enhanced ant colony optimization with dynamic mutation and

Research ArticleEnhanced Ant Colony Optimization withDynamic Mutation and Ad Hoc Initialization for Improvingthe Design of TSK-Type Fuzzy System

Chi-Chung Chen and Yi-Ting Liu

Department of Electrical Engineering National Chiayi University 300 Syuefu Road Chiayi City 60004 Taiwan

Correspondence should be addressed to Chi-Chung Chen chichungmailncyuedutw

Received 9 November 2017 Accepted 14 December 2017 Published 15 January 2018

Academic Editor Cheng-Jian Lin

Copyright copy 2018 Chi-Chung Chen and Yi-Ting Liu This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

This paper proposes an enhanced ant colony optimization with dynamic mutation and ad hoc initialization ACODM-I forimproving the accuracy of Takagi-Sugeno-Kang- (TSK-) type fuzzy systems design Instead of the generic initialization usuallyused in most population-based algorithms ACODM-I proposes an ad hoc application-specific initialization for generating theinitial ant solutions to improve the accuracy of fuzzy system design The generated initial ant solutions are iteratively improvedby a new approach incorporating the dynamic mutation into the existing continuous ACO (ACOR) The introduced dynamicmutation balances the exploration ability and convergence rate by providing more diverse search directions in the early stage ofoptimization process Application examples of two zero-order TSK-type fuzzy systems for dynamic plant tracking control andone first-order TSK-type fuzzy system for the prediction of the chaotic time series have been simulated to validate the proposedalgorithm Performance comparisons with ACOR and different advanced algorithms or neural-fuzzy models verify the superiorityof the proposed algorithm The effects on the design accuracy and convergence rate yielded by the proposed initialization andintroduced dynamic mutation have also been discussed and verified in the simulations

1 Introduction

In contrast to the Mamdani-type fuzzy systems having goodinterpretability Takagi-Sugeno-Kang- (TSK-) type fuzzy sys-tems usually are employed for the accuracy-oriented applica-tions Since each rule of a TSK-type fuzzy system has a crispoutput and the aggregated system output is computed viaweighted average thus avoiding time-consuming and math-ematically intractable defuzzification operation the TSK-type fuzzy system is a popular candidate for sample-basedfuzzymodeling [1] However because fuzzymodeling usuallyrequires high precision the TSK-type fuzzy systems usuallyare designed or refined by using the optimization techniquesto satisfy the requirement

The optimization of the TSK-type fuzzy system design isto find the antecedent parameters characterizing the fuzzysets for the inputs and the consequent parameters for theoutput by minimizing or maximizing the objective function

One of the major categories of solving such optimizationproblem is the gradient-based search methods in whichthe search direction is derived based on the gradient ofthe objective function with respect to the parameters Tocalculate the gradient the gradient-based method needs thetraining input-output data pair of the fuzzy system whichhowever usually is a difficulty for fuzzy control problemsbecause the target fuzzy outputs responding to the inputs arenot available in advance Another traditional difficulty for thegradient-based method is that it is easily trapped in a localoptimum when it is applied to the optimization problemshaving multiple peaks in the design space such as the designof fuzzy system

To avoid the issues encountered in the gradient-basedapproaches many population-based computational tech-niques such as genetic algorithm [2] and swarm intelligencealgorithms [3] have been proposed for solving optimizationproblems These computational techniques are derivative

HindawiComputational Intelligence and NeuroscienceVolume 2018 Article ID 9485478 15 pageshttpsdoiorg10115520189485478

2 Computational Intelligence and Neuroscience

free so they can easily be applied to any optimizationproblems including the fuzzy control problems and theproblems with nondifferentiable objective function Since thesearch direction in these techniques is stochastic they aremore likely to traverse across the highly nonlinear designspace thus avoiding to be trapped into a local optimumFurthermore these approaches evaluate many candidatesolutions concurrently so they have better chance to find thebetter solution

Genetic algorithm was inspired by the process of naturalselection and developed based on the principle of survival offittest Genetic algorithms use bioinspired crossover opera-tion between two selected parent solutions to produce childsolutions and the mutation operation for further exploitingthe generated individual child solutions for improving theperformance Genetic fuzzy systems [4ndash8] are the evolu-tionary fuzzy systems designed by genetic algorithms Forexample GA was used to design neural-fuzzy system fortemperature control in [5] and fuzzy controller for mobilerobots in [7]

Particle swarm optimization (PSO) is one of swarmintelligence models PSO was inspired by the social behaviorof fish schooling and bird flocking [9] In the PSO eachparticle represents a candidate solution to a problem and fliesin the hyperspace according to its flying velocity vector whichis stochastically determined by its previously personal bestand swarm best experiences In the end of PSO operationthe experienced swarm best solution is the finally obtainedsolution To address the encountered issues and improveover the parent PSO [9] many advanced PSO variants [10ndash15] have been proposed some of which were applied todesign the TSK-type fuzzy systems [14 15] These algorithmsinclude the PSO with time-varying acceleration coefficients(PSO-TVAC) [10] a self-organizing hierarchical particleswam optimizer with TVAC (HPSO-TVAC) [10] PSO withcontrollable random-exploration velocity (PSO-CREV) [11]enhanced PSO by incorporating a weighted particle [12] thehybrid of GA and PSO (HGAPSO) [13] two-phase swarmintelligence algorithm (TPSIA) [14] and ant and particleswarm cooperative optimization (APSCO) [15]

Another type of swarm intelligence model is ant colonyoptimization (ACO) [16ndash19] The ACO technique wasinspired by foraging behavior of the real ant colony andwas proposed initially for solving discrete combinationaloptimization problems such as travelling salesman problem(TSP) The ACO algorithms have been successfully appliedto optimize the fuzzy systems for mobile robot control[20 21] In those studies in order to apply the discreteACO for optimization the parameters charactering the fuzzycontroller were first discretized thus sacrificing the precision

To overcome the precision issue some ant-related algo-rithms for the optimization problems with real-value param-eters have been proposed [22ndash24] Among these algorithmsthis paper focuses the interest on ACOR [24] because it ismost related to the discrete ACO and has achieved goodperformances on continuous optimization of the benchmarkfunctions as demonstrated in [24] ACOR thus is advanta-geous on the application problems requiring high precisionsuch as the design of fuzzy controllers for dynamic systems

because their inputs and outputs are usually continuouscases Since the proposal of ACOR some variants have beenproposed and applied to design fuzzy systems for accuracy-oriented problems [25ndash28] In [25] a modified continuousACO algorithm (RCACO) was proposed for the design offuzzy-rule-based systems in order to achieve considerablelearning accuracy The paper [26] proposed a cooperativecontinuous ACO (CCACO) with multiple colonies of pop-ulations each colony of which is only responsible for opti-mizing a single fuzzy rule Although the simulation resultsdemonstrated that the performance of multicolony basedCCACO is better than that of the single-colony RCACO thecomputation of CCACO is much more complex than thatof RCACO Inspired by the cognitive psychology conceptsthe study in [27] proposed an assimilation-accommodationmixed continuous ant colony optimization (ACACO) for thedesigns of feed-forward fuzzy systems In addition the elite-guided continuous ACO (ECACO) was proposed to designrecurrent fuzzy systems [28] whichwas claimed to be the firstapplication of continuous ACO on recurrent fuzzy systemdesign Although those variants indeed helped accomplishthe accurate design of fuzzy systems there is still possibleroom for further improvement especially on the balancebetween the exploration and convergence rate of solutionsThis paper introduces the dynamic mutation [29] into ACORto enhance such balance

For population-based algorithms such as PSO and ACOanother major issue is population initialization [30ndash34]Startingwith a population of initial solutions the population-based algorithm improves the solutions iteratively in orderto find the better solution Therefore the population ini-tialization affects the performance In general however theinitial population solutions are generated randomly becauseof no a priori information The work in [31] suggesteduse of Sobol sequence generator for generating initial PSOparticles because the generated initial solutions are uniformlydistributed into the search space The study in [32] reportedthat the initialization using the nonlinear simplex method(NSM) helped improve the convergence rate and successrate of PSO By using the generators of centroidal Voronoitessellations (CVT) as the starting point as suggested in [33]the initial solutions can be more evenly distributed through-out the high-dimensional problem space and improve PSOperformanceThe study in [34] proposed a center-based sam-pling for the initialization of population-based algorithmsBy simulations that paper showed that the points in centerregion have higher chances to be closer to an unknownsolution and thus suggested the center region is a promisingregion for the initialization However those initializationapproaches [31ndash34] did not take into account application-specific information Based on [34] and the observations ofsome accuracy-oriented fuzzy controller designs this paperproposes an ad hoc population initialization method forinitial ant solutions to improve the design accuracy

This paper mainly contributes to propose an enhancedcontinuous ACO algorithm incorporating dynamic muta-tions and ad hoc initialization ACODM-I for the designof TSK-type fuzzy systems ACODM-I can be regardedas a population-based evolutionary algorithm Instead of

Computational Intelligence and Neuroscience 3

random generation the proposed population initialization inACODM-I takes the observations of well-performed fuzzysystems into account The introduced dynamic mutation inACODM-I initially provides more diverse search directionsfor exploring solutions to avoid being trapped into a localoptimum in the early stage The performance superiority ofACODM-I to the parent ACOR and different advanced algo-rithms or neural-fuzzy models is verified by the simulationresults of TSK-type fuzzy systems for the problems of trackingcontrol and chaotic time series predication Furthermore theeffects on the convergence rate and design accuracy yieldedby the proposed initialization and introduced dynamicmuta-tion are verified by the simulation results

This paper is organized as follows The next sectiondescribes the zero-order and first-order TSK-type fuzzysystems Section 3 introduces basic concepts of discrete ACOand ACOR and proposes ACODM-I for TSK-type fuzzy sys-tem design Section 4 presents the simulation results of TSK-type fuzzy systems designed by ACODM-I for the trackingcontrol of dynamic plant and the prediction of chaotic timeseries Section 4 also compares the ACODM-I performancewith the ones of ACOR with different values of parameter andadvanced algorithms or models Furthermore the effects ofthe proposed initialization and dynamic mutation are alsodiscussed and verified in Section 4 Ultimately conclusionremarks are given in Section 5

2 TSK-Type Fuzzy System

In contrast toMadami-typemodel having good interpretabil-ity TSK-type fuzzy systems focus on themodel accuracyThemain difference is on the consequent part of fuzzy rules Ina TSK-type fuzzy system the consequence of fuzzy rule isdefined as the linear combination of the fuzzy inputs The 119894thfuzzy rule of a TSK-type fuzzy system is described as follows

Rule 119894 If 1199091 (119896) is 119860 1198941 119909119899 (119896) is 119860 119894119899Then 119910 (119896) is 119891119894 (1199091 1199092 119909119899) (1)

where 119896 is the time step 1199091 1199092 119909119899 are the input variablesand 119910 is the output variable of the fuzzy system In theantecedent part of fuzzy rule 119894 119860 119894119895 is a fuzzy set for the input119909119895 and is characterized by a membership function If thefunction119891119894(1199091 1199092 119909119899) in the consequent part is a constant119891119894 (1199091 1199092 119909119899) = 119886119894 (2)

the system is denoted as a zero-order TSK-type fuzzy systemIf the function 119891119894(1199091 1199092 119909119899) is defined by

119891119894 (1199091 1199092 119909119899) = 1198861198940 + 119899sum119895=1

119886119894119895119909119895 (3)

it is a first-order TSK-type fuzzy system In this study aswidely used in fuzzy system designs [15ndash17 27ndash30] Gaussianfunction is selected to characterize the fuzzy set 119860 119894119895 and isdefined by

119872119894119895 (119909119895) = expminus(119909119895 minus 119898119894119895119887119894119895 )2 (4)

where 119898119894119895 and 119887119894119895 are the center and the width of fuzzy set119860 119894119895 respectively For a Gaussian function the parameters119898119894119895 and 119887119894119895 are independent variables and its function outputcorresponding to any input value is never zero which willmake the design task easier

For a TSK-type fuzzy system consisting of 119877 rules theoutput of the fuzzy system through the inference engine iscalculated by

119910 = sum119877119894=1 120601119894 (997888rarr119909) sdot 119891119894 (997888rarr119909)sum119877119894=1 120601119894 (997888rarr119909) 120601119894 (997888rarr119909) = 119899prod

119895=1

expminus(119909119895 minus 119898119894119895119887119894119895 )2 (5)

where 120601119894(997888rarr119909) is the firing strength of rule 119894 excited by agiven input dataset 997888rarr119909 = (1199091 1199092 119909119899) Thus to constructan 119877-rule TSK-type fuzzy system with 119899 input variables alldecision variables represented by997888rarr119904 = [11989811 11988711 1198981119899 1198871119899 1198861 11989821 11988721 1198982119899 1198872119899 1198862 1198981198771 1198871198771 119898119877119899 119887119877119899 119886119877] equiv [1199041 1199042 119904119863] (6)

for a zero-order TSK-type fuzzy system or997888rarr119904 = [11989811 11988711 1198981119899 1198871119899 11988610 11988611 1198861119899 11989821 11988721 11989821198991198872119899 11988620 11988621 1198862119899 1198981198771 1198871198771 119898119877119899 119887119877119899 1198861198770 1198861198771 119886119877119899] equiv [1199041 1199042 119904119863] (7)

for a first-order TSK-type fuzzy system are to be determinedHowever such design task of a TSK-type fuzzy system canbe treated as an optimization problem that finds the freeparameters represented in (6) or (7) such that the task-dependent objective function is optimized Following thistransformation the optimization algorithms can be usedto solve such optimization problem for accomplishing thedesign of TSK-type fuzzy system

3 Proposed ACODM-I forFuzzy System Design

This section presents the proposed enhanced continuousACO with dynamic mutation and ad hoc initialization(ACODM-I) for the design of TSK-type fuzzy systems Sincethe proposed ACODM-I is inspired and related to the ACOframework this section first reviewed the basic concept of thediscrete ACO and ACOR The simulation results optimizedby the ACOR will also be presented as the benchmark forcomparison in Section 4

