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Convergence, eigen fuzzy sets and stability analysis of relational matrices
FU||Y
sets and systems Fuzzy Sets and Systems 81 (1996) 227 234
ELSEVIER
Convergence, eigen fuzzy sets and stability analysis of relational matrices Mary M. Bourke, D. Grant Fisher* Department of Chemical Engineering, University of Alberta, Edmonton, Canada T6G 2G6 Received March 1994; revised June 1995
Abstract
If stable control applications are formulated with a max-product composition, then instability in the relational matrix under any conditions is undesirable. This paper reviews the stability analysis of relational matrices combined with the max-min composition [19] and then presents an analysis of the stability of relational matrices combined with the max-product composition. This analysis includes results defining the convergence properties of the relational matrix, determination of the eigen fuzzy sets of the stable matrices and algorithmic solutions for several important cases. A method is also presented to stabilize unstable relational matrices for some control applications.
Keywords: Operators; Fuzzy relations; Fuzzy control; Convergence; Eigen fuzzy sets
!. Introduction
In his original work on fuzzy set theory, Zadeh [22] suggested several methods of combining fuzzy sets. The most popular and widely used connectives are maximum (max) for union and minimum (rain) for intersection. Triangular norms (or, t-norms), other than rain, can also be used to model the intersection connective. One such t-norm, also suggested by Zadeh [22], is the algebraic product (product). Oden [9], Thole et al. [18], Zimmermann and Zysno [23], Pedrycz [121, Dubois and Prade [5"1, and, Xu and Lu [21] reported that the rain connector is not always the best choice for the
*Corresponding author. 0165-0114/96/$15.00 ,,~: 1996 Elsevier Science B.V. All rights reserved SSDI 0 1 6 5 - 0 1 1 4 t 9 5 ) 0 0 2 1 2 - X
intersection operation. In the situations referenced, the max-product composition gave results better or equivalent to the max-rain composition. Studies by Gupta and Qi [7] to determine the effects of t-operators on controller performance also show that the max-product composition results in a faster response than the max-min composition. The max-min composition has been used in almost every fuzzy logic controller design, as well as in the modeling of other decision making processes [6]. While much of the current research on fuzzy relational equations deals specifically with the maxrain composition solutions, there is a growing amount of work which generalizes the use of the max-t-norm composition, to which both the maxmin and the max-product belong, for example, see [3, 4, 8, l l, 13, 14].
228
M.M. Bourke, D.G. Fisher / Fuz~ Sets and Systems 81 (1996) 227-234
The areas of convergence and eigen fuzzy sets, as applied to the max-product composition, are of particular interest and concern due to the fact that for some cases the relational matrix, R, may be unstable. That is: LimR" = [0].
(1)
For control applications, degradation of the relational matrix is unacceptable because it results in unstable control. Thus, the ability to avoid or at least to test for such degradation is a requirement for any practical control system. Once stability is established, determination of the family of eigen fuzzy sets of the matrix R, or alternatively determining the range of R required to maintain a desired output, can be considered. This paper analyzes these stability problems, provides algorithmic solutions for several problem scenarios and presents a method of stabilizing an unstable relational matrix for a specific control situation.
If an exact solution to the inverse of (5) exists for the max-rain composition, it can be computed by the algorithm proposed by Pappis and Sugeno [10]. For the max-product composition, the solution algorithm is the same, but the ~- and fl-operators must be redefined as follows: Definition 1. For a and be[0, 1] and the maxproduct composition, the a-operator is redefined as 1 aab=
b/a
if a ~ b , if a > b .
Definition 2. For a and bE[0, 1] and the maxproduct composition the fl-operator is redefined as a fi b
fO b/a
if a < b, if a ~> b.
Alternatively, consider determining the stable fuzzy outputs, y, produced by the control action, Uss. For the first order non-delay system [15]: Yk = R ° Uss ° y k _ 1'
2. The control problem
(6)
Combine Uss and R: Let R(y,y, u) denote a finite (n x n x m) fuzzy relational matrix between the output, Yk, and the past input, Uk- ~, and past output, yk- 1, and let o denote either the max-product or the max-rain composition. Stabilization of the closed-loop system by proper choice of the control action, u, is a key issue for relational-based fuzzy control systems. Consider the f u z z y first order system without delay. Yk = R o Uk- 1 ° Y k - 1.
(7)
Substituting (7) into (6) results in: Yk = P ° Y k - 1'
(8)
(3)
At steady state Yk = Y k - a and the collection of stable output states produced by Us~is the family of all eigen fuzzy sets of P. The theory on convergence and stability of the relational matrix, R, for the max-min composition was presented by Thomason [19]. The theory required to determine the existence of the eigen fuzzy sets for this problem, and if it exists, the greatest element of the same was developed by Sanchez [16], for the max-rain composition. These theories are redeveloped for the max-product composition in the next section.
(4)
3. Convergence theory
(5)
The concept of structure or measure of shape of the fuzzy relational matrix will be utilized in this
(2)
The control action for this fuzzy system which produces an optimal or desired output defined by the fuzzy set Yopt, requires determination of a steady-state control u = u~ such that u~ satisfies the following equation [1]: Yopt -----R o Us s OYopt.
P = R o U~s.
Combining Yopt and R: Q = R °Yop t and substituting (4) into (3) results in: Yop, = Q ° u~s.
