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

Transcription

1 Control Theory Tech Vol 2 No 2 pp May 204 Control Theory and Technology Solving type-2 fuzzy relation equations via semi-tensor product of matrices Yongyi YAN 2 Zengqiang CHEN 2 Zhongxin LIU 2 College of Computer and Control Engineering Nankai University Tianjin China; 2Tianjin Key Laboratory of Intelligent Robotics Nankai University Tianjin China Received 22 August 203; revised 2 February 204; accepted 24 February 204 Abstract: 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; an r-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 Keywords: Fuzzy control system; Type-2 fuzzy logic system; Type-2 fuzzy relation; Type-2 fuzzy relation equation; Semitensor product of matrices DOI 0007/s Introduction The theory of semi-tensor product of matrices (STP) which was proposed by Cheng [ 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 [0] multi- and mixed-vaulted networks [2 3] graphic theory [9] and finite automata [4] 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 dynamics Corresponding author yyyan@mailnankaieducn Tel: This work was partially supported by the Natural Science Foundation of China (No ) the Tianjin Natural Science Foundation of China (No 3JCYBJC7400) and the Program for New Century Excellent Talents in University of China (No NCET ) 204 South China University of Technology Academy of Mathematics and Systems Science CAS and Springer-Verlag Berlin Heidelberg

2 74 Y Yan et al / Control Theory Tech Vol 2 No 2 pp May 204 of 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 [5] However until STP was proposed most existing algorithms only can get some particular solutions such as the minimum or maximum solution [6] or only provide a theoretical description for the solutions which is hard or unable to be carried out [7] 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 [8] 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 of type- and of interval valued fuzzy sets Such sets are fuzzy sets whose membership grades themselves are type- fuzzy sets; they are especially useful when the membership functions of type- fuzzy sets are hard to determine For example type- fuzzy sets are helpless in modeling linguistic uncertainties Type-2 fuzzy sets and fuzzy logic systems have been increasingly used in various areas [9 2] 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- fuzzy relation equations in the theory of type- 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 T 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 set U = {u u m } V = {v v n } W = {w w s } Let F (U V) and F (U V) denote the sets of type- and type-2 fuzzy set on the product space U V 2 Preliminaries and problem formulation 2 Type-2 FRs and their compositions 2 From type- fuzzy relations to type-2 ones A type- FR in U V is a type- fuzzy subset of U V and a type-2 FR in U V is a type-2 fuzzy subset of U V Type- and type-2 fuzzy relations are usually expressed in matrix form as μ R (u v ) μ R (u v 2 ) μ R (u v n ) μ R (u 2 v ) μ R (u 2 v 2 ) μ R (u 2 v n ) R = μ R (u m v ) μ R (u m v 2 ) μ R (u m v n ) and μ R(u v ) μ R(u v 2 ) μ R(u v n ) μ R(u 2 v ) μ R(u 2 v 2 ) μ R(u 2 v n ) R = () μ R(u m v ) μ R(u m v 2 ) μ R(u m v n ) where μ R (u i v j ) are crisp numbers in [0] and μ R(u i v j ) are fuzzy numbers (fuzzy sets) in [0] As the extension form type- fuzzy sets to type-2 ones we can obtain a type-2 fuzzy relation by adding additional uncertainty information to a type- one The