31 Basic Concept of Discrete Ant Colony Optimization Thediscrete ACO algorithms [16ndash19] were inspired and devel-oped by the behavior of real ant colonies and now are largelyemployed to find the solution of discrete combinationaloptimization problems (COP) When ants foraged for foodthey deposited the pheromone on the trail to guide other ants


middot middot middot








rarrs1rarrs2

rarrsi

rarrsN

s11

s12

s1i

s1N sjN

sj1

sj2

sji

sDN

sD1

sD2

sDi

E(rarrs1)

E(rarrs2)

E(rarrsi )

E(rarrsN)

w1

w2

wi

wN

Figure 1 The solutions archive in ACOR where the weights 1199081 ge1199082 ge sdot sdot sdot ge 119908119873 and the objective values 119864(997888rarr119904 1) le 119864(997888rarr119904 2) le sdot sdot sdot le119864(997888rarr119904 119873) [24]Since the pheromone will evaporate with time the path witha higher pheromone level is the most possibly a shorter oneto the food source Ant System (AS) [16] is the first discreteACOalgorithm and proposed to solve the travelling salesmanproblem (TSP) In Ant System for TSP 119875119896119894119895(119905) represents theprobability that ant k situated at city i at the time t chooses tovisit city j and is defined by

119875119896119894119895 (119905) = (120591119894119895 (119905))120572 (120578119894119895)120573sum119898isin119881119896119894

(120591119894119897 (119905))120572 (120578119894119897)120573 if 119895 isin 119881119896119894 0 otherwise (8)

where 120591119894119895(119905) is the pheromone trail on the link between cities119894 and 119895 120578119894119895 is the corresponding a priori heuristic informationof the link and 119881119896119894 is the set of allowed neighborhood citiesant 119896 can move from city 119894 The relative significance of thepheromone trail and heuristic information is determined bythe values of 120572 and 120573 Each ant is assumed to visit each cityonly once and it will construct a feasible solution to TSP afterit visited all cities After all ants accomplished their visits allfeasible solutions are gathered to update the pheromone levelsfor the next ant cycle The discrete ACO algorithm repeatssuch procedure to find the shorter path Since that proposalsome variants of discrete ACO algorithms were proposed[17 18] In addition by discretizing the parameters in fuzzysystems the applications of the discrete ACO to optimize thefuzzy controllers for mobile robots have been demonstrated[20 21]

32 Basic Concept of ACO119877 Among many proposed con-tinuous ACO algorithms the ACOR in [24] is one of themost promising algorithms and is most related to the orig-inal discrete ACO In [24] ACOR has demonstrated goodperformances for continuous optimization of the benchmarkfunctionsThebasic concept ofACOR is to extend the discreteprobability distributions (8) utilized in discrete ACO tocontinuousGaussian probability density functions (PDFs) InACOR these Gaussian PDFs are derived from a maintainedsolution archive as shown in Figure 1 and the sampledvalues from the chosen PDFs are gathered to construct newsolutions

In Figure 1 a feasible solution (ant path) to the opti-mization problem is represented by a row vector 997888rarr119904 119894 in thearchive and its quality 119864(997888rarr119904 119894) is measured by the value of

the predefined objective function All feasible solutions aresorted ranked from the best to the worst and maintained inthe fixed-size archive table By doing this the solution997888rarr119904 119897 thushas rank 119897 Then the rank of each ant solution in the archivedetermines its probability of being chosen to follow in thenext ant cycle The operation of ACOR operation is detailedas follows

ACOR usually initializes all N solutions in the archiveeach of which is a D-dimensional row vector 997888rarr119904 =[1199041 1199042 119904119863] by generating random numbers within therange of search space All initialized solutions then areevaluated sorted according to the evaluation values andranked in the solution archive Each solution 997888rarr119904 119897 of the rank lin the sorted archive is assigned with a weight 119908119897

119908119897 = 1119902119873radic2120587 expminus(119897 minus 1)2211990221198732 (9)

where 119902 is a parameter of the ACOR To generate a newcandidate solution the ACOR firstly chooses one leadingsolution 997888rarr119904 119897 among 119873 solutions in the archive according tothe probability distribution

119901119897 = 119908119897sum119873119898=1 119908119898 119898 = 1 2 119873 (10)

It indicates clearly from (10) that the better-ranked solutionhas higher probability being chosen as the leading solutionand the value of q in (9) controls the tendency for exploringthe archive solutions Once a leading solution 997888rarr119904 119897 is chosena new candidate solution is constructed by sampling thederived Gaussian PDF 119892119895

119897(119904 120583119895119897 120590119895119897) in a sequence of 119895 =1 2 119863 with the mean 120583119895

119897= 119904119895119897and the standard deviation120590119895

119897

119892119895119897(119904 120583119895119897 120590119895119897) = 1120590119895119897radic2120587 exp

minus(119904 minus 120583119895

119897)22 (120590119895119897)2

120590119895119897= 120576 119873sum119898=1

10038161003816100381610038161003816119904119895119898 minus 119904119895119897 10038161003816100381610038161003816119873 minus 1 (11)

where the pheromone evaporation rate 120576 is a positive param-eter A higher value of 120576 provides more exploration in searchthus converging slower while a lower value of 120576 providesmoreexploitation in search thus converging faster By repeatingsuch process 119871 times 119871 new candidate solutions are gener-ated Then these 119871 new candidate solutions are evaluatedTogether with the original 119873 solutions in the previous antcycle the total (119873 + 119871) solutions are sorted again The ACORonly reserved the 119873-top-performed solutions for next antcycle The ACOR repeats such ant cycle until the terminationcondition is satisfied


33 ACO119877 Using Dynamic Mutation andAd Hoc Initialization (ACODM-I)

331 ACO119877 with Dynamic Mutation (ACODM) In ACORthe value of 119902 in (9) controls the exploration ability thusaffecting convergence rate of the algorithm When the valueof 119902 is small ACOR strongly prefers the best-ranked solutionsas the leading solution which focuses on exploiting the best-ranked solutions locally This will increase the convergencerate but the chance of convergence result being trappedinto the local optimum is also increased When 119902 value islarge the probability for each solution being chosen as theleading solution becomes nearly uniform This can enhancethe exploration ability for possibly obtaining globally bettersolution but the convergence rate will be decreased

In order to avoid or lessen this issue this paper introducesmutation technique into the original ACOR to balance theexploration ability and the convergence rate In addition tothe Gaussian sampling technique the introduced mutationprovides another option for changing (generating) a newlyconstructed solution component by ldquojumpingrdquo to the neigh-boring of the other archive solutions Since the probabilityfor the mutation is not fixed which will be clearly seenthis modified algorithm is named as ACOR with dynamicmutation ACODM and its operation is detailed as follows

In ACODM without loss of generation the range of eachdecision variable to be identified is assumed in the interval[0 1] If a leading solution 997888rarr119904 119897 is chosen the value of the dthcomponent of a new candidate solution 997888rarr119904 119894 is generated

If rand119889119894 gt 119901119889(119905)119904119889119894 (119905 + 1) = Sampl (119892119889119897 (119904 119904119889119897 (119905) 120590119889119897 (119905))) (12)

else

119904119889119894 (119905 + 1) = 119904119889119903 (119905) + 119904119889119903 (119905) sdot 119880 [minus01 01] if 119904119889119903 (119905) le 05119904119889119894 (119905 + 1) = 119904119889119903 (119905) + (1 minus 119904119889119903 (119905)) sdot 119880 [minus01 01]if 119904119889119903 (119905) gt 05

(13)

In (12) the Sampl(119892119889119897 (119904 119904119889119897 (119905) 120590119889119897 (119905))) denotes the sampledvalue from a Gaussian PDF 119892119889119897 (119904 119904119889119897 (119905) 120590119889119897 (119905)) with the mean119904119889119897 (119905) and the standard deviation 120590119889119897 (119905) The ranges of theuniform randomnumbers rand119889119894 and119880[minus01 01] are in [0 1]and [minus01 01] respectively The index 119903 in (13) is a uniformrandom integer number in [1119873] Finally the mutationprobability 119901119889(119905) for the 119889th component is set to be linearlyproportional to the biased standard deviation 120590119889(119905) among allants and is calculated by

119901119889 (119905) = ℎ sdot 120590119889 (119905) = ℎ sdot ( 119873sum119894=1

(119904119889119894 (119905) minus 119904119889 (119905))2119873 )12

(14)

where ℎ is an adjustable parameter determining the mutationprobability and 119904119889(119905) is the average value among all the dthcomponents in the ant population

In ACODM in addition to the exploitation using Gaus-sian sampling in (12) the dynamic mutation in (13) canexplore more diverse search solutions to avoid being trappedinto a local optima in the early stage of the optimizationprocess Moreover themutation probability for each solutioncomponent inACODMdepends on the convergence status ofthe ant population so its value is not fixed but dynamic andis not the same for each solution component In (14) a highervalue of h suggests a possibly larger probability for dynamicmutation by directly jumping to the neighboring of the otherarchive solutions Therefore the higher the value of h is thelower the convergence rate is Finally the change of value forthe mutation of ACODM as presented in (13) is dynamic incontrary to the fixed change used in themutation operation ofgeneral genetic algorithms Equation (13) indicates the pointsaround the center region have larger possible deviation thanthose points close to the boundary when they are chosenas the neighbors for learning The initial motivation forthis thought is to hope the central points can wander andexplore in more diverse search directions in the early stageof optimization process

The ACODM algorithm repeats the selection of leadingsolution using (10) followed by the Gaussian sampling in(12) or dynamic mutation (13) to generate L new candidatesolutions These L new candidate solutions are evaluated andsorted together with theN solutions in the previous ant cycleSimilarly only the N-top-best solutions are reserved for nextant cycle ACODM repeats such ant cycle to find the bettersolution until the termination condition is met

332 Ad Hoc Population Initialization for Fuzzy SystemDesign The initialization of the ant solutions in ACOR isanother issue which is also a major problem for all otherpopulation-based algorithms [30ndash35] Starting with a popu-lation of initial ant solutions ACOR improves the solutionsiteratively to find the better solution Therefore the initial-ization of ant solutions in ACOR affects the performance Ingeneral good initialization can help achieve better optimumwhile bad initialization usually ends on a poor local optimumHowever because no a priori information is available ingeneral the initial ant solutions are generated randomly Asmentioned in Section 1 some advanced random initializationapproaches over the multiple-dimensional search space wereproposed [31ndash34] However those initialization approachesdid not take into account application-specific informationand thus they are regarded as generic initialization

The study in [34] proposed a center-based samplingfor the initialization of population-based algorithms Bysimulations that paper showed that the points in [02 08]in the search space [0 1] have higher chances to be closerto an unknown solution and thus the center region is apromising region for the initialization In addition in ourobservation of fuzzy systems especially accuracy-orientedfuzzy controllers one of the membership functions for eachinput in its antecedent part is usually designed at the location


Initialize the evaporation rate 120576 and the archive size119873Generate119873 solutions in (6) or (7) with119898119894119895 and 119887119894119895 with random values in [045 055] and 119886119894119895 in [0 1]Evaluate and sort the initial119873 solutionswhile the number of ant cycles is less than prescribed value do

Calculate the mutation probability 119901119889(119905) in (14)for 119896 = 1 119871 (generate candidate solutions 997888rarr119904 119873+119896)Choose a leading solution 997888rarr119904 119897 according to the probability distribution (10)for 119889 = 1 119863 (generate solution components 119904119889119873+119896)

if rand119889119896 gt 119901119889(119905) Gaussian sampling using (12)else dynamic mutation using (13)

endendEvaluate 119871 new candidate solutions and sort (119873 + 119871) solutionsUpdate population to keep119873 solutions in the archive

end while

Algorithm 1 ACODM-I algorithm for TSK-type fuzzy system design

around the input value arising more frequently or mostconcerned which is usually at the center of the search rangeTherefore this paper proposes an ad hoc central initializationrange [045 055] for initializing the parameters119898119894119895 and 119887119894119895 inthe antecedent part of fuzzy rule for the fuzzy system designsThe rest of free parameters 119886119894119895 in the consequent part of aTSK-type fuzzy system are initially generated uniformly inthe search space because of no a priori information

If ACODM generates the initial ant solutions using theproposed initialization method the resultant algorithm isdenoted by ACODM-I For clarity the pseudocode of theACODM-I algorithm is shown in Algorithm 1

4 Simulations

Three application examples of the designs of TSK-type fuzzysystems are demonstrated in this section to validate theproposed algorithm Two zero-order TSK-type fuzzy systemsare designed for nonlinear dynamic plant control and onefirst-order TSK-type fuzzy system is optimized for the pre-diction of the chaotic time series In the simulations thepopulation size N is 20 and the number of newly generatedtemporary solutions L is 20 The value of 120576 is 085 as usedin [24 25] The value of ℎ = radic12 is set to have themutation probability of 10 when the decision variable ineach dimension is uniformly distributed among all antsThe performances of fuzzy controllers and fuzzy predictoroptimized by ACODM-I are validated and compared withthose by ACOR with different values of q In the simulationsfor ACOR each initial parameter value in the fuzzy systemswas generated randomly and uniformly within its searchspace In addition the resulting ACODM-I performance isalso compared with the reported results of some advancedpopulation-based algorithms or neural-fuzzy models on thesame problem The advantage on the optimization accuracyyielded by the proposed initialization and dynamic mutationare also discussed through simulation results The personalcomputer for conducting all simulations possesses an IntelCore i5 28GHz dual-core-processor and runs onWindows 7

Example 1 As the first example a zero-order TSK fuzzysystem is designed to control the nonlinear plant as taken in[14 25] and described by

119910 (119896 + 1) = 119910 (119896)1 + 1199102 (119896) + 1199063 (119896) (15)

The initial state119910(0) of the system is assumed to zero andminus1 le119906(119896) le 1 is the control input of the plant The objective forthe fuzzy system (controller) to be optimized is to control theplant output to track the reference trajectory 119910119889(119896) as givenby

119910119889 (119896) = sin(12058711989650 ) cos(12058711989630 ) 1 le 119896 le 250 (16)

