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747 2002 216:Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science

D T Pham and M CastellaniAction aggregation and defuzzification in Mamdani-type fuzzy systems

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Action aggregation and defuzzi� cation inMamdani-type fuzzy systems

D T Pham* and M CastellaniManufacturing Engineering Centre, School of Engineering, University of Wales, Cardiff, Wales, UK

Abstract: This paper discusses the issues of action aggregation and defuzzi� cation in Mamdani-typefuzzy systems. The paper highlights the shortcomings of defuzzi� cation techniques associated with thecustomary interpretation of the sentence connective ‘and ’ by means of the set union operation. Theseinclude loss of smoothness of the output characteristic and inaccurate mapping of the fuzzy response.The most appropriate procedure for aggregating the outputs of different fuzzy rules and convertingthem into crisp signals is then suggested. The advantages in terms of increased transparency andmapping accuracy of the fuzzy response are demonstrated.

Keywords: fuzzy logic, output aggregation, defuzzi� cation, COG method, MOM method, weightedaverage

NOTATION

aY semi-support of membershipfunction Y

a1(a), a2(a) extremes of the a-cutA , B fuzzy termsAn area of nth fuzzy term of a fuzzy

space partitionAY area of fuzzy term YCOGn centre of gravity of nth fuzzy term of

a fuzzy space partitionf(x) functionh activation degreehY activation degree of fuzzy action YT (x) transformation functionx variableXn, Y n fuzzy terms2aY support of triangular fuzzy term Y

a, b, g, d parameters·X(x), nX(x) membership value of element x of

universe of discourse X

1 INTRODUCTION

In the design of fuzzy logic (FL) [1] systems, theMamdani model [2, 3] has its strong points in itscloseness to Zadeh’s method of fuzzy reasoning andfor its human-like representation of the response policy.Being close to Zadeh’s de� nition of FL, it allows anatural extension to the fuzzy domain of the familiarcrisp modus ponens logical inferencing rule. As opposedto Takagi and Sugeno’s model [3, 4], Mamdani’s modelexpresses the output using fuzzy terms instead ofmathematical combinations of the input variables. TheMamdani model has its main shortcoming in itsunsuitability for the analysis of closed-loop systemstability. However, in many applications, this issue isnot critical. For the above reasons, the Mamdani modelis still the basis for many industrial FL applications anda full understanding of its properties would be of valuein realizing further successful implementations anddevelopments.

The focus of the paper is on the interactions betweenthe output space partition, the rule aggregation operatorand the defuzzi� cation procedure. The basic propertiesof the aggregation and the defuzzi� cation operations aresummarized and a survey of output defuzzi� cationprocedures is presented. The limitations and advantagesof the current procedures are discussed with particularregard to their transparency and mapping accuracy.Following the discussion, the most appropriate proce-dure for converting the output of the fuzzy rules intocrisp signals is described. The suggested procedureallows full � exibility in the de� nition of the output

The M S was received on 18 December 2000 and was accepted afterrevision for publication on 21 M arch 2002.* Corresponding author: M anufacturing Engineering Centre, School ofEngineering, University of W ales, Cardiff, PO Box 688, Newport Road,Cardiff CF24 0YH, Wales, UK.

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partition, maintaining the transparency and predict-ability of the behaviour of the system. Moreover, theaggregation and defuzzi� cation algorithm adopted inthis work has a modular structure that lends itself todirect expression as a fuzzy neural network (FNN)architecture.

2 AN ANALYSIS OF MAMDANI-TYPE FLSYSTEMS

A block diagram of the information � ow in a generalMamdani FL system is given in F ig. 1. The lower sectionof the ‘FL System’ block represents the fuzzy knowl-edge, while the upper part contains the knowledgeprocessing operators.

The fuzzy knowledge base (KB) can be acquired fromhuman expertise or automatically generated via machinelearning techniques [5]. This static information isexpressed in terms of fuzzy production rules mappinga set of conditions on to a set of actions, where eachcondition (action) is de� ned by the linguistic value of aninput (output) variable. Individual rules are joinedtogether by the sentence connective ‘else’ to form theoverall rule base (RB) [2]. Because rules involvingmultiple outputs can always be decomposed into a setof single-output rules [5], in the rest of this paper onlyrules de� ned over a one-dimensional output space willbe considered. Accordingly, each rule consequent ismeant to de� ne one single-output action.

