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Reductionoffuzzyrulebaseviasingularvaluedecomposition

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120 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 7, NO. 2, APRIL 1999

Reduction of Fuzzy Rule Base ViaSingular Value Decomposition

Yeung Yam,Member, IEEE,Peter Baranyi, and Chi-Tin Yang

Abstract—This paper introduces a singular value-based methodfor reducing a given fuzzy rule set. The method conducts singularvalue decomposition of the rule consequents and generates certainlinear combinations of the original membership functions to formnew ones for the reduced set. The present work characterizesmembership functions by the conditions of sum normalization(SN), nonnegativeness (NN), and normality (NO). Algorithmsto preserve the SN and NN conditions in the new membershipfunctions are presented. Preservation of the NO condition relatesto a high-dimensional convex hull problem and is not alwaysfeasible in which case a closed-to-NO solution may be sought. Theproposed method is applicable regardless of the adopted inferenceparadigms. With product-sum-gravity inference and singletonsupport fuzzy rule base, output errors between the full and re-duced fuzzy set are bounded by the sum of the discarded singularvalues. The present work discusses three specific applications offuzzy reduction: fuzzy rule base with singleton support, fuzzyrule base with nonsingleton support (which includes the caseof missing rules), and the Takagi–Sugeno–Kang (TSK) model.Numerical examples are presented to illustrate the reductionprocess.

Index Terms—Fuzzy systems, reduced order systems, singularvalue decomposition.

I. INTRODUCTION

T HE advantage of fuzzy control lies in its ability to mimicand implement the actions of expert operator(s) without

the need of accurate mathematical models. The drawback,however, is that there is no standardized framework regardingthe design, optimality, reducibility, and partitioning of a fuzzyrule set. A fuzzy rule base, be it generated from expertoperators or by some learning or identification schemes, maycontain redundant, weakly contributing, or outright inconsis-tent components. Moreover, in pursuit of good approximation,one may be tempted to overly assign the number of antecedentsets, thereby resulting in large fuzzy rule bases and problemsin computation time and storage space. A formal approach toextracting the more pertinent elements of a given rule set is,hence, highly desirable. The present work is an attempt in thisdirection.

Manuscript received June 27, 1997; revised August 26, 1998. This work wassupported by the Hong Kong Research Grant Council (RGC) CUHK4138/97E,the Hungarian Ministry of Culture and Education (MKM) FKFP 0422/1997,and the Hungarian National Science Research Foundation (OTKA) T019671.

Y. Yam and C. T. Yang are with the Department of Mechanical andAutomation Engineering, The Chinese University of Hong Kong, Shatin, N.T.,Hong Kong.

P. Baranyi is with the Computer and Automation Institute, HungarianAcademy of Science and the Department of Automation, Technical Universityof Budapest, Budafoki u. 8, H-1111 Hungary.

Publisher Item Identifier S 1063-6706(99)02800-3.

In this paper, a method to reducing a fuzzy rule set bycapturing a large extent of its input/output characteristics isintroduced. The method is based on conducting singular valuedecomposition (SVD) of the rule consequents and generatingproper linear combinations of the original membership func-tions to form new ones for the reduced set. Instead of assumingcertain specific shapes for the membership functions, here wecharacterize the membership functions by the conditions ofsum normalization (SN), nonnegativeness (NN), and normality(NO). Algorithms to preserving the SN and NN conditions inthe new membership functions are presented. Preservation ofthe NO condition, however, involves the solution to a convexhull problem and is not always achievable.

The proposed approach can be applied regardless of theinference paradigm adopted for the fuzzy rule base. Forproduct-sum-gravity inference and singleton support rule base,output error bound between the original set and the reducedset is readily expressible as the sum of the discarded singularvalues. The method is applicable to fuzzy rule set with anynumber of input variables, but is illustrated here mostly withthe example of a two input fuzzy system. Three specific casesof fuzzy reduction are discussed. They are fuzzy rule basewith singleton support case, fuzzy rule base with nonsingletonsupport case (which includes the case of missing rules), andthe Takagi–Sugeno–Kang (TSK) model [1]. These three casesrepresent situations where the output of a fuzzy rule is definedby one, two, or more parameters.

The present work constitutes a detailed investigation of thepreliminary approaches outlined in the works of [2]–[4], andgives rigorous solutions to the problems posed thereof. Thealgorithms to be utilized here are mostly developed in [5]for fuzzy approximation of general functions. Application ofSVD to fuzzy reduction has been investigated recently byWang et al. [6]. Their approach utilizes SVD to reduce thedimensionality of the input space and then performing systemreduction based on optimizing a certain objective functions.Their work uses B-splines as the membership functions. Thepresent work, on the other hand, applies SVD directly to therule consequents of the given rule set, and works with themembership functions constrained only by the conditions ofSN, NN, and NO.

