T-S Fuzzy Modelling Using Advanced Genetic Algorithms
T-S Fuzzy Modelling Using Advanced Genetic AlgorithmsPAVEL ŠIŠPERA, M.Sc.1Prof. MIROSLAV POKORNÝ, Ph.D.1JAN ROUPEC, Ph.D.21Department of Measurement and ControlFaculty of Electrical Engineering and Computer Science, VSB Technical University Ostrava17. listopadu 15, 708 33 OstravaCZECH REPUBLIC2Department of Automatic Control and Computer ScienceFaculty of Mechanical Engineering, Technical University of BrnoTechnická 2, 616 69 BrnoCZECH REPUBLICAbstract:This paper introduces a soft-computing oriented approach to Takagi-Sugeno fuzzy modelling using theevolutionary principles. Genetic algorithms are applied to optimize fuzzy input variables space through genetic fuzzyclustering procedure and to identify the fuzzy model. Some advanced procedures e.g. individuals lifetime limitationand redundant genes application are used. The presented algorithm allows also the determination of the relevantinputs variables of fuzzy model from theirs potential candidates. To clarify the advantages of the proposed approachesthe numerical example of modelling of fuzzy non-linear system is also introduced.Key-words:Takagi-Sugeno Fuzzy Model, Input Variables Selection, Fuzzy Clustering, Advanced Genetic Algorithm,Numerical Example1. IntroductionNew research works focused on soft-computingmethods namely on exploitation of evolutionapproaches in fuzzy modelling and control were startedat the Department of Measurement and Control FEIVSB Technical University Ostrava. The aim of thiscontribution is to provide information concerningresults of investigation effort aimed on geneticalgorithms application and fuzzy model identificationimprovement.In fuzzy modelling technology the suitable fuzzydiversifications of input variable space as well as thenumber of linguistic rules determination and theirstructure and/or parameters identification are of crucialimportance. To investigate soft-computing methods intasks of Takagi-Sugeno (T-S) predictive model [8], [9]identification of the genetic algorithms (GA) wasapplied to find out the optimal distribution of clustercentres in data processed and proper T-S fuzzy modelidentification.2. Genetic Algorithm in T-S FuzzyModel Identification2.1 Advanced Genetic Algorithm (A-GA)The basic idea of a genetic algorithm is quite simple[5]. GA works not only with one solution in time butwith the whole population of solutions. The populationcontains many (ordinary several hundreds) individuals– bit strings representing solutions. The mechanism ofGA involves only elementary operations like stringscopying, partially bit swapping or bit value changing.GA starts with a population of strings and thereaftergenerates successive populations using the followingthree basic operations: reproduction, crossover, andmutation. Reproduction is the process by which
individual strings are copied according to an objectivefunction value (fitness). Copying of strings accordingto their fitness value means that strings with a highervalue have a higher probability of contributing one ormore offspring to next generation. This is an artificialversion of natural selection. Mutation is an occasional(with a small probability) random alteration of thestring position value. Mutation is needed since, in spiteof reproduction and crossover effectively searching andrecombining the existing representations, theyoccasionally become overzealous and lose somepotentially useful genetic material. The mutationoperator prevents such an irrecoverable loss. Therecombination mechanism allows mixing of parentalinformation while passing it to their descendants, andmutation introduces innovation into the population.In spite of simple principles, the design of GA forsuccessful practical using is surprisingly complicated.GA has many parameters that depend on the problem tobe solved. In the first, it is the size of population.Larger populations usually decrease the number ofiterations needed, but dramatically increase thecomputer time for any iteration. The factors increasingdemands on the size of population are the complexityof the problem being solved and the length of theindividuals. Every individual contains one or morechromosomes containing value of potential solution.Chromosomes consist of genes. The gene in our versionof Advanced GA (A-GA) is a structure representingone bit of solution value. It is usually advantageous touse some redundancy in genes and so the physicallength of our genes is greater than one bit. This type ofredundancy was introduced by Ryan [2]. The structureof gene we use is in Figure 1.population size as well as increasing the speed ofconvergence. It is necessary to store the best solutionobtained separately – the corresponding individual neednot to be always present in the population because ofthe limited lifetime.Many GAs are implemented on a population consistingof haploid individuals (each individual contains onechromosome). However, in nature, many livingorganisms have more than one chromosome and thereare mechanisms used to determine dominant genes.Sexual recombination generates an endless variety ofgenotype combination that increases the evolutionarypotential of the population. Because it increases thevariation among the offspring produced by anindividual, it improves the change that some of themwill be successful in varying and often-unpredictableenvironments they will encounter. Using diploid or“multiploid” individuals can often decrease demandson the population size. However the use of multiploidGA with sexual