The partition of the input (output) space into a set oflinguistic terms generates a symbolic description of thephysical space. The ‘link’ between the real world and itslinguistic expression is given by fuzzy membershipfunctions (MFs) [1], which characterize the interpreta-tion of the fuzzy input–output relationship.

The operators represented in the upper part of the ‘FLSystem’ block perform the symbolic processing ofdynamic knowledge and the direct and inverse transfor-mations between linguistic terms and numerical data. Adesirable property of the fuzzy operators is a consistentbehaviour irrespective of the nature of the storedknowledge. The absence of interactions between theprocessing algorithms and the KB is particularlyimportant for the transparency and predictability offuzzy systems.

The symbolic processing of fuzzy information isperformed using the compositional rule of inference,which allows the determination of the activation of eachrule consequent according to the degree of matching ofthe antecedent. Zadeh’s sup-min [6] and Larsen’ s sup-prod [5] are the most popular composition operators.The overall fuzzy output is normally created by super-position of the individual rule actions. This implies theinterpretation of the rule connective ‘else’ with the unionoperation, which is normally implemented through thepointwise max operator [5]. Consequently, only infor-mation having the maximum activation value is used forgenerating the crisp signal, sometimes discarding thecontribution of entire rules.

Two operators are needed, one to convert the inputnumerical data into qualitative information and theother to perform the inverse process on the qualitativeoutput of the system [6]. In F ig. 1, these two operatorshave been represented as two logical blocks, the fuzzi� erand the defuzzi� er, added respectively at the input andthe output of the fuzzy system. In the fuzzi� er block,each numerical observation is mapped on to a fuzzy set(direct mapping). Most commonly, this fuzzy set is afuzzy singleton. The fuzzi� ed input is then matched withthe rule antecedents and a set of fuzzy actions isgenerated. The overall fuzzy action is expressed as apossibility (truth) distribution over the universe of

Fig. 1 Mamdani FL System: information � ow

D T PHAM AND M CASTELLANI748

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discourse and it needs to be converted into somenumerical output.

There is not yet a commonly accepted procedure forconverting qualitative information into a numericalform (inverse mapping). Unlike the fuzzi� cation process,which is a one-to-one mapping from the input space tothe possibility interval, the defuzzi� cation procedureposes the problem of � nding a suitable many-to-onemapping from a possibility distribution to the outputspace. The choice of the crisp value best representingsuch a possibility distribution is normally affected byfactors such as problem requirements and implementa-tion constraints. The two most commonly used defuzzi-� cation procedures in FL are the centre of gravity(COG) method and the mean of max ima (MOM )method [5, 7], both having advantages and shortcomingsthat will be discussed later. Alternative methods includeYager and F ilev’s basic defuzzi� cation distribution(BADD) [8, 9], semi-linear defuzzi� cation (SLIDE) andmodi� ed-SL IDE (M-SLID E) [10], Jiang and Li’s Gaus-sian distribution transformation-based defuzzi� cation(GTD), polynomial transformation-based defuzzi� cation(PTD) [11] and multimode-oriented PTD (M-PTD)[12, 13], Runkler and G lesner’s ex tended centre of area(XCOA) [14] and Saade’ s approach unifying defuzzi� -cation with the comparison of fuzzy sets [15]. Mizumoto[16] compares the in� uence of several rule aggregationand output defuzzi� cation methods on the controlresponse of a simulated plant, and presents twoalgorithms, the height method and the area method,that bring together aggregation and defuzzi� cation offuzzy actions. The height method has also been adoptedby Cherkassky and Mulier [17] under the name additivedefuzzi� cation.

The overall crisp relationship mapped by the ‘FLSystem’ block of Fig. 1 is therefore not only dependenton the KB de� nition but also on the mathematicaloperators chosen to manipulate it [18].

3 AGGREGATION AND DEFUZZIFICATION OFRULE ACTIONS

3.1 Aggregation of fuzzy actions

In a fuzzy system, once the conditions of the rules havebeen matched, a set of actions is activated. Each rulewhose antecedent has a non-zero matching degree willcontribute an output with an activation value equal tothe matching degree of the antecedent. The maxoperator is by far the most common implementationof the rule aggregation operation. According to thisprocedure, the overall fuzzy output is calculated fromthe set of individual outputs taking the maximum truthvalue where one or more terms overlap. F igure 2 showsan example of max aggregation for two overlappingactions A and B with activation degrees of 0.3 and 0.8respectively.