3 Y Yan et al / Control Theory Tech Vol 2 No 2 pp May following example demonstrates the extension Example Consider the type- FR R F(X Y) : x is close to y where X = {x x 2 x 3 } and Y = {y y 2 } are set as X = {5 7} Y = {6 6} and R = x x 2 x 3 y y Consider another type- FR S F(Y Z) : y is much smaller than z where Z = {z z 2 z 3 } = { } and z z 2 z 3 S = y y Adding some additional uncertainties to these two type- FRs we may obtain the following membership grades (For saving space we sometimes represent a type-2 fuzzy relation in the following form ie the solidus (/) are replaced by built-up fractions): and S= R = y y 2 x x 2 x 3 y y (2) z z 2 z (3) Both of them are standard type-2 fuzzy relations 22 Composition of type-2 fuzzy relations Definition [27] If R and S (or R and S) are two T (or T2) FRs on U V and V W respectively the membership of any (u w) u U w W is non-zero iff there is at least one v V so that μ R (u v) 0 (or μ R(u v) /0) and μ S (v w) 0 (or μ S (v w) /0) in which /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 corresponding to the primary membership of 0 and all other secondary memberships equal to 0 For the composition of T FRs the definition is equivalent to the following sup-star composition: μ R S (u w) = sup[μ R (u v) μ S (v w)] (4) v V and for that of T2 FRs the condition is equivalent to the following extended version of the sup-star composition μ R S (u w) = [μ R(u v) μ S (v w)] (5) v V where and are meet and join operations respectively which are defined as follows [25] Let à and B be two type-2 fuzzy sets in X and let μã(x) and μ B(x) be the membership grades of à and B denoted as μã(x) = f x(u)/u and μ B(x) = g x(w)/w u w respectively where u w J x are the primary memberships of x f x (u) g x (w) [0 ] are the secondary memberships of x By Zadeh s extension principle [28] the membership grades for intersection and union of type-2 FSs à and B are defined as μã B μã B (x) = μã(x) μ B(x) = ( f x (u) g x (w))/(u w) u w (6) (x) = μã(x) μ B(x) = ( f x (u) g x (w))/(u w) u w (7) in which denotes the maximum s-norm and denotes a t-norm In (6) if more than one computation of u and w generates the same point u w then 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- (or type-2) fuzzy relations are defined as (4) (or (5) (7)) The following Example 2 shows the compositions Example 2 Consider the fuzzy relations in Example It is known that the expression x is close to y and y is much smaller than z indicates the composition R S (or R S) For the type- case using equation(4) we have μ R S (x i z j )= [μ R (x i y ) μ S (y z j )] [μ R (x i y 2 ) μ S (y 2 z j )] [μ R (x i y 3 ) μ S (y 3 z j )] (8)

4 76 Y Yan et al / Control Theory Tech Vol 2 No 2 pp May 204 where i = 2 3 and j = 2 3 We then obtain R S = (9) For the type-2 case using equation(5) we have μ R S (x i z j )= [μ R(x i y ) μ S (y z j )] [μ R(x i y 2 ) μ S (y 2 z j )] [μ R(x i y 3 ) μ S (y 3 z j )] (0) where i = 2 3 and j = 2 3 Using equations(0) (2) (3) (6) and (7) we obtain R S= STP method of solving type- FREs 22 Semi-tensor product of matrices Definition 2 [2] For M M m n and N M p q their STP denoted by M N is defined as follows: M N := (M I s/n )(N I s/p ) () where s is the least common multiple of n and p and is the Kronecker product Remark The following are some basis properties of STP which will be used in the sequel ) Let A B M m n and C M p q Then (A + B) C = A C + B C (2) C (A + B) = C A + C B 2) Let A M m n B M p q and C M r s Then (A B) C = A (B C) (3) 222 Algebraic expression of logical operators The following notations will be used in this paper )The range of k-valued logics: D k := { 0 k k 2 k } k 2 When k = 2 D 2 := {0 } is Boolean range; k = D := {α 0 α } is a fuzzy range 2) δ i n: the ith column of the identity matrix I n 3) Δ n := {δ i n i = 2n} 4) Let x D k (k< ) say x = i/(k ) we identify i/(k ) with δ k i i = 0 k Then x Δ k k δ k i k is the vector form of x 5) A matrix L M m n is called a logic matrix if the columns of L which is denoted by Col(L) are of the form of δ k n That is Col (L) Δ n Let L n r denote the set of n r logic matrices 6) If L L n r by definition it can be expressed as L = [δ i n δ i 2 n δ i r n] For compactness it is briefly denoted as L = δ n [i i 2 i r ] Theorem [5] ) Let f : D k D k be an r-ary k- } {{ } valued logic operator (or function) if the logic variables are of vector forms f can be expressed as f : D k D k } {{ } r r Δ k (4) 2) Let f be an r-ary k-valued logic operator (or function) then there is a unique logic matrix M f L k k r such that f (x x r ) = M f x x r (5) which is called the algebraic form of f and M f is called the structure matrix of f How to calculate M f and how to convert the algebraic form back to its logical form please refer to [5] 223 Using STP to solve type- FREs This subsection summarizes an STP method of solving type- FREs proposed by Cheng [5] which is the basis of our proposed algorithms of solving type-2 FREs Consider the following T FRE A X = B (6) where A F(U V) B F(U W) X F(V W) and is defined as (4) Equation (6) can be converted into canonical linear algebraic equations A X i = B i i = 2s (7) where X i and B i are the ith and ith columns of X and B respectively Collect different values of the elements in A and B and use them to construct a set S = { a ij b pq i = 2m; j = 2n;

