SPRINGER BRIEFS IN APPLIED SCIENCES AND TECHNOLOGY · COMPUTATIONAL INTELLIGENCE Leticia Amador Oscar Castillo Optimization of Type-2 Fuzzy Controllers Using the Bee Colony Algorithm SpringerBriefs in Applied Sciences and Technology Computational Intelligence Series editor Janusz Kacprzyk, Polish Academy of Sciences, Systems Research Institute, Warsaw, Poland About this Series The series “Studies in Computational Intelligence” (SCI) publishes new develop- mentsandadvancesinthevariousareasofcomputationalintelligence—quicklyand with a high quality. The intent is to cover the theory, applications, and design methods of computational intelligence, as embedded in the fields of engineering, computer science, physics and life sciences, as well as the methodologies behind them. 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More information about this series at http://www.springer.com/series/10618 Leticia Amador Oscar Castillo (cid:129) Optimization of Type-2 Fuzzy Controllers Using the Bee Colony Algorithm 123 Leticia Amador Oscar Castillo Division of Graduate Studies Division of Graduate Studies TijuanaInstitute of Technology TijuanaInstitute of Technology Tijuana, BajaCalifornia Tijuana, BajaCalifornia Mexico Mexico ISSN 2191-530X ISSN 2191-5318 (electronic) SpringerBriefs inApplied SciencesandTechnology ISBN978-3-319-54294-2 ISBN978-3-319-54295-9 (eBook) DOI 10.1007/978-3-319-54295-9 LibraryofCongressControlNumber:2017933057 ©TheAuthor(s)2017 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilarmethodologynowknownorhereafterdeveloped. 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Printedonacid-freepaper ThisSpringerimprintispublishedbySpringerNature TheregisteredcompanyisSpringerInternationalPublishingAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Preface Thisbookfocusesonthefieldsoffuzzylogicandbioinspiredalgorithms,especially the bee colony optimization algorithm, and also considers the fuzzy control area. The main idea is that these areas together can solve various control problems and obtainbetterresults.Wetesttheproposedmethodusingtwobenchmarkproblems: filling a water tank and controlling the trajectory in an autonomous mobile robot. WhenanIntervalType-2fuzzylogicsystemisimplementedtomodelthebehavior ofsystems,theresultsshowbetterstabilization,becausetheanalysisofuncertainty isbetter.Forthisreasoninthisbookweconsidertheproposedmethodusingfuzzy systems, fuzzy controllers, and the bee colony optimization algorithm to improve the behavior of complex control problems. Thisbookisintended tobeareferenceforscientistsandengineersinterestedin applyingfuzzylogictechniquestosolveproblemsinintelligent control.Thisbook can also be used as a reference for graduate courses including: soft computing, swarm intelligence, bioinspired algorithms, intelligent control, and fuzzy control, amongothers.Weconsiderthatthisbookcanalsobeusedtoinspirenovelideasfor new linesofresearch,ortocontinue thelinesofresearch proposedby theauthors. In Chap. 1, we begin by offering a brief introduction of the potential use of the optimization strategies in different real-world applications. We describe the use ofthebeecolonyoptimizationalgorithmusingIntervalType-2fuzzylogicsystems foraggregationofresultsinproblemsofintelligentcontrolofnonlinearplants.We also mention other possible applications of the proposed control approach. We describe in Chap. 2 the basic concepts, notation, and theory offuzzy logic, and thefuzzy controller.This chapter overviews thebackground, main definitions, and basic concepts useful for the development of this research work. Chapter3describesthetwoproblemstatementsthatareusedofcomplexplants, such as the characteristics and design of the proposed fuzzy logic system and the implementation of the problem with the bee colony optimization. The particular controlproblemsthatareusedtotesttheproposedmethodareexplained.Ahybrid systemcomposedoftwointelligenttechnologiesisusedasanewmethodforglobal control.Thisiscriticalforcomplexcontrolproblemsthatcanbesolvedbydividing them into several simple controllers. v vi Preface Chapter4isdevotedtodescribingthebeecolonyoptimizationalgorithmandthe proposed method in the dynamic adaptation of the parameters using fuzzy sets for control when applied to the particular nonlinear plants that are considered for validating the proposed approach, in particular, the water tank problem and an autonomous mobile robot problem. We offer in Chap. 5 the simulation results with the original bee colony opti- mization and the proposed method using the interval Type-2 fuzzy logic systems; various performance indices are used to show improvement of the proposed method.Inaddition,fuzzyandoriginalbeecolonyoptimizationalgorithmsareused to optimize the fuzzy controller and achieve a fair comparison. We explained in Chap. 6 the statistical tests and comparison of the results that show the advantage of the proposed method for control. We describe in Chap. 7 the conclusions of this work, as well as some future research work we envision. Basically, a new fuzzy bee colony optimization for control was proposed, and then different cases of control were studied. The first case was the water control and the second case was the autonomous mobile robot control. To know if the proposed method could give good results, first the control problems were studied with more detail. These details were obtained working first withtheiroptimizationusingtheoriginalbeecolonyoptimizationusingType-1and Interval Type-2 fuzzy systems and later with the proposed