In this example the fuzzy controller is fed with two signalsthe current plant output 119910(119896) and the target output 119910119889(119896 + 1)As the response to these two inputs the produced output 119906(119896)of the fuzzy system is used to control the nonlinear plant (15)For a designed fuzzy controller its performance is evaluatedby the root mean square error (RMSE) between the plantoutput and the reference trajectory over the 250 time stepsand is calculated by

RMSE = (250sum119896=1

(119910119889 (119896) minus 119910 (119896))2250 )12 (17)

The fuzzy controller in this example consists of fivefuzzy rules in order to compare with other algorithms laterThen the values of the free parameters 119898119894119895 isin [minus1 1]119887119894119895 isin [0 1] and 119886119894 isin [minus1 1] for constructing a fuzzysystem are searched using ACODM-I such that the errorin (17) is minimized For each single run of optimizationprocess 10000 evaluations were performed to conclude asolution Over 50 runs of simulations the average best-so-far RMSE at each performance evaluation of ACODM-I is shown in Figure 2 The learning results of the parentACOR with different values of 119902 are also shown in Figure 2


2000 4000 6000 8000 100000Evaluation

minus16

minus14

minus12

minus1

minus08

minus06

minus04

LＩＡ10

(RM

SE)

２ (q = 001)

２ (q = 01)

２ (q = 10)

２ (q = 10)

ACODM-I (q = 01)

Figure 2 The average best-so-far RMSE at each performanceevaluation for the evolutionary fuzzy controllers optimized byACODM-I and ACOR in Example 1

for comparison The learned statistical numerical results ofRMSE errors are shown in Table 1 The results show that theaverage RMSE minimum RMSE and maximum RMSE ofACODM-I are smaller than the ones of ACOR algorithmswith different values of 119902

The previous study [25] reported the performance resultsof some advance population-based evolutionary algorithmswhen they were applied to the same fuzzy control problemThese algorithms include a hierarchical PSO-TVAC (HPSO-TVAC) [10] a PSO with controllable random-explorationvelocity PSO (PSO-CREV) [11] a hybrid of GA and PSO(HGAPSO) [13] a two-phase swarm intelligence algorithm(TPSIA) using discrete ACO in the first phase and PSO inthe second phase [14] and fuzzy-rule-based continuous antcolony optimization (RCACO) [25] The operations of thesealgorithms and their settings of parameters used on thiscontrol problem were detailed in [14 25] The fuzzy systemoptimized by each of these algorithms was also composedof five rules to ensure the same number of free parametersas that by ACODM-I Moreover for each single run eachof these algorithms also performed the same number ofevaluations as that by ACODM-I Table 2 presents thereported performance results of these algorithmsThe resultsshow that ACODM-I achieved the smaller average error thanall other algorithms in comparison

The performance results of the fuzzy system optimizedby ACOR ACODM and ACOR-I (representing the ACORusing the proposed initialization) are also provided in orderto discuss or validate the effectiveness of the proposedinitialization and dynamic mutation Table 3 presents theselearned statistical numerical results when the value of 119902 is 01Simulation results show that ACODM achieved the smalleraverage error than ACOR The average error of ACODM-I is also smaller than that of the ACOR-I This indicates

ACODMACODM-I

2000 4000 6000 8000 100000Evaluation

minus16

minus14

minus12

minus1

minus08

minus06

minus04

LＩＡ10

(RM

SE)

２

ACOR-I

Figure 3 The average best-so-far RMSE at each performanceevaluation for the evolutionary fuzzy controllers optimized byACOR ACODM ACOR-I and ACODM-I in Example 1

that the dynamic mutation can improve the optimizationaccuracy thus validating the effectiveness of the introduceddynamic mutations The main reason to this is the abilityof the dynamic mutation providing more diverse searchdirections for the ant colony when the value of 119902 is notlarge Similarly the effectiveness of the proposed initializationon improving the accuracy of the fuzzy system design isalso validated by the observation that the average errorsof ACOR-I and ACODM-I are smaller than the ones ofACOR and ACODM respectively Moreover the learningcurves of ACOR ACODM ACOR-I and ACODM-I areshown in Figure 3 It is also observed that the algorithmsincorporating dynamic mutation ACODM and ACODM-I converged slower than the ones without incorporatingdynamic mutation ACOR and ACOR-I respectively Thepossibly best explanation is that the exploration on moresearch directions by the dynamic mutation brings the accu-racy improvement at the expense of the convergence rateFinally ACODM-I achieved the smallest error The trackingresults controlled by ACODM-I optimized fuzzy system arepresented in Figure 4 The results show that the controlledplant output is very close to the reference trajectory

Example 2 Thenonlinear plant for control using a zero-orderTSK-type fuzzy system as taken in [15 25] is described by

119910 (119896 + 1) = 119910 (119896) 119910 (119896 minus 1) (119910 (119896) + 25)1 + 1199102 (119896) + 1199102 (119896 minus 1) + 119906 (119896) (18)

The initial states 119910(minus1) and 119910(0) are assumed to minus6 andminus12 le 119906(119896) le 12 is the control input of the plant Theobjective of the zero-order TSK-type fuzzy controller is to


Table1Perfo

rmanceso

fthe

fuzzycontrollerd

esignedby

theA

CODM-IandAC

ORin

Exam

ple1

Algorith

ms

ACO

R(119902=0

01)AC

OR(119902=0

1)AC

OR(119902=0

2)AC

OR(119902=0

4)AC

OR(119902=0

6)AC

OR(119902=0

8)AC

OR(119902=1

0)AC

OR(119902=1

0)ACO

DM-I(119902=0

1)Av

erageR

MSE

00525

00350

00343

00357

00354

00392

00374

004

03002

56ST

D00241

00071

000

6700057

000

49000

6600056

00050

00032

Minim

um00273

00220

00231

00261

00269

00300

00280

00276

002

01Maxim

um01835

00524

00491

00472

00454

00575

00490

00518

00372


Table 2 Performances of fuzzy controller designed by the ACODM-I and different algorithms in Example 1

Algorithms HPSO-TVAC HGAPSO PSO-CREV TPSIA RCACO ACODM-IAverage RMSE 0039 0040 0041 0033 00260 00256STD 0014 0008 0012 0012 00047 00032

Table 3 Performances of fuzzy controller designed by ACOR ACODM ACOR-I and ACODM-I in Example 1

Algorithms ACOR ACOR-I ACODM ACODM-IAverage RMSE 00350 00275 00299 00256STD 00071 00053 00061 00032Minimum 00220 00215 00217 00201Maximum 00524 00441 00433 00372

ActualDesired

50 100 150 200 2500Time step

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Am

plitu

de

Figure 4 Fuzzy control results using ACODM-I in Example 1

control the plant output 119910(119896) to track the reference trajectory02119910119889(119896) where 119910119889(119896) is governed by the reference model

119910119889 (119896 + 1) = 06119910119889 (119896) + 02119910119889 (119896 minus 1) + 119903 (119896) 0 le 119896 lt 250119903 (119896) = 02 sin(211989612058725 ) + 04 sin(11989612058732 )

(19)

where the initial states of the reference model 119910119889(minus1) =119910119889(0) = minus6 are assumed Two previous system outputs119910(119896) 119910(119896 minus 1) and the target output 02119910119889(119896 + 1) are the threeinput variables of the fuzzy controller The produced output119906(119896) of the fuzzy system is to control the nonlinear plant(18) The performance evaluation of a designed controller isdefined as the sum of absolute error (SAE) between the plantoutput and reference trajectory over 250 time steps and iscomputed by

SAE = 250sum119896=1

100381610038161003816100381602119910119889 (119896) minus 119910 (119896)1003816100381610038161003816 (20)

1 2 3 4 5 60Evaluation times104

0

02

04

06

08

1

12

14

16

18

LＩＡ10

(SA

E)

２ (q = 001)

２ (q = 01)

２ (q = 10)

２ (q = 10)ACODM-I (q = 01)

Figure 5 The average best-so-far SAE at each performance evalua-tion for the evolutionary fuzzy controllers optimized by ACODM-Iand ACOR in Example 2

The fuzzy system to be optimized consists of four fuzzyrules The values of the free parameters 119898119894119895 isin [minus12 12]119887119894119895 isin [0 12] and 119886119894 isin [minus12 12] are searched to find a better-performed fuzzy controller yielding smaller SAE The 60000evaluations were performed to conclude a solution in eachrun of optimization process Over 50 runs of simulations theaverage best-so-far SAE at each performance evaluation ofeach algorithm for fuzzy system design is shown in Figure 5The learned numerical results of SAE errors are shown inTable 4The results show thatACODM-I achieved the smallererror than ACOR algorithms with different values of q

ACODM-I performance is also compared to the reportedresults of HPSO-TVAC HGAPSO PSO-CREV RCACO andant and particle swarm cooperative optimization (APSCO)using the parallel combination of ACO and PSO [15] whenthey were applied to the same fuzzy control problem Thecomparison is shown in Table 5 The results show that


Table4Perfo

rmanceso

fthe

fuzzycontrollerd

esignedby

ACODM-IandAC

ORin

Exam

ple2

Algorith

ms

ACO

R(119902=0

01)AC

OR(119902=0

1)AC

OR(119902=0

2)AC

OR(119902=0

4)AC

OR(119902=0

6)AC

OR(119902=0

8)AC

OR(119902=1

0)AC

OR(119902=1

0)AC

ODM-I(119902=0

1)Av

erageS

AE

96332

29983

23056

25632

21049

24515

22922

23184

1229

2ST

D86435

26633

17033

16424

08539

15094

08167

13302

07454

Minim

um16

844

0722

05101

07012

0838

09103

0686

09544

048

79Maxim

um584138

130973

1212

7197

500

37552

99463

344

0390

152

28749



Algorithms HPSO-TVAC HGAPSO PSO-CREV APSCO RCACO ACODM-IAverage SAE 664 533 40 373 192 123STD 564 289 12 165 106 075

times1041 2 3 4 5 60

Evaluation

0

02

04

06

08

1

12

14

16

18

LＩＡ10

(SA

E)

ACODMACODM-I

２

ACOR-I

Figure 6 The average best-so-far SAE at each performance eval-uation for the evolutionary fuzzy controllers optimized by ACORACODM ACOR-I and ACODM-I in Example 2

ACODM-I achieved smaller average error than all otheralgorithms in comparison

In this example when the value of q is 01 simulationresults of the zero-order TSK-type fuzzy controller optimizedby ACOR ACODM ACOR-I and ACODM-I are shown inTable 6 and Figure 6 In a similar way to Example 1 the effectsof the proposed initialization and dynamic mutation on thedesign accuracy and convergence rate are easily identifiedin the simulation results The tracking results of the fuzzysystem optimized by ACODM-I are shown in Figure 7 Thesimulation results show the controlled plant output is veryclose to the desired reference trajectory

Example 3 In this example a first-order TSK-type fuzzysystem is used to predict the chaotic time series The timeseries to be predicted is generated by thewell-knownMackey-Glass chaotic system which is defined by the following time-delay differential equation119889119909 (119905)119889119905 = 02119909 (119905 minus 120591)1 + 11990910 (119905 minus 120591) minus 01119909 (119905) (21)

where 120591 is set as 17 the initial point 119909(0) is assumed to be12 and 119909(119905) = 0 for 119905 lt 0 as in the previous studies[26 27] Four past values of 119909(119905 minus 24) 119909(119905 minus 18) 119909(119905 minus 12)and 119909(119905 minus 6) are the inputs of the first-order TSK-type fuzzysystem The produced output of the fuzzy system excited bythese four past inputs is the prediction of 119909(119905) The fuzzy

ActualDesired

50 100 150 200 2500Time step

minus12

minus1

minus08

minus06

minus04

minus02

0

02

04

06

Am

plitu

de


predictor is optimized by ACODM-I and evaluated in thefollowing experiment First the solution of (21) was solvedusing Runge-Kutta numerical method Secondly the timeseries data were generated by sampling the solution everysecond The sampled data patterns from 119905 = 124 to 623 wereused as the training set for optimizing the fuzzy system whilethe remaining patterns from 119905 = 624 to 1123 were used as testset for validating the designed fuzzy model

The first-order TSK-type fuzzy system consists of fourrules The values of the free parameters 119898119894119895 isin [0 2] 119887119894119895 isin[0 1] and 119886119894119895 isin [minus2 2] are searched to find a better-performed fuzzy predictor The objective function is definedas the RMSE between the sampled output of the chaoticsystem governed by (21) and the predicted output by the fuzzysystem The 300000 evaluations were performed in a singlerun of training The learning results of the average RMSE ateach evaluation over 50 runs are shown in Figure 8 Table 7shows the learned numerical results of training and testingRMSE errors The results show that ACODM-I achieved thesmaller prediction error than the ACOR algorithms withdifferent values of 119902

Similar to the two previous examples simulation resultsof the first-order TSK-type fuzzy systems optimized byACOR ACODM ACOR-I and ACODM-I are provided inTable 8 and Figure 9 to verify the effects of the proposedinitialization and dynamic mutation The results indicatethat the proposed initialization and dynamic mutation inACODM-I as applied to the design of first-order TSK-type


Table 6 Performances of fuzzy controllers designed by ACOR ACODM ACOR-I and ACODM-I in Example 2

Algorithms ACOR ACOR-I ACODM ACODM-IAverage SAE 29983 18273 21049 12292STD 26633 09141 0879 07454Minimum 0722 05465 05174 04879Maximum 130973 32088 4179 28749

minus22

minus2

minus18

minus16

minus14

minus12

minus1

minus08

minus06

minus04

LＩＡ10

(RM

SE)

times10521 30 2505 15

Evaluation

２ (q = 001)

２ (q = 01)

２ (q = 10)

２ (q = 10)

ACODM-I (q = 01)

Figure 8 The average best-so-far RMSE at each performance eval-uation for the evolutionary fuzzy predictor optimized by ACODM-Iand ACOR in Example 3

fuzzy system have the same effects as those used in the zero-order TSK-type fuzzy systems for control problems It isalso observed that the proposed initialization helps improvemuch more on design accuracy than dynamic mutation Thepredication results of the first-order TSK-type fuzzy systemoptimized by ACODM-I are shown in Figure 10 The resultsshow the prediction values by the fuzzy system are very closeto the desired ones