5 Y Yan et al / Control Theory Tech Vol 2 No 2 pp May p = 2m; q = 2s} Add and (or 0) to S if they are (or it is) not in S the result is an ordered set as Ξ={ξ i i=r; ξ =0<ξ 2 <<ξ r <ξ r =} Let x [0 ] define mappings π :[0 ] Ξ and π :[0 ] Ξ as [5] π (x) = max{ξ i Ξ ξ i x} (8) i π (x) = min{ξ i Ξ ξ i x} (9) i Theorem 2 [5] X = (x ij ) D n s is a solution to(6) iff π (X) = (π (x ij )) and π (X) = (π (x ij )) are solutions to (6) 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 [5] If equation (7) has a solution in Ξ n then it has a largest one in Ξ n To find the solutions to (6) we only need to solve equations in (7) That is it is sufficient to develop a method to solve each equation in (7) which is simply denoted by A x = b (20) where x R n and b R m stand for X i and B i in (7) respectively Consider the jth equation in (20) which is (a j x ) (a j2 x 2 ) (a jn x n ) = b j (2) where x i and b j are the ith and jth elements of x and b respectively i n j m Using the technique converting a logic expression into an algebraic expression the left hand of (2) can be transformed in vector form as LHS = (M r d )n (M r ca j x ) (M r ca jn x n ) = (M r d )n M r ca j [I r (M r ca j2 )] [I r n (M r ca jn )] n i= x i := L j x (22) where r is described in the above Ξ all the matrix products are the STP is omitted for compactness and L j = (M r d )n M r ca j [I r (M r ca j2 )] [I r n (M r ca jn )] L r r n x = n i= x i Then equation (2) becomes L j x = b j j = 2m (23) Multiplying both sides of m equations of (23) equation (20) can be expressed as L x = b (24) where L = L L 2 L m L r m r n and b = m j= b j ( is Khatri-Rao Product of matrices [28]) Precisely Col i (L) = Col i (L ) Col i (L 2 ) Col i (L m ) i = 2r n Since L is a logic matrix and b Δ r m x Δ r n equation (24) has solutions if and only if Now set b Col(L) (25) Λ = {λ Col λ (L) = b} we then obtain the solution set of (24) as {x λ = δ λ rn λ Λ} Finally we convert x back to (x x n ) Ξ n (x x n ) T is just the solution toequation (20) 23 Problem formulation In practice two sorts of T2 FREs are considered One is commonly used in designing fuzzy controllers (the following Model ); the other one is used for a problem similar to diagnosing diseases by symptoms (the following Model 2) Model Let à F (U V) and B F (U W) be two known type-2 fuzzy relations we are searching a T2 FR X F (V W) such that à X = B (26) Model 2 Let R F (V W) and B F (U W) be two known type-2 fuzzy relations we are searching a T2 FR X F (U V) such that X R = B (27) Recall that the composition is very similar to the usual product of matrices making a transpose on both sides of (27) we then obtain R T X T = B T (28)