This paper is organized as follows. Section II first definesthe various concepts to be used in the proposed method.Section III presents the basic operations of SVD reductionfor the example of a two input fuzzy rule set. Section IVextends the SVD operations to fuzzy rule set of three inputvariables. Extension to fuzzy set with a general number of

1063–6706/99$10.00 1999 IEEE

YAM et al.: REDUCTION OF FUZZY RULE BASE VIA SINGULAR VALUE DECOMPOSITION 121

inputs then follows readily. Section V applies fuzzy reductionto the cases of singleton support, nonsingleton support, and theTSK model, with a numerical example for each. And finally,Section VI presents the conclusions.

II. DEFINITION OF BASIC CONCEPTS

This section introduces the basic concepts to be used in theensuing SVD operations. These concepts are first utilized in[5] for fuzzy approximation of general functions. Let ,

be functions of variable defined on somecompact domain . The following properties canbe defined for the function set .

Sum Normalization (SN):A set of functionis SN if for any value of within the domain

of interest

(1)

Without ambiguity, a matrix is SN if

...(2)

where denotes the column vector obtained by sum-ming over the rows of matrix .

Nonnegativeness (NN):A set of functionis NN if for any value of within the domain

of interest

(3)

for each . Likewise, a matrix is NN if everyone of its elements is greater than or equal to zero.

The concepts of SN and NN are consistent with the usualdefinition of membership function. Note that

being SN and NN implies thatfor all within the domain of interest. Similarly, matrixbeing SN and NN implies for all of its element.Another property of the membership functions is normality.

Normality (NO): A set of functionis NO if it is SN and NN and that each of the functions

attains value one at some point within the domain of. Correspondingly, a matrix is NO if it is SN and NN and

that each of its column contains the value one as an element.For a set of membership functions, the NO condition

implies a certain localization property, i.e., when one of thefunctions achieves membership degree of approximate unity,the remaining ones would yield close to zero membershipdegrees. Hence, the membership functions take turns domi-nating different regions within the domain of interest. As aresult, linguistic labelings are readily assigned to membershipfunctions satisfying the NO condition.

The above definitions enable the following result to bestated.

Theorem 1: Given a set of functionand a matrix of dimension by , one forms newfunctions as

(4)

Then

1) the set of functions , is SN if, and are SN;

2) the set of functions , is NN if, and are NN;

3) the set of functions , is NO if, and are NO.

Proof: For any within the domain of interest, one has

which proves 1). The proof of 2) follows straightforwardlyas each constitutes the sums and products of positivequantities and . To prove 3) since and areNO and, hence, SN and NN; then by 1) and 2), is alsoSN and NN. This implies within the domain of

. To show that attains the value of one at some, onenotes the following. Since is NO, there exists a valuewithin the domain of such that . Similarly, since

is NO, there exists integer, , which gives thelocation of the element in theth column of attaining thevalue of one, i.e., . As a result

and, at ,

(5)

Now using the fact that , is SN andNN, one has at

Since , this implies for , ,, , , . Hence, (5) yields at .

This proves 3).

122 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 7, NO. 2, APRIL 1999

III. SINGULAR VALUE-BASED REDUCTION

This section outlines the procedures to conduct singularvalue-based reduction in terms of SN and NN, and possiblyNO, matrices. Details of the method can be found in [5]. Theprocedures are presented here with a two-dimensional (2-D)matrix and then extended to higher dimension in the nextsection.

Consider a 2-D matrix in SVD form

(6)

where is by and and are by and by ,respectively. Matrices and are orthogonal, i.e.,

and . Here, denotes the byidentity matrix. For later reference, we also introduce asthe by matrix of zeros, and as the by matrix ofones. The by matrix contains the singular values of

in decreasing magnitude. The maximum number of nonzerosingular values is . The singular valuesindicate the importance of the corresponding columns ofand in the formation of . A close approximation tocan be obtained by keeping those components having largesingular values.

Let be the number of singular values to keep and,hence, an approximation of

(7)

where and contain the columns of andcorresponding to the retained singular values in .

Approximation is exact if all singular values are kept,. Matrices and are in general neither SN nor

NN. However, they can be converted into SN and NN matricesby the following procedures. Take , as an example, andconsider the SN condition first; we have Theorem 2.

Theorem 2: Let be an by orthogonal matrixpartitioned as , where and have,respectively, and columns. Let beany by matrix satisfying the constraint

. Then, if , the bymatrix satisfies SN, where

(8)

and if , the by matrixsatisfies SN, where

(9)

Theorem 2 characterizes the extra column one needs, if atall, to supplement the matrix so as to achieve the SNcondition. The proof of the theorem can be found in [5] andis omitted here. Note that the theorem does not requireto be invertible. In theory, may contain zeroelements, which implies that some columns of are notnecessary to satisfy the SN condition. If so, a noninvertible

can actually be used for (8) or (9). For the present work,however, we prefer to retain all columns of andso as to facilitate a close approximation of in (7). Hence,

we desire to be invertible matrix. We adopt the followingscheme for constructing : if does not containzero elements, the matrix is chosen as

(10)

and if contains zero element(s), is chosen as

(11)

where one ensures that theth entry of isnonzero. After is chosen, one can proceed to compute

For

For (12)

Similar expressions hold for , , and as results ofapplying the above procedures to matrix. The approximation(7) can then be written as

(13)for the case when and

. Other cases correspondingto and/or

follow in a straightforward manner. The matrices

and are SN. They have either orcolumns depending on whether or .