M.M. Bourke, D.G. Fisher / Fuzzy Sets and Systems 81 (1996) 227 234
section for the convergence theory and, in the next section, for the theory of eigen fuzzy sets. Tong [20] proposed the position of the peaks in the membership function, which defines the fuzzy set, as a measure of structure and found it to be a satisfactory definition for developing the concepts of stability and controllability. The following definitions, taken from Tong [20], will formalize the concept of a peak pattern for the following theoretical development. Definition 3. A peak pattern, PP, of a fuzzy set x on a finite set X is a binary mapping of the discrete membership function of X and ktx with ifttx#/~ ....
PPx(x)=O =
1
i f ]l x :
],1. . . .
where //max is the maximum value of the membership function (i.e. ]~max = l ) .
229
Definition 8. The powers of R are defined recursively by R 0 ~ 1~
where I is the n x n unit diagonal matrix and R k =
R k-
1 o
for k = 1, 2, 3. . . . .
R,
The work presented by Thomason [19] showed that for the max-rain composition the powers of R: (i) converge to an idempotent W or (ii) oscillate with a finite period z. When the max-product composition is being considered, an element of instability is encountered. The following analysis pertains to the max-product composition. Proposition 1. For a max-product composition, the powers o f R (i) converge to the null matrix [0] or (ii) converge to an idempotent R ~ or (iii) oscillate with a finite period r.
Definition 4. A peak pattern covers an element x~Xif
P P ( x ) = 1.
Definition 5. Two fuzzy sets xl and x 2 are equivalent if they have the same peak pattern. Definition 6. A fuzzy set with a peak pattern which covers only one element in X is called a singular set. Definition 7. A relation R is a maximal relation if each row has at least one element of value 1.
Proof. (i) If rij < I for all i,j <~ n then lim r p ~ 0.
p~c~
(ii) If a dominant factor of 1 is present in the same position in all powers of R, then the matrix will converge to a finite R c. For example, the powers of the diagonal elements are all functions of themselves, so if they are set to the value 1, they will remain at the value 1. ~i = r~i + rljrjirini-2 +
"'"
3.1. Convergence o f the relation matrix, R
Let R denote a finite n x n fuzzy relational matrix. The max-product composition of two square fuzzy matrices is written as a conventional matrix product with fuzzy operations, that is, the multiplication is the same, but the summation is replaced by the max operator. For the max-min composition of two square matrices, the multiplication is replaced by the rain operator and the summation is replaced by the max operator. The analysis for the convergence properties of a matrix, R, applies to matrices which are ~< maximal.
(iii) If the dominate factor, 1, does not occur in the same position in all powers of R, then the matrix will oscillate. For example in a 3 × 3 matrix,
R=
R3=
0
0
0
1
,
0 1
R2=
.
[]
0
1
0
0
,
heLM Bourke, D.G. Fisher / Fuzzy Sets and Systems 81 (1996) 227-234
230
Proposition 2. I f the peak pattern equivalent o f
Proposition 3. I f R >~I then the powers o f R con-
a matrix, R, converges then the matrix R also converges•
verge to R e and R c = adj(R), or r~j = a O.
Proof. Given the n × n matrix R, Proof. In the peak pattern equivalent, all matrix elements < 1 are replaced by zero (0), and matrix multiplication quickly eliminates all non-dominant factors• Only those values of 1 remain that can be sustained in repeated powers. This same elimination of non-dominant factors takes place in the original matrix, R, however, convergence is slower than for the peak pattern equivalent. []
Theorem 1. T h e powers o f a matrix R converge to an idempotent Re f o r a finite c i f there exists at least one i <~n such that r , = l and there does not exist a j, k <~ n such that rjk = rkj = 1, where j, k ~ i.
R=
r21
1
•
rnl
.
rn2
.
"'" .
n
-..
consider p iterations of an ijth element of R and show that these elements are equivalent to the adjoint of R for p sufficiently large. r p : rij -4- ri3 • r 3 j -~ ri4 • i'4j -~ ri5 " r 5 j -t- ...
+ rl." r.~ + (terms ~ O} aij = rlj + ri3"r3j + ri4"r4j + r~5"rsj + "" + r,..'r.j
Proof. Directly from Proposition l(ii) and (iii).
[]
Theorem 2. The powers o f a matrix R oscillate with a finite period z if: (i) there exist an i <<.n such that rii = 1 and there exists a j, k,i <~ n such that r~k = rkj = 1, where j, k ~ i or (ii) the matrix is maximal and there does not exist an i <~ n such that ru -- 1.
Proof. Directly from Proposition l(iii).
[]
Theorem 3. T h e powers o f a matrix, R, converge to the null matrix [0] /f (i) all rij < 1 f o r all i, j <~ n or
(ii) all rii < 1 f o r all i <~ n
• ~
= a~j.
[]
Proposition 4. L e t d be the number o f diagonal l's o f R. I f d is such that 1 < d < n, then the rows and columns o f R ~ which intersect the diagonal l's converge to stable values f o r s <~ n - 1. Proof. Based on the symmetry of the matrix combination, for an n × n matrix, all possible basic combinations of the row and column entries of the diagonal l's have been exhausted by s = n - 1 powers of R. For s ~> n - I all additional combinations are multiples of previously defined combinations, so all future entries for these rows and columns are strictly less than the basic combinations defined for s ~< n - 1. []
and R < maximal.
Proof. Directly from Proposition l(i) and Theorem 2.