The performance of fuzzy system designed by ACODM-I is also compared to the reported results of differentalgorithm and neural-fuzzy systems that were applied to thesame prediction problem [28] These neural-fuzzy systemsinclude a Hybrid Neural-Fuzzy Inference System (HyFIS)[35] a Dynamic Fuzzy Neural Network (D-FNN) [36] aSubsethood-Product Fuzzy Neural Inference System (SuP-FuNIS) [37] and a Generalized Fuzzy Neural Network (G-FNN) [38]The algorithm in comparison is amultiple-colonytopology based Cooperative Continuous ACO- (CCACO-)designed neural-fuzzy system [26] The comparison resultsare shown in Table 9 The results show the fuzzy predictoroptimized by ACODM-I achieved the smaller test errorthan the most of reported neural-fuzzy systems Though theACODM-I performance is very close to the G-FNN andCCACO ACODM-I optimized fuzzy system uses smaller

minus22

minus2

minus18

minus16

minus14

minus12

minus1

minus08

minus06

minus04

LＩＡ10

(RM

SE)

times10521 30 2505 15

Evaluation

２

ACOR-IACODMACODM-I

Figure 9 The average best-so-far RMSE at each performanceevaluation for the evolutionary fuzzy predictor optimized by ACORACODM ACOR-I and ACODM-I in Example 3

ActualDesired

700 750 800 850 900 950 1000 1050 1100650Time step

02

04

06

08

1

12

14

16

Am

plitu

de

Figure 10 Prediction results of the fuzzy system optimized byACODM-I in Example 3

number of rules thanG-FNNAs reported in [26] the averagetraining RMSE of the CCACO is 00094 which is larger


Table7Perfo

rmanceso

fthe

fuzzypredictord

esignedby

theA

CODM-IandAC

ORin

Exam

ple3

Algorith

ms

ACO

R(119902=0

01)AC

OR(119902=0

1)AC

OR(119902=0

2)AC

OR(119902=0

4)AC

OR(119902=0

6)AC

OR(119902=0

8)AC

OR(119902=1

0)AC

OR(119902=1

0)ACO

DM-I(119902=0

1)Training

Average

00944

00391

00364

00173

00187

00184

00183

00188

000

78TestAv

erage

00934

00386

00359

00171

00184

00182

00180

00186

000

78TestST

D00262

0035

00347

000

67000

4400039

00037

00039

00012

TestMinim

um00364

00087

00102

00079

000

9500123

000

9700113

000

58TestMaxim

um01553

00962

00962

00532

00337

00266

00294

00281

00119


Table 8 Performances of fuzzy predictor designed by ACOR ACODM ACOR-I and ACODM-I in Example 3

Algorithms ACOR ACOR-I ACODM ACODM-ITest Average 00386 00084 00303 00078Test STD 00350 00012 00314 00012Test Minimum 00087 00057 00079 00058Test Maximum 00962 00131 00962 00119

Table 9 Performances of ACODM-I CCACO and different neural-fuzzy systems in Example 3

Models HyFIS D-FNN SuPFuNIS G-FNN CCACO ACODM-IRule number 16 5 4 10 4 4Test RMSE 00100 00131 00075 00056 00061 00058

than 00078 of ACODM-I Moreover the CCACO algorithmis a multiple-colony algorithm whose computation is morecomplex than the simple single-colony algorithm such asACODM-I

5 Conclusions

This paper proposes an enhanced ant colony optimizationwith dynamicmutation and ad hoc initialization ACODM-Ifor improving the accuracy of TSK-type fuzzy systems designACODM-I is developed based on ACOR and is regardedas a new population-based evolutionary optimization algo-rithm Based on the observations of some accuracy-orientedfuzzy controllers this paper proposes an ad hoc populationinitialization for initial ant solutions to improve the designaccuracy This is an application-specific initialization ratherthan generic initialization currently used inmost population-based algorithms ACODM-I incorporates the dynamicmutation technique into ACOR to balance exploration abilityand convergence rate In addition to exploiting local searcharound the chosen leading solution using Gaussian samplingthe dynamic mutation probabilistically provides more searchdirections by ldquojumpingrdquo to the neighboring of other archivesolutions in order to avoid being trapped into a local opti-mum

To validate the proposed algorithm three applicationexamples of the TSK-type fuzzy system designs have beensimulated two zero-order TSK-type fuzzy systems for non-linear dynamic plant control and one first-order TSK-typefuzzy system for the prediction of the chaotic time seriesThe simulation results have demonstrated that ACODM-Iachieves smaller error than the ones by the ACOR algorithmswith different values of q The effects on the design accuracyand convergence rate yielded by the proposed initializationand introduced dynamic mutation have also been discussedand verified in the simulations In addition the comparisonwith some advanced population-based algorithms also showsthat the error achieved by ACODM-I is smaller than thoseby those algorithms used for comparison including HPSO-TVAC PSO-CREV HGAPSO TPSIA APSCO and RCACOfor the design of zero-order TSK-type fuzzy controller andCCACO for the design of first-order TSK-type fuzzy predic-tor

In the future the performances of ACODM-I for thedesigns of various types of feed-forward or recurrent fuzzysystems will be studied The adaptive mechanism or algo-rithm of adjusting the value of ℎ in (14) determining themutation probability will be investigated in the future studyThe hybridization of ACODM-I with other computationaltechniques may further improve optimization accuracy thusit is possibly worthy of being studied in the future

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work was supported by Ministry of Science and Tech-nology Taiwan under Grant MOST 105-2221-E-415-018

References

[1] J-S R Jang C T Sun and E Mizutani Neuro-Fuzzy and SoftComputing A computational Approach to Learning andMachineIntelligence Prentice Hall Upper Saddle River NJ USA 1997

[2] D E Goldberg ldquoGenetic Algorithms in Search Optimizationand Machine Learningrdquo Addison-Wesley Reading Mass USA1989

[3] J Kennedy R Eberhart and Y Shi Swarm Intelligence MorganKaufmann San Mateo Calif USA 2001

[4] O Cordon F Gomide F Herrera F Hoffmann and L Mag-dalena ldquoTen years of genetic fuzzy systems Current frameworkand new trendsrdquo Fuzzy Sets and Systems vol 141 no 1 pp 5ndash312004

[5] C-J Lin ldquoA GA-based neural fuzzy system for temperaturecontrolrdquo Fuzzy Sets and Systems vol 143 no 2 pp 311ndash3332004

[6] C-H Chou ldquoGenetic algorithm-based optimal fuzzy controllerdesign in the linguistic spacerdquo IEEE Transactions on FuzzySystems vol 14 no 3 pp 372ndash385 2006

[7] M Mucientes D L Moreno A Bugarın and S Barro ldquoDesignof a fuzzy controller in mobile robotics using genetic algo-rithmsrdquoApplied Soft Computing vol 7 no 2 pp 540ndash546 2007


[8] F Hoffmann D Schauten and S Holemann ldquoIncremental evo-lutionary design of TSK fuzzy controllersrdquo IEEETransactions onFuzzy Systems vol 15 no 4 pp 563ndash577 2007

[9] R C Eberhart and J Kennedy ldquoA new optimizer using particleswarm theoryrdquo in Proceedings of the 6th International Sympo-sium on Micro Machine and Human Science (MHS rsquo95) pp 39ndash43 Nagoya Japan October 1995

[10] A Ratnaweera S K Halgamuge and H C Watson ldquoSelf-organizing hierarchical particle swarm optimizer with time-varying acceleration coefficientsrdquo IEEE Transactions on Evolu-tionary Computation vol 8 no 3 pp 240ndash255 2004

[11] X Chen and Y M Li ldquoA modified PSO structure resulting inhigh exploration ability with convergence guaranteedrdquo IEEETransactions on Systems Man and Cybernetics Part B Cyber-netics vol 37 no 5 pp 1271ndash1289 2007

[12] N-J Li W-J Wang C-C James Hsu W Chang H-GChou and J-W Chang ldquoEnhanced particle swarm optimizerincorporating a weighted particlerdquo Neurocomputing vol 124pp 218ndash227 2014

[13] C F Juang ldquoA hybrid of genetic algorithm and particle swarmoptimization for recurrent network designrdquo IEEE Transactionson Systems Man and Cybernetics Part B Cybernetics vol 34no 2 pp 997ndash1006 2004

[14] C-F Juang and C Lo ldquoZero-order TSK-type fuzzy systemlearning using a two-phase swarm intelligence algorithmrdquoFuzzy Sets and Systems vol 159 no 21 pp 2910ndash2926 2008

[15] C-F Juang and C-Y Wang ldquoA self-generating fuzzy systemwith ant and particle swarm cooperative optimizationrdquo ExpertSystems with Applications vol 36 no 3 pp 5362ndash5370 2009

[16] MDorigo VManiezzo andA Colorni ldquoAnt system optimiza-tion by a colony of cooperating agentsrdquo IEEE Transactions onSystems Man and Cybernetics Part B Cybernetics vol 26 no1 pp 29ndash41 1996

[17] M Dorigo and L M Gambardella ldquoAnt colony system a coop-erative learning approach to the traveling salesman problemrdquoIEEETransactions on Evolutionary Computation vol 1 no 1 pp53ndash66 1997

[18] T Stutzle and H H Hoos ldquoMax-min ant systemrdquo FutureGeneration Computer Systems vol 16 no 8 pp 889ndash914 2000

[19] M Dorigo and T Stutzle Ant Colony Optimization MIT PressCombridge MA USA 2004

[20] M Mucientes and J Casillas ldquoQuick design of fuzzy controllerswith good interpretability in mobile roboticsrdquo IEEE Transac-tions on Fuzzy Systems vol 15 no 4 pp 636ndash651 2007

[21] O Castillo H Neyoy J Soria P Melin and F Valdez ldquoAnew approach for dynamic fuzzy logic parameter tuning in antcolony optimization and its application in fuzzy control of amobile robotrdquoApplied SoftComputing vol 28 pp 150ndash159 2015

[22] G Bilchev and I C Parmee ldquoThe ant colony metaphor forsearching continuous design spacesrdquo in Proceedings of the AISBWorkshop on Evolutionary Computation T C Fogarty Ed vol993 pp 25ndash39 Springer-Verlag Berlin Germany 1995

[23] J Dre and P Siarry ldquoA new ant colony algorithm using theheterarchical concept aimed at optimization of multiminimacontinuous functionsrdquo in Proceedings of the Third InternationalWorkshop on Ant Algorithms (ANTS2002) M Dorigo G DiCaro and M Sampels Eds vol 2463 pp 216ndash221 Springer-Verlag Berlin Germany 2005

[24] K Socha and M Dorigo ldquoAnt colony optimization for contin-uous domainsrdquo European Journal of Operational Research vol185 no 3 pp 1155ndash1173 2008

[25] C-F Juang and P-H Chang ldquoDesigning fuzzy-rule-based sys-tems using continuous ant-colony optimizationrdquo IEEE Transac-tions on Fuzzy Systems vol 18 no 1 pp 138ndash149 2010

[26] C-F Juang C-W Hung and C-H Hsu ldquoRule-based coop-erative continuous ant colony optimization to improve theaccuracy of fuzzy system designrdquo IEEE Transactions on FuzzySystems vol 22 no 4 pp 723ndash735 2014

[27] C-C Chen L P Shen C-F Huang and B-R ChangldquoAssimilation-accommodation mixed continuous ant colonyoptimization for fuzzy system designrdquo Engineering Computa-tions (Swansea Wales) vol 33 no 7 pp 1882ndash1898 2016

[28] C-F Juang and P-H Chang ldquoRecurrent fuzzy system designusing elite-guided continuous ant colony optimizationrdquoAppliedSoft Computing vol 11 no 2 pp 2687ndash2697 2011

[29] C-CChen andL P Shen ldquoRecurrent fuzzy systemdesign usingmutation- Aided elite continuous ant colony optimizationrdquo inProceedings of the 2016 IEEE International Conference on FuzzySystems FUZZ-IEEE 2016 pp 1642ndash1648 Canada July 2016

[30] C Maurice Initialisations for Particle Swarm OptimizationWiley Estados Unidos USA 2006 httpclercmauricefreefrpso

[31] K E Parsopoulos and M N Vrahatis ldquoModification of theparticle swarm optimizer for locating all the global minimardquoin Artificial Neural Nets and Genetic Algorithms V KurkovaN C Steele R Neruda and M Karny Eds pp 289ndash294IASTEDACTA Press 2001

[32] K E Parsopoulos and M N Vrahatis ldquoInitializing the particleswarm optimizer using the nonlinear simplex methodsrdquo inAdvances in Intelligent Systems Fuzzy Systems EvolutionaryComputation A Grmela and N E Mastorakis Eds pp 216ndash221 WSEAS Press Evolutionary Computation 2002

[33] M Richards andD Ventura ldquoChoosing a starting configurationfor particle swarm optimizationrdquo in Proceedings of the 2004IEEE International Joint Conference on Neural Networks pp2309ndash2312 Budapest Hungary July 2004

[34] S Rahnamayan and G G Wang ldquoCenter-based sampling forpopulation-based algorithmsrdquo in Proceedings of the 2009 IEEECongress on Evolutionary Computation CEC 2009 pp 933ndash938Trondheim Norway May 2009

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

[36] S Wu and M J Er ldquoDynamic fuzzy neural networks - a novelapproach to function approximationrdquo IEEE Transactions onSystems Man and Cybernetics Part B Cybernetics vol 30 no2 pp 358ndash364 2000

[37] S Paul and S Kumar ldquoSubsethood-product fuzzy neuralinference system (SuPFuNIS)rdquo IEEE Transactions on NeuralNetworks and Learning Systems vol 13 no 3 pp 578ndash599 2002

[38] Y Gao and M J Er ldquoNARMAX time series model predictionFeedforward and recurrent fuzzy neural network approachesrdquoFuzzy Sets and Systems vol 150 no 2 pp 331ndash350 2005

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free so they can easily be applied to any optimizationproblems including the fuzzy control problems and theproblems with nondifferentiable objective function Since thesearch direction in these techniques is stochastic they aremore likely to traverse across the highly nonlinear designspace thus avoiding to be trapped into a local optimumFurthermore these approaches evaluate many candidatesolutions concurrently so they have better chance to find thebetter solution