It is possible, still, that and contain negativeelements and, hence, not satisfying the NN condition. Thefollowing gives a set of procedures to generate from a matrixsatisfying the SN condition another one of the same dimensionsatisfying both SN and NN conditions. Let be a matrix ofdimension by satisfying the SN condition. Thennote the following.

1) Look for the minimum element , of . Set

parameter

if

otherwise. (14)

2) Form a by matrix

......

. . ....

(15)

YAM et al.: REDUCTION OF FUZZY RULE BASE VIA SINGULAR VALUE DECOMPOSITION 123

3) The matrix product satisfies the SN and NNconditions.

The subscript in is adopted to stress the dependence ofon . To see that is SN, one notes that

and, hence,as is SN to begin with. To see that is also

NN, one notes that theth column of is given by thescalar multiplying a column vector that is thesum of and times the th column of , andthe parameter ensures proper scaling to result in positivevalues for all entries.

Applying these results to (13), one has

(16)

with

(17)

(18)

(19)

The matrices and are by and by ,respectively. They satisfy the SN and NN conditions. The sameprocedures can be applied to other cases of

and/or . Theresulting and will have or columns,depending on different cases.

The NO condition is now considered. Following the ap-proach for SN and NN, we desire to have invertible matrices

and of appropriate dimensions such that (16) can beexpressed as

(20)

and that the matrix products and are SN, NN, andNO. However, while it is always possible to tailor the matricesinvolved to satisfying the SN and NN conditions, the same isnot true for the NO condition. Successful incorporation of theNO condition depends very much on the specific matrix athand. The following gives a set of procedures which conductsa tight bounding operation on the matrix and yields, if possible,a NO matrix or otherwise, a close-to-NO matrix. Here, a close-to-NO matrix is taken to be one that is SN and NN but not allof its columns containing the value one as element.

Take as an example and let its column dimension be,one first notes that each of the rows of corresponds toa point lying on a -dimensional space. Furthermore, since

is SN, the points from the rows of actually lieon a hyperplane of dimension. The procedures theninclude the following steps.

1) Project the points in the -dimensional space ontothe -dimensional hyperplane satisfying the SNcondition.

2) Obtain the convex hull of the points on the -dimensional hyperplane. Algorithms to treat convex hullproblem in a general dimensional space are discussedin, e.g., [8] and [9].

3) Check the convex hull. If the convex hull has exactlyvertices, successful incorporation of the NO con-

dition by is possible. In this case the matrixcan be obtained as inverse of the matrix containing the

rows of associated with the convex hull. This iscalled the tightest bounding case.

4) If the convex hull has more than vertices, however,determination of to strictly satisfying the NO con-dition is generally not possible. In this case, we haveto search for a relaxed bounding with vertices notall of which coming from the points of . Thecorresponding is then determined according to these

vertices. This leads in general to a close-to-NOmatrix product .

A more detail discussions of these steps can be found in [5].Carrying out the NO procedures for and , the approx-

imation (20) now becomes

(21)

where

(22)

and and are SN, NN, and NO, or close-to-NO.Several points are worth noticing here. First, The NO

condition is always possible when equals to two or .For , the hyperplane is a straight-line and one canalways find (in this case) a convex hull of two points. For

, the full set of the points constitute the vertices ofthe convex hull. As such, and . Thiscorresponds to the trivial case where there is actually no SVDreduction. In general, however, NO condition is not possibleand one has to adopt a relaxed bounding and settle with close-to-NO condition. Moreover, the choice of a relaxed boundingis not unique and exact choosing will depend on the matrix athand. One may even consider a bounding convex hull whichslightly violates the NN condition at a few points in exchangefor readily bounding on a majority of the points. Note thatthe scheme of testing various combinations of taking outof rows of works only if a tightest bound of the pointsdo exist. The scheme would not be applicable for the relaxedbounding case.

Second, the reduction, , and of (22) is not unique. Infact, given that and are SN, NN, and NO, the choice of

is another valid reduction if the invertible matrices andof appropriate dimensions are SN, NN, and NO.

Third, one observes that the choice of constructingandto satisfy the SN and NN conditions is not unique. An

issue of interest is whether there exists and in what way a moreefficient method to choosing them. In this regards, choices withbetter results do exist. The point to note, however, is that no

124 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 7, NO. 2, APRIL 1999

Fig. 1. SVD reduction for 3-D matrix.

matter the initial choices of and , they could actuallyyield the same final result after the bounding procedures,as topology would dictate similar relative distribution of thepoints and bounding convex hull. Interested readers may referto [5] for more detail discussion of the above issues.