Alternative aggregation operations have been pro-posed generally based on a different implementation ofthe union operation. The most common examplesreplace the max operator with other triangular co-norms such as the algebraic sum or the bounded product[5, 16]. As the ordering of the rules is unimportant, anyoperation possessing the properties of commutativityand associativity is a candidate for implementing the elseconnective. In practice, the quality of the choice is

Fig. 2 sup composition

ACTION AGGREGATION AND DEFUZZIFICATION IN MAMDANI-TYPE FUZZY SYSTEMS 749



normally affected by the type of composition rule used.Even though several studies support the association ofthe union operation to the sup-star composition rule (seereference [5] for an overview), successful results havebeen reported using an additive procedure [16, 17].

3.2 Output defuzzi� cation

The overall fuzzy output generally constitutes a multi-modal non-zero truth distribution of possible crispvalues over a subset of the output space. In thedefuzzi� cation stage, one of those possible crisp valueshas to be selected as the output crisp signal. The designof a sound defuzzi� cation method is important as it willaffect the interpretation of the fuzzy response policy.The � rst aim is always to be able to generate a crispvalue that will be representative of the output possibilitydistribution.

For this purpose, some desirable defuzzi� cationproperties are outlined in reference [7], of which thebasic ones are consistency, section invariance andmonotonicity. A consistent defuzzi� cation methodmaps convex crisp sets to their centroid. F rom this, itfollows that the empty set and the universal set are bothdefuzzi� ed to their centre and a fuzzy singleton isdefuzzi� ed to its sole non-zero truth element. Sectioninvariance guarantees that the defuzzi� ed value isuniquely dependent on the output space elements havinga non-zero truth value. Outside this set, any modi� ca-tion of the fuzzy universe of discourse does not affect thedefuzzi� ed value. Monotonicity requires that, for anydecrease (increase) in the truth degree of a single-outputspace element, the defuzzi� cation result remainsunchanged or is moved away from (closer to) thatelement. This property implies that the contribution ofeach single-output space element to the � nal defuzzi� edvalue increases with the degree of truth of that element.

A desirable defuzzi� cation procedure should alsorequire a low computational effort to allow itsimplementation in real-time applications. At the sametime, it should allow a smooth response and mappingaccuracy to be obtained over all or most of the outputspace.

Finally, an ideal defuzzi� cation method should easethe design of the fuzzy system and keep the decision-making logic transparent to the user. The manipulationoperators should not add any overhead to the systemdesign and analysis.

Any defuzzi� cation method possessing the above-mentioned properties is a good candidate for imple-mentation in an FL expert system. The following sectionprovides a critical overview of some of the mostcommon defuzzi� cation methods. For an application-oriented overview of defuzzi� cation methods the readeris referred to reference [19].

4 MAIN DEFUZZIFICATION TECHNIQUES

4.1 Maxima methods

Once the shape of the output possibility distribution hasbeen determined, a quick and simple defuzzi� cationprocedure is to pick up one of the crisp values having amaximum truth degree (maxima methods) [7, 20].Possible choices are the � rst (smallest), the last (largest)or, in the case of a unimodal possibility distribution, themedian value. By far the most common maxima methodis to select the mean value of elements with maximumtruth degrees (MOM method) [5, 7].

The strength of maxima methods lies in theirsimplicity and speed of execution, but their majorweakness is in not being truly fuzzy. They are sectioninvariant and monotonous, but as a consequence of onlyconsidering elements of highest membership degrees,information not related to rules of maximal activation isignored. This causes a loss of the smooth outputcharacteristic generated by the gradual transitionsfrom input space areas where different rules prevail[18]. The crisp response curve does not retain thecontinuous and gradual nature of the input–outputfuzzy relationship and it is characterized by briskdiscontinuities of the kind produced by a multilevelrelay system. The functional identity between a multi-level relay and an FL system using symmetrical MFs,the sup-star composition rule and the MOM defuzzi� ca-tion procedure has been demonstrated in reference [21].