[]
Definition 9. The n × n adjoint matrix of R, adj(R), is defined as:
Once the dominant rows and columns have stabilized, the rest of the matrix entries of R converge to their final values with each additional iteration. Thus for any matrix with 1 < d < n the remaining matrix entries of R c are functions of a dominant stable row and stable column of R s. Corollary 1. I f R >>.I then R converges to R c, and c<<.n - 1 .
where aij is an element of the ad](R).
Proof. This 4. []
follows directly from Proposition
M.M. Bourke, D.G. Fisher / Fuzzy Sets and Systems 81 (1996) 227 234
Assume R has a peak pattern that converges, so that R converges.
3.2. Convergence o f a fuzzy set, D The convergence analysis, for the max-product composition, is concerned with the convergence of an initial fuzzy set, D, with elements d~s [0, 1], to a final, non-zero, non-oscillatory state, D ¢.
Definition 10. Define the fuzzy state process as: D 1 =D
D k:R:D
~-l,
fork=2,3
....
Theorem 4. D converges to D ¢, iff R converges to R e. Proof.
~
D 2 :-ROD
1,
D 3 = R O D z - - R z o D I. So D ~+1 = R O O D , Dc+Z=_R~+loD=R
231
e°D
Theorem 5. 1f R converges then DGEvs exists. Proof. F r o m T h e o r e m 4, since D c exists for a matrix R, and since the m a x i m u m value of the elements of D is b o u n d e d (i.e. di ~< 1), then there exists a largest D or DGEVS. [] The greatest eigen fuzzy set can be calculated using a brute force m e t h o d o l o g y starting with the c o l u m n vector Do, = [1 1 .,- 1] t. Repeating the convergence analysis of T h e o r e m 4, when Do, converges to D c, then D c = DGEVS. Since the intermediate values of D i are all decreasing and bounded, and, since D c is the first instance of an eigen fuzzy set, it is the greatest eigen fuzzy set of R. Using the same brute force approach, one can save iterations by noting that any element of DGEVS c a n n o t be greater than the greatest element in the corresponding row of R, so that:
since R has converged to R ~. (di)" = maxrii • . D e+l = O c+2
Dc+ I = R,~ D~ = RC o D
Dc+Z:RoDC+t but D e =
D e+I
= Re+ l o D :
D ~+2
since D has converged to D e • . R e - t = R ~ converges.
for all i ~< n.
i
converges.
[]
The above convergence analysis of D leads directly to the eigen fuzzy analysis covered in the next section.
4. Eigen fuzzy set theory
Continuing with this analysis, i f R is maximal, then D m = D c = D G E F S ----- [-1 1 1 -.- 1] t. Since the brute force m e t h o d o l o g y of determining the 9reatest eigenfuzzy set can be time consuming, a more efficient algorithm for this calculation is presented in the next section along with solution algorithms for several i m p o r t a n t convergence situations.
5, Algorithmic solutions Algorithmic solutions to eigen fuzzy set and convergence problems, for the m a x - p r o d u c t operator, are developed in this section.
Problem 1. Algorithm to determine R e given R.
This section presents the theory associated with the greatest eigen fuzzy set (GEFS) associated with R. An eigen fuzzy set, D, of a relational matrix R is such that:
(i) If R is maximal and the l's are along the diagonal, then R c can be determined directly using P r o p o s i t i o n 3.
D : R ,~D.
R c =- a d j ( R ) .
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M.M. Bourke, D.G. Fisher / Fuzzy Sets and Systems 81 (1996) 227-234
(ii) I f R has only a single diagonal element with the value of 1 and non-diagonal l's such that R converges, then R ~ can be determined as follows: (1) Calculate R"- 1. (2) Calculate R* by deleting all entries from R"except for the row and column that contain a diagonal 1. (3) Calculate R ~ = (R*)c(R*)r, where (R*)r is the row of R* containing the diagonal 1, and, (R*)~ is the column of R* containing the diagonal 1. (iii) Other more general formulations of R result in too many interactions for a simplified solution algorithm. Solutions in these situations are obtained by repeated iteration as discussed in the previous section.
Problem 2. Algorithm to determine D c given D and R (i) If the matrix, R, has only one diagonal 1, then the following algorithm generates the D c for the given D: (1) Calculate R e. (2) Calculate R* by deleting all entries from R 2 except for the row and column that contain a diagonal 1. (3) Calculate c~ = (R*)r°D, where (R*)r is the row of R* containing the diagonal 1. (4) Calculate D e = ~'(R*)c where (g*)¢ is the column of R* containing the diagonal 1. (ii) If the matrix R has more than one diagonal 1, then the D's must be calculated as follows: (1) Calculate R* = R with rows without diagonal l's zeroed. (2) Calculate D* = R* oD. (3) Calculate D ¢ = ROD*.
Problem 3. Algorithm to determine D6Evs given R (i) I f R is maximal, then D c E v s = [1, 1. . . . . 1] t. (ii) If R is non-maximal, then D6EFS is calculated as follows: (1) Calculate R* = R with all columns without diagonal l's zeroed. (2) Calculate D* = max((R*)- x)t. (3) Calculate D6Evs = RoD*. Other eigen fuzzy vectors can be linear multiples of D6EFS. If DGEVS contains only one element with the value of 1, than all other eigen fuzzy vectors are linear multiples of D6EVS.
