav and M Gupta ldquoDesign of fractional order differen-tiators and integrators using indirect discretization approachrdquoin Proceedings of the 2nd International Conference on Advances

in Recent Technologies in Communication and Computing (ART-Com rsquo10) pp 126ndash130 Kottayam India October 2010

[26] R Yadav and M Gupta ldquoDesign of fractional order differen-tiators and integrators using indirect discretization schemerdquoin Proceedings of the India International Conference on PowerElectronics (IICPE rsquo10) New Delhi India January 2011

[27] G E Carlson and C A Halijak ldquoSimulation of the fractionalderivative operator (1119904)

12 and the fractional integral operator(1119904)12rdquo Kansas State University Bulletin vol 45 pp 1ndash22 1961

[28] S K Steiglitz ldquoComputer-aided design of recursive digitalfiltersrdquo IEEE Transactions Audio and Electroacoustics vol AU-18 no 2 pp 123ndash129 1970

[29] A N Khovanskii The Application of Continued Fractions andtheir Generalizations to Problems in Approximation Theory PNoordhoff Ltd 1963 Translate by Peter Wynn

[30] S C Roy ldquoSome exact steady-state sinewave solutions ofnonuniform RC linesrdquo British Journal of Applied Physics vol 14pp 378ndash380 1963

[31] A Oustaloup F Levron B Mathieu and F M NanotldquoFrequency-band complex noninteger differentiator charac-terization and synthesisrdquo IEEE Transactions on Circuits andSystems I vol 47 no 1 pp 25ndash39 2000

[32] G Maione ldquoLaguerre approximation of fractional systemsrdquoElectronics Letters vol 38 no 20 pp 1234ndash1236 2002

[33] N Papamarkos and C Chamzas ldquoA new approach for thedesign of digital integratorsrdquo IEEE Transactions on Circuits andSystems I vol 43 no 9 pp 785ndash791 1996

[34] M Gupta M Jain and B Kumar ldquoRecursive wideband digitalintegrator and differentiatorrdquo International Journal of CircuitTheory and Applications vol 39 no 7 pp 775ndash782 2011

[35] N Jain M Gupta and M Jain ldquoLinear phase second orderrecursive digital integrators and differentiatorsrdquo Radioengineer-ing Journal vol 21 pp 712ndash717 2012

[36] C-C Hse W-Y Wang and C-Y Yu ldquoGenetic algorithm-derived digital integrators and their applications in discretiza-tion of continuous systemsrdquo in Proceedings of the CEC Congresson Evolutionary Computation pp 443ndash448 Honolulu HawaiiUSA 2002

[37] S Das B Majumder A Pakhira I Pan S Das and AGupta ldquoOptimizing continued fraction expansion based iirrealization of fractional order differ-integrators with geneticalgorithmrdquo in Proceedings of the International Conference onProcess Automation Control and Computing (PACC rsquo11) pp 1ndash6Coimbatore India July 2011

[38] G K Thakur A Gupta and D Upadhyay ldquoOptimizationalgorithmapproach for errorminimization of digital integratorsand differentiatorsrdquo in Proceedings of the International Con-ference on Advances in Recent Technologies in Communicationand Computing (ARTCom rsquo09) pp 118ndash122 Kottayam IndiaOctober 2009

[39] M A Al-Alaoui ldquoClass of digital integrators and differentia-torsrdquo IET Signal Processing vol 5 no 2 pp 251ndash260 2011

[40] D K Upadhyay and R K Singh ldquoRecursive wideband digitaldifferentiator and integratorrdquo Electronics Letters vol 47 no 11pp 647ndash648 2011

[41] D K Upadhyay ldquoClass of recursive wide band digital differen-tiators and integratorsrdquo Radioengineering Journal vol 21 no 3pp 904ndash910 2012

[42] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks pp 1942ndash1948 December 1995

Journal of Optimization 11

[43] Y-T Kao and E Zahara ldquoA hybrid genetic algorithm andparticle swarm optimization formultimodal functionsrdquoAppliedSoft Computing Journal vol 8 no 2 pp 849ndash857 2008

[44] Y Shi and R C Eberhart ldquoEmpirical study of particle swarmoptimizationrdquo in Proceedings of the IEEE international Con-ference Evolution Computer pp 1945ndash1950 Washington DCUSA 1999

