DECSAI Deparment of Computer Scienceand Arti�cial IntelligenceGenetic Algorithms Applications to Fuzzy LogicBased SystemsF. Herrera, M. Lozano and J.L. Verdegay

Technical Report #DECSAI-93116October, 1993Proc. 9TH Polish-Italian and 5TH Polish-Finnish Symposium onSYSTEMS ANALYSIS AND DECISION SUPPORT INECONOMICS AND TECHNOLOGY.(R. Kulikowski, K. Szkatula, J. Kacprzyk, Eds., Omnitech Press)Radziejowice near Warsaw, Poland, October 25-29, 1993, 125-134.ETS de Ingenier��a Inform�atica. Universidad de Granada.18071 Granada - Spain - Phone +34.58.244019, Fax +34.58.243317

Genetic Algorithms Applicationsto Fuzzy Logic Based Systems�F. Herrera, M. Lozano, J.L. VerdegayDept. of Computer Science and Arti�cial InteligenceUniversity of Granada 18071 - Granada, Spaine-mail: herrera, lozano, verdegay@robinson.ugr.esKeywords: Soft Computing, Genetic Algorithms, Fuzzy Logic, Fuzzy Logic Based Systems.AbstractIn this paper the integration of Fuzzy Logic and Genetic Algorithms is discussed. Somepotencial Genetic Algorithms applications to fuzzy logic based systems are presented: thegeneration of the structure of fuzzy IF-THEN rules, the tuning of a fuzzy rules base, andthe fuzzy classi�er systems.1 Introduction"Soft Computing is concerned with modes of computing in which precision is traded fortractability, robustness and ease of implementation" [Zad92]. As is known, Soft Computingcontains Fuzzy Logic and Genetic Algorithms among its components.From a very broad point of view a Fuzzy System (FS) is any Fuzzy Logic Based System. FuzzyLogic (FL) can be used either as the basis for the representation of divers forms of knowledgesystems, or to model the interactions and relationships among the system variables. Fuzzy logiccontrol systems (FLCS) can be seen as a special case of this more general class of FS.Genetic Algorithms (GA) are search algorithms that use operations found in natural geneticsto guide the trek through a search space. GA are theoretically and empirically proven to providerobust search in complex spaces, giving a valid approach to problems requiring e�cient ande�ective search ([Gol89]).Each of them, FL and GA, have di�erent advantages, for example, on the one hand, non-linearity and explicit knowledge expression of FS, and, on the another hand learning capability,global and local search approach of GA.Recently, there is an increasing number of publications about the combination of these twotopics, as the interface between GA and FL may produce useful results. Here, we study the� This research has been supported by DGICYT PB92-0933.

integration of both topics, focusing the discussion in some GA applications to fuzzy systems withmain emphasis on tuning parameters and learning. This paper mainly introduces our researchin this topic, together with the concept of fuzzy classi�er systems [Gey92, 93, Val91a, b].To do that, a description of GA is given in section two. Section three presents GA applica-tions to FS. We shall end with some �nal remarks.2 Genetic AlgorithmsGA starts with a population of randomly generated solutions, chromosomes, and advancetoward better solutions by applying genetic operators, modeled on the genetic processes occurringin nature. In these algorithms we maintain a population of solutions for a given problem; thispopulation undergoes evolution in a form of natural selection. In each generation, relativelygood solutions reproduce to give o�spring that replace the relatively bad solutions which die. Anevaluation of �tness function plays the role of the environment to distinguish between good andbad solutions.Although there are many possible variants of the basic GA, the fundamental underlying mech-anism operates on a population of chromosomes (representing possible solutions to the problem)and consists of three operations:(1) evaluation of individual �tness,(2) formation of a gene pool, and(3) recombination and mutation.The next �gure shows the structure of a simple GA,Procedure genetic algorithmbegin (1)t = 0;inicialize P (t);evaluate P (t);While (Not termination-condition) dobegin (2)t = t + 1;select P (t) from P (t � 1);recombine P (t);evaluate P (t);end (2)end (1)Figure 1. Structure of a Genetic AlgorithmThus, it is generally accepted that any GA must take into account the �ve following pointsto solve a problem:1. A genetic representation of solutions to the problem,2. a way to create an initial population of solutions,3. an evaluation function which gives the �tness of each individual,4. genetic operators that alter the genetic composition of children during reproduction, and5. values for the parameters that the GA uses (population size, probabilities of applying geneticoperators, etc.).