6 78 Y Yan et al / Control Theory Tech Vol 2 No 2 pp May 204 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- FREs (described in Subsection 223) 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 Symmetrical-valued T2 FRs and their properties Definition 3 The following discussed type- fuzzy sets are defined in a finite universe of discourse ) In a type- fuzzy set the element having membership grade equal to is called a principal element of the fuzzy set If all the membership grades are not equal to we uniformly scale the membership grades such that the largest one equal to 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- fuzzy sets having one principal element 2) A symmetrical-valued type- fuzzy set is a type- 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) If a type- fuzzy set has 0 (or ) 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 ) 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- fuzzy sets If all the numbers of elements of these symmetrical-valued type- fuzzy sets are less than or equal to r (r is a positive integer) the symmetrical-valued type-2 fuzzy relation is called an r-ary symmetrical-valued type-2 fuzzy relation The type-2 fuzzy relations (2) and (3) in Subsection 2 are both 3-ary symmetrical-valued type-2 fuzzy relations Next we define a partial order on F (U V) Definition 4 Let R = ( μ R (u i v j ) ) F (U V) R 2 = ( μ R 2 (u i v j ) ) F (U V) ) We say R R 2 if μ R (u i v j ) μ R 2 (u i v j ) i = 2m; j = 2n where μ R (u i v j ) μ R 2 (u i v j ) means the principal element of μ R (u i v j ) is greater than or equal to that of μ R 2 (u i v j ) 2) If R R 2 and R R 2 then we say R R 2 3) Let Θ be a subset of F (U V) R Θ is called a maximum (minimum) element if there is no R 2 in Θ such that R 2 R ( R R 2 ) 4) R Θ is called the largest (smallest) element if R R 2 R 2 Θ (or R 2 R R 2 Θ) In order to facilitate calculating the composition of type-2 fuzzy relations we have the following results Theorem 3 Let A = {a a 2 a m } and B = {b b 2 b n } be two sets of real number in [0] Define then A B = {a i b j a i A b j B} A B = {a i b j a i A b j B} min{m n} A B m + n (29) min{m n} A B m + n (30) where A stands for the number of elements of A Proof Proofs of (29) and (30) are very similar We just give the proof of (29) The total number of elements of A and B is m + n The smallest element in A B will be excluded during the process of pair-wise comparison thus we have A B m + n where = holds if a i and b j (i = 2m j = 2n) are of derangement For example a i and b j are arranged as a b a 2 b 2 a m b n Next we prove the left hand of (29) Without loss of generality we assume that m n For each a i A (i = 2m) we have a i B where obviously = holds only if a i b j (j = 2n) which indicates that a i B = a i Therefore we have A B m where = holds if a i b j (j = 2n) for any a i A The proof is completed

7 Y Yan et al / Control Theory Tech Vol 2 No 2 pp May Using mathematical induction Theorem 3 can be easily extended to the case of multiple sets Corollary Let A i = {a i a i2 a im } (i = 2n) be real number sets in [0] Define then n A i = {a k a 2k a nk a ik A i } n A i = {a k a 2k a nk a ik A i } i= i= min{i m i = 2n} min{i m i = 2n} Proposition 2 [0] then n A i ( n i m ) n + i= i= n A i ( n i m ) n + i= i= Let x and y be two real numbers in x (x y) = x x (x y) = x Proof The proof can be done by a straightforward computation Proposition 3 For symmetrical-valued type- fuzzy sets A = f /a + + /a k + + f m /a m (3) B = g /b + + /b l + + g n /b n ) If a b then the entry f /a will appear in A B and will disappear in A B 2)If b n a m then the entry f n /b n will appear in A B and will disappear in A B Proof We only prove the first conclusion the other one can be got by a similar way Since B is a symmetrical-valued type- fuzzy set we have b < b 2 < b l <<b n Because a b it is evident that based on which we have a b < b 2 <<b n a b j = a (j = 2n) According to Proposition 2 we know that ( f g ) ( f g 2 ) f ( f g n ) = f Thus f /a remains unchanged in the pair-wise comparisons that is f /a will appear in A B On the other hand since a b < b 2 <<b n then a b j a (j = 2n) which indicates A B does not contain a Therefore the conclusion follows The following Proposition 4 is an immediate consequence of Proposition 3 Proposition 4 For symmetrical-valued type- fuzzy sets If a m b then A = f /a + + /a k + + f m /a m B = g /b + + /b l + + g n /b n A B = A A B = B Theorem 4 Let A and B be two symmetrical-valued type- fuzzy sets as described in (3) The principal element of A B (or A B) is determined only by the principal elements of A and B Proof The 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 32 Obviously among the pair-wise comparisons the membership grade only occurs at a k b l (or a k b l ) If a k b l (or a k b l ) is unique in A B (or A B) we are done Otherwise say a i b j = a k b l (or a i b j = a k b l ) since ( f i g j ) = the membership grade of a k b l (or a k b l ) is still equal to The conclusion is proved Remark 2 Theorem 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 32 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- FR Then a principal sub-equation model is established for introducing the STP method to solve type-2 FREs 32 Decomposition of type-2 fuzzy relations Definition 5 ) A principal sub-matrix of the type- 2 fuzzy relation () denoted as R p is a fuzzy matrix (type- fuzzy relation) whose elements are the principal