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Journal of Optimization

image and its fractional differential filter implementationrdquoScience in China F vol 51 no 9 pp 1319ndash1339 2008

[10] S Das I Pan S Das andA Gupta ldquoImprovedmodel reductionand tuning of fractional-order PI120582D120583 controllers for analyticalrule extraction with genetic programmingrdquo ISA Transactionsvol 51 no 2 pp 237ndash261 2012

[11] S Das I Pan K Halder S Das and A Gupta ldquoImpact offractional order integral performance indices in LQR basedPID controller design via optimum selection of weightingmatricesrdquo in Proceedings of the International Conference onComputer Communication and Informatics (ICCCI rsquo12) pp 1ndash6Coimbatore India January 2012

[12] Y Chen and B M Vinagre ldquoA new IIR-type digital fractionalorder differentiatorrdquo Signal Processing vol 83 no 11 pp 2359ndash2365 2003

[13] Y Chen B M Vinagre and I Podlubny ldquoContinued frac-tion expansion approaches to discretizing fractional orderderivatives-an expository reviewrdquo Nonlinear Dynamics vol 38no 1ndash4 pp 155ndash170 2004

[14] Y Q Chen and K L Moore ldquoDiscretization schemes forfractional-order differentiators and integratorsrdquo IEEE Transac-tions on Circuits and Systems I vol 49 no 3 pp 363ndash367 2002

[15] M A Al-Alaoui ldquoNovel class of digital integrators and differ-entiatorsrdquo IEEE Transactions on Signal Processing p 1 2009

[16] M A Al-Alaoui ldquoAl-Alaoui operator and the new transfor-mation polynomials for discretization of analogue systemsrdquoElectrical Engineering vol 90 no 6 pp 455ndash467 2008

[17] M Gupta M Jain and N Jain ldquoA new fractional orderrecursive digital integrator using continued fraction expansionrdquoin Proceedings of the India International Conference on Powerelectronics (IICPE) pp 1ndash3 New Delhi India 2010

[18] M Gupta P Varshney and G S Visweswaran ldquoDigitalfractional-order differentiator and integrator models based onfirst-order and higher order operatorsrdquo International Journal ofCircuitTheory and Applications vol 39 no 5 pp 461ndash474 2011

[19] F Leulmi and Y Ferdi ldquoAn improvement of the rationalapproximation of the fractional operator 119904

120572rdquo in Proceedingsof the Saudi International Electronics Communications andPhotonics Conference (SIECP rsquo11) pp 1ndash6 Riyadh Saudi Arabia2011

[20] Y Ferdi ldquoComputation of fractional order derivative andintegral via power series expansion and signal modellingrdquoNonlinear Dynamics vol 46 no 1-2 pp 1ndash15 2006

[21] G S Visweswaran P Varshney and M Gupta ldquoNew approachto realize fractional power in z-domain at low frequencyrdquo IEEETransactions on Circuits and Systems II vol 58 no 3 pp 179ndash183 2011

[22] P Varshney M Gupta and G S Visweswaran ldquoSwitchedcapacitor realizations of fractional-order differentiators andintegrators based on an operator with improved performancerdquoRadioengineering vol 20 no 1 pp 340ndash348 2011

[23] B T Krishna and K V V Reddy ldquoDesign of fractional orderdigital differentiators and integrators using indirect discretiza-tionrdquo An International Journal for Theory and Applications vol11 pp 143ndash151 2008

[24] B T Krishna and K V V Reddy ldquoDesign of digital differentia-tors and integrators of order 12rdquo World Journal of Modellingand Simulation vol 4 pp 182ndash187 2008

[25] R Yadav and M Gupta ldquoDesign of fractional order differen-tiators and integrators using indirect discretization approachrdquoin Proceedings of the 2nd International Conference on Advances

in Recent Technologies in Communication and Computing (ART-Com rsquo10) pp 126ndash130 Kottayam India October 2010

[26] R Yadav and M Gupta ldquoDesign of fractional order differen-tiators and integrators using indirect discretization schemerdquoin Proceedings of the India International Conference on PowerElectronics (IICPE rsquo10) New Delhi India January 2011