Genetic algorithm was inspired by the process of naturalselection and developed based on the principle of survival offittest Genetic algorithms use bioinspired crossover opera-tion between two selected parent solutions to produce childsolutions and the mutation operation for further exploitingthe generated individual child solutions for improving theperformance Genetic fuzzy systems [4ndash8] are the evolu-tionary fuzzy systems designed by genetic algorithms Forexample GA was used to design neural-fuzzy system fortemperature control in [5] and fuzzy controller for mobilerobots in [7]

Particle swarm optimization (PSO) is one of swarmintelligence models PSO was inspired by the social behaviorof fish schooling and bird flocking [9] In the PSO eachparticle represents a candidate solution to a problem and fliesin the hyperspace according to its flying velocity vector whichis stochastically determined by its previously personal bestand swarm best experiences In the end of PSO operationthe experienced swarm best solution is the finally obtainedsolution To address the encountered issues and improveover the parent PSO [9] many advanced PSO variants [10ndash15] have been proposed some of which were applied todesign the TSK-type fuzzy systems [14 15] These algorithmsinclude the PSO with time-varying acceleration coefficients(PSO-TVAC) [10] a self-organizing hierarchical particleswam optimizer with TVAC (HPSO-TVAC) [10] PSO withcontrollable random-exploration velocity (PSO-CREV) [11]enhanced PSO by incorporating a weighted particle [12] thehybrid of GA and PSO (HGAPSO) [13] two-phase swarmintelligence algorithm (TPSIA) [14] and ant and particleswarm cooperative optimization (APSCO) [15]

Another type of swarm intelligence model is ant colonyoptimization (ACO) [16ndash19] The ACO technique wasinspired by foraging behavior of the real ant colony andwas proposed initially for solving discrete combinationaloptimization problems such as travelling salesman problem(TSP) The ACO algorithms have been successfully appliedto optimize the fuzzy systems for mobile robot control[20 21] In those studies in order to apply the discreteACO for optimization the parameters charactering the fuzzycontroller were first discretized thus sacrificing the precision

To overcome the precision issue some ant-related algo-rithms for the optimization problems with real-value param-eters have been proposed [22ndash24] Among these algorithmsthis paper focuses the interest on ACOR [24] because it ismost related to the discrete ACO and has achieved goodperformances on continuous optimization of the benchmarkfunctions as demonstrated in [24] ACOR thus is advanta-geous on the application problems requiring high precisionsuch as the design of fuzzy controllers for dynamic systems

because their inputs and outputs are usually continuouscases Since the proposal of ACOR some variants have beenproposed and applied to design fuzzy systems for accuracy-oriented problems [25ndash28] In [25] a modified continuousACO algorithm (RCACO) was proposed for the design offuzzy-rule-based systems in order to achieve considerablelearning accuracy The paper [26] proposed a cooperativecontinuous ACO (CCACO) with multiple colonies of pop-ulations each colony of which is only responsible for opti-mizing a single fuzzy rule Although the simulation resultsdemonstrated that the performance of multicolony basedCCACO is better than that of the single-colony RCACO thecomputation of CCACO is much more complex than thatof RCACO Inspired by the cognitive psychology conceptsthe study in [27] proposed an assimilation-accommodationmixed continuous ant colony optimization (ACACO) for thedesigns of feed-forward fuzzy systems In addition the elite-guided continuous ACO (ECACO) was proposed to designrecurrent fuzzy systems [28] whichwas claimed to be the firstapplication of continuous ACO on recurrent fuzzy systemdesign Although those variants indeed helped accomplishthe accurate design of fuzzy systems there is still possibleroom for further improvement especially on the balancebetween the exploration and convergence rate of solutionsThis paper introduces the dynamic mutation [29] into ACORto enhance such balance

For population-based algorithms such as PSO and ACOanother major issue is population initialization [30ndash34]Startingwith a population of initial solutions the population-based algorithm improves the solutions iteratively in orderto find the better solution Therefore the population ini-tialization affects the performance In general however theinitial population solutions are generated randomly becauseof no a priori information The work in [31] suggesteduse of Sobol sequence generator for generating initial PSOparticles because the generated initial solutions are uniformlydistributed into the search space The study in [32] reportedthat the initialization using the nonlinear simplex method(NSM) helped improve the convergence rate and successrate of PSO By using the generators of centroidal Voronoitessellations (CVT) as the starting point as suggested in [33]the initial solutions can be more evenly distributed through-out the high-dimensional problem space and improve PSOperformanceThe study in [34] proposed a center-based sam-pling for the initialization of population-based algorithmsBy simulations that paper showed that the points in centerregion have higher chances to be closer to an unknownsolution and thus suggested the center region is a promisingregion for the initialization However those initializationapproaches [31ndash34] did not take into account application-specific information Based on [34] and the observations ofsome accuracy-oriented fuzzy controller designs this paperproposes an ad hoc population initialization method forinitial ant solutions to improve the design accuracy

This paper mainly contributes to propose an enhancedcontinuous ACO algorithm incorporating dynamic muta-tions and ad hoc initialization ACODM-I for the designof TSK-type fuzzy systems ACODM-I can be regardedas a population-based evolutionary algorithm Instead of









119891119894 (1199091 1199092 119909119899) = 1198861198940 + 119899sum119895=1

119886119894119895119909119895 (3)


119872119894119895 (119909119895) = expminus(119909119895 minus 119898119894119895119887119894119895 )2 (4)




119895=1

expminus(119909119895 minus 119898119894119895119887119894119895 )2 (5)
















rarrs1rarrs2

rarrsi

rarrsN

s11

s12

s1i

s1N sjN

sj1

sj2

sji

sDN

sD1

sD2

sDi

E(rarrs1)

E(rarrs2)

E(rarrsi )

E(rarrsN)

w1

w2

wi

wN


119875119896119894119895 (119905) = (120591119894119895 (119905))120572 (120578119894119895)120573sum119898isin119881119896119894









119901119897 = 119908119897sum119873119898=1 119908119898 119898 = 1 2 119873 (10)




119897

119892119895119897(119904 120583119895119897 120590119895119897) = 1120590119895119897radic2120587 exp

minus(119904 minus 120583119895

119897)22 (120590119895119897)2

120590119895119897= 120576 119873sum119898=1

10038161003816100381610038161003816119904119895119898 minus 119904119895119897 10038161003816100381610038161003816119873 minus 1 (11)







If rand119889119894 gt 119901119889(119905)119904119889119894 (119905 + 1) = Sampl (119892119889119897 (119904 119904119889119897 (119905) 120590119889119897 (119905))) (12)

else


(13)


119901119889 (119905) = ℎ sdot 120590119889 (119905) = ℎ sdot ( 119873sum119894=1

(119904119889119894 (119905) minus 119904119889 (119905))2119873 )12

(14)











end while




4 Simulations



119910 (119896 + 1) = 119910 (119896)1 + 1199102 (119896) + 1199063 (119896) (15)


119910119889 (119896) = sin(12058711989650 ) cos(12058711989630 ) 1 le 119896 le 250 (16)


RMSE = (250sum119896=1

(119910119889 (119896) minus 119910 (119896))2250 )12 (17)



2000 4000 6000 8000 100000Evaluation

minus16

minus14

minus12

minus1

minus08

minus06

minus04

LＩＡ10

(RM

SE)

２ (q = 001)

２ (q = 01)

２ (q = 10)

２ (q = 10)

ACODM-I (q = 01)





ACODMACODM-I

2000 4000 6000 8000 100000Evaluation

minus16

minus14

minus12

minus1

minus08

minus06

minus04

LＩＡ10

(RM

SE)

２

ACOR-I




119910 (119896 + 1) = 119910 (119896) 119910 (119896 minus 1) (119910 (119896) + 25)1 + 1199102 (119896) + 1199102 (119896 minus 1) + 119906 (119896) (18)



Table1Perfo

rmanceso

fthe

fuzzycontrollerd

esignedby

theA

CODM-IandAC

ORin

Exam

ple1

Algorith

ms

ACO

R(119902=0

01)AC

OR(119902=0

1)AC

OR(119902=0

2)AC

OR(119902=0

4)AC

OR(119902=0

6)AC

OR(119902=0

8)AC

OR(119902=1

0)AC

OR(119902=1

0)ACO

DM-I(119902=0

1)Av

erageR

MSE

00525

00350

00343

00357

00354

00392

00374

004

03002

56ST

D00241

00071

000

6700057

000

49000

6600056

00050

00032

Minim

um00273

00220

00231

00261

00269

00300

00280

00276

002

01Maxim

um01835

00524

00491

00472

00454

00575

00490

00518

00372






ActualDesired

50 100 150 200 2500Time step

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Am

plitu

de




(19)


SAE = 250sum119896=1

100381610038161003816100381602119910119889 (119896) minus 119910 (119896)1003816100381610038161003816 (20)


0

02

04

06

08

1

12

14

16

18

LＩＡ10

(SA

E)

２ (q = 001)

２ (q = 01)

２ (q = 10)

２ (q = 10)ACODM-I (q = 01)





Table4Perfo

rmanceso

fthe

fuzzycontrollerd

esignedby

ACODM-IandAC

ORin

Exam

ple2

Algorith

ms

ACO

R(119902=0

01)AC

OR(119902=0

1)AC

OR(119902=0

2)AC

OR(119902=0

4)AC

OR(119902=0

6)AC

OR(119902=0

8)AC

OR(119902=1

0)AC

OR(119902=1

0)AC

ODM-I(119902=0

1)Av

erageS

AE

96332

29983

23056

25632

21049

24515

22922

23184

1229

2ST

D86435

26633

17033

16424

08539

15094

08167

13302

07454

Minim

um16

844

0722

05101

07012

0838

09103

0686

09544

048

79Maxim

um584138

130973

1212

7197

500

37552

99463

344

0390

152

28749




times1041 2 3 4 5 60

Evaluation

0

02

04

06

08

1

12

14

16

18

LＩＡ10

(SA

E)

ACODMACODM-I

２

ACOR-I






ActualDesired

50 100 150 200 2500Time step

minus12

minus1

minus08

minus06

minus04

minus02

0

02

04

06

Am

plitu

de








minus22

minus2

minus18

minus16

minus14

minus12

minus1

minus08

minus06

minus04

LＩＡ10

(RM

SE)

times10521 30 2505 15

Evaluation

２ (q = 001)

２ (q = 01)

２ (q = 10)

２ (q = 10)

ACODM-I (q = 01)




minus22

minus2

minus18

minus16

minus14

minus12

minus1

minus08

minus06

minus04

LＩＡ10

(RM

SE)

times10521 30 2505 15

Evaluation

２

ACOR-IACODMACODM-I


ActualDesired

700 750 800 850 900 950 1000 1050 1100650Time step

02

04

06

08

1

12

14

16

Am

plitu

de




Table7Perfo

rmanceso

fthe

fuzzypredictord

esignedby

theA

CODM-IandAC

ORin

Exam

ple3

Algorith

ms

ACO

R(119902=0

01)AC

OR(119902=0

1)AC

OR(119902=0

2)AC

OR(119902=0

4)AC

OR(119902=0

6)AC

OR(119902=0

8)AC

OR(119902=1

0)AC

OR(119902=1

0)ACO

DM-I(119902=0

1)Training

Average

00944

00391

00364

00173

00187

00184

00183

00188

000

78TestAv

erage

00934

00386

00359

00171

00184

00182

00180

00186

000

78TestST

D00262

0035

00347

000

67000

4400039

00037

00039

00012

TestMinim

um00364

00087

00102

00079

000

9500123

000

9700113

000

58TestMaxim

um01553

00962

00962

00532

00337

00266

00294

00281

00119







5 Conclusions






Acknowledgments


References













































Advances in

FuzzySystems


Volume 2018













Journal of

Journal of



Hindawi


Advances in

Multimedia


Biomedical Imaging





RoboticsJournal of









Volume 2018



Advances in












119891119894 (1199091 1199092 119909119899) = 1198861198940 + 119899sum119895=1

119886119894119895119909119895 (3)


119872119894119895 (119909119895) = expminus(119909119895 minus 119898119894119895119887119894119895 )2 (4)




119895=1

expminus(119909119895 minus 119898119894119895119887119894119895 )2 (5)
















rarrs1rarrs2

rarrsi

rarrsN

s11

s12

s1i

s1N sjN

sj1

sj2

sji

sDN

sD1

sD2

sDi

E(rarrs1)

E(rarrs2)

E(rarrsi )

E(rarrsN)

w1

w2

wi

wN


119875119896119894119895 (119905) = (120591119894119895 (119905))120572 (120578119894119895)120573sum119898isin119881119896119894









119901119897 = 119908119897sum119873119898=1 119908119898 119898 = 1 2 119873 (10)




119897

119892119895119897(119904 120583119895119897 120590119895119897) = 1120590119895119897radic2120587 exp

minus(119904 minus 120583119895

119897)22 (120590119895119897)2

120590119895119897= 120576 119873sum119898=1

10038161003816100381610038161003816119904119895119898 minus 119904119895119897 10038161003816100381610038161003816119873 minus 1 (11)







If rand119889119894 gt 119901119889(119905)119904119889119894 (119905 + 1) = Sampl (119892119889119897 (119904 119904119889119897 (119905) 120590119889119897 (119905))) (12)

else


(13)


119901119889 (119905) = ℎ sdot 120590119889 (119905) = ℎ sdot ( 119873sum119894=1

(119904119889119894 (119905) minus 119904119889 (119905))2119873 )12

(14)











end while




4 Simulations



119910 (119896 + 1) = 119910 (119896)1 + 1199102 (119896) + 1199063 (119896) (15)


119910119889 (119896) = sin(12058711989650 ) cos(12058711989630 ) 1 le 119896 le 250 (16)


RMSE = (250sum119896=1

(119910119889 (119896) minus 119910 (119896))2250 )12 (17)



2000 4000 6000 8000 100000Evaluation

minus16

minus14

minus12

minus1

minus08

minus06

minus04

LＩＡ10

(RM

SE)