For control applications, the type of multirelaycontrol characteristic induced by maxima proceduresmay also show some of the limitations of such systems.Suboptimal control performances in terms of steadystate error, minimization of control effort and plant� uctuations have occasionally been associated with theuse of the MOM method [22, 23]. To retain the smoothoutput transitions typical of FL control, it is necessaryfor the defuzzi� cation procedure to consider also valueswhose membership degree is other than maximal. Themost popular alternative to maxima methods is tocalculate the � nal crisp value from the area of the outputpossibility distribution (area-based methods).

4.2 Area-based methods

A well-known area-based procedure is to select thedefuzzi� cation result as the centroid of the outputpossibility distribution COG method [5, 7]. A closevariant is the centre of area (COA) method [7], whichselects the crisp value as the position where the outputarea can be split into two equal halves. If the outputdistribution is symmetrical, the two methods giveidentical results.

The COG defuzzi� cation method is section invariant,monotonous and consistent, and its deterministic




response curve is characterized by a smooth andcontinuous behaviour. In control applications, com-pared to maxima methods, the COG rule gives asuperior steady state performance and a reduction ofcontrol effort and plant oscillations [22, 23]. On theother hand, a better transient performance using theMOM defuzzi� cation procedure has been reported [22].This is probably because the output of the MOMmethod is characterized by fewer but larger variations,corresponding to the boundaries between neighbouringrules. Given the same RB, this allows larger corrections,which should explain the superior transient response.

4.3 Other methods

Modelled on the application of the COG procedure,several learning algorithms have been proposed foroutput defuzzi� cation. The general idea underlying allthese methods is to perform some transformation of theoutput possibility distribution according to an auto-matically generated set of parameters. A generictransformation on a fuzzy term X can be written as [11]

nX x ·X x T x 1

where nX x is the new membership value of the elementx [ X , ·X x is the original value and T x is anytransformation function. Some examples of such func-tions are:

BADD method [8, 9]:

T x ·X x g¡1 2

where g is an automatically learned parameter.SLIDE method [10]:

T x1 ¡ b, x 4 a1, x a

»3

where a and b are parameters to be learned.M-PTD method [11, 12]:

T xN

j 0

bj ·X x ¡ 0 5 j

2

4

where bj are parameters that are adaptively tuned and·X x and T x are discrete functions.

In all of the above procedures, the kind of transfor-mation induced by T x is a distortion of the outputpossibility distribution in order to increase or decreasethe weight of elements of higher membership values. Themore the transformation magni� es the differencesbetween low and high membership values, the morethe defuzzi� cation method will approximate the MOMcharacteristic. In the limiting case, all values except themaximal will be brought to zero and the defuzzi� cationprocedure will correspond to the MOM method. On the

other hand, if the transformation is the identityfunction, the defuzzi� cation procedure will correspondto the COG method. A similar approach has beenadopted in reference [15], where the output possibilitydistribution is defuzzi� ed to the value

y1

0da1 a 1 ¡ d a2 a Š da 5

In equation (5), d is a parameter set by the designer anda1 a and a2 a are the two extremes of an a-cut X a

a [ U ·X a 5a [3] of output possibility distribution Xde� ned over universe of discourse U. The aim ofthis procedure is to magnify the contribution of theelements on one side of the centroid of the outputdistribution. In this way, more or less drastic actions canbe achieved.

These algorithms can help to tune the output of thesystem but they also introduce a considerable computa-tional overhead for the determination of the transfor-mation parameters and in some cases in thedefuzzi� cation procedure. The action of some of theparameters is also not immediately obvious and this canaffect the transparency of the fuzzy system. It ispreferable whenever possible to tune the behaviour ofthe system directly by acting on the shape of the outputMFs. The response curve of the system can be mademore or less smooth by adjusting the overlap of theoutput terms, and efforts towards an adaptive behaviourshould be focused in the same direction. Changes in thebehaviour of the system should re� ect variations in theresponse policy that should be encoded in the RB andthe input and output space partitions. Keeping the fuzzyknowledge conceptually separate from the operators canconsiderably improve the transparency and predictabil-ity of the fuzzy system.