Problem 4. Algorithm to determine RGEVS given D This is the straight inverse of the problem D =RoD
Rc~vs -- (D ~ D t) -
6. Matrix stabilization for control applications From a control perspective, set-point tracking requires both relational matrix stability and eigen fuzzy set determination, with the latter being impossible if the relational matrix is unstable. As evident from this paper and the work by Thomason [19], the requirements assigned to the relational matrix to ensure stability are rigorous. In many cases, these requirement may be impossible to ensure while maintaining acceptable control performance. So can practical fuzzy logic applications be formulated to meet control objectives and overcome instability caused by the powers of R? The following discussion explains how fuzzy logic applications maintain stability during setpoint tracking and offers a method to ensure stability for control situations when the output must be defuzzifled for a discrete result. In their work on fuzzy learning and identification, Shaw and Krfiger [17] introduced the term leakage and presented a method to overcome the problem that this distortion causes. Leakage occurs when a relational matrix, R, introduces additional non-zero terms into the calculated fuzzy output data that are not present in the actual output data. In these instances, the operations of fuzzification and defuzzification are no longer reciprocal. This phenomenon of leakage necessitates the use of approximate defuzzification techniques, such as max of maximum (MOM) or center of area (COA). To overcome the problem of leakage in their identification algorithm, Shaw and Kriiger [17] suggested defuzzifying the calculated output and then refuzzifying it before applying the value as a state estimator into the first order fuzzy system model. This same methodology can be used to stabilize control systems when the servo requirements of the system generate powers of R. In most identification and control situations, the methods of fuzzification and defuzzification are not
M.M. Bourke, D.G. Fisher / Fuzzy Sets and Systems 81 (1996) 227 234
reciprocal. N o w consider (8) and the stabilization of fuzzy outputs, y, (i.e. Yk = Yk- ~) by the control action Uss. Yk = P"Yk-- 1.
(8)
Feeding back the defuzzified and then refuzzified previously calculated output, Yk-~, eliminates the problem of instability caused by the powers of R, because it prevents increasing powers of R from being calculated since the fuzzified values ofyk and Yk-1 are not equal, The discrete or defuzzified values of yk and Y k - t are equal at steady state so there is no effect from this procedure on the measured output. The presence of system noise can also eliminate this stability problem in practical discrete control applications. Consider (8) with y~p = Yk, and Ysp calculated by the m e t h o d of fuzzification. The value of y k - ~ is the previous actual discrete output, fuzzified by the same m e t h o d as y~p, so there is a possibility that yk-~ = Y s p . In applications with noise, when the actual o u t p u t is tracking in an acceptable region a b o u t the setpoint, the values ofyk andy~p m a y not be equal for a sustained length of time, and therefore the powers of R generation is limited and model instability is prevented. This m e t h o d of defuzzifying and fuzzifying the previous output, Y k - a , prevents both the m a x - p r o d u c t and max-rain compositions from oscillating~ and, prevents the relational matrix of the m a x - p r o d u c t composition from converging to zero,
7. Conclusions C o n t r o l system stability is a concern regardless of the operator used to c o m p o s e the fuzzy sets. The m a x - p r o d u c t composition is often superior to max-rain in performance, however, stability is a key issue since studies indicate that this o p e r a t o r is subject to greater instability problems. This paper identifies those relational matrices which, when c o m p o s e d with the m a x - p r o d u c t operator, are subject to instability or oscillation. Algorithms to determine stable or critical relational matrices, as well as, eigen fuzzy sets, are then developed for the design of stable control systems using the max-
233
p r o d u c t composition. Finally, a m e t h o d of stabilizing the powers of R, for control applications, is presented for those systems which require defuzzifled o u t p u t data. The stabilization is applicable to both the max-rain and the m a x - p r o d u c t compositions.
Acknowledgements The authors gratefully acknowledge N S E R C support while completing this work.
References [1] E. Czogala and W. Pedrycz, Fuzzy rules generation for fuzzy control, Cybernet. Systems 13 {1982) 275 293. [2] A. Di Nola, W. Pedrycz and S. Sessa, Some theoretical aspects of fuzzy relation equations describing fuzzy systems, 1nJorm. Set. 34 (1984) 241 264. [3] A. Di Nola, W. Pedrycz, S. Sessa and W. Pei-Zhuang, Fuzzy relation equations under a class of triangular norms: a survey and new results, Stochastiea 8 (1984) 99-145. [4] A. Di Nola, S. Sessa, W. Pedrycz and E. Sanchez, Fuzzy Relational Equations and Their Applications to Knowled~te Enqineerin9 (Kluwer Academic, Norwell, 19891.
[5] D. Dubois and H. Prade, New results about properties and semantics of fuzzy set-theoretic operators, in: P.P. Wang and S.K. Chang, Eds,, Fuzzy Sets (Plenum Press, New York, 1986) 59 75. [6] M.M. Gupta and J. Qi, Theory of l-norms and fuzzy inference methods, Fuzzy Sets and Systems 40 (1991) 431~451. [7] M.M. Gupta and J. Qi, Design of fuzzy logic controllers based on generalized t-operators, Fuzzy Sets and Systems 40 11991)473~486. [8] M. Miyakoshi and M. Shimbo, Solutions of composite fuzzy relational equations with triangular norms. Fuzzy Sets and Systems 16 (1985) 53 63. [9] G.C. Oden, Integration of fuzzy logical information, J. Experiment. Psycholoqy (Human Pereeption aml PetJormance) 106 {1977) 565 575.