２ (q = 001)

２ (q = 01)

２ (q = 10)

２ (q = 10)

ACODM-I (q = 01)





ACODMACODM-I

2000 4000 6000 8000 100000Evaluation

minus16

minus14

minus12

minus1

minus08

minus06

minus04

LＩＡ10

(RM

SE)

２

ACOR-I




119910 (119896 + 1) = 119910 (119896) 119910 (119896 minus 1) (119910 (119896) + 25)1 + 1199102 (119896) + 1199102 (119896 minus 1) + 119906 (119896) (18)



Table1Perfo

rmanceso

fthe

fuzzycontrollerd

esignedby

theA

CODM-IandAC

ORin

Exam

ple1

Algorith

ms

ACO

R(119902=0

01)AC

OR(119902=0

1)AC

OR(119902=0

2)AC

OR(119902=0

4)AC

OR(119902=0

6)AC

OR(119902=0

8)AC

OR(119902=1

0)AC

OR(119902=1

0)ACO

DM-I(119902=0

1)Av

erageR

MSE

00525

00350

00343

00357

00354

00392

00374

004

03002

56ST

D00241

00071

000

6700057

000

49000

6600056

00050

00032

Minim

um00273

00220

00231

00261

00269

00300

00280

00276

002

01Maxim

um01835

00524

00491

00472

00454

00575

00490

00518

00372






ActualDesired

50 100 150 200 2500Time step

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Am

plitu

de




(19)


SAE = 250sum119896=1

100381610038161003816100381602119910119889 (119896) minus 119910 (119896)1003816100381610038161003816 (20)


0

02

04

06

08

1

12

14

16

18

LＩＡ10

(SA

E)

２ (q = 001)

２ (q = 01)

２ (q = 10)

２ (q = 10)ACODM-I (q = 01)





Table4Perfo

rmanceso

fthe

fuzzycontrollerd

esignedby

ACODM-IandAC

ORin

Exam

ple2

Algorith

ms

ACO

R(119902=0

01)AC

OR(119902=0

1)AC

OR(119902=0

2)AC

OR(119902=0

4)AC

OR(119902=0

6)AC

OR(119902=0

8)AC

OR(119902=1

0)AC

OR(119902=1

0)AC

ODM-I(119902=0

1)Av

erageS

AE

96332

29983

23056

25632

21049

24515

22922

23184

1229

2ST

D86435

26633

17033

16424

08539

15094

08167

13302

07454

Minim

um16

844

0722

05101

07012

0838

09103

0686

09544

048

79Maxim

um584138

130973

1212

7197

500

37552

99463

344

0390

152

28749




times1041 2 3 4 5 60

Evaluation

0

02

04

06

08

1

12

14

16

18

LＩＡ10

(SA

E)

ACODMACODM-I

２

ACOR-I






ActualDesired

50 100 150 200 2500Time step

minus12

minus1

minus08

minus06

minus04

minus02

0

02

04

06

Am

plitu

de








minus22

minus2

minus18

minus16

minus14

minus12

minus1

minus08

minus06

minus04

LＩＡ10

(RM

SE)

times10521 30 2505 15

Evaluation

２ (q = 001)

２ (q = 01)

２ (q = 10)

２ (q = 10)

ACODM-I (q = 01)




minus22

minus2

minus18

minus16

minus14

minus12

minus1

minus08

minus06

minus04

LＩＡ10

(RM

SE)

times10521 30 2505 15

Evaluation

２

ACOR-IACODMACODM-I


ActualDesired

700 750 800 850 900 950 1000 1050 1100650Time step

02

04

06

08

1

12

14

16

Am

plitu

de




Table7Perfo

rmanceso

fthe

fuzzypredictord

esignedby

theA

CODM-IandAC

ORin

Exam

ple3

Algorith

ms

ACO

R(119902=0

01)AC

OR(119902=0

1)AC

OR(119902=0

2)AC

OR(119902=0

4)AC

OR(119902=0

6)AC

OR(119902=0

8)AC

OR(119902=1

0)AC

OR(119902=1

0)ACO

DM-I(119902=0

1)Training

Average

00944

00391

00364

00173

00187

00184

00183

00188

000

78TestAv

erage

00934

00386

00359

00171

00184

00182

00180

00186

000

78TestST

D00262

0035

00347

000

67000

4400039

00037

00039

00012

TestMinim

um00364

00087

00102

00079

000

9500123

000

9700113

000

58TestMaxim

um01553

00962

00962

00532

00337

00266

00294

00281

00119







5 Conclusions






Acknowledgments


References













































Advances in

FuzzySystems


Volume 2018













Journal of

Journal of



Hindawi


Advances in

Multimedia


Biomedical Imaging





RoboticsJournal of









Volume 2018



Advances in













rarrs1rarrs2

rarrsi

rarrsN

s11

s12

s1i

s1N sjN

sj1

sj2

sji

sDN

sD1

sD2

sDi

E(rarrs1)

E(rarrs2)

E(rarrsi )

E(rarrsN)

w1

w2

wi

wN


119875119896119894119895 (119905) = (120591119894119895 (119905))120572 (120578119894119895)120573sum119898isin119881119896119894









119901119897 = 119908119897sum119873119898=1 119908119898 119898 = 1 2 119873 (10)




119897

119892119895119897(119904 120583119895119897 120590119895119897) = 1120590119895119897radic2120587 exp

minus(119904 minus 120583119895

119897)22 (120590119895119897)2

120590119895119897= 120576 119873sum119898=1

10038161003816100381610038161003816119904119895119898 minus 119904119895119897 10038161003816100381610038161003816119873 minus 1 (11)







If rand119889119894 gt 119901119889(119905)119904119889119894 (119905 + 1) = Sampl (119892119889119897 (119904 119904119889119897 (119905) 120590119889119897 (119905))) (12)

else


(13)


119901119889 (119905) = ℎ sdot 120590119889 (119905) = ℎ sdot ( 119873sum119894=1

(119904119889119894 (119905) minus 119904119889 (119905))2119873 )12

(14)











end while




4 Simulations



119910 (119896 + 1) = 119910 (119896)1 + 1199102 (119896) + 1199063 (119896) (15)


119910119889 (119896) = sin(12058711989650 ) cos(12058711989630 ) 1 le 119896 le 250 (16)


RMSE = (250sum119896=1

(119910119889 (119896) minus 119910 (119896))2250 )12 (17)



2000 4000 6000 8000 100000Evaluation

minus16

minus14

minus12

minus1

minus08

minus06

minus04

LＩＡ10

(RM

SE)

２ (q = 001)

２ (q = 01)

２ (q = 10)

２ (q = 10)

ACODM-I (q = 01)





ACODMACODM-I

2000 4000 6000 8000 100000Evaluation

minus16

minus14

minus12

minus1

minus08

minus06

minus04

LＩＡ10

(RM

SE)

２

ACOR-I




119910 (119896 + 1) = 119910 (119896) 119910 (119896 minus 1) (119910 (119896) + 25)1 + 1199102 (119896) + 1199102 (119896 minus 1) + 119906 (119896) (18)



Table1Perfo

rmanceso

fthe

fuzzycontrollerd

esignedby

theA

CODM-IandAC

ORin

Exam

ple1

Algorith

ms

ACO

R(119902=0

01)AC

OR(119902=0

1)AC

OR(119902=0

2)AC

OR(119902=0

4)AC

OR(119902=0

6)AC

OR(119902=0

8)AC

OR(119902=1

0)AC

OR(119902=1

0)ACO

DM-I(119902=0

1)Av

erageR

MSE

00525

00350

00343

00357

00354

00392

00374

004

03002

56ST

D00241

00071

000

6700057

000

49000

6600056

00050

00032

Minim

um00273

00220

00231

00261

00269

00300

00280

00276

002

01Maxim

um01835

00524

00491

00472

00454

00575

00490

00518

00372






ActualDesired

50 100 150 200 2500Time step

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Am

plitu

de




(19)


SAE = 250sum119896=1

100381610038161003816100381602119910119889 (119896) minus 119910 (119896)1003816100381610038161003816 (20)


0

02

04

06

08

1

12

14

16

18

LＩＡ10

(SA

E)

２ (q = 001)

２ (q = 01)

２ (q = 10)

２ (q = 10)ACODM-I (q = 01)





Table4Perfo

rmanceso

fthe

fuzzycontrollerd

esignedby

ACODM-IandAC

ORin

Exam

ple2

Algorith

ms

ACO

R(119902=0

01)AC

OR(119902=0

1)AC

OR(119902=0

2)AC

OR(119902=0

4)AC

OR(119902=0

6)AC

OR(119902=0

8)AC

OR(119902=1

0)AC

OR(119902=1

0)AC

ODM-I(119902=0

1)Av

erageS

AE

96332

29983

23056

25632

21049

24515

22922

23184

1229

2ST

D86435

26633

17033

16424

08539

15094

08167

13302

07454

Minim

um16

844

0722

05101

07012

0838

09103

0686

09544

048

79Maxim

um584138

130973

1212

7197

500

37552

99463

344

0390

152

28749




times1041 2 3 4 5 60

Evaluation

0

02

04

06

08

1

12

14

16

18

LＩＡ10

(SA

E)

ACODMACODM-I

２

ACOR-I






ActualDesired

50 100 150 200 2500Time step

minus12

minus1

minus08

minus06

minus04

minus02

0

02

04

06

Am

plitu

de








minus22

minus2

minus18

minus16

minus14

minus12

minus1

minus08

minus06

minus04

LＩＡ10

(RM

SE)

times10521 30 2505 15

Evaluation

２ (q = 001)

２ (q = 01)

２ (q = 10)

２ (q = 10)

ACODM-I (q = 01)




minus22

minus2

minus18

minus16

minus14

minus12

minus1

minus08

minus06

minus04

LＩＡ10

(RM

SE)

times10521 30 2505 15

Evaluation

２

ACOR-IACODMACODM-I


ActualDesired

700 750 800 850 900 950 1000 1050 1100650Time step

02

04

06

08

1

12

14

16

Am

plitu

de




Table7Perfo

rmanceso

fthe

fuzzypredictord

esignedby

theA

CODM-IandAC

ORin

Exam

ple3

Algorith

ms

ACO

R(119902=0

01)AC

OR(119902=0

1)AC

OR(119902=0

2)AC

OR(119902=0

4)AC

OR(119902=0

6)AC

OR(119902=0

8)AC

OR(119902=1

0)AC

OR(119902=1

0)ACO

DM-I(119902=0

1)Training

Average

00944

00391

00364

00173

00187

00184

00183

00188

000

78TestAv

erage

00934

00386

00359

00171

00184

00182

00180

00186

000

78TestST

D00262

0035

00347

000

67000

4400039

00037

00039

00012

TestMinim

um00364

00087

00102

00079

000

9500123

000

9700113

000

58TestMaxim

um01553

00962

00962

00532

00337

00266

00294

00281

00119







5 Conclusions






Acknowledgments


References













































Advances in

FuzzySystems


Volume 2018













Journal of

Journal of



Hindawi


Advances in

Multimedia


Biomedical Imaging





RoboticsJournal of









Volume 2018



Advances in









If rand119889119894 gt 119901119889(119905)119904119889119894 (119905 + 1) = Sampl (119892119889119897 (119904 119904119889119897 (119905) 120590119889119897 (119905))) (12)

else


(13)


119901119889 (119905) = ℎ sdot 120590119889 (119905) = ℎ sdot ( 119873sum119894=1

(119904119889119894 (119905) minus 119904119889 (119905))2119873 )12

(14)











end while




4 Simulations



119910 (119896 + 1) = 119910 (119896)1 + 1199102 (119896) + 1199063 (119896) (15)


119910119889 (119896) = sin(12058711989650 ) cos(12058711989630 ) 1 le 119896 le 250 (16)


RMSE = (250sum119896=1

(119910119889 (119896) minus 119910 (119896))2250 )12 (17)



2000 4000 6000 8000 100000Evaluation

minus16

minus14

minus12

minus1

minus08

minus06

minus04

LＩＡ10

(RM

SE)

２ (q = 001)

２ (q = 01)

２ (q = 10)

２ (q = 10)

ACODM-I (q = 01)





ACODMACODM-I

2000 4000 6000 8000 100000Evaluation

minus16

minus14

minus12

minus1

minus08

minus06

minus04

LＩＡ10

(RM

SE)

２

ACOR-I




119910 (119896 + 1) = 119910 (119896) 119910 (119896 minus 1) (119910 (119896) + 25)1 + 1199102 (119896) + 1199102 (119896 minus 1) + 119906 (119896) (18)



Table1Perfo

rmanceso

fthe

fuzzycontrollerd

esignedby

theA

CODM-IandAC

ORin

Exam

ple1

Algorith

ms

ACO

R(119902=0

01)AC

OR(119902=0

1)AC

OR(119902=0

2)AC

OR(119902=0

4)AC

OR(119902=0

6)AC

OR(119902=0

8)AC

OR(119902=1

0)AC

OR(119902=1

0)ACO

DM-I(119902=0

1)Av

erageR

MSE

00525

00350

00343

00357

00354

00392

00374

004

03002

56ST

D00241

00071

000

6700057

000

49000

6600056

00050

00032

Minim

um00273

00220

00231

00261

00269

00300

00280

00276

002

01Maxim

um01835

00524

00491

00472

00454

00575

00490

00518

00372






ActualDesired

50 100 150 200 2500Time step

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Am

plitu

de




(19)


SAE = 250sum119896=1

100381610038161003816100381602119910119889 (119896) minus 119910 (119896)1003816100381610038161003816 (20)


0

02

04

06

08

1

12

14

16

18

LＩＡ10

(SA

E)

２ (q = 001)

２ (q = 01)

２ (q = 10)

２ (q = 10)ACODM-I (q = 01)





