Solving type-2 fuzzy relation equations via semi-tensor product of matrices

ongyi YAN,Zengqiang CHEN,Zhongxin LIU

1.College of Computer and Control Engineering,Nankai University,Tianjin 300071,China;

2.Tianjin Key Laboratory of Intelligent Robotics,Nankai University,Tianjin 300071,China

Solving type-2 fuzzy relation equations via semi-tensor product of matrices

1.College of Computer and Control Engineering,Nankai University,Tianjin 300071,China;

2.Tianjin Key Laboratory of Intelligent Robotics,Nankai University,Tianjin 300071,China

The problem of solving type-2 fuzzy relation equations is investigated.In order to apply semi-tensor product of matrices,a new matrix analysis method and tool,to solve type-2 fuzzy relation equations,a type-2 fuzzy relation is decomposed into two parts as principal sub-matrices and secondary sub-matrices;anr-ary symmetrical-valued type-2 fuzzy relation model and its corresponding symmetrical-valued type-2 fuzzy relation equation model are established.Then,two algorithms are developed for solving type-2 fuzzy relation equations,one of which gives a theoretical description for general type-2 fuzzy relation equations;the other one can find all the solutions to the symmetrical-valued ones.The results can improve designing type-2 fuzzy controllers,because it provides knowledge to search the optimal solutions or to find the reason if there is no solution.Finally some numerical examples verify the correctness of the results/algorithms.

Fuzzy control system;Type-2 fuzzy logic system;Type-2 fuzzy relation;Type-2 fuzzy relation equation;Semitensor product of matrices

1 Introduction

The theory of semi-tensor product of matrices(STP),which was proposed by Cheng[1,2],is a powerful matrix analysis tool.STP is well established and is gaining more and more attentions.Many different areas to which STP has been successfully applied include:Boolean networks(modeling,analysis,and control)[3-5],Boolean algebra and Boolean calculus[6],nonlinear systems(analysis and control)[7,8],multi-agent systems[9],fuzzy control systems[10,11],multi-and mixed-vaulted networks[12,13],graphic theory[9],and finite automata[14],etc.

One of the advantages of STP is that STP can represent the dynamics of networks in a discrete mode.Thus,matrices can be used to analyze and control the dynamicsof networks or systems.Take Boolean control networks for example,STP has been established as a basic analytic tool for modeling,analyzing,and controlling[5].STP's advantages also lie in abilities to constructively prove descriptive propositions.The proof itself usually provides a specific algorithm to solve the corresponding problems[9];the algorithm is easily implemented by computers.

In the area of fuzzy logic,STP can convert logic equations to algebraic ones,thus the problem of solving logic equations is converted to solve algebraic equations.Since we have well-established methods to solve the latter,using STP we can find all the solutions to logic equations[15].However,until STP was proposed,most existing algorithms only can get some particular solutions,such as,the minimum or maximum solution[16],or only provide a theoretical description for the solutions,which is hard or unable to be carried out[17].Moreover,using STP,general logic operators(functions)can be expressed in the form of matrices,which can greatly simplify the proof of some logic propositions and can further extend logic operators to the case of multi-or mixed-valued logic.This provides a foundation for analyzing and designing a fuzzy logic system,and provides a chance for developing numerical algorithms for fuzzy logic control systems[18].

In the field of type-2 fuzzy sets and systems,the problem of solving type-2 fuzzy relation equations(FREs)is one of the most important issues.Type-2 fuzzy sets(T2 FSs)are extensions oftype-1 and of interval valued fuzzy sets.Such sets are fuzzy sets whose membership grades themselves are type-1 fuzzy sets;they are especially useful when the membership functions of type-1 fuzzy sets are hard to determine.For example,type-1 fuzzy sets are helpless in modeling linguistic uncertainties.Type-2 fuzzy sets and fuzzy logic systems have been increasingly used in various areas[19-21].