reproduction brings somecomplications, the advantage of multiploidity can beoften substitute by the “death” operator and redundantgenes coding.New individuals are created by operation calledcrossover. In the simplest case crossover meansswapping of two parts of two chromosomes split inrandomly selected point (so called one point crossover).In A-GA we use the uniform crossover on the bit levelis used.The strategy of selection individuals for crossover isvery important. It strongly determines the behaviour ofGA. For genetic clustering the ranking selection withelite brings satisfactory results.We must consider the mechanisms linking geneticalgorithm to the solved problem. It is very important tofind a good encoding of solutions to the bit strings inchromosomes. The Gray code is usually used toeliminate the Hamming barrier. The second mechanismis represented by an objective function. The evaluationof this function is the link between the geneticalgorithm and the problem to be solved. Most of thecomputer time in GAs is spent by evaluating objectivefunctions.Fig.1 - The Structure of GeneTo prevent degeneration and the deadlock in localextreme the limited lifetime of individual in A-GA canbe used. Limited lifetime is realized by the “death”operator [3], which represents something like continualrestart of GA. This operator enables decreasing ofGenetic algorithms commonly use heuristic andstochastic approaches. From the theoretical viewpoint,the convergence of heuristic algorithms is notguaranteed for the most of application cases. That iswhy the definition of the stopping rule of the GA bringsa new problem. It can be shown [4] that while using aproper version of GA the typical number of iterationscan be determine.
1. Generation of the initial population: At thebeginning the whole population is generatedrandomly, the members are sorted by the fitness(in descendent order).2. Mutation: The mutation is applied to each genewith the same probability, all GAs described hereuse pmut 0.05. The mutation of the gene meansthe inversion of one randomly selected bit in thegene.3. Death: Classical GA uses two main operations –crossover and mutation (the other operation shouldbe migration). In A-GA described in this paper, weuse the third operation – death. Every individualhas the additional information – age. A simplecounter that is incremented in each of GAiterations represents the age. If the age of anymember reaches the preset lifetime limit LT, thismember "dies" and is immediately replaced by anew randomly generated member. The age is notmutated nor crossed over. The age of newindividuals (incl. individuals created by crossover)is set to zero. Lifetime limit in our application isset to 5.4. Sorting by the fitness.5. Crossover: Uniform crossover is used for all genes(each bit of the offspring gene is selectedseparately from corresponding bits of bothparent’s genes).6. Go to step 2.In crossover, we do not replace all members of thepopulation. The crossover generates the number ofindividuals corresponding to the quarter of thepopulation only. Created individuals are sorted into thecorresponding places in the population according totheir fitness in such a way that the size of thepopulation remains the same. Newly created offspringof low fitness do not have to be involved in thepopulation.sets (eight bits for each fuzzy set, the total number ofbits depends on the dimension of the space of inputvariables and on the number of clusters, in our casesixteen bits for each cluster). In our example we use thetriangular fuzzy sets and the parameters means theshape of fuzzy set. Fuzzy sets parameters use the Graycode.We denote the input data points, x1 , x 2 , K x n , n is thenumber of data points, the number of clusters is k . Theobjective function we use has the formf (x ) (n) 1 max µ j (xi ) 1,jkK i 1 2(1)µ j (xi ) is the membership value of the point x i to the jcluster. We search for minimum of f (x ) .The objective function described above does not affectall disadvantageous situations possible and we must usesome penalization. The unacceptable case occurs, whenwe can observe point that does not belong to anycluster. The penalty value is 100 for each point. Theextreme situation occurs when no point belong to anycluster, the objective function value for this improbablecase is 1099. The penalization is also used, when wehave cluster that does not contain any data point (orwhen any cluster contains very low number of points –the value depends on input data). In this case, theobjective function is multiplied by 4 for each “empty”cluster.Because we have relatively short chromosomes in ourexample, we can use small population (200 individuals)and the convergence is very fast. The typical course ofconvergence is shown in Figure 2.5000450040003500FitnessA-GA we use has the following scheme:300025002000150010005002.2 A-GA in Task of Fuzzy Model Identification00510152025303540GenerationThe method of fuzzy genetic clustering is inspired by[5]. In the beginning the centre of clusters aregenerated. We do not use randomly generated centres,but the centres equidistantly cover the space of inputdata. The number of centres is the parameter of the taskand is entered manually. The second parameter is ashape of fuzzy sets. The value bits in chromosomesrepresents if the corresponding cluster is used or not(one bit for each cluster) and the parameters of fuzzyFig. 2 - The Convergence of A-GANext, the above-mentioned A-GA genetic algorithm weuse in task of non-linear fuzzy system modelling [7].