Table4Perfo

rmanceso

fthe

fuzzycontrollerd

esignedby

ACODM-IandAC

ORin

Exam

ple2

Algorith

ms

ACO

R(119902=0

01)AC

OR(119902=0

1)AC

OR(119902=0

2)AC

OR(119902=0

4)AC

OR(119902=0

6)AC

OR(119902=0

8)AC

OR(119902=1

0)AC

OR(119902=1

0)AC

ODM-I(119902=0

1)Av

erageS

AE

96332

29983

23056

25632

21049

24515

22922

23184

1229

2ST

D86435

26633

17033

16424

08539

15094

08167

13302

07454

Minim

um16

844

0722

05101

07012

0838

09103

0686

09544

048

79Maxim

um584138

130973

1212

7197

500

37552

99463

344

0390

152

28749




times1041 2 3 4 5 60

Evaluation

0

02

04

06

08

1

12

14

16

18

LＩＡ10

(SA

E)

ACODMACODM-I

２

ACOR-I






ActualDesired

50 100 150 200 2500Time step

minus12

minus1

minus08

minus06

minus04

minus02

0

02

04

06

Am

plitu

de








minus22

minus2

minus18

minus16

minus14

minus12

minus1

minus08

minus06

minus04

LＩＡ10

(RM

SE)

times10521 30 2505 15

Evaluation

２ (q = 001)

２ (q = 01)

２ (q = 10)

２ (q = 10)

ACODM-I (q = 01)




minus22

minus2

minus18

minus16

minus14

minus12

minus1

minus08

minus06

minus04

LＩＡ10

(RM

SE)

times10521 30 2505 15

Evaluation

２

ACOR-IACODMACODM-I


ActualDesired

700 750 800 850 900 950 1000 1050 1100650Time step

02

04

06

08

1

12

14

16

Am

plitu

de




Table7Perfo

rmanceso

fthe

fuzzypredictord

esignedby

theA

CODM-IandAC

ORin

Exam

ple3

Algorith

ms

ACO

R(119902=0

01)AC

OR(119902=0

1)AC

OR(119902=0

2)AC

OR(119902=0

4)AC

OR(119902=0

6)AC

OR(119902=0

8)AC

OR(119902=1

0)AC

OR(119902=1

0)ACO

DM-I(119902=0

1)Training

Average

00944

00391

00364

00173

00187

00184

00183

00188

000

78TestAv

erage

00934

00386

00359

00171

00184

00182

00180

00186

000

78TestST

D00262

0035

00347

000

67000

4400039

00037

00039

00012

TestMinim

um00364

00087

00102

00079

000

9500123

000

9700113

000

58TestMaxim

um01553

00962

00962

00532

00337

00266

00294

00281

00119







5 Conclusions






Acknowledgments


References













































Advances in

FuzzySystems


Volume 2018













Journal of

Journal of



Hindawi


Advances in

Multimedia


Biomedical Imaging





RoboticsJournal of









Volume 2018



Advances in









end while




4 Simulations



119910 (119896 + 1) = 119910 (119896)1 + 1199102 (119896) + 1199063 (119896) (15)


119910119889 (119896) = sin(12058711989650 ) cos(12058711989630 ) 1 le 119896 le 250 (16)


RMSE = (250sum119896=1

(119910119889 (119896) minus 119910 (119896))2250 )12 (17)



2000 4000 6000 8000 100000Evaluation

minus16

minus14

minus12

minus1

minus08

minus06

minus04

LＩＡ10

(RM

SE)

２ (q = 001)

２ (q = 01)

２ (q = 10)

２ (q = 10)

ACODM-I (q = 01)





ACODMACODM-I

2000 4000 6000 8000 100000Evaluation

minus16

minus14

minus12

minus1

minus08

minus06

minus04

LＩＡ10

(RM

SE)

２

ACOR-I




119910 (119896 + 1) = 119910 (119896) 119910 (119896 minus 1) (119910 (119896) + 25)1 + 1199102 (119896) + 1199102 (119896 minus 1) + 119906 (119896) (18)



Table1Perfo

rmanceso

fthe

fuzzycontrollerd

esignedby

theA

CODM-IandAC

ORin

Exam

ple1

Algorith

ms

ACO

R(119902=0

01)AC

OR(119902=0

1)AC

OR(119902=0

2)AC

OR(119902=0

4)AC

OR(119902=0

6)AC

OR(119902=0

8)AC

OR(119902=1

0)AC

OR(119902=1

0)ACO

DM-I(119902=0

1)Av

erageR

MSE

00525

00350

00343

00357

00354

00392

00374

004

03002

56ST

D00241

00071

000

6700057

000

49000

6600056

00050

00032

Minim

um00273

00220

00231

00261

00269

00300

00280

00276

002

01Maxim

um01835

00524

00491

00472

00454

00575

00490

00518

00372






ActualDesired

50 100 150 200 2500Time step

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Am

plitu

de




(19)


SAE = 250sum119896=1

100381610038161003816100381602119910119889 (119896) minus 119910 (119896)1003816100381610038161003816 (20)


0

02

04

06

08

1

12

14

16

18

LＩＡ10

(SA

E)

２ (q = 001)

２ (q = 01)

２ (q = 10)

２ (q = 10)ACODM-I (q = 01)





Table4Perfo

rmanceso

fthe

fuzzycontrollerd

esignedby

ACODM-IandAC

ORin

Exam

ple2

Algorith

ms

ACO

R(119902=0

01)AC

OR(119902=0

1)AC

OR(119902=0

2)AC

OR(119902=0

4)AC

OR(119902=0

6)AC

OR(119902=0

8)AC

OR(119902=1

0)AC

OR(119902=1

0)AC

ODM-I(119902=0

1)Av

erageS

AE

96332

29983

23056

25632

21049

24515

22922

23184

1229

2ST

D86435

26633

17033

16424

08539

15094

08167

13302

07454

Minim

um16

844

0722

05101

07012

0838

09103

0686

09544

048

79Maxim

um584138

130973

1212

7197

500

37552

99463

344

0390

152

28749




times1041 2 3 4 5 60

Evaluation

0

02

04

06

08

1

12

14

16

18

LＩＡ10

(SA

E)

ACODMACODM-I

２

ACOR-I






ActualDesired

50 100 150 200 2500Time step

minus12

minus1

minus08

minus06

minus04

minus02

0

02

04

06

Am

plitu

de








minus22

minus2

minus18

minus16

minus14

minus12

minus1

minus08

minus06

minus04

LＩＡ10

(RM

SE)

times10521 30 2505 15

Evaluation

２ (q = 001)

２ (q = 01)

２ (q = 10)

２ (q = 10)

ACODM-I (q = 01)




minus22

minus2

minus18

minus16

minus14

minus12

minus1

minus08

minus06

minus04

LＩＡ10

(RM

SE)

times10521 30 2505 15

Evaluation

２

ACOR-IACODMACODM-I


ActualDesired

700 750 800 850 900 950 1000 1050 1100650Time step

02

04

06

08

1

12

14

16

Am

plitu

de




Table7Perfo

rmanceso

fthe

fuzzypredictord

esignedby

theA

CODM-IandAC

ORin

Exam

ple3

Algorith

ms

ACO

R(119902=0

01)AC

OR(119902=0

1)AC

OR(119902=0

2)AC

OR(119902=0

4)AC

OR(119902=0

6)AC

OR(119902=0

8)AC

OR(119902=1

0)AC

OR(119902=1

0)ACO

DM-I(119902=0

1)Training

Average

00944

00391

00364

00173

00187

00184

00183

00188

000

78TestAv

erage

00934

00386

00359

00171

00184

00182

00180

00186

000

78TestST

D00262

0035

00347

000

67000

4400039

00037

00039

00012

TestMinim

um00364

00087

00102

00079

000

9500123

000

9700113

000

58TestMaxim

um01553

00962

00962

00532

00337

00266

00294

00281

00119







5 Conclusions






Acknowledgments


References













































Advances in

FuzzySystems


Volume 2018













Journal of

Journal of



Hindawi


Advances in

Multimedia


Biomedical Imaging





RoboticsJournal of









Volume 2018



Advances in





2000 4000 6000 8000 100000Evaluation

minus16

minus14

minus12

minus1

minus08

minus06

minus04

LＩＡ10

(RM

SE)

２ (q = 001)

２ (q = 01)

２ (q = 10)

２ (q = 10)

ACODM-I (q = 01)





ACODMACODM-I

2000 4000 6000 8000 100000Evaluation

minus16

minus14

minus12

minus1

minus08

minus06

minus04

LＩＡ10

(RM

SE)

２

ACOR-I




119910 (119896 + 1) = 119910 (119896) 119910 (119896 minus 1) (119910 (119896) + 25)1 + 1199102 (119896) + 1199102 (119896 minus 1) + 119906 (119896) (18)



Table1Perfo

rmanceso

fthe

fuzzycontrollerd

esignedby

theA

CODM-IandAC

ORin

Exam

ple1

Algorith

ms

ACO

R(119902=0

01)AC

OR(119902=0

1)AC

OR(119902=0

2)AC

OR(119902=0

4)AC

OR(119902=0

6)AC

OR(119902=0

8)AC

OR(119902=1

0)AC

OR(119902=1

0)ACO

DM-I(119902=0

1)Av

erageR

MSE

00525

00350

00343

00357

00354

00392

00374

004

03002

56ST

D00241

00071

000

6700057

000

49000

6600056

00050

00032

Minim

um00273

00220

00231

00261

00269

00300

00280

00276

002

01Maxim

um01835

00524

00491

00472

00454

00575

00490

00518

00372






ActualDesired

50 100 150 200 2500Time step

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Am

plitu

de




(19)


SAE = 250sum119896=1

100381610038161003816100381602119910119889 (119896) minus 119910 (119896)1003816100381610038161003816 (20)


0

02

04

06

08

1

12

14

16

18

LＩＡ10

(SA

E)

２ (q = 001)

２ (q = 01)

２ (q = 10)

２ (q = 10)ACODM-I (q = 01)





Table4Perfo

rmanceso

fthe

fuzzycontrollerd

esignedby

ACODM-IandAC

ORin

Exam

ple2

Algorith

ms

ACO

R(119902=0

01)AC

OR(119902=0

1)AC

OR(119902=0

2)AC

OR(119902=0

4)AC

OR(119902=0

6)AC

OR(119902=0

8)AC

OR(119902=1

0)AC

OR(119902=1

0)AC

ODM-I(119902=0

1)Av

erageS

AE

96332

29983

23056

25632

21049

24515

22922

23184

1229

2ST

D86435

26633

17033

16424

08539

15094

08167

13302

07454

Minim

um16

844

0722

05101

07012

0838

09103

0686

09544

048

79Maxim

um584138

130973

1212

7197

500

37552

99463

344

0390

152

28749




times1041 2 3 4 5 60

Evaluation

0

02

04

06

08

1

12

14

16

18

LＩＡ10

(SA

E)

ACODMACODM-I

２

ACOR-I






ActualDesired

50 100 150 200 2500Time step

minus12

minus1

minus08

minus06

minus04

minus02

0

02

04

06

Am

plitu

de








minus22

minus2

minus18

minus16

minus14

minus12

minus1

minus08

minus06

minus04

LＩＡ10

(RM

SE)

times10521 30 2505 15

Evaluation

２ (q = 001)

２ (q = 01)

２ (q = 10)

２ (q = 10)

ACODM-I (q = 01)




minus22

minus2

minus18

minus16

minus14

minus12

minus1

minus08

minus06

minus04

LＩＡ10

(RM

SE)

times10521 30 2505 15

Evaluation

２

ACOR-IACODMACODM-I


ActualDesired

700 750 800 850 900 950 1000 1050 1100650Time step

02

04

06

08

1

12

14

16

Am

plitu

de




Table7Perfo

rmanceso

fthe

fuzzypredictord

esignedby

theA

CODM-IandAC

ORin

Exam

ple3

Algorith

ms

ACO

R(119902=0

01)AC

OR(119902=0

1)AC

OR(119902=0

2)AC

OR(119902=0

4)AC

OR(119902=0

6)AC

OR(119902=0

8)AC

OR(119902=1

0)AC

OR(119902=1

0)ACO

DM-I(119902=0

1)Training

Average

00944

00391

00364

00173

00187

00184

00183

00188

000

78TestAv

erage

00934

00386

00359

00171

00184

00182

00180

00186

000

78TestST

D00262

0035

00347

000

67000

4400039

00037

00039

00012

TestMinim

um00364

00087

00102

00079

000

9500123

000

9700113

000

58TestMaxim

um01553

00962

00962

00532

00337

00266

00294

00281

00119







5 Conclusions






Acknowledgments


References













































Advances in

FuzzySystems


Volume 2018













Journal of

Journal of



Hindawi


Advances in

Multimedia


Biomedical Imaging





RoboticsJournal of









Volume 2018



Advances in





Table1Perfo

rmanceso

fthe

fuzzycontrollerd

esignedby

theA

CODM-IandAC

ORin

Exam

ple1

Algorith

ms

ACO

R(119902=0

01)AC

OR(119902=0

1)AC

OR(119902=0

2)AC

OR(119902=0

4)AC

OR(119902=0

6)AC

OR(119902=0

8)AC

OR(119902=1

0)AC

OR(119902=1

0)ACO

DM-I(119902=0

1)Av

erageR

MSE

00525

00350

00343

00357

00354

00392

00374

004

03002

56ST

D00241

00071

000

6700057

000

49000

6600056

00050

00032

Minim

um00273

00220

00231

00261

00269

00300

00280

00276

002

01Maxim

um01835

00524

00491

00472

00454

00575

00490

00518

00372






ActualDesired

50 100 150 200 2500Time step

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Am

plitu

de




(19)


SAE = 250sum119896=1

100381610038161003816100381602119910119889 (119896) minus 119910 (119896)1003816100381610038161003816 (20)


0

02

04

06

08

1

12

14

16

18

LＩＡ10

(SA

E)

２ (q = 001)

２ (q = 01)

２ (q = 10)

２ (q = 10)ACODM-I (q = 01)