A type-2 fuzzy relation(T2 FR),which is essentially a type-2 fuzzy set,is one way to increase the fuzziness of a relation,and,according to Hisdal,'increased fuzziness in a description means increased ability to handle inexact information in a logically correct manner[22]'.As the role of type-1 fuzzy relation equations in the theory of type-1 fuzzy sets and fuzzy logic systems,type-2 fuzzy relation equations play a key role in the theory of type-2 ones,and have found a wide variety of practical applications,such as,the design and optimization of type-2 fuzzy controllers,type-2 fuzzy inference,image compression,medical diagnosis[23-25],etc.Therefore,solving type-2 fuzzy relation equations is of both theoretical and practical significance.

Authors of this paper have proposed an algorithm to solve a simple kind of type-2 fuzzy relation equations,singleton type-2 fuzzy relation equations[26].However,to date,there have been no results available on solving general T2 FREs,to the best of our knowledge.In this paper,we investigate the problem by STP based on the Cheng's approach to T1 FREs.First,we decompose a type-2 fuzzy relation into two parts as principal sub-matrix and secondary sub-matrix.Then,the concept of principal sub-equation of a type-2 fuzzy relation equation,and a symmetrical-valued type-2 fuzzy relation equation model are introduced.Based on this,two algorithms are developed to solve type-2 fuzzy relation equations.One of which is for general ones,the other one is for the symmetrical-valued ones.

Throughout this paper,we focus only on some finite universes of discourse.Particularly,we setU=Let F(UXV)and?F(UXV)denote the sets of type-1 and type-2 fuzzy set on the product spaceUXV.

2 Preliminaries and problem formulation

2.1 Type-2 FRs and their compositions

2.1.1From type-1 fuzzy relations to type-2 ones

A type-1 FR inUXVis a type-1 fuzzy subset ofUXVand a type-2 FR inUXVis a type-2 fuzzy subset ofUXV.Type-1 and type-2 fuzzy relations are usually expressed in matrix form as

and

where μR(ui,vj)are crisp numbers in[0,1],and μ?R(ui,vj)are fuzzy numbers(fuzzy sets)in[0,1].

As the extension form type-1 fuzzy sets to type-2 ones,we can obtain a type-2 fuzzy relation by adding additional uncertainty information to a type-1 one.The following example demonstrates the extension.

Example 1Consider the type-1 FRR∈F(XXY):'xis close toy',whereX={x1,x2,x3}andY={y1,y2}are set asX={5,11,17},Y={6,16}and

Consider another type-1 FRS∈F(YXZ):'yis much smaller thanz',whereZ={z1,z2,z3}={17,20,30},and

Adding some additional uncertainties to these two type-1 FRs,we may obtain the following membership grades(For saving space,we sometimes represent a type-2 fuzzy relation in the following form,i.e.,the solidus(/)are replaced by built-up fractions):

and

Both of them are standard type-2 fuzzy relations.

2.1.2Composition of type-2 fuzzy relations

Definition 1[27]IfRandS(orandare two T1(or T2)FRs onUXVandVXW,respectively,the membership of any(u,w),u∈U,w∈W,is non-zero iff there is at least onev∈Vso that μR(u,v) ≠ 0(orand(orin which 1/0 denotes the concept of zero membership grades in the case of type-2 fuzzy sets.An element having a zero membership in a type-2 set means it has a secondary membership equal to 1 corresponding to the primary membership of 0,and all other secondary memberships equal to 0.

For the composition of T1 FRs,the definition is equivalent to the following sup-star composition:

and,for that of T2 FRs,the condition is equivalent to the following 'extended' version of the sup-star composition.

in which ∨ denotes the maximum s-norm and★denotes a t-norm.In(6),if more than one computation ofuandwgenerates the same pointthen the one with the largest membership grade is kept in the union.We do the same things for(7).In this paper,we set★as minimum t-norm,denoted as∧,and assume that the composition of type-1(or type-2)fuzzy relations are defined as(4)(or(5)-(7)).The following Example 2 shows the compositions.

Example 2Consider the fuzzy relations in Example 1.

It is known that the expression'xis close toyandyis much smaller thanz'indicates the compositionR?S(or).