3. Case StudyThe A-GA procedures were tested through modellingand identification of simple non-linear fuzzy systemwith three inputs and one output variable. The eightrules of initial T-S fuzzy system were determined(Figure 3) and input/output data set was generated.IF (x1 is S) AND (x2 is S) AND (x3 is S) THENy1 3.2x1 2.2x2 0.05x3 1IF (x1 is S) AND (x2 is H) AND (x3 is S) THENy2 2.5x1 2.3x2 0.06x3 4IF (x1 is H) AND (x2 is S) AND (x3 is S) THENy3 0.3x1 3.0x2 0.05x3 5IF (x1 is H) AND (x2 is H) AND (x3 is S) THENy4 0.9x1 0.9x2 0.06x3 7IF (x1 is S) AND (x2 is S) AND (x3 is H) THENy5 3.2x1 2.2x2 0.06x3 1IF (x1 is S) AND (x2 is H) AND (x3 is H) THENy6 2.5x1 2.3x2 0.05x3 4IF (x1 is H) AND (x2 is S) AND (x3 is H) THENy7 0.3x1 3.0x2 0.06x3 5IF (x1 is H) AND (x2 is H) AND (x3 is H) THENy8 0.9x1 0.9x2 0.05x3 7Fig. 3 – Initial Three-Inputs Fuzzy SystemInput 1 (x1): S (0, 0, 8), H (4, 10, 10). Input 2 (x2):S (0, 0, 7), H (3, 12, 12). Input 3 (x3): S (0, 0, 6),H (5, 11, 11).Firstly the procedure of relevant system inputs selectionwas applied.3.1 A Relevant Input Variables SelectionIf more input variable candidates exist for the inputs tothe system, we can determine relevant pre-definednumber of them through procedure using acombinatorial approach. We take a heuristic method toselect some inputs and increase the number ofcombinatorial steps watching a regularity criterion.Let us consider a fuzzy system with n inputs and oneoutput (which is representing a nonlinear function) anda set of output values of this system for randomlychosen inputs. In general, let X be a set of possibleinput candidates x1, x2, . , xn then the total number ofcases is the number of subsets except an empty subsetof X, i.e., 2n - 1. Here we take a heuristic method toselect some inputs from among the candidates. Weincrease the number of inputs one by one, watching asuitable criterion [6]. First, we divide the data into twogroups: A (first half of the data) and B (second half ofthe data). As a criterion to this purpose, we use the socalled regularity criterion RC, in GMDH (groupmethod of data handling), which is defined as follows[6]: kARC y iA y iAB i 1()2kB(/ k A y iB y iBAi 1)2 / kB / 2 (2)where kA and kB are the number of data of the groups Aand B, yiA and yiB are the output data of the groups Aand B. Next the yAB represents the model output for thegroup A input estimated by the model identified usingthe group B data and yBA the model output for the groupB input estimated by the model identified using thegroup A data. Therefore, we build two models for twodata groups at each stage of the identification. First, webegin with a fuzzy model with one input. We make nmodels: one model for one particular input. After theidentification of the models using the data groups A andB, we calculate RC of each model and select one modelto minimize RC from among the one-input models.Next, we fix the one input selected above and addanother input to our fuzzy model from among theremaining three candidates. Our fuzzy model has twoinputs at this stage. We select the second input as we doat the first step, according to the value of RC. Wecontinue the process until the value of RC increases. Ifthe value of RC becomes bigger in all cases in somestep then in the previous one, the process is terminatedand the combination of inputs with the smallest RCvalue is the result of our selection.In our case, we find out 2 relevant input variables from3 possible candidates. The minimum of RC criterionwas calculated (RC 1.9303) and relevant inputs x1 andx2 were determined for future model identification(Figure 4).IF (x1 is S) AND (x2 is S) THENy1 3.2x1 2.2x2 1IF (x1 is S) AND (x2 is H) THENy2 2.5x1 2.3x2 4IF (x1 is H) AND (x2 is S) THENy3 0.3x1 3.0x2 5IF (x1 is H) AND (x2 is H) THENy4 0.9x1 0.9x2 7Fig. 4 – Rules of Initial Reduced Fuzzy SystemInput 1 (x1): S (0, 0, 8), H (4, 10, 10). Input 2 (x2):S (0, 0, 7): H (3, 12, 12).The shape of input/output dependence of determinedtwo-dimensional non-linear system is shown in Figure5.