To solve a problem with GA, potential solutions have �rst to be encoded as chromosomes andan evaluation function has to be de�ned in order to measure the performance.Considering the coding issue, chromosomes have been generally string of bits, but also otheralphabets can be used. It would seem particularly natural to represent the genes directly as realnumbers for optimization problems of parameters with variables on continuous domains. Then achromosome vector is a vector of oating point numbers. The use of real parameters is e�cientin problems with a lot of variables, and the real representation of the solutions is very close tothe natural representation of many problems, [Mic92, Esh93, Her93b].In applications 3.1 and 3.2 special real coded genetic algorithms (RCGA) are used. Next wedescribe shortly their two classical genetic operators: mutation and crossover.We propose to use the non-uniform mutation proposed by Z. Michalewicz, [Mic92]. If Ctv =(c1; :::; ck; :::; cH) is a chromosome and the element ck was selected for mutation (the domain ofck is [ckl; ckr]). The result is a vector Ct+1v = (c1; :::; c0k; :::; cH), withc0k = ( ck +4(t; ckr � ck) if a random digit is 0,ck �4(t; ck � ckl) if a random digit is 1.where the function 4(t; y) returns a value in the range [0; y] such that the probability of 4(t; y)being close to 0 increases as t increases. This property causes this operator makes a uniformsearch into the initial space when t is small, and very locally at later stages.In relation with the crossover operator, next we describe two of them:1. Simple crossover [Wri91, Mic92]: It is de�ned in the usual way with a crossover point. IfCtv = (c1; :::; ck; :::; cH) and Ctw = (c01; :::; c0k; :::; c0H) are to be crossed, and assume that thecrossover point is (randomly) selected before the k-th gene, the two resulting o�spring are:Ct+1v = (c1; :::ck�1; c0k; :::; c0H)Ct+1w = (c01; :::c0k�1; ck; :::; cH)2. Max-Min-Arithmetical crossover [Her93a, b]: If Ctv and Ctw are to be crossed, we generate:Ct+11 = aCtw + (1� a)Ctv Ct+12 = aCtv + (1� a)CtwCt+13 with ct+13k = minfck; c0kg Ct+14 with ct+14k = maxfck; c0kgThis operator can use a parameter a which is either a constant, or varies with regard to thenumber of generations made. The two most promising o�spring from these four substitutetheir parents in the population.In [Her93b] it may be found a complete description of crossover operators and o�springselection methods for real coded GA, di�erent crossover operators are proposed using fuzzy con-nectives.3 Genetic Algorithms Applications to Fuzzy SystemsThere are di�erent aspects on the design of FS and FLCS, as for instance, how to get the setof fuzzy control rules, to decide the number of fuzzy rules, to decide the shape of the membershipfunctions, to tune the fuzzy rules base, etc, in which GA have been successfully applied [Bon93,Cas93, Gey92, 93, Gon93, Her93a, c, Kar91a, b ,c, Kro93, Lee93a, Nom92, Sur93, Tak93,Thr91, Val91a, b].As it is known, the problem of managing a fuzzy rules base (adquisition, learning, tuning) isof utmost importance in the development of fuzzy systems. In a general learning process we candistinguish di�erent components:

1. A generation method of desirable fuzzy rules able to include the complete knowledge of theset of examples.2. A simplify method:2.1. Solving con icts among the generated rules,2.2. Finding the �nal set of fuzzy rules able to approximate the input-output behavior of areal system.3. A tuning method of the set of rules obtained in the step 2.Next we present two approaches based on GA for generating rules and tuning a fuzzy rulesbase. On other hand, we present the concept of fuzzy classi�er system [Gey92, 93, Val91a, b], aGA approach of fuzzy machine learning.3.1 Generating fuzzy IF-THEN rulesWe describe shortly a generation method of desirable fuzzy rules able to include the completeknowledge of a set of examples E = fe1; :::; epg. We use a RCGA where a chromosome representsa fuzzy rule, and it is evaluated by means of a frequency method ([Del93a, b]).The RCGA will �nd the best rule in every running over the set of examples, and with aniterative process it is obtained a set of rules covering the set of examples.In the population of the RCGA a candidate solution Cr, r = 1; :::;M , represents a fuzzy ruleIF x1 is Ar1 ... and xn is Arn THEN y1 is Br1 and ... and ym is Brmwhere the real values a1rj, a2rj, a3rj, a4rj, b1rh, b2rh, b3rh, b4rh characterize the membership functionsof Arj and Brh j = 1; :::; n, h = 1; :::; m, respectively. Thus, Cr codes the vector values:(a1r1; a2r1; a3r1; a4r1; :::; a1rn; a2rn; a3rn; a4rn; b1r1; b2r1; b3r1; b4r1; :::; b1rm; b2rm; b3rm; b4rm)The initial gene pool is created partially using the initial set of examples, and partially ran-domly, and to de�ne the �tness function the following concepts must be taken into account.The frequency of any fuzzy rule Ri through the set of p examples Ep (Ep = fek = (exk; eyk);k = 1; :::; pg) is Ep(Ri) = Ppk=1Ri(ek)pwith Ri(ek) a compatibility degree between the rule Ri and the example ek,Ri(ek) = �(Ai(exk); Bi(eyk))Ai(exk) = �(Ai1(exk1); :::; Ain(exkn))Bi(eyk) = �(Bi1(eyk1); :::; Bim(eykn))where � is a t-norm.For a fuzzy rule Ri we de�ne:a) The set of positive examples with compatibility degree greater than � as:E+(Ri) = fek 2 Ep=Ri(ek) > �gwith n+Ri = jE+(Ri)j; GRi =Pek2E+(Ri)Ri(ek)=n+Ri