5 COMPARISON OF DEFUZZIFICATIONRESULTS

To illustrate the in� uence of different defuzzi� cationprocedures on the overall system response, a generalsingle-input–single-output Mamdani-type fuzzy systemis used for modelling a linear crisp relationship. The taskhas been chosen for the straightforwardness of the KBdesign and the ease of detecting any divergence from thedesired behaviour. As multidimensional fuzzy spaces aregenerated by performing the Cartesian product of theirone-dimensional components [5], the results achieved inthe example can be readily extended to more complexmulti-input fuzzy rules.

The system adopts Zadeh’s sup-min composition ruleof inference and interprets the logical connectives elseand and respectively using the max and min operators.By varying the defuzzi� cation method and keeping the




rest of the system unaltered, it is possible to compare theeffect of different defuzzi� cation procedures.

The approximate fuzzy mapping is de� ned bypartitioning the input and output spaces each into � vefuzzy terms, respectively X0–X4 and Y 0–Y 4, delimited bytriangular MFs equally spaced between consecutivepeaks. Differently from many FL implementations, inthis experiment the two terms at the extremes of theoutput universe of discourse keep a symmetricaltriangular shape and their peaks do not correspond tothe space boundaries. This con� guration avoids certaindeformations of the input–output characteristic that willbe discussed later. The partition of the input and outputspaces is normalized [3]; i.e. at any point of the universeof discourse no more than two fuzzy sets overlap and thesum of their membership degrees is always equal to 1.The fuzzy mapping is realized by de� ning the � ve ruleslisted in Table 1.

The overall fuzzy associative memory (FAM) [24] andthe desired linear crisp behaviour are illustrated in F ig.3. It is possible to see that the de� nition of the fuzzy maprepresents a direct fuzzi� cation of the crisp linearrelationship. Ideally, the fuzzy system should thereforebe able to reproduce the desired response curve.

Figure 4 displays the crisp input–output relationshipsproduced by two fuzzy systems using respectively theMOM and COG rules. The input and output fuzzypartitions are shown next to the input–output coordi-nate axes and the desired linear characteristic is plottedfor reference. The � gure illustrates the stepwise outputresponse obtained using the MOM method and alreadydocumented in reference [21] together with the smootherresponse given by the COG rule.

The application of the COG method allows a betterreproduction of the desired output, but introduces anoscillatory behaviour around the reference curve. Thereason for this inaccuracy is that the contribution ofeach term to the � nal crisp value is determined by theportion of its area aggregated into the overall fuzzyoutput (see F ig. 2). Fuzzy output terms having a highactivation level will contribute with larger sections oftheir total area, therefore driving the � nal crisp outputcloser to their centre. Unfortunately, the section ofactivated area does not increase linearly with the � ringstrength of an output term, but in convex MFs it growsmore quickly for low � ring strengths. In the case oftriangular MFs, as used in the example of F ig. 4, theactivated area contributing to the total fuzzy output is,for each term Y i, equivalent to

AY i 2 ¡ hY i aY i hY i ¡aY i h2Y i

2aY i hY i 6

where A Y i is the activated area, hY i is the activation levelof Y i and aY i is half the support of the MF. According toequation (6), the weighting factor (i.e. the portion ofarea) for the contribution of each term to the defuzzi� edoutput grows parabolically with the activation degree.

Table 1 Rule base

Rules In Out

1 X 0 Y 0

2 X 1 Y 13 X 2 Y 24 X 3 Y 3

5 X 4 Y 4

Fig. 3 Fuzzy map and desired behaviour




The contribution of each rule consequent will there-fore rise quickly and fade slowly as the input valuecrosses the ‘attraction basin’ of the rule. This determinesthe distortion of the response of the system towards theaverage values between the centroids of the activatedterms. In the example of F ig. 4, the response is biasedtowards the intermediate values between consecutivecentroids, alternately overshooting or undershooting thereference output.