As a final topic in this section, we derive an expressionfor the error due to the SVD reduction. One notes here thatincorporating the SN and NN conditions and carrying out theNO procedures do not actually change the approximation of

by . This is because

(23)

Hence, the approximation error can be expressed as

(24)

Absolute value is not required for in (24) as singularvalues are always positive. Since the columns of and

are having Euclidean norm of unity, absolute value oftheir elements must be bounded by one. Thus

(25)where denotes theth singular value in . Hence, the errorintroduced into each elements of using the approximation(21) is less than or equal to sum of discarded singular values.

IV. GENERALIZATION TO HYPERDIMENSIONAL MATRIX

The previous section gives the SVD-based procedures forconstructing an approximation when is a 2-Dmatrix, with and being SN, NN, and NO, or close-to-NO.The approximation can be expressed in terms of the elementsof the various matrices

(26)

where and are the number of columns of the matricesand . By previous results, and equal or ,depending on the specific case at hand.

The procedures developed above can be readily extended tomatrix of higher dimension. In [5], extension of the proceduresto a three-dimensional (3-D) matrix is given. The extendedprocedures convert a 3-D matrix into a sequence of 2-D oneswhere SVD can be applied. To provide a basic idea of theprocess involved, a pictorial depiction of the procedures isgiven in Fig. 1. Upon given a 3-D by by matrix[step (i)], the extended procedures call for spreading the matrixin the -index direction to form a 2-D by matrix[step (ii)], and then apply the SVD reduction of Section III to

[step (iii)]. This yields

(27)

where is now SN, NN, and NO, or close-to-NO. Thedimension of is by where or ,as the case may be. The by matrix denotes the

YAM et al.: REDUCTION OF FUZZY RULE BASE VIA SINGULAR VALUE DECOMPOSITION 125

resultant matrix to the right of . One can now restackto become a 3-D by by matrix [step (iv)], which canthen be spreaded, this time in the-index direction, to forma 2-D by matrix [step (v)]. Again, applying theSVD reduction procedures to [step (vi)], one obtains

(28)

where is SN, NN, and NO, or close-to-NO. There areor columns in . The matrix

is by . Restacking into a by by matrix[step (vii)] and spreading it in the-index direction to form

[step (viii)], one has, once more, after SVD reduction on[step (ix)]

(29)

The matrix is SN, NN, and NO, or close-to-NO, and theremay be or columns in . Thematrix is by . It can be restacked into a 3-Dby by matrix [step (x)]. This completes the processof generating all necessary quantities for the reduction of thegiven matrix . The resulting approximation to the elementsof , , can be expressed as

(30)

Equation (30) is a 3-D generalization of (26). One can alsowork out an approximation error bound for ,which is

(31)

Here, , , and are sum of the discarded singularvalues in the SVD reduction of matrices , , and .They represent the errors induced in the process of obtaining

, , and . Interested readers are referred to [5] for themathematical details of these results.

The above procedures structures a 3-D matrix so thatSVD reduction can be performed in stages. At each stage,approximation and conditioning of the matrix is conductedfor one of the dimensions and a certain singular value erroris generated. With additional stages and proper indexing, theprocedures can be extended to matrix of a general number of

dimensions.

V. APPLICATION TO FUZZY RULE-BASE REDUCTION

The methodology developed above is now applied to theactual reduction of fuzzy rule bases. For convenience, fuzzyrule bases of two input variables are utilized for illustrationof the various cases. As mentioned before, application can bereadily extended to fuzzy rule base with a general number ofinputs.

A. Fuzzy Rule Base with Singleton Support

Consider a fuzzy rule base with two inputsand and asingle output ,

If and (32)

where and , , , aremembership functions of the variableand , respectively,and is the rule consequent of the th fuzzy rule. Toperform rule-base reduction, one forms a by matrixwith as the elements

......

......

(33)

and then applies the SVD procedures to obtain

(34)

where , , and are, respectively, by , by , andby matrices. A reduced rule base can then be obtained as

If and

with and . Here, the newmembership functions , are given by

(35)

(36)

and the new rule consequent is given by the elementof ,

(37)

The reduced rule base has fuzzy rules as comparedto of the original one. Since is definitely SN andNN, by Theorem 1 the membership function set ,

is SN and NN if the original set ,is SN and NN. If is NO as well, the new

membership functions are also NO if the original membershipfunctions are NO. If is only close-to-NO, however, ,

will only be SN and NN, but not NO ingeneral. The new membership functions and rule consequentscan be interpreted as thede facto fuzzy rule componentsembedded in the operation of the original rule base. Theseresults apply to the nonsingleton support and the TSK modelcases to follow as well.

