Fuzzy Modeling With Emphasis On Analog Hardware Implementation

Fuzzy Modeling with Emphasis on Analog HardwareImplementation – Part I: DesignSHAKTI KUMARHaryana Engineering CollegeJagadhari, YamunanagarINDIAK.K. AGGARWALGGS Indraprastha UniversityDelhi – 110 006INDIAARUN KHOSLADepartment of Electronics and Communication EngineeringNational Institute of TechnologyJalandhar – 144 011INDIAAbstract: - In this paper, a new approach of fuzzy modeling with emphasis on analog hardwareimplementation is suggested. In the suggested approach, the input space of each input variable ispartitioned independently through the modified form of fuzzy c-means (FCM) clusteringalgorithm to generate the membership functions. By considering all the possible combinationsbetween the membership functions of the input variables and the cluster centers, the completerule-base is generated through reference to the available input-output data. The outputmembership functions are taken to be singletons for easy and straightforward implementation inanalog domain. The created rule-base is then optimized through the exhaustive search techniquessuggested by the authors. The suggested approach has been applied on fuzzy controller data forrapid Nickel-Cadmium batteries charger developed [1]. The data for the batteries charger hasbeen obtained through experimentation with an objective to charge the batteries as fast aspossible. Special purpose analog modules, designed by the authors and realized with operationalamplifiers (op-amps), diodes, resistors are considered for hardware implementation. Theproposed fuzzy modeling design approach is described in Part I. Part II describes the synthesisand analog implementation.Key Words: - fuzzy modeling, FCM, membership functions, analog modules1IntroductionFuzzy models are generally identifiedthrough following two approaches:1) Knowledge-drivendesign,inwhich the fuzzy system isidentified using the humanknowledge2)Data-driven design, in which thefuzzy system is derived from theinput-output dataDesign and implementation of afuzzy system depends on the applicationtowards which it is addressed. Non-realtime applications like data-analysis or1

decision-makingaresoftwareimplemented on a personal computerand provide a high flexibility since theysupport fuzzy systems with an arbitrarynumber of rules, without any limitationsconcerning number and types ofmembership functions and wide range ofinferencing mechanisms. Fuzzy logictoolbox for Matlab is such an applicationpackage .On the other hand real-timeapplications are generally implementedusinggeneral-purposeprocessors.Although the fuzzy systems areinherently parallel systems, but generalpurpose processors are exclusivelysequential and the three processingstages of a fuzzy system viz.fuzzification,inferenceanddefuzzification are performed seriallythus resulting in a high response time.Fuzzy chips employing parallelarchitectures [2] have been proposed inorder to reduce the processing time andboth analog and digital techniques areused for implementation. These chips arecostly and don’t provide any flexibilityfor accommodating more number ofinput & output variables and fuzzy rulesthan the specified numbers.Analog circuits have been found tovery attractive for implementingarithmetic functions used in fuzzycontrollers. Parallelism can be employedthrough analog circuits and hence theyoffer the advantage of high speed.Analog fuzzy modules are generalmodules for fuzzification such asmembership function generating circuits,modules related to inference engine,such as fuzzy operator implementationcircuits and defuzzification circuits. Themodules can be organized by designer toimplement fuzzy controller specific tohis particular application.The implementation through analogfuzzy modules is very flexible, since it iscapable of accommodating any numberof input and output variables in contrastto fuzzy chips, where number of inputsand outputs are fixed.This paper presents a new fuzzymodeling approach with emphasis onanaloghardwareimplementation.General-purpose analog fuzzy modulesare considered for implementation.This paper is organized as follows. InSection 2, the algorithm for buildingfuzzy model from the available inputoutput data is explained. The createdfuzzy model may contain certainredundant rules, which can be identifiedand removed through the exhaustivesearch technique suggested by theauthors. The exhaustive search techniqueis discussed in Section 3, which isapplied on the rule-base generated.Section 4 compares the hardwarerequirements of the reduced system withthe system considering all the rules.Finally, conclusions are drawn inSection 5.2The Proposed AlgorithmSTEPI:GenerationoftheMembership functions and clustercenters through the modified FCMalgorithmFuzzy clustering algorithms [3, 4, 5] areunsupervised algorithms used forpartitioning the data into a pre-definednumber of clusters with fuzzyboundaries and are used extensively forclassification, approximation problemsand recognition of geometrical shapes inimage processing.Consider the data matrix Z consists ofvectors zk, k 1, 2, .,N, contained in itscolumn and each vector is n-tuple. Thesevectors are partitioned into c clusters,2