method. Then the pro- posedmethodwasappliedusingType-1andIntervalType-2fuzzysystemsforthe two problems applied in fuzzy control. We end this preface by giving thanks to all the people who have helped or encouraged us during the writing of this book. First of all, we would like to thank ourcolleagueandfriendProf.JuanRamónCastroforalwayssupportingourwork, and for motivating us to write up our research work. We would also like to thank our colleagues working in soft computing, who are too many to mention each by name.Ofcourse,weneedtothankoursupportingagencies,CONACYTandTNM, in our country for their help during this project. We have to thank our institution, theTijuanaInstituteofTechnology,foralwayssupportingourprojects.Finally,we thank our respective families for their continuous support during the time that we spent on this project. Tijuana, Mexico Dr. Leticia Amador Prof. Oscar Castillo Contents 1 Introduction .. .... .... .... ..... .... .... .... .... .... ..... .. 1 References .... .... .... .... ..... .... .... .... .... .... ..... .. 4 2 Theory and Background .... ..... .... .... .... .... .... ..... .. 7 2.1 Fuzzy Inference System.. ..... .... .... .... .... .... ..... .. 7 2.1.1 Type-1 Fuzzy Logic Systems. .... .... .... .... ..... .. 7 2.1.2 Interval Type-2 Fuzzy Logic Systems .. .... .... ..... .. 8 2.2 Fuzzy Controllers... .... ..... .... .... .... .... .... ..... .. 9 References .... .... .... .... ..... .... .... .... .... .... ..... .. 10 3 Problem Statements .... .... ..... .... .... .... .... .... ..... .. 13 3.1 Water Tank Controller... ..... .... .... .... .... .... ..... .. 13 3.1.1 Model Equations of the Water Tank ... .... .... ..... .. 13 3.1.2 Design of the Fuzzy Controller ... .... .... .... ..... .. 14 3.2 Autonomous Mobile Robot Controller.... .... .... .... ..... .. 16 3.2.1 General Description of Problem... .... .... .... ..... .. 16 3.2.2 Fuzzy Logic Control Design . .... .... .... .... ..... .. 18 References .... .... .... .... ..... .... .... .... .... .... ..... .. 20 4 Bee Colony Optimization Algorithm.... .... .... .... .... ..... .. 23 4.1 Proposed Method... .... ..... .... .... .... .... .... ..... .. 28 References .... .... .... .... ..... .... .... .... .... .... ..... .. 31 5 Simulation Results for the Proposed Methods .... .... .... ..... .. 33 5.1 Simulation Results for the Water Tank Controller... .... ..... .. 34 5.1.1 Results for the Original Bee Colony Optimization Algorithm... .... ..... .... .... .... .... .... ..... .. 34 5.1.2 Results for the Fuzzy Bee Colony Optimization Algorithm for the Water Tank Controller.... .... ..... .. 34 vii viii Contents 5.2 Results for the Original BCO for an Autonomous Mobile Robot.. .... .... ..... .... .... .... .... .... ..... .. 40 5.2.1 Results for the Fuzzy BCO for an Autonomous Mobile Robot.... ..... .... .... .... .... .... ..... .. 48 6 Statistical Analysis and Comparison of Results ... .... .... ..... .. 55 7 Conclusions... .... .... .... ..... .... .... .... .... .... ..... .. 57 Appendix.... .... .... .... .... ..... .... .... .... .... .... ..... .. 59 Bibliograpy.. .... .... .... .... ..... .... .... .... .... .... ..... .. 69 Index... .... .... .... .... .... ..... .... .... .... .... .... ..... .. 71 Chapter 1 Introduction Nowadays the use of fuzzy logic controllers is more common, because of the manner of processing information. Interval Type-2 fuzzy logic controllers are pri- marily used because they can handle uncertainty and they are considered robust compared with other types of controllers. This book focuses on the use of optimization strategies as applied to Interval Type-2 fuzzy logic controllers, such as particle swarm optimization, ant colony optimization, and genetic algorithms, among others, making them more attractive. With the optimization of the Interval Type-2 fuzzy systems the problem of processing time arises, which can be solved by processing, but in a physical implementation in particular, this methodology is not solved with bee colony op- timization (BCO). For this reason we propose an implementation of the opti- mization of the Interval Type-2 fuzzy logic controllers with the bee colony optimization algorithm; a modification of the original BCO is also applied in this book. Fuzzy logic is a branch of computational intelligence that can handle vague or uncertain information. Fuzzy logic is born when you stop thinking that all the phenomena observed in the world can be described between well-established borders,andwhenyoubegintoadmitthepossibilityofhalfphenomena,whichare formally resolved for this time. Lofti A. Zadeh originally proposed fuzzy sets (FS) in 1965 [1], and his vision was to give more control over decision making. With his fuzzy logic an immea- surable amount of decision-making situations could be easily modeled whereas hardlogic,trueorfalse,couldnot.Thisopenedaneweraindecisionmakingwith FSthathasbeenevolvingsinceitsinitialdays,firststartingoutwithType-1fuzzy logic systems (T1FLS), and then coming into Interval Type-2 fuzzy logic systems (IT2FLS). In recent years, much work has been performed on control system stabilization [2–5]. However, all these control design methods require the exact mathematical models of the physical systems, which may not be available in practice. On the otherhand,fuzzycontrolhasbeensuccessfullyappliedforsolvingmanynonlinear ©TheAuthor(s)2017 1 L.AmadorandO.Castillo,OptimizationofType-2FuzzyControllers UsingtheBeeColonyAlgorithm,SpringerBriefsinComputationalIntelligence DOI10.1007/978-3-319-54295-9_1