[27] G E Carlson and C A Halijak ldquoSimulation of the fractionalderivative operator (1119904)

12 and the fractional integral operator(1119904)12rdquo Kansas State University Bulletin vol 45 pp 1ndash22 1961

[28] S K Steiglitz ldquoComputer-aided design of recursive digitalfiltersrdquo IEEE Transactions Audio and Electroacoustics vol AU-18 no 2 pp 123ndash129 1970

[29] A N Khovanskii The Application of Continued Fractions andtheir Generalizations to Problems in Approximation Theory PNoordhoff Ltd 1963 Translate by Peter Wynn

[30] S C Roy ldquoSome exact steady-state sinewave solutions ofnonuniform RC linesrdquo British Journal of Applied Physics vol 14pp 378ndash380 1963

[31] A Oustaloup F Levron B Mathieu and F M NanotldquoFrequency-band complex noninteger differentiator charac-terization and synthesisrdquo IEEE Transactions on Circuits andSystems I vol 47 no 1 pp 25ndash39 2000

[32] G Maione ldquoLaguerre approximation of fractional systemsrdquoElectronics Letters vol 38 no 20 pp 1234ndash1236 2002

[33] N Papamarkos and C Chamzas ldquoA new approach for thedesign of digital integratorsrdquo IEEE Transactions on Circuits andSystems I vol 43 no 9 pp 785ndash791 1996

[34] M Gupta M Jain and B Kumar ldquoRecursive wideband digitalintegrator and differentiatorrdquo International Journal of CircuitTheory and Applications vol 39 no 7 pp 775ndash782 2011

[35] N Jain M Gupta and M Jain ldquoLinear phase second orderrecursive digital integrators and differentiatorsrdquo Radioengineer-ing Journal vol 21 pp 712ndash717 2012

[36] C-C Hse W-Y Wang and C-Y Yu ldquoGenetic algorithm-derived digital integrators and their applications in discretiza-tion of continuous systemsrdquo in Proceedings of the CEC Congresson Evolutionary Computation pp 443ndash448 Honolulu HawaiiUSA 2002

[37] S Das B Majumder A Pakhira I Pan S Das and AGupta ldquoOptimizing continued fraction expansion based iirrealization of fractional order differ-integrators with geneticalgorithmrdquo in Proceedings of the International Conference onProcess Automation Control and Computing (PACC rsquo11) pp 1ndash6Coimbatore India July 2011

[38] G K Thakur A Gupta and D Upadhyay ldquoOptimizationalgorithmapproach for errorminimization of digital integratorsand differentiatorsrdquo in Proceedings of the International Con-ference on Advances in Recent Technologies in Communicationand Computing (ARTCom rsquo09) pp 118ndash122 Kottayam IndiaOctober 2009

[39] M A Al-Alaoui ldquoClass of digital integrators and differentia-torsrdquo IET Signal Processing vol 5 no 2 pp 251ndash260 2011

[40] D K Upadhyay and R K Singh ldquoRecursive wideband digitaldifferentiator and integratorrdquo Electronics Letters vol 47 no 11pp 647ndash648 2011

[41] D K Upadhyay ldquoClass of recursive wide band digital differen-tiators and integratorsrdquo Radioengineering Journal vol 21 no 3pp 904ndash910 2012

[42] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks pp 1942ndash1948 December 1995

Journal of Optimization 11

[43] Y-T Kao and E Zahara ldquoA hybrid genetic algorithm andparticle swarm optimization formultimodal functionsrdquoAppliedSoft Computing Journal vol 8 no 2 pp 849ndash857 2008

[44] Y Shi and R C Eberhart ldquoEmpirical study of particle swarmoptimizationrdquo in Proceedings of the IEEE international Con-ference Evolution Computer pp 1945ndash1950 Washington DCUSA 1999

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Optimization 11

[43] Y-T Kao and E Zahara ldquoA hybrid genetic algorithm andparticle swarm optimization formultimodal functionsrdquoAppliedSoft Computing Journal vol 8 no 2 pp 849ndash857 2008

[44] Y Shi and R C Eberhart ldquoEmpirical study of particle swarmoptimizationrdquo in Proceedings of the IEEE international Con-ference Evolution Computer pp 1945ndash1950 Washington DCUSA 1999

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of