For the type-1 case,using equation(4),we have

wherei=1,2,3 andj=1,2,3.We then obtain

For the type-2 case,using equation(5),we have

圖10為從位移場(chǎng)計(jì)算出的I型裂紋緊湊拉伸試樣4的J積分隨時(shí)間的演化數(shù)據(jù)關(guān)系，由圖可見(jiàn)J積分在初始加載時(shí)緩慢增加，約20s后開(kāi)始快速增大，88s時(shí)達(dá)到臨界載荷23kN，此時(shí)J積分約293.4kJ/m2，記為JiDic，即為光學(xué)方法測(cè)量得到的實(shí)際起裂斷裂韌度，低于標(biāo)準(zhǔn)GB/T 21143的計(jì)算值。

wherei=1,2,3 andj=1,2,3.Using equations(10),(2),(3),(6)and(7),we obtain

2.2 STP method of solving type-1 FREs

2.2.1Semi-tensor product of matrices

Definition 2[2]ForM∈MmXnandN∈MpXq,their STP,denoted byM■N,is defined as follows:

wheresis the least common multiple ofnandp,and?is the Kronecker product.

Remark 1The following are some basis properties of STP,which will be used in the sequel.

1)LetA,B∈MmXn,andC∈MpXq.Then,

2)LetA∈MmXn,B∈MpXq,andC∈MrXs.Then,

2.2.2Algebraic expression of logical operators

The following notations will be used in this paper.

Whenk=2,D2:={0,1}is Boolean range;k=∞,is a fuzzy range.

4)Letsayx=i/(k-1),we identifywith.Then,is the vector form ofx.

5)A matrixL∈MmXnis called a logic matrix if the columns ofL,which is denoted by Col(L),are of the form ofThat is Col(L)? Δn.Let LnXrdenote the set ofnXrlogic matrices.

6)Ifby definition,it can be expressed asFor compactness,it is briefly denoted asL=δn[i1i2...ir].

2)Letfbe anr-aryk-valued logic operator(or function),then there is a unique logic matrixMf∈LkXkr,such that

which is called the algebraic form off,andMfis called the structure matrix off.

How to calculateMfand how to convert the algebraic form back to its logical form,please refer to[5].

2.2.3Using STP to solve type-1 FREs

This subsection summarizes an STP method of solving type-1 FREs proposed by Cheng[15],which is the basis of our proposed algorithms of solving type-2 FREs.Consider the following T1 FRE

whereand'?'is defined as(4).

Equation(16)can be converted into canonical linear algebraic equations

whereXiandBiare theith andith columns ofXandB,respectively.

Collect different values of the elements inAandB,and use them to construct a set

Add 1 and(or 0)toSif they are(or it is)not inS,the result is an ordered set as

Letx∈ [0,1],define mappings π?:[0,1] → Ξ and π?:[0,1]→ Ξ as[15]

Theorem 2[15]is a solution to(16),iffandare solutions to(16).

Theorem 2 gives a complete picture for the set of solutions.The following proposition ensures the existence of the largest solution in the set of solutions.

Proposition 1[15] If equation(17)has a solution in Ξn,then it has a largest one in Ξn.

To find the solutions to(16),we only need to solve equations in(17).That is,it is sufficient to develop a method to solve each equation in(17),which is simply denoted by

wherex∈Rnandb∈Rmstand forXiandBiin(17),respectively.

Consider thejth equation in(20),which is

wherexiandbjare theith andjth elements ofxandb,respectively,1≤i≤n,1≤j≤m.

Using the technique converting a logic expression into an algebraic expression,the left hand of(21)can be transformed in vector form as

whereris described in the above Ξ,all the matrix products are the STP,is omitted for compactness,and

Then,equation(21)becomes

Multiplying both sides ofmequations of(23),equation(20)can be expressed as

where,andKhatri-Rao Product of matrices[28]).Precisely,

SinceLis a logic matrix,andb∈Δrm,x∈Δrn,equation(24)has solutions if and only if

Now,set

we then obtain the solution set of(24)as

Finally,we convertxback to(x1,...,xn)∈Ξn,(x1,...,xn)Tis just the solution toequation(20).

2.3 Problem formulation

In practice,two sorts of T2 FREs are considered.One is commonly used in designing fuzzy controllers(the following Model 1);the other one is used for a problem similar to diagnosing diseases by symptoms(the following Model 2).