A more severe drawback follows from scaling thecontribution of each action according to the area of itspossibility distribution. Considering, for example, twonon-overlapping actions triggered by a certain set ofconditions, the defuzzi� ed value will be the centroid ofthe system composed of their two possibility distribu-tions. F rom a well-known property of the COG , this isequivalent to the centroid of a system composed of twoelements, each placed on the centroid of one of the twodistributions and having a possibility equal to the areaof that distribution:

COGCOG 1A 1 COG 2A 2

A 1 A 27

where COG, COG1, A1, COG2 and A2 are respectivelythe overall centroid position, the centroid position andarea of the � rst distribution, and the centroid positionand area of the second distribution. From the aboveformula, it is clear that the defuzzi� ed value will becloser to the action whose possibility distribution has the

largest area. The rationale for this is the attempt to bringthe output closer to the action of the rule of maximalactivation; this will happen as long as the output spacehas been partitioned by a set of MFs of equal areas. Inthis case, the action whose possibility distribution hasthe largest area will correspond to the action belongingto the rule of maximal activation.

However, if different output linguistic terms havepossibility distributions of different areas, the COGmethod will no longer guarantee a defuzzi� ed valueclose to the action of maximal activation. Actions whoseMF encompasses a large area will dominate the responsecurve in a way that is proportional to the size of theirarea. A consequence is that output terms having MFsde� ned with a high vagueness (fuzziness) will contributewith a large area and consequently dominate the outputof the fuzzy system. Considering, for example, an actionY de� ned by a triangular MF of base 2aY (i.e. thesupport) and � ring strength hY, its contribution AY tothe � nal crisp output is determined by equation (6). AY

increases with hY in a parabolic way, but it also growsproportionally to aY . In a triangular MF, aY gives ameasure of the fuzziness of the linguistic term—thefuzzier the de� nition, the larger aY . For instance, if twoactions have the same � ring strength and A1 is fourtimes larger than A2, the defuzzi� ed output will be fourtimes closer to COG1 than to COG2. If A1 is four timeslarger than A2 and the � ring strength h2 of the secondrule is equal to 1, then according to equation (6) a � ringstrength h1 equal to 0.13 would be enough for the � rst

Fig. 4 MOM method and COG rule




rule to bring the defuzzi� ed output to the mid-pointbetween COG1 and COG 2.

In standard Mamdani-type FL systems where themax operator is customarily used for action aggrega-tion, the application of the COG method has thereforethe undesirable effect of weighting the linguisticoutputs proportionally to their imprecision. Moreover,the smoothness of the output transition between twoneighbouring rules is affected by the MF areamismatch between their rule actions. Rule actionshaving possibility distributions of large area willprevail until their activation level becomes very low,at which point the output of the system brisklychanges to the value indicated by the action of theneighbouring rule.

Figure 5 illustrates this for an input–output relation-ship similar to the one plotted in F ig. 4. The numbers ofinput and output MFs have been decreased to three, thereference output has been kept as a straight line and theRB has been reduced to the � rst three rules of Table 1.Compared to the MF of the other two actions, the MFof Y 1 in the second rule encompasses an area four timeslarger. The output of the MOM method has been keptto show the boundaries between the areas wheredifferent rules prevail. The plot reveals the dominanceof the second output term well beyond the points wherethe ‘steps’ in the MOM response mark the end of thearea of prevalence of the second rule.

Saade [15] has pointed out the fact that the totality ofthe crisp output range is not used as a weakness of theCOG method. The closest the output can approach theboundaries of its interval of de� nition are in fact thecentroids of the two MFs at the extremes of the fuzzypartition. Unless fuzzy singletons are employed, thesevalues will never coincide with the output rangeextremes. Therefore, there are always two bands ofvalues at the bottom and at the top of the output rangethat are not ‘reachable’ by the fuzzy system (see alsoreference [7]). F igure 6 shows this situation for a fuzzysystem partitioning the input and the output spaces intothree linguistic terms each and using the � rst three rulesof Table 1 to model the usual linear relationship. Tomaximize the response interval, the peak values of thetwo MFs at the extremes of the action range have beenset to the output interval extremes. The area under thesecond output MF is twice the area of the other twoterms in order to maintain a normalized partition of theoutput universe of discourse.

In an approximate mapping, the feature of rarelyrecommending actions close to the extremes of theoutput range can generally be considered desirable, as itavoids extreme responses in the presence of uncertainty.However, problems arise when attempting to compen-sate a possible slower transient response by adjustmentsof the fuzzy mapping. To enlarge the action range, it isnecessary to bring the centroid of the extremal MFs

Fig. 5 Output curve distortion: MF of Y 1 is 4 times wider than those of Y 0 and Y 2




closer to the limits of the output range. This can beaccomplished by reducing the width and hence the areaof the extreme MFs. Unfortunately, as shown above, theresponse characteristic is affected by the relative areasencompassed by the output MFs. Reducing the areas ofthe two extreme terms will therefore cause a distortionof the response curve.