8 80 Y Yan et al / Control Theory Tech Vol 2 No 2 pp May 204 elements of the entries of () with the original order See Example 6 for a demonstration 2) A secondary sub-matrix of the type-2 fuzzy relation () denoted as R s is the remaining part that the principal elements and their membership grades of the entries of () are removed Example 6 gives a demonstration 3) A principal sub-equation of the type-2 fuzzy relation equation (26) is a type- fuzzy relation equation which is constituted by the principal sub-matrices of the corresponding parts in (26) that is à p X p = B p See equations (44) and (5) 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) of X Example 3 Consider the type-2 fuzzy relations R and S in Example Their principal and secondary submatrices are as follows: R p = S p = R s = S s = It is obvious that the solution to the type-2 fuzzy relation equation (26) X is completely determined by its principal sub-matrix X p and secondary sub-matrix X s 322 Solution of type-2 fuzzy relation equations Consider the general type-2 fuzzy relation equation (26) in which à = a a 2 a n a i à X = B a i2 a in a m a m2 a mn X = x x 2 x s x i x i2 x is x n x n2 x ns B = b b 2 b s b i b i2 b is b m b m2 b ms where a ij x ij and b ij are all type- fuzzy sets ie a ij = f ij /a ij + f 2 ij /a2 ij + + f r ij ij /ar ij ij i = 2m; j = 2n (32) x ij = h ij /x ij + h2 ij /x2 ij + + hs ij ij /xs ij ij i = 2n; j = 2s (33) b ij = g ij /b ij + g2 ij /b2 ij + + gt ij ij /bt ij ij i = 2m; j = 2s (34) We have known that to solve equation(26) we only have to solve à X j = B j (35) where X j and B j are the jth columns of X and B respectively That is we only need to consider the following equations which are expansions of equation(35) (a i x j ) (a i2 x 2j ) (a in x nj )=b ij (36) Algorithm To find the solutions to (36) we can take the following steps Step Expand the left hand of (36) using equation (5) The resulting expansion equation (37) is a logical expression of the elements a k ij xl and their membership ij grades f k ij hl which are connected by the logical operators and where k = 2r ij ; l = 2s ij ij where d k α k c k β k k=n c c 2 c n d d 2 d n (37) α k = { a ik x kj a ik x2 kj a ik xs kj kj a r ik ik x kj ar ik ik x2 kj ar ik ik xs kj kj } β k = { f ik h kj f ik h2 kj f ik hs kj kj f r ik ik h kj f r ik ik h2 kj f r ik ik hs kj kj } According to Theorem 3 there are m entries in (37) where m satisfies n p k m n (r ik + s kj ) (38) k= k=

9 Y Yan et al / Control Theory Tech Vol 2 No 2 pp May in which p k = min{r ik s kj } i = 2n; j = 2s Step 2 Collect a t ij -permutation of the entries in (37) and set other entries membership grades to zero (Note that t ij is described in (34) and should be less than m 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 a i x j =b ij ; a7 i4 x7 4j =b2 ij ; ; ad iq xd qj =bt ij ij ; (39) f i h j = g ij ; f 7 i4 h7 4j = g2 ij ; ; f d iq hd qj = gt ij ij ; (40) where d max{r ik r kj k = 2n} Step 3 According to (5) or (24) equations (39) and (40) can be converted into their algebraic forms: where L x = b (4) K y = c (42) x = x j x7 4j xd qj b = b ij b2 ij bt ij ij y = h j h7 4j hd qj c = g ij g2 ij gt ij ij Step 4 Solve the equations (4) and (42) by the STP method which is described in Subsection 223 The obtained solutions is a solution set of equation (36) Step 5 Apply steps -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 3 ) Choosing different permutations in Step 2 will produce different solution sets for equation (35) 2) Since m in (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 33 Solving symmetrical-valued type-2 FREs Algorithm 2 Assume 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 Establish the principal sub-equation of (35) and use the STP method solving type- FREs to find its solutions These solutions are the elements of the principal sub-matrices of X j Step 2 Since equation (35) is a symmetrical-valued one the elements (not the membership grades) of the secondary sub-matrix of X j can be determined according to the solutions obtained in Step Step 3 Establish 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 X j can also be found by the STP method Step 4 Apply steps 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 4 The essence of Algorithms and 2 is that T2 FREs are first decomposed into T FREs by collecting every permutation of the entries of equation (36) all the solutions to these T FREs can be obtained by the Cheng s method in corresponding k-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 that S is 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 4 Consider the following type-2 fuzzy relation equation: à X = B (43) where à =