b) The set of negative example as:E�(Ri) = fek 2 Ep=Ri(ek) = 0 and Ai(exk) > 0gwith n�Ri = jE�(Ri)j.An example is considered negative for a rule when it matches better with some other rulehaving the same antecedent but di�erent consequent.An evaluation function of the rule Ri and therefore a �tness function to the associated chro-mosome Ci can be de�ned as:F (Ci) = ( 0 if n�Ri > 0g(n+Ri;Ep(Ri); GRi) otherwisewith g an increasing function in its variables, with regard to maximize the �tness function.A complete description of the generation method together with the covering process of a setof examples may be found in [Her93c]. In [Cas93] (section III.3) it was proposed a simplifymethod to �nd the �nal set of fuzzy rules able to approximate the input-output behavior of a realsystem.3.2 Tuning a fuzzy rules baseAs is known the performance of a FLCS depends on its linguistic rules and membershipfunctions, and it is usual to determine the membership functions of the linguistic variablesparticipating in the process. From this point of view it should be noted that the correct choiceof the membership functions plays an essential role on the performance of an FLCS. It is veryimportant to adjust the parameters of the membership functions, that is, to tune the fuzzy rulesbase to make the FLCS behaves as closely as possible to the operator or expert behavior.We propose a tuning method for obtaining high-performance fuzzy control rules by means ofan special GA.The tuning method using GA �ts the membership functions of the fuzzy rules dealing withthe parameters of the membership functions, and obtaining high-performance, that is, makingminimum an error function de�ned by means of an input-output data set for evaluation.When applying GA to a problem two basic decisions are to be made: (1) how to code thepossible solutions of the problem, and (2) how to evaluate the merit of each string, the �tnessfunction.A linguistic rules base is represented as a chromosome, and we try to obtain the chromosomewith best adaptation. To do that we use RCGA, with the special genetic operators de�ned in theprevious section.The rules have the formRi: IF x1 is Ai1 ... and xn is Ain THEN y1 is Bi1 and ... and ym is Bimwhere x1; :::; xn, and y1; :::; ym are variables representing the process state variables and thecontrol variables respectively; and Ai1; :::; Ain, Bi1; :::; Bim are fuzzy sets in the universes ofdiscourse U1; :::; Un, V1; :::; Vm .These fuzzy sets are characterized by their membership functionsAij(Bih) : Uj(Vh)! [0; 1]; j = 1; :::; n; h = 1; ::; m

As usual trapezoidal membership functions with a parametric representation givin by meansof the 4-tuples (a1ij ; a2ij; a3ij ; a4ij), (b1ih; b2ih; b3ih; b4ih) are considered to represent the variables.A rule Ri is represented by a piece of chromosome Cri,Cri = (a1i1; a2i1; a3i1; a4i1; :::; a1in; a2in; a3in; a4in; b1i1; b2i1; b3i1; b4i1; :::; b1im; b2im; b3im; b4im);Therefore a base of t rules, R, is represented by CrCr = Cr1 Cr2 ::: Crtwith t the number of rules.The initial gene pool is created from the initial base of fuzzy rules. For every gene an intervalof performance is de�ned, that is, the interval of adjustment of this variable. Then we createa population of chromosomes with C1 as the initial base of fuzzy rules, and the rest initializedrandomly, being each gene into its respective interval of adjustment.Using a training input-output data set (consisting of the values that the variables take duringan experiment in which the system is controlled by an expert) we de�ne the �tness function asthe quadratic error of the FLCS over this set. The GA process will be to make minimum the�tness function. The genetic operators are non-uniform mutation, [Mic92] and simple crossoveror max-min-arithmetical crossover [Her93a,b].To test this approach we have carried out experiments with a simulation of the invertedpendulum problem. A pendulum with 5 kg in weight, 5 m in length has been considered in areal simulation, applying the force to the gravity center, during a constant time 10 ms. Underthese parameters the universes of discourse of the variables angle, angular speed and force arethe followings:� 2 [�0:5240; 0:5240] rad! 2 [�0:8580; 0:8580] rad/scf 2 [�2980:0; 2980:0] NwInitially the membership functions corresponding to the linguistic labels must be de�ned. Wehave considered the discretization of the discourse universes presented in [Lia91], and the nextset of fuzzy control rules, [Yam89]: A N G L ENL NM NS ZR PS PM PLA NLNG NMUL NS NS ZRAR ZR NM ZR PMS PS ZR PSPE PMED PLAny optimum values always depend on the reasoning model. The results that we will presenthave been obtained using the Lukaciewicz implication function, I(x; y) = min(1; 1� x+ y), the