Figure 6 shows the kind of deformation introducedwhen the support of the two extreme MFs is half thewidth of the other term. To obtain a smoother outputcharacteristic, a � ner space partition has now to beintroduced to reduce the mismatch between the area ofthe extreme output terms and the area of the third term.The designer will therefore have to enlarge the size of theRB, increasing the complexity of the system andaffecting therefore its transparency and speed ofexecution.

The use of fuzzy singletons to enlarge the outputrange would limit the � exibility of the system design. Inparticular, it would be impossible to modulate theresponse of a single rule according to its activationdegree by conveniently shaping the MFs of the ruleactions. Moreover, there is a sizeable amount ofliterature featuring non-singleton fuzzy MFs where theintroduction of such a constraint would imply a majorreview of the interpretation of the fuzzy algorithm.

The BADD, SLIDE and M-PTD defuzzi� cationmethods will approach the behaviour of either theMOM or COG method according to the tuning of thetransformation parameters. Saade’s method is essen-

tially an area-based procedure and as such it will behavesimilarly to the COG method. None of the procedureslisted in Section 4.3 will therefore overcome the short-comings affecting the MOM and COG methods.

6 AN IMPROVED INTERPRETATION OF FUZZYRULE ACTIONS

The above analysis showed the inadequacy of maximaprocedures and the drawbacks of area-based methodswhen combined with the implementation of actionaggregation through the max operator. For the processof output defuzzi� cation, area-based methods seem tobe the most appropriate choice due to their properties ofconsistency, section invariance and monotonicity andthe straightforwardness of the algorithm. Nonetheless, itis necessary to � nd an alternative aggregation procedureto avoid the weighting of the contribution of eachoutput according to the area of its possibility distribu-tion.

It is clear that changing the implementation of theunion operation does not serve the purpose, as it wouldonly modify the way of forming the pointwise combina-tions of the possibility values of the actions in the overalloutput distribution. It is therefore necessary to changethe process of MF superposition related to the oper-ation of set union. To this end, Mizumoto [16]and Cherkassky and Mulier [17] proposed that the

Fig. 6 COG defuzzi� cation range




aggregation operator be implemented through anadditive procedure.

The idea behind these methods is to compute the � nalcrisp value from the properties of each separate actionrather than from an aggregated fuzzy output [16]. Eachaction term is � rst defuzzi� ed into a representative pointand then combined with the values of the other actionsto work out the � nal crisp output. The COG defuzzi-� cation method is still used for the choice of each actionrepresentative and for the combination of the actioncentroids in the overall crisp output. Each actionrepresentative is weighted by a factor related to the� ring strength of the rule. The overall process cantherefore be divided into two cascaded operations ofcentroid defuzzi� cation. The � rst defuzzi� cation isapplied to obtain the discrete fuzzy set composed ofthe action centroids and their activation degrees, whilethe second defuzzi� cation operation yields the � naloutput out of that set. The defuzzi� ed output COG canbe written as

COG i COG i f hi

i f hi8

where COG i and f hi are respectively the centroidposition and a general function of the activation degreeof the ith term. If the function f is the activation-dependent area of the output possibility distribution, thedefuzzi� cation becomes Mizumoto’s area method. If fcorresponds to the identity function, the operation is

referred to as the height method [16] or additivedefuzzi� cation [17].

Because the scaling factor of each rule action is stillthe area of its possibility distribution, the area methodcannot be expected to overcome the drawbacks of otherarea-based methods. The height method is instead agood algorithm for the process of action aggregationand defuzzi� cation.

In the computation of the crisp output, the heightmethod does not take into account the areas of the MFsof the actions. Only the rule activation degree appears inthe calculation of the contribution of each action.Therefore, terms of wide possibility distribution willnot dominate the output of the controller and will notaffect the smoothness of the response curve. At the sametime, at the level of each individual rule, the centroid-based procedure allows the tuning of the location of therepresentative point of the action according to thedegree of activation of the rule. De� ning the outputterms through asymmetric possibility distributionsallows the shifting of the COG of the rule outputaccording to the activation degree.