[10] C.P. Pappis and M. Sugeno, Fuzzy relational equations and the inverse problem, Fuzzy Sets and Systems 15 (1985) 79-90. [11] W. Pedrycz, Some applicational aspects of fuzzy relational equations in systems analysis, lnternat..I. General Systems 9(1983) 125 132. [12] W. Pedrycz, An identification algorithm in fuzzy relation systems, Fuzzy Sets and Systems 13 (1984) 153 167. [13] W. Pedrycz, Identification in fuzzy systems. IEEE Trans. Systems Man Cybernet. 14 (1984) 361 366.
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M.M. Bourke, D.G. Fisher / Fuzzy Sets and Systems 81 (1996) 227-234
[14] W. Pedrycz, On generalized fuzzy relational equations and their applications, J. Math. Anal. Appl. 107 (1885) 520-536. [15] W. Pedrycz, FuzzyControlandFuzzySystems(Wiley, New York, 1989). [16] E. Sanchez, Resolution ofeigen fuzzy sets equations, Fuzzy Sets and Systems 1 (1978) 69-74. [17] I.S. Shaw and J.J. Kriiger, New fuzzy learning model with recursive estimation for dynamic systems, Fuzzy Sets and Systems 48 (1992) 217~229. [18] U. Thole, H.-J. Zimmermann and P. Zysno, On the suitability of minimum and product operators for intersection of fuzzy sets, Fuzzy Sets and Systems 2 (1979) 167-180.
[19] M.G. Thomason, Convergence of powers of a fuzzy matrix, J. Math. Anal. Appl. 57 (1977) 476-480. [20] R.M, Tong, Analysis and control of fuzzy systems using finite discrete relations, lnternat. J. Control 27 (1978) 431-440. [21] C.W. Xu and Y.-Z. Lu, Fuzzy model identification and self-learning for dynamic systems, IEEE Trans. Systems, Man Cybernet. 17 (1987) 683-689. [22] L.A. Zadeh, Fuzzy Sets, Inform. and Control 8 (1965) 338-353. [23] H.-J. Zimmermann and P. Zysno, Latent connectives in human decision-making, Fuzzy Sets and Systems 4 (1980) 37 51.
sets and systems Fuzzy Sets and Systems 81 (1996) 227 234
ELSEVIER
Convergence, eigen fuzzy sets and stability analysis of relational matrices Mary M. Bourke, D. Grant Fisher* Department of Chemical Engineering, University of Alberta, Edmonton, Canada T6G 2G6 Received March 1994; revised June 1995
Abstract
If stable control applications are formulated with a max-product composition, then instability in the relational matrix under any conditions is undesirable. This paper reviews the stability analysis of relational matrices combined with the max-min composition [19] and then presents an analysis of the stability of relational matrices combined with the max-product composition. This analysis includes results defining the convergence properties of the relational matrix, determination of the eigen fuzzy sets of the stable matrices and algorithmic solutions for several important cases. A method is also presented to stabilize unstable relational matrices for some control applications.
Keywords: Operators; Fuzzy relations; Fuzzy control; Convergence; Eigen fuzzy sets
!. Introduction
In his original work on fuzzy set theory, Zadeh [22] suggested several methods of combining fuzzy sets. The most popular and widely used connectives are maximum (max) for union and minimum (rain) for intersection. Triangular norms (or, t-norms), other than rain, can also be used to model the intersection connective. One such t-norm, also suggested by Zadeh [22], is the algebraic product (product). Oden [9], Thole et al. [18], Zimmermann and Zysno [23], Pedrycz [121, Dubois and Prade [5"1, and, Xu and Lu [21] reported that the rain connector is not always the best choice for the
*Corresponding author. 0165-0114/96/$15.00 ,,~: 1996 Elsevier Science B.V. All rights reserved SSDI 0 1 6 5 - 0 1 1 4 t 9 5 ) 0 0 2 1 2 - X
intersection operation. In the situations referenced, the max-product composition gave results better or equivalent to the max-rain composition. Studies by Gupta and Qi [7] to determine the effects of t-operators on controller performance also show that the max-product composition results in a faster response than the max-min composition. The max-min composition has been used in almost every fuzzy logic controller design, as well as in the modeling of other decision making processes [6]. While much of the current research on fuzzy relational equations deals specifically with the maxrain composition solutions, there is a growing amount of work which generalizes the use of the max-t-norm composition, to which both the maxmin and the max-product belong, for example, see [3, 4, 8, l l, 13, 14].
228
M.M. Bourke, D.G. Fisher / Fuz~ Sets and Systems 81 (1996) 227-234
The areas of convergence and eigen fuzzy sets, as applied to the max-product composition, are of particular interest and concern due to the fact that for some cases the relational matrix, R, may be unstable. That is: LimR" = [0].
(1)
For control applications, degradation of the relational matrix is unacceptable because it results in unstable control. Thus, the ability to avoid or at least to test for such degradation is a requirement for any practical control system. Once stability is established, determination of the family of eigen fuzzy sets of the matrix R, or alternatively determining the range of R required to maintain a desired output, can be considered. This paper analyzes these stability problems, provides algorithmic solutions for several problem scenarios and presents a method of stabilizing an unstable relational matrix for a specific control situation.
If an exact solution to the inverse of (5) exists for the max-rain composition, it can be computed by the algorithm proposed by Pappis and Sugeno [10]. For the max-product composition, the solution algorithm is the same, but the ~- and fl-operators must be redefined as follows: Definition 1. For a and be[0, 1] and the maxproduct composition, the a-operator is redefined as 1 aab=
b/a
if a ~ b , if a > b .