The above scheme can be applied to any fuzzy rule baseregardless of the adopted inference paradigm. Compact ex-pression for the output error bound, however, can be obtainedif product-sum-gravity (PSG) inference is adopted. Given that

, , the output error is given by

(38)

where the fact that membership functions , , ,satisfying the SN condition has been used. The first

and second term on the right-hand side of (38) correspond,

126 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 7, NO. 2, APRIL 1999

TABLE IFUZZY RULE CONSEQUENT ri; j OF THE ORIGINAL RULE BASE

respectively, to the PSG-inferred outputs of the original andthe reduced rule base. Substituting (35)–(37) into (38)

(39)

and, with (25)

(40)

Hence, the sum of the discarded singular values constitutes anupper bound to the output error incurred by the reduction. Ifone is to include all nonzero singular values in the reduction,the reduced rule base would yield the same output as theoriginal one with PSG inference. Error bounds for otherinference paradigms can also be derived but the final formis not as pleasing.

Example 1: Consider one of the examples from [10] witha fuzzy rule base of input variablesand . The membershipfunctions of are overlapping isosceles triangles of basewidth0.08 centering from to at separationsof 0.04. The membership function of are similar but withbasewidth of 0.6 and centering from to

at separations of 0.3. The original example in [10]utilizes min–max inference paradigm and assigns triangularmembership functions to the fuzzy outputs. In this example,however, we adopt the PSG inference to illustrate the reductionprocess. As a result, the fuzzy rules can be expressed as

If and

where ’s are the center points of the output membershipfunctions in the original example of [10] and are tabulatedin Table I. They form the matrix on which the proposedreduction method is performed.

The singular values of are determined as 22.6767,22.6767, 0.9835, 0.9835, 0.5523, 0.5523, 0.3433, 0.3433, 0.Keeping only the two largest singular values and going throughthe SN and NN processes yield three membership functionsfor and for . The convex hull problem then amounts tofinding a bounding triangle on a 2-D plane and can be readilycarried out. Fig. 2 shows the membership function for

. The membership functions ofare similar except

Fig. 2. Membership functions ofa for the reduced rule base.

TABLE IIFUZZY RULE CONSEQUENT

i; jOF THE REDUCED RULE BASE

for the domain of interest. In this case, only close-to-NOcondition can be achieved, leading to only two out of the threemembership functions for and attaining value of one withintheir ranges. The rule consequents are given in Table II.

The reduced rule base contains nine rules, as compared tothe original set of 81, in the form of

If and

For comparison, Fig. 3 shows the PSG-inferred output usingthe original fuzzy rule base of 81 rules (top plot) and thereduced rule base of nine rules (bottom plot). To checkhow close the reduced rule set duplicates the output of theoriginal set, the rule bases are applied to controlling aninverted pendulum given in [10]. The dynamics equation of

YAM et al.: REDUCTION OF FUZZY RULE BASE VIA SINGULAR VALUE DECOMPOSITION 127

(a)

(b)

Fig. 3. PSG-inferred control surfaces due to (a) original and (b) reducedrule bases.

the pendulum is

(41)

where is the angular position, is the angular velocity, andis the control input. Fig. 4 shows the closed-loop responses

due to the original and reduced rule bases for the initialconditions (0.08, 0) and (0.12,0.8). Here, andare used as fuzzy variables in the inference of. Note thatthe responses have been scaled down by a factor of 0.1in the figure. It can be observed that closed-loop responsesproduced with the reduced rule base are quite close to that ofthe original one.

The present numerical example yields three membershipfunctions for the reduced set. As a result, the boundingprocedures is readily executed in a 2-D plane. In the casewhere the number of membership functions is larger thanthree, the bounding procedures require a solution to a hyper-dimensional convex hull problem, which can be quite tedious.

Fig. 4. Closed-loop responses due to original and reduced rule bases.

The SN and NN conditions, however, can always be achievedregardless the number of membership functions.

B. Fuzzy Rule Base with Nonsingleton Support

In the nonsingleton support case, each of the fuzzy rulesin (32) is associated with a support factor , which canbe interpreted as a firing strength or a reliability coefficient.Given that and , and assuming PSG inference,the inferred output is now generated as

(42)

Equation (42) indicates that reduction in this case involves anumerator part and a denominator part, in contrast to just thenumerator part in the singleton support case. To proceed withthe reduction, one forms a 3-D by by two matrixwith elements

(43)

and then applies the extended procedures in Section IV asfollows. Referring to Fig. 1, one first carries out steps (i)–(iii)to obtain a by matrix for variable , and then steps(iv)–(vi) to obtain a by matrix for variable . In theremaining steps (vii)–(ix), however, one keeps both singularvalues of without any reduction, i.e., one has .As such, (30) becomes in this case

(44)

for where matrices and are SN, NN, and NO,or close-to-NO, and the matrix is by by two. Onecan now write a reduced fuzzy rule base as

If and

128 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 7, NO. 2, APRIL 1999

TABLE IIIRULE CONSEQUENT i; j OF THE GIVEN FUZZY RULE BASE WITH MISSING RULES

where andare the new membership functions. The new rule conse-

quent and support factor are computed as

Again, the reduced set has fuzzy rules as compared tothe original of rules.