Figure 7 shows the response curves of two fuzzysystems using respectively the COG and the heightmethods. The same single-input–single-output exampleof F ig. 4 was employed to enable the results to beplotted. However, the procedure adopted is general andvalid for any n-dimensionally complex input space. It ispossible to see that the output characteristic of the FLsystem using the height method coincides with the

Fig. 7 COG rule and height method




desired straight line. In particular, its response curve isnot affected by the oscillatory behaviour caused by thecombination of the COG method with the maxaggregation operator.

Moreover, the system response is now not distortedtowards the centroids of MFs with larger areas. F igure 8shows the results of applying the height method and thestandard union aggregation and COG defuzzi� cationprocedure to the same example as F ig. 6. Even thoughthe MF of the second rule action encompasses an areatwice as large as the other two output terms, theresponse curve obtained using the height methodremains undistorted and follows a straight line.

The defuzzi� cation range of the height method doesnot improve upon the one obtainable by combiningunion aggregation and COG defuzzi� cation. Thesystem’s output still does not cover the full output scaleand the extreme output values are not reached. Becauseof this limited output range, the slope of the straight lineplotted in F ig. 8 is smaller than the desired one.

Nevertheless, it is now possible to widen the outputrange, while restricting the width of the extremal MFswithout further distorting the output characteristic.F igure 9 shows the result of reducing the support ofthe extreme MFs by a factor of 10. The responseobtained using the height method is compared with theoutput of a standard fuzzy system using max aggrega-tion and COG defuzzi� cation. It is possible to see howthe former is now almost identical to the desiredbehaviour, while the latter has an almost � at character-

istic with two sharp steps at the extremes of the samplingspace.

The height method therefore gives substantial advan-tages in action aggregation and defuzzi� cation in FLsystems. It is interesting to note that none of the authorswho investigated this method have noted such advan-tages. In reference [16] various aggregation and defuzzi-� cation methods were compared on the basis of theresults obtained for the fuzzy control of a simulatedplant. The study was limited to the performance of thefuzzy controller, without giving any analysis of thereasons underlying the results. Cherkassky and Mulier[17] proposed the height method for its ease ofimplementation and system analysis. This is particularlytrue when symmetrical output MFs are used. In thiscase, the representative point of each fuzzy action isconstant and equal to the middle point of the support ofthe term. Each action term can therefore be describedthrough a single parameter corresponding to the centreof its MF , allowing a considerable simpli� cation of thearchitecture of the system. Under this form, the fuzzyinferencing becomes equivalent to a special case of theTakagi–Sugeno model, where the linear combinations ofthe input variables are replaced by constant values in therule consequents. Because of its ease of implementation,this particular representation has often been adopted inFNN applications.

The main limitation of the height method incomparison with the union aggregation and COGdefuzzi� cation procedure is in the reduced use of the

Fig. 8 COG rule and height method: defuzzi� cation range




information related to the MF shape. This informationis in fact utilized now only to determine the representa-tive point for each individual term. In particular, sincethe contribution of each action is no longer dependenton the area of its possibility distribution, differencesbetween the MF shape of different actions cannot nowbe used to tune the fuzzy output.

7 CONCLUSIONS AND FURTHER WORK

This paper has focused on the operations of aggregationand defuzzi� cation of fuzzy rule actions in Mamdani-type fuzzy systems. The superior transparency andmapping accuracy of the height method with respectto the usual combination of the union of fuzzy sets withmaxima or area-based methods have been demon-strated. Because of the modular way in which the crispoutput is formed, the height method is also well suited toan FNN implementation.

The above conclusions were drawn independently ofthe implementation of the fuzzy inferencing operatorsand the compositional rule of inference (e.g. sup-max ,sup-prod, etc). The results presented are therefore validfor a wide range of problems. The only constraint inapplying the proposed procedure is that the sentenceconnective else should be generally interpreted throughthe union operation.

Further work should be directed at studying theeffects of the shape of the output MFs on the overallcrisp output and its interaction with the actionaggregation and defuzzi� cation procedures. Additionalstudies should also be conducted on the issue of closed-loop stability when applying the proposed FL methodwith asymmetrical output MFs.

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