Definition 2. For a and bE[0, 1] and the maxproduct composition the fl-operator is redefined as a fi b
fO b/a
if a < b, if a ~> b.
Alternatively, consider determining the stable fuzzy outputs, y, produced by the control action, Uss. For the first order non-delay system [15]: Yk = R ° Uss ° y k _ 1'
2. The control problem
(6)
Combine Uss and R: Let R(y,y, u) denote a finite (n x n x m) fuzzy relational matrix between the output, Yk, and the past input, Uk- ~, and past output, yk- 1, and let o denote either the max-product or the max-rain composition. Stabilization of the closed-loop system by proper choice of the control action, u, is a key issue for relational-based fuzzy control systems. Consider the f u z z y first order system without delay. Yk = R o Uk- 1 ° Y k - 1.
(7)
Substituting (7) into (6) results in: Yk = P ° Y k - 1'
(8)
(3)
At steady state Yk = Y k - a and the collection of stable output states produced by Us~is the family of all eigen fuzzy sets of P. The theory on convergence and stability of the relational matrix, R, for the max-min composition was presented by Thomason [19]. The theory required to determine the existence of the eigen fuzzy sets for this problem, and if it exists, the greatest element of the same was developed by Sanchez [16], for the max-rain composition. These theories are redeveloped for the max-product composition in the next section.
(4)
3. Convergence theory
(5)
The concept of structure or measure of shape of the fuzzy relational matrix will be utilized in this
(2)
The control action for this fuzzy system which produces an optimal or desired output defined by the fuzzy set Yopt, requires determination of a steady-state control u = u~ such that u~ satisfies the following equation [1]: Yopt -----R o Us s OYopt.
P = R o U~s.
Combining Yopt and R: Q = R °Yop t and substituting (4) into (3) results in: Yop, = Q ° u~s.
M.M. Bourke, D.G. Fisher / Fuzzy Sets and Systems 81 (1996) 227 234
section for the convergence theory and, in the next section, for the theory of eigen fuzzy sets. Tong [20] proposed the position of the peaks in the membership function, which defines the fuzzy set, as a measure of structure and found it to be a satisfactory definition for developing the concepts of stability and controllability. The following definitions, taken from Tong [20], will formalize the concept of a peak pattern for the following theoretical development. Definition 3. A peak pattern, PP, of a fuzzy set x on a finite set X is a binary mapping of the discrete membership function of X and ktx with ifttx#/~ ....
PPx(x)=O =
1
i f ]l x :
],1. . . .
where //max is the maximum value of the membership function (i.e. ]~max = l ) .
229
Definition 8. The powers of R are defined recursively by R 0 ~ 1~
where I is the n x n unit diagonal matrix and R k =
R k-
1 o
for k = 1, 2, 3. . . . .
R,
The work presented by Thomason [19] showed that for the max-rain composition the powers of R: (i) converge to an idempotent W or (ii) oscillate with a finite period z. When the max-product composition is being considered, an element of instability is encountered. The following analysis pertains to the max-product composition. Proposition 1. For a max-product composition, the powers o f R (i) converge to the null matrix [0] or (ii) converge to an idempotent R ~ or (iii) oscillate with a finite period r.
Definition 4. A peak pattern covers an element x~Xif
P P ( x ) = 1.
Definition 5. Two fuzzy sets xl and x 2 are equivalent if they have the same peak pattern. Definition 6. A fuzzy set with a peak pattern which covers only one element in X is called a singular set. Definition 7. A relation R is a maximal relation if each row has at least one element of value 1.
Proof. (i) If rij < I for all i,j <~ n then lim r p ~ 0.
p~c~
(ii) If a dominant factor of 1 is present in the same position in all powers of R, then the matrix will converge to a finite R c. For example, the powers of the diagonal elements are all functions of themselves, so if they are set to the value 1, they will remain at the value 1. ~i = r~i + rljrjirini-2 +
"'"
3.1. Convergence o f the relation matrix, R
Let R denote a finite n x n fuzzy relational matrix. The max-product composition of two square fuzzy matrices is written as a conventional matrix product with fuzzy operations, that is, the multiplication is the same, but the summation is replaced by the max operator. For the max-min composition of two square matrices, the multiplication is replaced by the rain operator and the summation is replaced by the max operator. The analysis for the convergence properties of a matrix, R, applies to matrices which are ~< maximal.
(iii) If the dominate factor, 1, does not occur in the same position in all powers of R, then the matrix will oscillate. For example in a 3 × 3 matrix,
R=
R3=
0
0
0
1
,
0 1
R2=
.
[]
0
1
0
0
,
heLM Bourke, D.G. Fisher / Fuzzy Sets and Systems 81 (1996) 227-234
230
Proposition 2. I f the peak pattern equivalent o f
Proposition 3. I f R >~I then the powers o f R con-
a matrix, R, converges then the matrix R also converges•
verge to R e and R c = adj(R), or r~j = a O.
Proof. Given the n × n matrix R, Proof. In the peak pattern equivalent, all matrix elements < 1 are replaced by zero (0), and matrix multiplication quickly eliminates all non-dominant factors• Only those values of 1 remain that can be sustained in repeated powers. This same elimination of non-dominant factors takes place in the original matrix, R, however, convergence is slower than for the peak pattern equivalent. []
Theorem 1. T h e powers o f a matrix R converge to an idempotent Re f o r a finite c i f there exists at least one i <~n such that r , = l and there does not exist a j, k <~ n such that rjk = rkj = 1, where j, k ~ i.