Model 1Letandbe two known type-2 fuzzy relations,we are searching a T2 FRsuch that

Model 2Letandbe two known type-2 fuzzy relations,we are searching a T2 FRsuch that

Recall that the composition'?'is very similar to the usual product of matrices,making a transpose on both sides of(27),we then obtain

whose form is the same as equation(26).

The purpose of this paper is to solve type-2 FRE(26)with the help of the STP method which can find all the solutions to type-1 FREs(described in Subsection 2.2.3).

3 Main results

In this section,we investigate the problem of solving type-2 FREs,and present the main results of this paper.First,we propose the concept of symmetrical-valued type-2 fuzzy relation and explore some of its properties,which can simplify the calculation of composition of type-2 fuzzy relations and make solving type-2 FREs easier.

3.1 Symmetrical-valued T2 FRs and their properties

Definition 3The following discussed type-1 fuzzy sets are defined in a finite universe of discourse.

1)In a type-1 fuzzy set,the element having membership grade equal to 1 is called a principal element of the fuzzy set.If all the membership grades are not equal to 1,we uniformly scale the membership grades such that the largest one equal to 1.That is,the principal element is the element whose membership grade is largest.Such scaling makes the solving algorithm of type-2 fuzzy relation equations proposed in Section 4 more understandable and easier to calculate.This paper only considers finite type-1 fuzzy sets having one principal element.

2)A symmetrical-valued type-1 fuzzy set is a type-1 fuzzy set whose elements are symmetrically distributed on both sides of the principal element from small to large and followed by a difference of a constant(for example,0.1).If a type-1 fuzzy set has 0(or 1)as its element,it may also be considered as a symmetrical one,since we can think that the membership grades of the elements on the left(or right)of 0(or 1)are zero.

3)A symmetrical-valued type-2 fuzzy relation is a type-2 fuzzy relation in whose matrix form all entries are symmetrical-valued type-1 fuzzy sets.If all the numbers of elements of these symmetrical-valued type-1 fuzzy sets are less than or equal tor(ris a positive integer),the symmetrical-valued type-2 fuzzy relation is called anr-ary symmetrical-valued type-2 fuzzy relation.

The type-2 fuzzy relations(2)and(3)in Subsection 2.1 are both 3-ary symmetrical-valued type-2 fuzzy relations.

Next,we define a partial order'?'on?F(UXV).

In order to facilitate calculating the composition of type-2 fuzzy relations,we have the following results.

Theorem 3Let}andB={b1,b2,...,bn}be two sets of real number in[0,1].Define

then

where|A|stands for the number of elements ofA.

ProofProofs of(29)and(30)are very similar.We just give the proof of(29).The total number of elements ofAandBism+n.The smallest element inA∪Bwill be excluded during the process of pair-wise comparison,thus we have

where'='holds ifaiandbj(i=1,2,...,m,j=1,2,...,n)are of derangement.For example,aiandbjare arranged as

Next,we prove the left hand of(29).Without loss of generality,we assume thatm≤n.For eachai∈A(i=1,2,...,m),we have

where obviously '=' holds only ifai≥bj(j=1,2,...,n),which indicates that

Therefore,we have

where'='holds ifai≥bj(j=1,2,...,n)for anyai∈A.The proof is completed.

Using mathematical induction,Theorem 3 can be easily extended to the case of multiple sets.

Corollary 1LetAi={ai1,ai2,...,aim}(i=1,2,...,n)be real number sets in[0,1].Define

then

Proposition 2Letxandybe two real numbers in[0,1],then

ProofThe proof can be done by a straightforward computation.

Proposition 3For symmetrical-valued type-1 fuzzy sets

1)Ifa1≤b1,then the entryf1/a1will appear inand will disappear in

2)Ifbn≥am,then the entryfn/bnwill appear inand will disappear in

ProofWe only prove the first conclusion,the other one can be got by a similar way.

SinceBis a symmetrical-valued type-1 fuzzy set,we have

Becausea1≤b1,it is evident that

based on which,we have

According to Proposition 2,we know that

Thus,f1/a1remains unchanged in the pair-wise comparisons,that isf1/a1will appear in

On the other hand,since,then

which indicatesA■Bdoes not containa1.Therefore,the conclusion follows.