Two special cases are worth noticing here. First, whenthe supports are singleton, i.e., when , the resultshere reduce to that of the singleton support case. This canbe observed from the fact that (44) with can bedecomposed into the following two parts

(45)

(46)

where one recognizes that (45) is the reduction problem for thesingleton support case and, if one were to obtain the matrices

and and values from (45) only, then (46) wouldautomatically be satisfied with . This is becauseand are SN and NN so that the right-hand side of (46) wouldbecome one. In effect, (46) indicates that singleton support rulebase would be reduced to another singleton support rule baseand the reduction process would be dictated solely by (45),the same as for the singleton support case.

Second, the nonsingleton support case considered here in-cludes fuzzy rule base with missing rules as special case. Inthis case, the support factor is assigned value zero forthe missing rules and one for the included rules. This case isfurther explored by the use of following numerical example.

Example 2: Consider another example from [10]. Themembership functions of are now isosceles triangles ofbasewidth 0.16 with the centers located from to

at separations of 0.08. The membership functionsfor are also isosceles triangles but with basewidth of 0.12 andthe centers going from to at separationsof 0.6. The rule consequents are given in Table III, wherethe blank entries correspond to missing rules. The number offuzzy rules in this case is 17. The rule set can be expressed asa nonsingleton support system, with for the missingrules and for the rest.

Application of the SVD procedures yields the followingsingular values: 13.0860, 12.7475, 2.0633, 2.0, 0. Again,keeping only the two largest singular values, one obtains

Fig. 5. Membership functions ofa for the reduced rule base.

TABLE IVRULE CONSEQUENT r{; | AND SUPPORT

FACTOR s{; | FOR THE REDUCED RULE BASE

three membership functions forand for . Fig. 5 shows themembership function of , . Membershipfunctions of are again similar except for the different domainof interest. In this case, the bounding procedures are able toachieve the NO condition for the membership functions. Theresultant rule consequents and support factor for thereduced set are given in Table IV.

The reduced rule set in this case has nine rules and isexpressed as

If and

with support factor . Note that in this example negativesupport factors are obtained. To gain more insight, Fig. 6(a)shows the inferred outputs for the original and Fig. 6(b) showsthe reduced rule base adopting the inference of (42). Becauseof the missing rules, the output surface of the original rulebase is not defined in certain region on the domain.By comparison, the output surface for the reduced case isdefined on a larger region in the domain. Furthermore, becauseof the negative support, the reduced rule base actually yieldsinfinite outputs at certain values of . Fig. 6 has in factbeen constrained to show only the output surface with valuesbetween 6. One may thus claim that the SVD procedures hasextrapolated somewhat the original surface to a wider regionthan it is initially defined. There is, however, a limit to theextent of this extrapolation; the negative supports may causethe interpolated surface to go to infinity if one ventures too

YAM et al.: REDUCTION OF FUZZY RULE BASE VIA SINGULAR VALUE DECOMPOSITION 129

(a)

(b)

Fig. 6. PSG-inferred control surfaces due to (a) original and (b) reducedrule bases.

far out. The extrapolation property of the SVD proceduresand how it relates to existing interpolation techniques, say,e.g., [7], is a research topic of interest.

For further comparison, the two rule bases are again utilizedto controlling the inverted pendulum of (41). Fig. 7 plots theclosed-loop responses due to the original and reduced rulebases for the initial conditions of (0.08, 0) and (0.12,0.8).The first set of initial conditions and its subsequent responseslie inside the region where the original fuzzy rule base is welldefined. The second set of initial conditions, however, startsat a region where the original output surface is not defined.To carry out the simulation, we let the output be zero inthe undefined region. The idea here is to let the pendulumdynamics return itself to where the original output is defined.The response at around s shows a slight changein the decay rate which is indicative of this transition. Onthe other hand, both set of initial conditions and subsequenttrajectories remain in the extended well-defined region of the

Fig. 7. Closed-loop responses due to original and reduced rule bases.

reduced rule base. The closed-loop responses hence exhibitsmoother trajectories.