and clusters are represented by prototypevectors, vi [vi1, vi2, .,vin]T ℜn, i 1, c. Prototype matrix is representedby V and has vi in its ith-column.The fuzzy partition is represented bythe matrix U ℜc x N, whose element,µi,k [0,1] and represents themembership degree of the data vector zkin ith-cluster.FCM clustering algorithm partitionsthe data Z into c overlapping clusters soas to minimize the objective functiondefined in equation (1).cNJ ( Z ; V , U ) ( µ i ,k ) m d 2 ( z k , v i ) (1)i 1 k 1The exponent m (1, ) determinesthe fuzziness of the clusters and for mostoftheapplications,m 2[4].d ( z k , v i ) defines the distance of datavector zk from the cluster prototype vi.The minimization of equation (1) iscarried out subject to the constraintsdefined in equations (2) & (3).c µ i,k 1, k 1, .,N (2)i 1N0 µ i,k N ,i 1, .,cii)FCM finds local minima oftheobjectivefunction,because it is derived from thegradient of the objectivefunctioniii)The results of FCM not onlydepends on the parameters m& c, but also on the choice ofinitial prototypesFollowing parameters are to bedefined by the user for applying FCMalgorithm:1. Number of clusters (c)2. m ( 1)3. termination criteria, ε (say) 0In the modified clustering algorithm,the partitioning of each input variable isdone independently i.e data matrix Zrepresents one-dimensional data.The modified FCM algorithms islisted as below:Step 1: Initialize partition matrix U(0)(cxn)U [µik ]Step 2: At step r calculate cluster centerV(r) [ vij ], using (3)nk 1Equation (2) implies that themembership coefficients for each datapoint must add to unity and equation (3)ensures that the clusters are neitherempty nor contain all the points todegree 1.The application of FCM algorithmimplies the minimization of equation (1)subject to the constraints defined inequations (2) & (3) and leads to thefinding fuzzy partition matrix U, andprototype matrix V.Some important observations aboutFCM [6] are:i)FCM always converges form 1vij µk 1nmik µk 1x kj, j 1, 2, 3 . mmikStep 3: Update partition matrix for rthstep by 1 c2 /( m 1) µ d ik( r ) / d (jkr ) for Ik φ j 1 r 1µ ik 0, for all classes I ,where I Є Ik’where Ik {i 2 c n; dik(r ) 0}and Ik’ {1,2 .,c} - Ikr 1ik()Step 4: If U(r 1) - U(r) ε ,stop; elseset r r 1 and return to step 2.3

Step 5: Find the cluster centers V1 andV2where,V1 is the cluster center with minimumdistance from the first data pointV2 is the cluster center with maximumdistance from the first data pointStep 6:For V1:If µi1 0.0 then z c 1µzk 1- µik for xk V1else µzk 1 for xk V1where, k 1,2 .nFigure 1. Membership functions for theinput variable Temperature (T)For V2:If µin 0.0 then z c 1µzk 1- µik for xk V2else µzk 1 for xk V2where, k 1,2 .nThe purpose of this step is to placemembership functions with appropriateshape at the two extremes.The modified FCM algorithm is appliedon the battery charger data. Themembership functions generated areshown in figures 1 and 2 for the twoinput variables viz. temperature andtemperature gradient respectively. Forthe input variable temperature, 12clusters were defined and for the otherinput variable temperature gradient, 4clusters were taken. The cluster centersfor the two input variables are shown inTable 1. In an air-conditioningenvironment, the ambient temperature isclose to 25oC, the universe of discoursefor the variable temperature has beentaken from 25-50oC, in contrast to thetraining data, where the values arebetween 0-50oC.Figure 2. Membership functions for theinput variable Temperature gradient (dT)STEP II: Generating Rule-base fromthe cluster centers generated in theStep I and training dataFrom the cluster centers generated in theStep I and the training data available,rules can be generated. The number ofrules generated for the given case are12x4 48. The rules generated arerepresented as fuzzy associated memoryand are shown in Table. 2.The rules can be read as under:Rule 1If T is T1 and dT is dT1 thenCharging current is 8Rule 2If T is T2 and dT is dT1 thenCharging current is 8 4

Table 1: Cluster CentresTemperatureCluster number Cluster ure GradientCluster number Cluster Centre1.0.07592.0.35663.0.64104.0.9232This is in contrast to the approachsuggested by Mendel et. al [8, 9], wherea rule is generated corresponding toevery data point. A fuzzy rule base, be itgenerated from experts or by somelearning or identification schemes maycontain redundant, weakly contributingor outright inconsistent rules. It istherefore, highly desirable to extract themore pertinent elements of a given ruleset. Researchers have formulated varioustechniques for this objective, yet there isno uniformly accepted approach fordesigning a fuzzy rule set efficiently andeffectively.Variousorthogonaltransformation methods [10, 11, 12, 13,14] have been proposed for selectingimportant fuzzy rules from a given rulebase. K. Nozaki et.al [15] proposed amethod for automatically generatingfuzzy if-then rules from numerical data.Genetic algorithms have also been used[16, 17, 18, 19, 20, 21, 22, 23, 24] foroptimizing fuzzy membership functionsand fuzzy rule base. In this paper, theredundant rules are eliminated using theexhaustive search technique proposed bythe authors [7]. The technique isexplained in the following section.3ExhaustiveAlgorithm [7]SearchExhaustive Search Technique is basedon the identityX XY X,which means that the term XY iscontained in X and is not affecting theoutput and hence can be dropped.The proposed algorithm involvessearching for such rule combinationswhere variables can be dropped and therules can be merged to get a reducedrule-base.The exhaustive search algorithm iseasy to apply if all the membershipfunctions corresponding to inputvariables are listed in sequence. Thealgorithm searches such rule groupings,in which only one of the input variablesis changing and all the consequents aresimilar. If all the rules are listed insequence, then the rules forming thegroup based on the above criteria shallbe adjacent. Whenever such a groupingis found, one of the variables in thegrouping shall be dropped and all therules in the grouping can be merged toget a single rule. The basic step of thealgorithm can be understood from theTable 3.The suggested algorithm searches forsuch combinations as found for theabove three rules and is beingrepresentedby(1,2,3),thereby5