The following Proposition 4 is an immediate consequence of Proposition 3.

Proposition 4For symmetrical-valued type-1 fuzzy sets

Ifam≤b1,then

Theorem 4LetAandBbe two symmetrical-valued type-1 fuzzy sets as described in(31).The principal element of(or)is determined only by the principal elements ofAandB.

ProofThe conclusion can also be proved by α-cut decomposition theorem.Here,we provide an easier way to prove it,which is helpful for understanding the algorithm to solve type-2 FREs proposed in Subsection 3.2.

Obviously,among the pair-wise comparisons,the membership grade 1 only occurs at(or).If(or)is unique in(orwe are done.Otherwise,saysincethe membership grade of(oris still equal to 1.The conclusion is proved.

Remark 2Theorem 4 provides a picture how the principal elements of the intersection and union of type-2 fuzzy sets are produced,and can help us easily determine the principal elements of the intersection and union.

3.2 Solving type-2 fuzzy relation equations

In this subsection,we first decompose a type-2 FR into two parts as principal sub-matrix and secondary sub-matrix,where the principal sub-matrix is a type-1 FR.Then,a principal sub-equation model is established for introducing the STP method to solve type-2 FREs.

3.2.1Decomposition of type-2 fuzzy relations

Definition 51)A principal sub-matrix of the type-2 fuzzy relation(1),denoted asis a fuzzy matrix(type-1 fuzzy relation)whose elements are the principal elements of the entries of(1)with the original order.See Example 6 for a demonstration.

2)A secondary sub-matrix of the type-2 fuzzy relation(1),denoted as,is the remaining part that the principal elements and their membership grades of the entries of(1)are removed.Example 6 gives a demonstration.

3)A principal sub-equation of the type-2 fuzzy relation equation(26)is a type-1 fuzzy relation equation which is constituted by the principal sub-matrices of the corresponding parts in(26),that isSee equations(44)and(51)for an example.

Moreover,the membership grades(or elements)of the entries of a type-2 FR,X,are generally called the membership grades(or elements)ofX.

Example 3Consider the type-2 fuzzy relationsandin Example 1.Their principal and secondary submatrices are as follows:

It is obvious that the solution to the type-2 fuzzy relation equation(26),is completely determined by its principal sub-matrixand secondary sub-matrix

3.2.2Solution of type-2 fuzzy relation equations

Consider the general type-2 fuzzy relation equation(26),

in which

whereaij,xijandbijare all type-1 fuzzy sets,i.e.,

We have known that to solve equation(26)we only have to solve

That is,we only need to consider the following equations,which are expansions of equation(35),

Algorithm 1To find the solutions to(36),we can take the following steps.

Step 1Expand the left hand of(36)using equation(5).The resulting expansion,equation(37),is a logical expression of the elements,and their membership grades,which are connected by the logical operatorsandwhere

where

According to Theorem 3,there arementries in(37),wheremsatisfies

in whichpk=min{rik,skj},i=1,2,...,n;j=1,2,...,s.

Step 2Collect atij-permutation of the entries in(37),and set other entries-membership grades to zero.(Note thattijis described in(34)and should be less thanm,otherwise,equation(26)has no solution).Establish equations using the entries of the chosen permutation and equation(34),that is,make the entries with the same position in the permutation and equation(34)equal,respectively.See equations(48)and(55)of Example 4 in Section 4 for an example.Here,for statement ease,we assume that the resulting equations are

Step 3According to(15)or(24),equations(39)and(40)can be converted into their algebraic forms:

where

Step 4Solve the equations(41)and(42)by the STP method which is described in Subsection 2.2.3.The obtained solutions is a solution set of equation(36).

Step 5Apply steps 1-4 to each equation in(36),their solution sets can be found.The intersection of these solution sets is just a solution set of(35).

Remark 31)Choosing different permutations in Step 2 will produce different solution sets for equation(35).