C. Takagi–Sugeno–Kang (TSK) Model

The TSK model utilizes functions of the input variables asfuzzy outputs. For a fuzzy system with input variablesand, the fuzzy rules are expressed as

If and

To illustrate application of the reduction procedures, we hereconfine to be consisting of only three terms, aconstant term, and linear terms inand in , i.e.,

(47)

In this case, one forms a by by three matrix withelements

(48)

and applies the extended SVD procedures of Fig. 1. Similar tothe nonsingleton case, here we apply the reduction proceduresto obtain the by and by matrices and ,but in step (ix) we keep all three singular values for, i.e.,

. As a result, (30) becomes

(49)

for . The matrices and are SN, NN, and NO,or close-to-NO. The reduced TSK model can be written as

If and

where andare the new membership functions. The output functions

for the reduced set are given by

(50)

130 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 7, NO. 2, APRIL 1999

(a)

(b)

Fig. 8. Membership functions of (a)a and (b)b for the original rule base.

TABLE VPARAMETERS �i; j , �i; j , AND i; j FOR FUNCTION gi; j(a; b)

where

The reduced set now has fuzzy rules compared to theoriginal number of .

Example 3: In their work [1], Takagi and Sugeno consid-ered the following fuzzy rule base:

If and

with and . The functionsare of the form (47) with parameters , , and astabulated in Table V. The membership functions forandare as shown in Fig. 8. They do not satisfy the SN condition.Moreover, the example utilizes an inference paradigm, which,upon given and , generates the output as

(51)

The reduction procedure of Fig. 1, however, can be appliedjust the same. Two case studies are conducted here. From step(iii), the singular values of the matrix are: 15.9065, 3.6559,2.1562, 0. The first case study keeps all three nonzero singular

(a)

(b)

Fig. 9. Membership functions of (a)a for case study #1 and (b) case study#2.

TABLE VIPARAMETERS �{; |, �{; |, AND {; | FOR FUNCTION g{; |(a; b)

values of , as well as the two nonzero singular values ofin step (vi), which are 13.3853, 3.6433. This yields threemembership functions for variable and two for . The topplot of Fig. 9 shows the membership functions for thefirst case study. The membership functions are the sameas the original ones in Fig. 8. The corresponding function

is given by: , wherethe parameters , , and are shown in Table VI.

The top plot of Fig. 10 shows the output surface of thereduced TSK model for the first case study according to theinference of (51). The output surface turns out to be the sameas that due to the original rule set. Hence, the original TSKmodel of eight rules can be replaced by the presently reducedone of six rules without any output error.

Instead of retaining all nonzero singular values in thereduction process, the second case study keeps only the largestsingular value of and in steps (iii) and (vi). As a result,two membership functions each forand are obtained. Thebottom plot of Fig. 9 shows the membership functionsfor this case. Again, the corresponding membership functions

are identical to in Fig. 8. The reduced TSK modelis now comprised of four rules instead of the original eight.The parameters , , and giving rise to functions

in this case are tabulated in Table VII.The bottom plot of Fig. 10 shows the output surface of

the reduced TSK model for the second case study using the

YAM et al.: REDUCTION OF FUZZY RULE BASE VIA SINGULAR VALUE DECOMPOSITION 131

(a)

(b)

Fig. 10. Control surfaces due to reduced rule bases in (a) case study #1 and(b) case study #2.

TABLE VIIPARAMETERS �{; |, �

{; |, AND

{; |FOR FUNCTION g

{; |(a; b)

inference of (51). Comparing the output surface of the first casestudy, which is the same as the original one, it can be observedthat appreciable error is generated for the regionand .

VI. CONCLUSIONS

This paper introduces a new reduction method for fuzzyrule base. The method calls for forming a matrix with the ruleconsequents and then applying singular value decomposition.The resultant singular values and orthogonal matrices are thentailored to form linear combinations upon which membershipfunctions and rule consequents of the reduced set are generatedfrom the original ones. Given that the original membership

functions are sum normalized and nonnegative, the proceduresguarantee that membership functions for the reduced set aresum normalized and nonnegative as well. Normality or closed-to-normality condition for the reduced set, however, dependson the nature of the bounding solution to a convex hullproblem. The proposed method is applicable independentof the inference paradigm. In the special case when PSGinference is being used and that the fuzzy rule base hassingleton support, output error due to reduction is readilybounded by the sum of the discarded singular values. Themethod is presented here using a fuzzy system of two inputsbut is readily extended to rule base with a general number ofantecedent variables.

Three cases of fuzzy reduction are discussed in this paper.They include the singleton support case, the nonsingletonsupport case, and the TSK model case. These three casesrepresent situations where the number of parameters definingthe fuzzy system is gradually increased. For the singletonsupport case, the output of each rule is determined by just onenumerical value. For the nonsingleton support case two areneeded, namely, the rule consequent and the support factor.As for the TSK model, the numerical example here utilizesa fuzzy output function of three parameters, but, in general,the function can include any number of parameters. Fromthis perspective, the proposed method can be applied to avariety of other cases. Take the example where the outputmembership functions are trapezoidal; for instance, one mayapply the reduction procedures using the characteristic pointsof the trapezoids as the parameters.