10 82 Y Yan et al / Control Theory Tech Vol 2 No 2 pp May 204 X= h x h 2 x h3 x 2 x h3 2 x 2 x B= h 2 x 2 h 22 x h3 2 x 2 x h3 22 x 2 x h 3 x 3 h 23 x h3 3 x 2 x h3 23 x 2 x First we establish the principal sub-equation of (43) as where 09 0 Ã p = B p = Ã p X p = B p (44) X p = x 2 x2 2 x2 3 x 2 2 x2 22 x2 23 Using the STP method of solving type- FREs we can obtain the unique solution: S = which is the principal sub-matrix of X Since equation (43) is a symmetrical-valued type-2 one other elements of X can be determined as X= h h3 06 h h h h h h h h3 3 h h (45) Next we will look for the membership grades of X ie the h k s in (45) ij Consider the canonical linear algebraic equations of (43): Ã X i = B i i = 2 3 (46) The first equation in (46) is h h = h h (47) Expanding equation (47) by (5) Propositions 3 will make calculations much easier we have h h3 06 = h h3 06 = h h3 2 = (48) From (48) we can obtain the following equations: h = 04 h3 = h 2 = h3 2 = 06 (49) Solving (49) then we have h = 04 h3 = h 2 0 h Similarly we can obtain the solutions to the second and third equations in (45) which are h 2 = 03 h3 2 = 05 h 2 = 05 h3 2 = h 3 0 h h 23 = 06 h3 23 = 04 Therefore the solution set to equation (43) is X= α 0 + β γ η (50) where 05 α 0 β γ 0 η 05 Comparing (50) with (3) we observe that S in (3) belongs to the solution set (50)

11 Y Yan et al / Control Theory Tech Vol 2 No 2 pp May Example 5 Consider the following symmetricalvalued type-2 fuzzy relation equation: à X = B (5) where the first and second columns of à are Col 2 (Ã) = The third and fourth columns are Col 34 (Ã) = and X= f x f 2 x 2 f 3 x 3 f 4 x f 3 x 2 x f 3 2 x 2 x f 3 3 x 2 x f 3 4 x 2 x B= f 2 x 2 f 22 x 22 f 32 x 32 f 42 x f 3 2 x 2 x f 3 22 x 2 x f 3 32 x 2 x f 3 42 x 2 x f 3 x 3 f 23 x 23 f 33 x 33 f 43 x f 3 3 x 2 x f 3 23 x 2 x f 3 33 x 2 x f 3 43 x 2 x Equation (5) is a 3-ary symmetrical-valued type-2 fuzzy relation equation We use the algorithm proposed in sub-section 33 to solve it Step We establish the principal sub-equation of (5) as à p X p = B p (52) where à p = X p = B p = x 2 x2 2 x2 3 x 2 2 x2 22 x2 23 x 2 3 x2 32 x2 33 x 2 4 x2 42 x2 43 Using the STP method we can obtain each column of the solutions to (52) The first column is X p = [ ] X 2 = [ ] p X 3 p = [ ] X 4 = [ ] p X 5 p = [ ] X 6 = [ ] p X 7 p = [ ] X 8 = [ ] p X 9 p = [ ] X 0 = [ ] p X p = [ ] X 2 = [ ] p X 3 p = [ ] X 4 p X 5 p = [ ] X 6 p X 7 p = [ ] X 8 p X 9 p = [ ] X 20 p X 2 p = [ ] X 22 p = [ ] X 23 p The second column is = [ ] = [ ] = [ ] = [ ] = [ ] X = [ ] X 2 = [ ] X 3 = [ ] X 4 = [[ ] X 5 = [[ ] X 6 = [[ ] X 7 = [[ ] X 8 = [[ ] X 9 = [[ ] X 0 = [ ] X = [ ] X 2 = [ ] X 3 = [ ] X 4 X 5 = [ ] X 6 X 7 = [ ] X 8 = [ ] X 9 The third column is = [ ] = [ ] = [ ] X = [ ] X 2 = [ ] X 3 = [ ] X 4 = [ ] X 5 = [ ] X 6 = [ ]