2)Sincemin(38)may be very large,the computation is heavier.This will exert tremendous burden on the computer memory when calculated by computers.In practical applications,to reduce the burden,we propose,according to Theorem 4,the following algorithm to solve the symmetrical-valued type-2 FREs.

3)The method can find out all the solutions to T2 FREs theoretically,however,its computation complexity may be high,especially when a T2 FRE consists of a large number of elements.That is why we further introduce the following symmetrical-valued T2 FREs,which is a simpler kind of T2 FRE but can meet most requirements of some practical applications.

3.3 Solving symmetrical-valued type-2 FREs

Algorithm 2Assume that type-2 fuzzy relation equation(35)is a symmetrical-valued one.To find all the solutions,we can take the following steps.Example 5 in Section 4 is a demonstration.

Step 1Establish the principal sub-equation of(35),and use the STP method solving type-1 FREs to find its solutions.These solutions are the elements of the principal sub-matrices of?Xj.

Step 2Since equation(35)is a symmetrical-valued one,the elements(not the membership grades)of the secondary sub-matrix of?Xjcan be determined according to the solutions obtained in Step 1.

Step 3Establish equations using the membership grades with the same position in the left-and right-hand sides of the expansion of(36),that is,make them equal respectively.Then,convert the equations into their algebraic forms,their solutions,the membership grades of the secondary sub-matrix of?Xj,can also be found by the STP method.

Step 4Apply steps 1-3 to each equation in(36),we can find their solution sets.The intersection of these solution sets is just the solution set of(35).

Remark 4The essence of Algorithms 1 and 2 is that T2 FREs are first decomposed into T1 FREs by collecting every permutation of the entries of equation(36),all the solutions to these T1 FREs can be obtained by the Cheng's method in correspondingk-valued logic range.

4 Illustrative examples

This section gives two examples to demonstrate the proposed method for solving type-2 fuzzy relation equations.The first example is a simple one constructed from Example 2 by assuming thatis unknown,which is used to show the solving procedure.The second one is a symmetrical-valued type-2 fuzzy relation equation,which is for showing the structure of solutions to a type-2 fuzzy relation equation.

Example 4Consider the following type-2 fuzzy relation equation:

where

First,we establish the principal sub-equation of(43)as

where

Using the STP method of solving type-1 FREs,we can obtain the unique solution:

which is the principal sub-matrix of?X.

Since equation(43)is a symmetrical-valued type-2 one,other elements of?Xcan be determined as

Next,we will look for the membership grades of?X,i.e.,thein(45).

Consider the canonical linear algebraic equations of(43):

The first equation in(46)is

Expanding equation(47)by(5),Propositions 1-3 will make calculations much easier,we have

From(48),we can obtain the following equations:

Solving(49),then we have

Similarly,we can obtain the solutions to the second and third equations in(45),which are

Therefore,the solution set to equation(43)is

where0.5≤α≤1,0≤β≤0.6,0.8≤γ≤1,0≤η≤0.5.Comparing(50)with(3),we observe thatin(3)belongs to the solution set(50).

Example 5Consider the following symmetricalvalued type-2 fuzzy relation equation:

where the first and second columns ofare

The third and fourth columns are

and

Equation(51)is a 3-ary symmetrical-valued type-2 fuzzy relation equation.We use the algorithm proposed in sub-section 3.3 to solve it.

Step 1We establish the principal sub-equation of(51)as

where

Using the STP method,we can obtain each column of the solutions to(52).The first column is

The second column is

The third column is

We find totally 23X19X36=15732 solutions in the range of 9-valued logic for the principal sub-equation(52),in which the largest one is

a minimum one is

and an ordinary one is

Step 2Since equation(51)is a symmetrical-valued one,for each solution to(52),we can determine the elements of the secondary sub-matrix of(51).Take the largest one,for example,can be partially determined as

Step 3Establish equations by making the membership grades equal,which have the same position in the expansion of(51)where?Xis replaced by(53).To do this,we consider the canonical linear algebraic equations of(51):

The first equation in(54)can be expanded by(5)as follows:

By(55),we have the following equations

Solving(56),we obtain

Remark 51)It is worthwhile to note that equation(55)may have no solutions.For instance,ifb21inin(51)is 0.8/0.5+1/0.6+0.7/0.7,there is nosatisfying(55).