Table4Perfo

rmanceso

fthe

fuzzycontrollerd

esignedby

ACODM-IandAC

ORin

Exam

ple2

Algorith

ms

ACO

R(119902=0

01)AC

OR(119902=0

1)AC

OR(119902=0

2)AC

OR(119902=0

4)AC

OR(119902=0

6)AC

OR(119902=0

8)AC

OR(119902=1

0)AC

OR(119902=1

0)AC

ODM-I(119902=0

1)Av

erageS

AE

96332

29983

23056

25632

21049

24515

22922

23184

1229

2ST

D86435

26633

17033

16424

08539

15094

08167

13302

07454

Minim

um16

844

0722

05101

07012

0838

09103

0686

09544

048

79Maxim

um584138

130973

1212

7197

500

37552

99463

344

0390

152

28749




times1041 2 3 4 5 60

Evaluation

0

02

04

06

08

1

12

14

16

18

LＩＡ10

(SA

E)

ACODMACODM-I

２

ACOR-I






ActualDesired

50 100 150 200 2500Time step

minus12

minus1

minus08

minus06

minus04

minus02

0

02

04

06

Am

plitu

de








minus22

minus2

minus18

minus16

minus14

minus12

minus1

minus08

minus06

minus04

LＩＡ10

(RM

SE)

times10521 30 2505 15

Evaluation

２ (q = 001)

２ (q = 01)

２ (q = 10)

２ (q = 10)

ACODM-I (q = 01)




minus22

minus2

minus18

minus16

minus14

minus12

minus1

minus08

minus06

minus04

LＩＡ10

(RM

SE)

times10521 30 2505 15

Evaluation

２

ACOR-IACODMACODM-I


ActualDesired

700 750 800 850 900 950 1000 1050 1100650Time step

02

04

06

08

1

12

14

16

Am

plitu

de




Table7Perfo

rmanceso

fthe

fuzzypredictord

esignedby

theA

CODM-IandAC

ORin

Exam

ple3

Algorith

ms

ACO

R(119902=0

01)AC

OR(119902=0

1)AC

OR(119902=0

2)AC

OR(119902=0

4)AC

OR(119902=0

6)AC

OR(119902=0

8)AC

OR(119902=1

0)AC

OR(119902=1

0)ACO

DM-I(119902=0

1)Training

Average

00944

00391

00364

00173

00187

00184

00183

00188

000

78TestAv

erage

00934

00386

00359

00171

00184

00182

00180

00186

000

78TestST

D00262

0035

00347

000

67000

4400039

00037

00039

00012

TestMinim

um00364

00087

00102

00079

000

9500123

000

9700113

000

58TestMaxim

um01553

00962

00962

00532

00337

00266

00294

00281

00119







5 Conclusions






Acknowledgments


References













































Advances in

FuzzySystems


Volume 2018













Journal of

Journal of



Hindawi


Advances in

Multimedia


Biomedical Imaging





RoboticsJournal of









Volume 2018



Advances in









ActualDesired

50 100 150 200 2500Time step

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Am

plitu

de




(19)


SAE = 250sum119896=1

100381610038161003816100381602119910119889 (119896) minus 119910 (119896)1003816100381610038161003816 (20)


0

02

04

06

08

1

12

14

16

18

LＩＡ10

(SA

E)

２ (q = 001)

２ (q = 01)

２ (q = 10)

２ (q = 10)ACODM-I (q = 01)





Table4Perfo

rmanceso

fthe

fuzzycontrollerd

esignedby

ACODM-IandAC

ORin

Exam

ple2

Algorith

ms

ACO

R(119902=0

01)AC

OR(119902=0

1)AC

OR(119902=0

2)AC

OR(119902=0

4)AC

OR(119902=0

6)AC

OR(119902=0

8)AC

OR(119902=1

0)AC

OR(119902=1

0)AC

ODM-I(119902=0

1)Av

erageS

AE

96332

29983

23056

25632

21049

24515

22922

23184

1229

2ST

D86435

26633

17033

16424

08539

15094

08167

13302

07454

Minim

um16

844

0722

05101

07012

0838

09103

0686

09544

048

79Maxim

um584138

130973

1212

7197

500

37552

99463

344

0390

152

28749




times1041 2 3 4 5 60

Evaluation

0

02

04

06

08

1

12

14

16

18

LＩＡ10

(SA

E)

ACODMACODM-I

２

ACOR-I






ActualDesired

50 100 150 200 2500Time step

minus12

minus1

minus08

minus06

minus04

minus02

0

02

04

06

Am

plitu

de








minus22

minus2

minus18

minus16

minus14

minus12

minus1

minus08

minus06

minus04

LＩＡ10

(RM

SE)

times10521 30 2505 15

Evaluation

２ (q = 001)

２ (q = 01)

２ (q = 10)

２ (q = 10)

ACODM-I (q = 01)




minus22

minus2

minus18

minus16

minus14

minus12

minus1

minus08

minus06

minus04

LＩＡ10

(RM

SE)

times10521 30 2505 15

Evaluation

２

ACOR-IACODMACODM-I


ActualDesired

700 750 800 850 900 950 1000 1050 1100650Time step

02

04

06

08

1

12

14

16

Am

plitu

de




Table7Perfo

rmanceso

fthe

fuzzypredictord

esignedby

theA

CODM-IandAC

ORin

Exam

ple3

Algorith

ms

ACO

R(119902=0

01)AC

OR(119902=0

1)AC

OR(119902=0

2)AC

OR(119902=0

4)AC

OR(119902=0

6)AC

OR(119902=0

8)AC

OR(119902=1

0)AC

OR(119902=1

0)ACO

DM-I(119902=0

1)Training

Average

00944

00391

00364

00173

00187

00184

00183

00188

000

78TestAv

erage

00934

00386

00359

00171

00184

00182

00180

00186

000

78TestST

D00262

0035

00347

000

67000

4400039

00037

00039

00012

TestMinim

um00364

00087

00102

00079

000

9500123

000

9700113

000

58TestMaxim

um01553

00962

00962

00532

00337

00266

00294

00281

00119







5 Conclusions






Acknowledgments


References













































Advances in

FuzzySystems


Volume 2018













Journal of

Journal of



Hindawi


Advances in

Multimedia


Biomedical Imaging





RoboticsJournal of









Volume 2018



Advances in





Table4Perfo

rmanceso

fthe

fuzzycontrollerd

esignedby

ACODM-IandAC

ORin

Exam

ple2

Algorith

ms

ACO

R(119902=0

01)AC

OR(119902=0

1)AC

OR(119902=0

2)AC

OR(119902=0

4)AC

OR(119902=0

6)AC

OR(119902=0

8)AC

OR(119902=1

0)AC

OR(119902=1

0)AC

ODM-I(119902=0

1)Av

erageS

AE

96332

29983

23056

25632

21049

24515

22922

23184

1229

2ST

D86435

26633

17033

16424

08539

15094

08167

13302

07454

Minim

um16

844

0722

05101

07012

0838

09103

0686

09544

048

79Maxim

um584138

130973

1212

7197

500

37552

99463

344

0390

152

28749




times1041 2 3 4 5 60

Evaluation

0

02

04

06

08

1

12

14

16

18

LＩＡ10

(SA

E)

ACODMACODM-I

２

ACOR-I






ActualDesired

50 100 150 200 2500Time step

minus12

minus1

minus08

minus06

minus04

minus02

0

02

04

06

Am

plitu

de








minus22

minus2

minus18

minus16

minus14

minus12

minus1

minus08

minus06

minus04

LＩＡ10

(RM

SE)

times10521 30 2505 15

Evaluation

２ (q = 001)

２ (q = 01)

２ (q = 10)

２ (q = 10)

ACODM-I (q = 01)




minus22

minus2

minus18

minus16

minus14

minus12

minus1

minus08

minus06

minus04

LＩＡ10

(RM

SE)

times10521 30 2505 15

Evaluation

２

ACOR-IACODMACODM-I


ActualDesired

700 750 800 850 900 950 1000 1050 1100650Time step

02

04

06

08

1

12

14

16

Am

plitu

de




Table7Perfo

rmanceso

fthe

fuzzypredictord

esignedby

theA

CODM-IandAC

ORin

Exam

ple3

Algorith

ms

ACO

R(119902=0

01)AC

OR(119902=0

1)AC

OR(119902=0

2)AC

OR(119902=0

4)AC

OR(119902=0

6)AC

OR(119902=0

8)AC

OR(119902=1

0)AC

OR(119902=1

0)ACO

DM-I(119902=0

1)Training

Average

00944

00391

00364

00173

00187

00184

00183

00188

000

78TestAv

erage

00934

00386

00359

00171

00184

00182

00180

00186

000

78TestST

D00262

0035

00347

000

67000

4400039

00037

00039

00012

TestMinim

um00364

00087

00102

00079

000

9500123

000

9700113

000

58TestMaxim

um01553

00962

00962

00532

00337

00266

00294

00281

00119







5 Conclusions






Acknowledgments


References













































Advances in

FuzzySystems


Volume 2018













Journal of

Journal of



Hindawi


Advances in

Multimedia


Biomedical Imaging





RoboticsJournal of









Volume 2018



Advances in







times1041 2 3 4 5 60

Evaluation

0

02

04

06

08

1

12

14

16

18

LＩＡ10

(SA

E)

ACODMACODM-I

２

ACOR-I






ActualDesired

50 100 150 200 2500Time step

minus12

minus1

minus08

minus06

minus04

minus02

0

02

04

06

Am

plitu

de








minus22

minus2

minus18

minus16

minus14

minus12

minus1

minus08

minus06

minus04

LＩＡ10

(RM

SE)

times10521 30 2505 15

Evaluation

２ (q = 001)

２ (q = 01)

２ (q = 10)

２ (q = 10)

ACODM-I (q = 01)




minus22

minus2

minus18

minus16

minus14

minus12

minus1

minus08

minus06

minus04

LＩＡ10

(RM

SE)

times10521 30 2505 15

Evaluation

２

ACOR-IACODMACODM-I


ActualDesired

700 750 800 850 900 950 1000 1050 1100650Time step

02

04

06

08

1

12

14

16

Am

plitu

de




Table7Perfo

rmanceso

fthe

fuzzypredictord

esignedby

theA

CODM-IandAC

ORin

Exam

ple3

Algorith

ms

ACO

R(119902=0

01)AC

OR(119902=0

1)AC

OR(119902=0

2)AC

OR(119902=0

4)AC

OR(119902=0

6)AC

OR(119902=0

8)AC

OR(119902=1

0)AC

OR(119902=1

0)ACO

DM-I(119902=0

1)Training

Average

00944

00391

00364

00173

00187

00184

00183

00188

000

78TestAv

erage

00934

00386

00359

00171

00184

00182

00180

00186

000

78TestST

D00262

0035

00347

000

67000

4400039

00037

00039

00012

TestMinim

um00364

00087

00102

00079

000

9500123

000

9700113

000

58TestMaxim

um01553

00962

00962

00532

00337

00266

00294

00281

00119







5 Conclusions






Acknowledgments


References













































Advances in

FuzzySystems


Volume 2018













Journal of

Journal of



Hindawi


Advances in

Multimedia


Biomedical Imaging





RoboticsJournal of









Volume 2018



Advances in







minus22

minus2

minus18

minus16

minus14

minus12

minus1

minus08

minus06

minus04

LＩＡ10

(RM

SE)

times10521 30 2505 15

Evaluation

２ (q = 001)

２ (q = 01)

２ (q = 10)

２ (q = 10)

ACODM-I (q = 01)




minus22

minus2

minus18

minus16

minus14

minus12

minus1

minus08

minus06

minus04

LＩＡ10

(RM

SE)

times10521 30 2505 15

Evaluation

２

ACOR-IACODMACODM-I


ActualDesired

700 750 800 850 900 950 1000 1050 1100650Time step

02

04

06

08

1

12

14

16

Am

plitu

de




Table7Perfo

rmanceso

fthe

fuzzypredictord

esignedby

theA

CODM-IandAC

ORin

Exam

ple3

Algorith

ms

ACO

R(119902=0

01)AC

OR(119902=0

1)AC

OR(119902=0

2)AC

OR(119902=0

4)AC

OR(119902=0

6)AC

OR(119902=0

8)AC

OR(119902=1

0)AC

OR(119902=1

0)ACO

DM-I(119902=0

1)Training

Average

00944

00391

00364

00173

00187

00184

00183

00188

000

78TestAv

erage

00934

00386

00359

00171

00184

00182

00180

00186

000

78TestST

D00262

0035

00347

000

67000

4400039

00037

00039

00012

TestMinim

um00364

00087

00102

00079

000

9500123

000

9700113

000

58TestMaxim

um01553

00962

00962

00532

00337

00266

00294

00281

00119







5 Conclusions






Acknowledgments


References













































Advances in

FuzzySystems


Volume 2018













Journal of

Journal of



Hindawi


Advances in

Multimedia


Biomedical Imaging





RoboticsJournal of









Volume 2018



Advances in





Table7Perfo

rmanceso

fthe

fuzzypredictord

esignedby

theA

CODM-IandAC

ORin

Exam

ple3

Algorith

ms

ACO

R(119902=0

01)AC

OR(119902=0

1)AC

OR(119902=0

2)AC

OR(119902=0

4)AC

OR(119902=0

6)AC

OR(119902=0

8)AC

OR(119902=1

0)AC

OR(119902=1

0)ACO

DM-I(119902=0

1)Training

Average

00944

00391

00364

00173

00187

00184

00183

00188

000

78TestAv

erage

00934

00386

00359

00171

00184

00182

00180

00186

000

78TestST

D00262

0035

00347

000

67000

4400039

00037

00039

00012

TestMinim

um00364

00087

00102

00079

000

9500123

000

9700113

000

58TestMaxim

um01553

00962

00962

00532

00337

00266

00294

00281

00119







5 Conclusions






Acknowledgments


References













































Advances in

FuzzySystems


Volume 2018













Journal of

Journal of



Hindawi


Advances in

Multimedia


Biomedical Imaging





RoboticsJournal of









Volume 2018



Advances in










5 Conclusions






Acknowledgments


References













































Advances in

FuzzySystems


Volume 2018













Journal of

Journal of



Hindawi


Advances in

Multimedia


Biomedical Imaging





RoboticsJournal of









Volume 2018



Advances in









































Advances in

FuzzySystems


Volume 2018













Journal of

Journal of



Hindawi


Advances in

Multimedia


Biomedical Imaging





RoboticsJournal of









Volume 2018



Advances in




enhanced ant colony optimization with dynamic mutation and

Documents