R=
r21
1
•
rnl
.
rn2
.
"'" .
n
-..
consider p iterations of an ijth element of R and show that these elements are equivalent to the adjoint of R for p sufficiently large. r p : rij -4- ri3 • r 3 j -~ ri4 • i'4j -~ ri5 " r 5 j -t- ...
+ rl." r.~ + (terms ~ O} aij = rlj + ri3"r3j + ri4"r4j + r~5"rsj + "" + r,..'r.j
Proof. Directly from Proposition l(ii) and (iii).
[]
Theorem 2. The powers o f a matrix R oscillate with a finite period z if: (i) there exist an i <<.n such that rii = 1 and there exists a j, k,i <~ n such that r~k = rkj = 1, where j, k ~ i or (ii) the matrix is maximal and there does not exist an i <~ n such that ru -- 1.
Proof. Directly from Proposition l(iii).
[]
Theorem 3. T h e powers o f a matrix, R, converge to the null matrix [0] /f (i) all rij < 1 f o r all i, j <~ n or
(ii) all rii < 1 f o r all i <~ n
• ~
= a~j.
[]
Proposition 4. L e t d be the number o f diagonal l's o f R. I f d is such that 1 < d < n, then the rows and columns o f R ~ which intersect the diagonal l's converge to stable values f o r s <~ n - 1. Proof. Based on the symmetry of the matrix combination, for an n × n matrix, all possible basic combinations of the row and column entries of the diagonal l's have been exhausted by s = n - 1 powers of R. For s ~> n - I all additional combinations are multiples of previously defined combinations, so all future entries for these rows and columns are strictly less than the basic combinations defined for s ~< n - 1. []
and R < maximal.
Proof. Directly from Proposition l(i) and Theorem 2.
[]
Definition 9. The n × n adjoint matrix of R, adj(R), is defined as:
Once the dominant rows and columns have stabilized, the rest of the matrix entries of R converge to their final values with each additional iteration. Thus for any matrix with 1 < d < n the remaining matrix entries of R c are functions of a dominant stable row and stable column of R s. Corollary 1. I f R >>.I then R converges to R c, and c<<.n - 1 .
where aij is an element of the ad](R).
Proof. This 4. []
follows directly from Proposition
M.M. Bourke, D.G. Fisher / Fuzzy Sets and Systems 81 (1996) 227 234
Assume R has a peak pattern that converges, so that R converges.
3.2. Convergence o f a fuzzy set, D The convergence analysis, for the max-product composition, is concerned with the convergence of an initial fuzzy set, D, with elements d~s [0, 1], to a final, non-zero, non-oscillatory state, D ¢.
Definition 10. Define the fuzzy state process as: D 1 =D
D k:R:D
~-l,
fork=2,3
....
Theorem 4. D converges to D ¢, iff R converges to R e. Proof.
~
D 2 :-ROD
1,
D 3 = R O D z - - R z o D I. So D ~+1 = R O O D , Dc+Z=_R~+loD=R
231
e°D
Theorem 5. 1f R converges then DGEvs exists. Proof. F r o m T h e o r e m 4, since D c exists for a matrix R, and since the m a x i m u m value of the elements of D is b o u n d e d (i.e. di ~< 1), then there exists a largest D or DGEVS. [] The greatest eigen fuzzy set can be calculated using a brute force m e t h o d o l o g y starting with the c o l u m n vector Do, = [1 1 .,- 1] t. Repeating the convergence analysis of T h e o r e m 4, when Do, converges to D c, then D c = DGEVS. Since the intermediate values of D i are all decreasing and bounded, and, since D c is the first instance of an eigen fuzzy set, it is the greatest eigen fuzzy set of R. Using the same brute force approach, one can save iterations by noting that any element of DGEVS c a n n o t be greater than the greatest element in the corresponding row of R, so that:
since R has converged to R ~. (di)" = maxrii • . D e+l = O c+2
Dc+ I = R,~ D~ = RC o D
Dc+Z:RoDC+t but D e =
D e+I
= Re+ l o D :
D ~+2
since D has converged to D e • . R e - t = R ~ converges.
for all i ~< n.
i
converges.
[]
The above convergence analysis of D leads directly to the eigen fuzzy analysis covered in the next section.
4. Eigen fuzzy set theory
Continuing with this analysis, i f R is maximal, then D m = D c = D G E F S ----- [-1 1 1 -.- 1] t. Since the brute force m e t h o d o l o g y of determining the 9reatest eigenfuzzy set can be time consuming, a more efficient algorithm for this calculation is presented in the next section along with solution algorithms for several i m p o r t a n t convergence situations.
5, Algorithmic solutions Algorithmic solutions to eigen fuzzy set and convergence problems, for the m a x - p r o d u c t operator, are developed in this section.
Problem 1. Algorithm to determine R e given R.
This section presents the theory associated with the greatest eigen fuzzy set (GEFS) associated with R. An eigen fuzzy set, D, of a relational matrix R is such that:
(i) If R is maximal and the l's are along the diagonal, then R c can be determined directly using P r o p o s i t i o n 3.
D : R ,~D.
R c =- a d j ( R ) .