12 84 Y Yan et al / Control Theory Tech Vol 2 No 2 pp May 204 X 7 = [ ] X 8 = [ ] X 9 = [ ] X 0 = [ ] X = [ ] X 2 = [ ] X 3 = [ ] X 4 = [ ] X 5 = [ ] X 6 = [ ] X 7 = [ ] X 8 = [ ] X 9 = [ ] X 20 = [ ] X 2 = [ ] X 22 = [ ] X 23 = [ ] X 24 = [ ] X 25 = [ ] X 26 = [ ] X 27 = [ ] X 28 = [ ] X 29 = [ ] X 30 = [ ] X 3 = [ ] X 32 = [ ] X 33 = [ ] X 34 = [ ] X 35 = [ ] X 36 = [ ] We find totally = 5732 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 X = X = X = Step 2 Since equation (5) is a symmetrical-valued one for each solution to(52) we can determine the elements of the secondary sub-matrix of (5) Take the largest one X for example X can be partially determined as f f 3 03 f f X= f f f f f f f f 3 22 f f f f f f f f f f 3 33 f f 3 43 (53) Step 3 Establish equations by making the membership grades equal which have the same position in the expansion of (5) where X is replaced by (53) To do this we consider the canonical linear algebraic equations of (5) : Ã X i = B i i = 2 3 (54) The first equation in (54) can be expanded by (5) as follows: f f = f f 3 4 = (55) (09 f ) f ( f 3 f 3 2 ) 03 = By (55) we have the following equations f 3 = 07 f 3 3 = f 4 = f 3 4 = 06 (09 f ) f 2 = 09 (07 f 3 ) f 3 2 = 07 Solving (56) we obtain f 3 = 07 f 3 3 = 06 f 4 06 f min{ f f } 09 2 max{ f 3 f 3 } 07 2 (56) Remark 5 ) It is worthwhile to note that equation (55) may have no solutions For instance if b 2 in B in (5) is 08/05+/06+07/07 there is no h 4 satisfying (55)

13 Y Yan et al / Control Theory Tech Vol 2 No 2 pp May ) We may have to resort to the STP method solving type- FREs to solve (55) when it is too complex to directly find the solution Similarly we can obtain other membership grades of X by solving the second and third equations in (54) which are and f 2 = 06 f 3 2 = 08 f f f 32 f f 42 f f f 33 f 3 33 f 0 43 min{ f 3 f 23 } = max{ f 3 3 f 3 } 0 23 Now for the largest principal solution X that is the solution to the principal sub-equation we find all the solutions for equation (5) All the other solutions can be obtained in a similar way Finally we summarize the solutions to (5) as follows There are totally 5732 principal solutions each of them determines some corresponding secondary submatrices of the solutions The following are some examples the largest solution is X = a minimum one is X = and an ordinary one is X= Remark 6 Finding 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- FREs is introduced to solve type-2 FREs We first propose an r-ary symmetrical-valued type-2 fuzzy relation mode according to how a type- fuzzy relation is extended to a type-2 one To facilitate the calculation of the composition of type-2 fuzzy sets some properties of the r-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 References [] D Cheng Semi-tensor product of matrices and its application to Morgen s problem Science in China Series F: Information Sciences (3): [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 202 [3] H Li Y Wang On reachability and controllability of switched Boolean control networks Automatica (): [4] D Cheng Disturbance decoupling of boolean control networks IEEE Transactions on Automatic Control 20 56(): 2 0 [5] D Cheng H Qi Y Zhao Analysis and control of Boolean networks: a semi-tensor product approach Acta Automatica Sinica 20 37(5):

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