Using results of genetic fuzzy clustering the premisesstructure and parameters of 4 rules was done andantecedent part of identified fuzzy T-S model wasobtained.3.3 Rule Consequent Parts DeterminationNext, the consequent part of T-S model was identifiedusing GA procedure of genetic toolbox of MATLABand final fuzzy T-S model was obtained (Figure 9).IF (x1 is S) AND (x2 is S) THENy1 3.507x1 1.736x2 0.1082IF (x1 is S) AND (x2 is H) THENy2 2.487x1 2.163x2 5.497IF (x1 is H) AND (x2 is S) THENy3 0.7395x1 4.142x2 1.019IF (x1 is H) AND (x2 is H) THENy4 0.1535x1 1.766x2 4.076Fig. 5 – Shape of Two-Dimensional FunctionUsing values of two inputs x1, x2 the initial twodimensional fuzzy input space was defined using datapoints in Figure 6.Fig. 9 – Rules of Final Fuzzy ModelInput 1 (x1): S (-7.808, 0.0362, 7.881), H (3.888, 9.999,16.11). Input 2 (x2): S (-8.985, 0.0063, 8.997),H (-2.763, 11.98, 26.72).Fig. 6 – Two-Dimensional Input Space3.2 Data Fuzzy Clustering and Rule Premise PartDeterminationThe data set of initial input space diversification(Figure 4) was investigated using fuzzy clustering AGA algorithm to find out the suitable number and fuzzyapproximation of input variable terms. The results ofthis procedure are shown in Figure 7 and Figure 8. Bothof two inputs variables x1, x2 are expressed using twolinguistic terms – namely SMALL (S) and HIGH (H).The GA algorithm of MATLAB we used started withnext parameters: size of population N 800individuals, generation number limit t 1500generations, information was coded into chromosomesas a chain of real numbers with 16 genes. As a selectionmethod the tournament selection between 2 individualswas used, as a crossover method the arithmeticcrossover and as a mutation method the uniformmutation were used. As a termination condition thegeneration limit reached is used. Fitness function hasthe following formulaN yFitness GAi yi(3)i 1where N is the number of samples. The value y is anoutput of the reference fuzzy model. yGA is an outputvalue of the fuzzy model being designed by the geneticalgorithm.Fig. 7 – Linguistic Terms of x1Fig. 8 – Linguistic Terms of x2The course of fitness function convergence we can seein Figure 10.