2)We may have to resort to the STP method solving type-1 FREs to solve(55),when it is too complex to directly find the solution.

Similarly,we can obtain other membership grades ofby solving the second and third equations in(54),which are

and

Now,for the largest principal solution,that is,the solution to the principal sub-equation,we find all the solutions for equation(51).All the other solutions can be obtained in a similar way.

Finally,we summarize the solutions to(51)as follows.There are totally 15732 principal solutions,each of them determines some corresponding secondary submatrices of the solutions.The following are some examples,the largest solution is

a minimum one is and an ordinary one is

Remark 6Finding all the solutions to a type-2 fuzzy relation equation is useful for designing a type-2 fuzzy controller,because it provides knowledge to search the optimal solutions or to find the reason if there is no solution.

5 Conclusions

The STP method,which can find all the solutions to the type-1 FREs,is introduced to solve type-2 FREs.We first propose anr-ary symmetrical-valued type-2 fuzzy relation mode according to how a type-1 fuzzy relation is extended to a type-2 one.To facilitate the calculation of the composition of type-2 fuzzy sets,some properties of ther-ary symmetrical-valued type-2 fuzzy relation are revealed.We then put forward the concept of symmetrical-valued type-2 fuzzy relation equation.Based on this,two algorithms are developed for solving type-2 fuzzy relation equations,one may be helpful for analyzing the solutions to general type-2 FREs,the other one can programmatically find all the solutions to the symmetrical-valued type-2 FREs.The approach proposed in this paper can be helpful for designing and optimizing type-2 fuzzy controllers.

[1]D.Cheng.Semi-tensor product of matrices and its application to Morgen-s problem.Science in China Series F:Information Sciences,2001,44(3):195-212.

[2]D.Cheng,H.Qi,Y.Zhao.An Introduction to Semi-tensor Product of Matrices and Its Applications.Singapore:World Scientific Publishing Co.Pte.Ltd.,2012.

[3] H.Li,Y.Wang.On reachability and controllability of switched Boolean control networks.Automatica,2012,48(11):2917-2922.

[4]D.Cheng.Disturbance decoupling of boolean control networks.IEEE Transactions on Automatic Control,2011,56(1):2-10.

[5]D.Cheng,H.Qi,Y.Zhao.Analysis and control of Boolean networks:a semi-tensor product approach.Acta Automatica Sinica,2011,37(5):529-540.

[6]H.Li,Y.Wang.Boolean derivative calculation with application to fault detection of combinational circuits via the semi-tensor product method.Automatica,2012,48(4):688-693.

[7]Z.Li,Y.Qiao,H.Qi,et al.Stability of switched polynomial systems.Journal of Systems Science and Complexity,2008,21(3):362-377.

[8]D.Cheng,H.Qi.Global stability and stabilization of polynomial systems.Proceedings of the 46th IEEE Conference on Decision and Control.New Orleans:IEEE,2007:1746-1751.

[9]Y.Wang,C.Zhang,Z.Liu.A matrix approach to graph maximum stable set and coloring problems with application to multi-agent systems.Automatica,2012,48(7):1227-1236.

[10]A.Ge,Y.Wang,A.Wei,et al.Control design for multi-variable fuzzy systems with application to parallel hybrid electric vehicles.IET Control Theory and Applications,2013,30(8):998-1004.

[11]H.Li,Y.Wang.A matrix approach to latticized linear programming with fuzzy-relation inequality constraints.IEEE Transactions on Fuzzy Systems,2013,21(4):781-788.

[12]F.Li,J.Sun.Stability and stabilization of multivalued logical networks.Nonlinear Analysis:Real World Applications,2011,12(6):3701-3712.

[13]Z.Liu,Y.Wang,H.Li.Disturbance decoupling of multi-valued logical networks.Proceedings of the 30th Chinese Control Conference.Yantai:IEEE,2011:93-96.

[14]X.Xu,Y.Hong.Matrix expression and reachability analysis of finite automata.Journal of Control Theory and Applications,2012,10(2):210-215.