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M.M. Bourke, D.G. Fisher / Fuzzy Sets and Systems 81 (1996) 227-234
(ii) I f R has only a single diagonal element with the value of 1 and non-diagonal l's such that R converges, then R ~ can be determined as follows: (1) Calculate R"- 1. (2) Calculate R* by deleting all entries from R"except for the row and column that contain a diagonal 1. (3) Calculate R ~ = (R*)c(R*)r, where (R*)r is the row of R* containing the diagonal 1, and, (R*)~ is the column of R* containing the diagonal 1. (iii) Other more general formulations of R result in too many interactions for a simplified solution algorithm. Solutions in these situations are obtained by repeated iteration as discussed in the previous section.
Problem 2. Algorithm to determine D c given D and R (i) If the matrix, R, has only one diagonal 1, then the following algorithm generates the D c for the given D: (1) Calculate R e. (2) Calculate R* by deleting all entries from R 2 except for the row and column that contain a diagonal 1. (3) Calculate c~ = (R*)r°D, where (R*)r is the row of R* containing the diagonal 1. (4) Calculate D e = ~'(R*)c where (g*)¢ is the column of R* containing the diagonal 1. (ii) If the matrix R has more than one diagonal 1, then the D's must be calculated as follows: (1) Calculate R* = R with rows without diagonal l's zeroed. (2) Calculate D* = R* oD. (3) Calculate D ¢ = ROD*.
Problem 3. Algorithm to determine D6Evs given R (i) I f R is maximal, then D c E v s = [1, 1. . . . . 1] t. (ii) If R is non-maximal, then D6EFS is calculated as follows: (1) Calculate R* = R with all columns without diagonal l's zeroed. (2) Calculate D* = max((R*)- x)t. (3) Calculate D6Evs = RoD*. Other eigen fuzzy vectors can be linear multiples of D6EFS. If DGEVS contains only one element with the value of 1, than all other eigen fuzzy vectors are linear multiples of D6EVS.
Problem 4. Algorithm to determine RGEVS given D This is the straight inverse of the problem D =RoD
Rc~vs -- (D ~ D t) -
6. Matrix stabilization for control applications From a control perspective, set-point tracking requires both relational matrix stability and eigen fuzzy set determination, with the latter being impossible if the relational matrix is unstable. As evident from this paper and the work by Thomason [19], the requirements assigned to the relational matrix to ensure stability are rigorous. In many cases, these requirement may be impossible to ensure while maintaining acceptable control performance. So can practical fuzzy logic applications be formulated to meet control objectives and overcome instability caused by the powers of R? The following discussion explains how fuzzy logic applications maintain stability during setpoint tracking and offers a method to ensure stability for control situations when the output must be defuzzifled for a discrete result. In their work on fuzzy learning and identification, Shaw and Krfiger [17] introduced the term leakage and presented a method to overcome the problem that this distortion causes. Leakage occurs when a relational matrix, R, introduces additional non-zero terms into the calculated fuzzy output data that are not present in the actual output data. In these instances, the operations of fuzzification and defuzzification are no longer reciprocal. This phenomenon of leakage necessitates the use of approximate defuzzification techniques, such as max of maximum (MOM) or center of area (COA). To overcome the problem of leakage in their identification algorithm, Shaw and Kriiger [17] suggested defuzzifying the calculated output and then refuzzifying it before applying the value as a state estimator into the first order fuzzy system model. This same methodology can be used to stabilize control systems when the servo requirements of the system generate powers of R. In most identification and control situations, the methods of fuzzification and defuzzification are not
M.M. Bourke, D.G. Fisher / Fuzzy Sets and Systems 81 (1996) 227 234
reciprocal. N o w consider (8) and the stabilization of fuzzy outputs, y, (i.e. Yk = Yk- ~) by the control action Uss. Yk = P"Yk-- 1.
(8)
Feeding back the defuzzified and then refuzzified previously calculated output, Yk-~, eliminates the problem of instability caused by the powers of R, because it prevents increasing powers of R from being calculated since the fuzzified values ofyk and Yk-1 are not equal, The discrete or defuzzified values of yk and Y k - t are equal at steady state so there is no effect from this procedure on the measured output. The presence of system noise can also eliminate this stability problem in practical discrete control applications. Consider (8) with y~p = Yk, and Ysp calculated by the m e t h o d of fuzzification. The value of y k - ~ is the previous actual discrete output, fuzzified by the same m e t h o d as y~p, so there is a possibility that yk-~ = Y s p . In applications with noise, when the actual o u t p u t is tracking in an acceptable region a b o u t the setpoint, the values ofyk andy~p m a y not be equal for a sustained length of time, and therefore the powers of R generation is limited and model instability is prevented. This m e t h o d of defuzzifying and fuzzifying the previous output, Y k - a , prevents both the m a x - p r o d u c t and max-rain compositions from oscillating~ and, prevents the relational matrix of the m a x - p r o d u c t composition from converging to zero,
7. Conclusions C o n t r o l system stability is a concern regardless of the operator used to c o m p o s e the fuzzy sets. The m a x - p r o d u c t composition is often superior to max-rain in performance, however, stability is a key issue since studies indicate that this o p e r a t o r is subject to greater instability problems. This paper identifies those relational matrices which, when c o m p o s e d with the m a x - p r o d u c t operator, are subject to instability or oscillation. Algorithms to determine stable or critical relational matrices, as well as, eigen fuzzy sets, are then developed for the design of stable control systems using the max-
233
p r o d u c t composition. Finally, a m e t h o d of stabilizing the powers of R, for control applications, is presented for those systems which require defuzzifled o u t p u t data. The stabilization is applicable to both the max-rain and the m a x - p r o d u c t compositions.
Acknowledgements The authors gratefully acknowledge N S E R C support while completing this work.
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