Fig. 10 – Course of Fitness ConvergenceThe shape of final model input/output non-linearfunction is shown in Figure11.sufficient convergence speed with initial equidistantdata clusters centre distribution.The results of genetic fuzzy clusteringprocedure and the suitable diversification of fuzzymodel input space is done including the number ofinput variable linguistic values and appropriateapproximation of their membership functions as well.The numerical example proved effectiveness ofproposed methods. The model identification algorithmsinclude the procedure of pre-defined relevant numberof input variables selection.The future research works will be focused onhierarchical and parallel GAs application to increasethe necessary computational effort.Acknowledgments:This work has been supported by the GACR project102/05/H525: “The Postgradual Study Rationalizationat Faculty of Electrical Engineering and ComputerScience VSB-TU Ostrava”.References:[1] Golgberg,D.E.: Genetic Algorithms in Search,Optimization and Machine Learning, Addison-WesleyPub. Comp., INC, 1989, ISBN 0-201-15767-5Fig. 11 – Shape of Final Function[2] Ryan, C.: Shades. Polygenic Inheritance Scheme. InProceedings of MENDEL ’97,Brno, 1997, pp.140-147.3.4 Final Fuzzy Model TestingThe final results of identification process can be judgedsubjectively to compare both of the function shapes inthe Figures 5 and Figure 11. To calculate the errorcriterionFCR 100 NNy GA yi 1y (4)[3] Roupec, J.: Design of Genetic Algorithm forOptimisation of Fuzzy Controllers Parameters. (InCzech) Ph.D. Thesis, Brno University of Technology,Brno, 2001.[4] Roupec, J. – Popela, P. – Ošmera, P.: TheAdditional Stopping Rule for Heuristic Algorithms. InProceedings of Mendel ’97, Brno, 1997. pp. 135–139we can obtain the relative error of final model of value1.659%.[5] Pistauer,M.: Clustering Algorithms for process dataAnalysis, Proc.15-th IASTED, Innsbruck, 1996,pp.137-1394. Conclusion[6] Sugeno,M.,Yasukawa,T.: A Fuzzy-Logic BasedApproach to Qualitative Modelling, IEEE Trans. onFuzzy Systems, 1, No.1, 1993, pp.7-31The main problem of fuzzy data clustering is to find outa suitable number and shape of data clusters to achievesatisfying results of next fuzzy modelling.In approach presented the input data fuzzypartition is modified in such a way that the suitablefuzzy clusters are determined using the advancedgenetic algorithm. Some effective procedures namelyindividual lifetime limitation and the redundant genesapplication are used. In this method, the computationaleffort is bigger when conventional
T-S Fuzzy Modelling Using Advanced Genetic Algorithms PAVEL ŠIŠPERA, M.Sc.1 Prof. MIROSLAV POKORNÝ, Ph.D.1 JAN ROUPEC, Ph.D.2 1Department of Measurement and Control Faculty of Electrical Engineering and Computer Science, VSB Technical University Ostrava
ing fuzzy sets, fuzzy logic, and fuzzy inference. Fuzzy rules play a key role in representing expert control/modeling knowledge and experience and in linking the input variables of fuzzy controllers/models to output variable (or variables). Two major types of fuzzy rules exist, namely, Mamdani fuzzy rules and Takagi-Sugeno (TS, for short) fuzzy .
fuzzy controller that uses an adaptive neuro-fuzzy inference system. Fuzzy Inference system (FIS) is a popular computing framework and is based on the concept of fuzzy set theories, fuzzy if and then rules, and fuzzy reasoning. 1.2 LITERATURE REVIEW: Implementation of fuzzy logic technology for the development of sophisticated
Different types of fuzzy sets [17] are defined in order to clear the vagueness of the existing problems. D.Dubois and H.Prade has defined fuzzy number as a fuzzy subset of real line [8]. In literature, many type of fuzzy numbers like triangular fuzzy number, trapezoidal fuzzy number, pentagonal fuzzy number,
Fuzzy Logic IJCAI2018 Tutorial 1. Crisp set vs. Fuzzy set A traditional crisp set A fuzzy set 2. . A possible fuzzy set short 10. Example II : Fuzzy set 0 1 5ft 11ins 7 ft height . Fuzzy logic begins by borrowing notions from crisp logic, just as
of fuzzy numbers are triangular and trapezoidal. Fuzzy numbers have a better capability of handling vagueness than the classical fuzzy set. Making use of the concept of fuzzy numbers, Chen and Hwang [9] developed fuzzy Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) based on trapezoidal fuzzy numbers.
ii. Fuzzy rule base: in the rule base, the if-then rules are fuzzy rules. iii. Fuzzy inference engine: produces a map of the fuzzy set in the space entering the fuzzy set and in the space leaving the fuzzy set, according to the rules if-then. iv. Defuzzification: making something nonfuzzy [Xia et al., 2007] (Figure 5). PROPOSED METHOD
2D Membership functions : Binary fuzzy relations (Binary) fuzzy relations are fuzzy sets A B which map each element in A B to a membership grade between 0 and 1 (both inclusive). Note that a membership function of a binary fuzzy relation can be depicted with a 3D plot. (, )xy P Important: Binary fuzzy relations are fuzzy sets with two dimensional
The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol. 34, Part XXX 3.2 Fuzzy inference system Fuzzy inference is the process of formulating the mapping from a given input to an output using fuzzy logic. The process of fuzzy inference involves: membership functions, fuzzy logic