The present work aims at establishing a methodology inextracting the essential elements of a given fuzzy rule base.This is important as there is yet no uniformly acceptedformulation for designing a fuzzy rule set efficiently andeffectively. As illustrated by the numerical examples, theproposed approach manages to generate a reduced rule basewhich quite reasonably approximates the operation of theoriginal one. This is the case especially in Example 1, when afuzzy rule base of 81 rules is effectively replaced by one withjust nine. Moreover, Example 2 demonstrates a certain abilityin the present approach to extrapolate the original output toregion where it is previously undefined. The third example inthis work demonstrates that the proposed method is capableof eliminating certain redundancy in the fuzzy rule set. Inthis case, the method reveals that the original rule base ofeight rules can actually be replaced by a rule base of six ruleswithout any output error.

REFERENCES

[1] T. Takagi and M. Sugeno, “Fuzzy identification of systems and itsapplications to modeling and control,”IEEE Trans. Syst., Man, Cybern.,vol. 15, pp. 116–132, 1985.

[2] P. Baranyi and Y. Yam, “Singular value-based fuzzy approximationwith nonsingleton support,” inProc. 7th Int. Fuzzy Syst. Assoc. WorldCongress,Prague, Czech Republic, June 1997, pp. 127–132.

[3] , “Singular value-based fuzzy approximation with Sugeno typefuzzy rule base,” inProc. 6th IEEE Int. Conf. Fuzzy Syst. (FUZZ-IEEE’97), Barcelona, Spain, July 1997, pp. 265–270.

[4] Y. Yam and C. T. Yang, “Singular value-based fuzzy approximator: Acase study,” inProc. Int. Panel Conf. Soft Intell. Comput., Budapest,Hungary, Sept. 30–Oct. 3, 1996, pp. 305–312

132 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 7, NO. 2, APRIL 1999

[5] Y. Yam, “Fuzzy approximation via grid point sampling and singularvalue decomposition,”IEEE Trans. Syst., Man, Cybern., vol. 27, pp.933–951, Dec. 1997.

[6] L. Wang, R. Langari, and J. Yen, “Principal components, B-splines, andfuzzy system reduction,” inFuzzy Logic for the Applications to ComplexSystems, W. Chiang and J. Lee, Eds. Singapore: World Scientific,1996, pp. 255–259.

[7] P. Baranyi, T. D. Gedeon, and L. T. K´oczy, “A general interpolationtechnique in fuzzy rules bases with arbitrary membership functions,”in IEEE Int. Conf. Syst., Man, Cybern. (SMC’96), Beijing, China, Oct.1996, pp. 510–515.

[8] J. O’Rourke,Computational Geometry in C.Cambridge, U.K.: Cam-bridge Univ. Press, 1994.

[9] H. Edelsbrunner,Algorithms in Combinatorial Geometry.Berlin, Ger-many: Springer-Verlag, 1987.

[10] F. Bouslama and A. Ichikawa, “Application of limit fuzzy controllersto stability analysis,”Fuzzy Sets Syst., vol. 49, pp. 103–120, 1992.

Yeung Yam (M’92) received the B.S. degree fromthe Chinese University of Hong Kong and the M.S.degree from the University of Akron, OH, both inphysics, in 1975 and 1977, respectively, and theM.S. and Sc.D. degrees in aeronautics and astronau-tics from the Massachusetts Institute of Technology,Cambridge, in 1979 and 1983, respectively.

From 1985 to 1992, he was a member of theTechnical Staff in the Control Analysis ResearchGroup of the Guidance and Control Section at theJet Propulsion Laboratory, Pasadena, CA. He joined

the Chinese University of Hong Kong in 1992 and is currently an AssociateProfessor in the Department of Mechanical and Automation Engineering.His research interests include analysis, design, and identification of controlsystems.

Peter Baranyi was born in Hungary in 1970. Hereceived both the M.Sc. (electrical engineering) andthe M.Sc. degrees (education of engineering sci-ences) from the Technical University of Budapest,Hungary, in 1994 and 1995, respectively. He iscurrently working toward the Ph.D. degree at theTechnical University of Budapest, Hungary.

Since 1998, he has been a Research Assistantat the Technical University of Budapest. He hasalso conducted research work at the CNRS LAASInstitute in Toulouse, France, in 1996, the Chinese

University of Hong Kong in 1996 and 1998, and the University of New SouthWales, Sydney, Australia, in 1997. His research interests include fuzzy andneural network techniques.

Mr. Baranyi is a member of the Hungarian Society of IFSA (InternationalFuzzy Systems Association), the Hungarian Neuman Janos Computer ScienceAssociation, and the Hungarian Elektrotechnic Association.

Chi-Tin Yang was born in Hong Kong in 1970.He received the B.S. and M.S. degrees in electri-cal engineering from the University of Wisconsin-Madison in 1992 and 1993, respectively.

Since 1995, he was a graduate student in theDepartment of Mechanical and Automation Engi-neering at the Chinese University of Hong Kong.His research interests include fuzzy control, identi-fication, and modeling.