[15]D.Cheng,J.Feng,H.Lv.Solving fuzzy relational equations via semitensor product.IEEE Transactions on Fuzzy Systems,2012,20(2):390-396.

[16]A.A.Molai,E.Khorram.An algorithm for solving fuzzy relation equations with max-T composition operator.Information Sciences,2008,178(5):1293-1308.

[17]B.Hu.Foundation of Fuzzy Theory.Wuhan:Wuhan University Press,2010.

[18]H.Qi,D.Cheng.Logic and logic-based control.Journal of Control Theory and Applications,2008,6(1):26-36.

[19]T.Dereli,A.Baykasoglu,K.Altun,et al.Industrial applications of type-2 fuzzy sets and systems:A concise review.Computers in Industry,2011,62(2):125-137.

[20]H.Li,X.Duan,Z.Liu.Three-dimensional fuzzy logic system for process modeling and control.Journal of Control Theory and Applications,2010,8(3):280-285.

[21]Z.Liang,C.Wang.Robust exponential stabilization of nonholonomic wheeled mobile robots with unknown visual parameters.Journal of Control Theory and Applications,2011,9(2):295-301.

[22]E.Hisdal.The IF THEN ELSE statement and interval-valued fuzzy sets of higher type.International Journal of Man-Machine Studies,1981,15(4):385-455.

[23]T.Chen,C.Chang,J.Lu.The extended QUALIFLEX method for multiple criteria decision analysis based on interval type-2 fuzzy sets and applications to medical decision making.European Journal of Operational Research,2013,226(3):615-625.

[24]P.R.Innocent,R.I.John.Computer aided fuzzy medical diagnosis.Information Sciences,2004,162(2):81-104.

[25]Q.Zhang,S.Jiang.A note on information entropy measures for vague sets and its applications.Information Sciences,2008,178(21):4184-4191.

[26]Y.Yan,Z.Chen.A semi-tensor product approach to solving singleton type-2 fuzzy relation equations.Proceedings of the 32th Chinese Control Conference.Xi'an:IEEE,2013:3434-3439.

[27]N.Karnik,J.Mendel.Operations on type-2 fuzzy sets.Fuzzy Sets and Systems,2001,122(2):327-348.

[28]L.Ljung,T.Sderstrm.Theory andPracticeof Recursive Identification.Dordrecht:Kluwer Academic Publisher,1983.

22 August 2013;revised 21 February 2014;accepted 24 February 2014

DOI10.1007/s11768-014-0137-7

?Corresponding author.

E-mail:yyyan@mail.nankai.edu.cn.Tel.:+86-22-23508547.

This work was partially supported by the Natural Science Foundation of China(No.61174094),the Tianjin Natural Science Foundation of China(No.13JCYBJC17400),and the Program for New Century Excellent Talents in University of China(No.NCET-10-0506).

?2014 South China University of Technology,Academy of Mathematics and Systems Science,CAS,and Springer-Verlag Berlin Heidelberg

Yongyi YANreceived the B.S.and M.S.degrees in Mathematics from Luoyang Normal University,Luoyang,China,in2005and 2008,respectively,and is currently pursuing the Ph.D.degree in Control Theory and Engineering at Nankai University,Tianjin,China.His current research interests are in the fields of modeling and optimization of complex systems,fuzzy control,intelligent predictive control.E-mail:yyyan@mail.nankai.edu.cn.

Zengqiang CHENwas born in Tianjin,China in 1964.He received the B.S.,M.E.and Ph.D.degrees from Nankai University,in 1987,1990,and 1997,respectively.He is currently a professor of Control Theory and Engineering of Nankai University,and Deputy Director of Institute of Robotics and Information Automation.His current research interests are in the fields of intelligent predictive control,chaotic systems and complex dynamic network,multi-agent system control.E-mail:chenzq@nankai.edu.cn.

Zhongxin LIUreceived his B.E.and D.E.degrees in Nankai University,in 1997 and 2002,respectively.He is currently a professor of Control Theory and Engineering of Nankai University,Tianjin,China.His current research interests include Multi-agent systems,complex and dynamic networks,computer control and management.E-mail:lzhx@nankai.edu.cn.