conjunction operator T = MIN , and the defuzzi�cation strategies: center of area (COA), meanof maximum (MOM), weight average of the center or area by the heights (WECOA), maximalheight methods: max-maximum criteria (MC), max-center of area (MCOA), [Miz89], [Zha91].A training set of examples has been obtained experimentally with 68 input-output data in theintervals [-0.275, 0.275], [-0.454, 0.454] and [1576.681, 1576.681] for �, ! and f respectively.We apply two GA for obtaining the high-performance fuzzy control rules base: GA1 with simplecrossover, GA2 with max-min-arithmetical crossover, for T = 5000 iterations. The next tablesshow the behavior of the initial FCR, the FCR-GA1 and the FCR-GA2, presenting the quadraticand linear error: INITIAL FCR FCR FCRQE GA1 - QE GA2 - QECOA 2.957.848.5000 320.560.4062 301.237.0312MOM 2.038.396.7500 193.374.7188 250.266.0312WECOA 2.028.396.2500 195.281.8281 215.555.5312MC 1.248.722.5000 686.036.0000 759.847.8125MCOA 1.248.722.2500 501.209.1250 522.959.0625Table 1. Quadratic errorINITIAL FCR FCR FCRQE GA1 - QE GA2 - QECOA 12.683.9043 4.425.7715 4.165.2378MOM 10.407.3691 3.270.4431 3.392.4426WECOA 10.407.3691 3.236.4832 3.253.7516MC 7.267.7412 5.581.0361 6.018.1968MCOA 7.267.7417 5.439.5615 5.571.3335Table 2: Linear errorAs it is shown by the experimental results, the obtained fuzzy control rules base improvegreatly the behavior of the FLCS. In [Her93a] it may be found a complete description of thetuning process together with a complete description of experimental results obtained applying thetuning method to the fuzzy control rules base of the inverted pendulum.3.3 Fuzzy Classi�er SystemsThe fuzzy classi�er system (FCS) merges credit assignment mechanisms of classi�er systemsand fuzzy systems.A classi�er system consists of three layers: a parallel rule-based message-passing system, thebucket-brigade learning algorithm, and the GA. The message-passing system is the fundamentalcomputational engine of the system, it consists of a database of condition/action rules, calledclassi�ers, that read and write messages on a short-term message list. At the second level is thebucket-brigade learning algorithm which manages credit assignment among competing classi�ers,distributing external reward to the rules that contribute to successful behavior. The bucket brigadeplays a role similar to that of back propagation in neural networks. Finally, at the highest levelare genetic operators that create new classi�er. [Gol89, For91, Hol86].The next �gure shows the information ow in a classi�er system.

Input

Message List

Environment

Classifier System

StateVariables Rules

Information Action

Classifiers Output

ControlVariablesFigure 2: Information ow in a classi�er systemThe FCS is a genetic based machine learning system which integrates a fuzzy rule base, thebucket-brigade learning algorithm, and GA.The FCS learns by creating fuzzy rules which relate the values of the input variables tointernal or output variables. It has credit assignment mechanisms which reassemble those ofcommon classi�er systems, but with a fuzzy nature. The FCS employs a GA to evolve adequatefuzzy rules ([Val91a,b]). Classi�ers represent fuzzy rules. Each classi�er is a binary string thatencodes the membership functions of the fuzzy sets involved in the represented fuzzy rule. A GAruns over the population of classi�ers, searching new improved classi�ers.More details about FCS can be found in [Val91a,b, Gey92].4 ConclusionsThis paper has been focused on some GA applications to fuzzy systems. The robustnessand simple mechanism of GA make them a potentially useful tool to search good parametercon�guration, what justify its use in some aspects of fuzzy logic: tuning a fuzzy rules base,learning fuzzy rules, GA approach to auto-design fuzzy systems, fuzzy classi�er systems, etc.In the same way, fuzzy logic techniques can be used to improve the behavior of GA. In [San93]it is proposed a fuzzy crossover of binary coded GA based on the use of templates and mergingoperators. In [Lee93b] it is show the use of fuzzy logic techniques to dynamic control of GA. In[Her93b] we show how to use the fuzzy connectives to design crossover operators.Finally to remark that the integration of both topics together other components of soft com-puting (neural networks, ...) may provide a basis for the achievement of greater performance ofa great variety of systems.References[Bon93] Bonarini, A., ELF: learning incomplete fuzzy rule sets for an autonomous robot. Proc.First European Congress on Fuzzy and Intelligent Technologies, Aachen, 1993, 69-75.

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