A unified parameterized formulation of reasoning in fuzzy modeling and control

Fuzzy Sets and Systems 108 (1999) 59–81 www.elsevier.com/locate/fss

A uni ed parameterized formulation of reasoning in fuzzy modeling and control Mohammad R. Emami a ; ∗ , I. Burhan Turksen b , Andrew A. Goldenberg c a Robotics

and Automation Laboratory, Department of Mechanical & Industrial Engineering, University of Toronto, 5 King’s College Road, Toronto, Ont., Canada M5S 3G8 b Intelligent Fuzzy Systems Laboratory, Department of Mechanical & Industrial Engineering, University of Toronto, Toronto, Ont., Canada M5S 1A1 c Robotics and Automation Laboratory, Department of Mechanical & Industrial Engineering, University of Toronto, Toronto, Ont., Canada M5S 1A1 Received May 1996; received in revised form March 1998

Abstract A uni ed parameterized formulation for the reasoning process in fuzzy modeling and control is developed in this paper. First, by selecting a suitable parameterized family of triangular functions and extending to the n-ary operation, the parameterized form of Mamdani’s approximation and formal logical reasoning approaches is introduced. Next, by unifying the inference mechanism for both approaches, a uni ed parameterized reasoning function is developed. It is also proved that for crisp input variables, the two methods of inference from a set of rules, i.e., rst-aggregate-theninfer (FATI) and rst-infer-then-aggregate (FITA), generate identical fuzzy outputs. The proposed reasoning formulation introduces four reasoning parameters. Depending on these parameters, the reasoning operation varies continuously among the extreme cases in each step of the inference. In order to reduce the computational eort, a fast algorithm for computing the parameterized family of triangular functions is suggested. Further, a simpli ed parameterized reasoning formulation is also suggested in which the defuzzi ed output can be calculated directly from the individual consequent fuzzy sets. This simpli ed formulation is comparable with Sugeno’s and Yager’s heuristic simpli ed reasoning functions. Some examples c 1999 Elsevier Science B.V. All rights reserved. demonstrate the validity of the results. Keywords: Fuzzy set theory; Approximate reasoning; Fuzzy systems; Fuzzy modeling

1. Introduction Fuzzy systems are generally referred to those resulting from fuzziÿcation of a conventional system such that the essential features of the system are represented by the apparatus of the fuzzy set theory. The decisionmaking capability of the fuzzy model depends on the rule base and the type of the fuzzy reasoning mechanism. ∗

Corresponding author. E-mail: [email protected].

c 1999 Elsevier Science B.V. All rights reserved. 0165-0114/99/$ – see front matter PII: S 0 1 6 5 - 0 1 1 4 ( 9 8 ) 0 0 0 7 6 - 1

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In the most general form, the encoded knowledge of a multi-input–multi-output (MIMO) system can be interpreted by a fuzzy model consisting of IF–THEN rules with multi-antecedent and multi-consequent variables as follows (with r antecedents, s consequents, and n rules): IF U1 is B11 AND U2 is B12 AND : : : AND Ur is B1r THEN V1 is D11 AND V2 is D12 AND : : : AND Vs is D1s ALSO :::

(1)

ALSO IF U1 is Bn1 AND U2 is Bn2 AND : : : AND Ur is Bnr THEN V1 is Dn1 AND V2 is Dn2 AND : : : AND Vs is Dns ; where U1 ; U2 ; : : : ; Ur are input variables, and V1 ; V2 ; : : : ; Vs are output variables; Bij (i = 1; : : : ; n; j = 1; : : : ; r) and Dik (i = 1; : : : ; n; k = 1; : : : ; s) are fuzzy sets of the universes of discourse X1 ; X2 ; : : : ; Xr and Y1 ; Y2 ; : : : ; Ys of input and output variables, respectively. The set of rules operating with linguistic values of input–output variables appears as an analogy to the system of equations used for expressing linear and nonlinear systems; the fuzzy sets Bij ’s and Dik ’s contain parameters of the fuzzy model. Conceptually, a system with multiple independent outputs can be considered as a collection of several groups of single output systems. Consequently, the general structure of a MIMO fuzzy system can also be presented as a collection of multi-input–single-output (MISO) fuzzy systems so that for a system with s outputs, each multi-consequent rule is broken into s single-consequent rules. Although the number of rules in the new fuzzy system will be increased, modeling and inference would be more straightforward for MISO fuzzy systems. That is the reason why the literature concentrates on multi-input single-output rules as a generic presentation of fuzzy systems. One major step in fuzzy-logic modeling is to decide about the reasoning mechanism. The process of reasoning in fuzzy modeling proceeds through the following steps: (1) fuzzy aggregation of antecedents in each rule (AND connective), (2) implication relation for each individual rule (IF–THEN connective), (3) aggregation of the rules (ALSO connective), (4) inference from the set of rules, using the crisp input to obtain the fuzzy output, (5) defuzzi cation of the output. In the current methods of fuzzy modeling, the connectives in all of the above steps are selected a priori to the modeling procedure without any theoretical basis. The number of selections are limited to a few known aggregation and implication operators [2]. In this paper, a general and uni ed framework for the reasoning process is developed. In Section 2, we start from the basic elements of reasoning, the connective operators (AND, ALSO, and IF–THEN), and adopt a special parameterized family of triangular functions which, due to its simplicity and symmetry, is appropriate for our purpose. We extend the binary operations of connectives to n-ary operations, and prove the validity of De Morgan laws for n-ary operations. In Sections 3 and 4, we derive a parametric formulation of fuzzy rule implication (step 2) and rule aggregation (step 3), respectively. In Section 5, for systems with crisp inputs, which is the case in most applications of fuzzy modeling and control, we rst illustrate the property of distributivity of triangular functions, and then introduce a uni ed reasoning mechanism for two extreme reasoning formulations that leads us to a uni ed parameterized fuzzy reasoning method. In Section 6, a parameterized formulation is also introduced for the defuzzi cation step. As a result, four reasoning parameters p; q; ; and ÿ are introduced whose variation will cause a continuous range of variations for reasoning mechanisms. Therefore, we are no longer restricted to the extremes in any

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step of the reasoning process, but in each case, it is the system itself that speci es what combination of the above parameters is more appropriate for expressing its behavior. Reasoning parameters can be optimized based on the input–output data obtained from the system [7]. In order to reduce the computational eort, in Section 2, a fast algorithm is suggested for the calculation of the parameterized family of triangular functions. Further, in Section 7, by providing an approximation of this family of functions, a simpli ed parameterized reasoning formulation is derived in which the defuzzi ed output can be calculated directly from the individual consequent fuzzy sets of the rule set. The validity of the proposed formulation and its simpli ed version is illustrated through several examples. The work is concluded in Section 8.

2. Fuzzy connectives and fuzzy aggregation At the computational level of expressions, natural linguistic connectives “and ”, “or”, and “not” are transformed into algebraic functions such as “Min”, Max”, or in a general form, triangular norm (t-norm), triangular conorm (t-conorm), and complementation operators [17]. In classical set theory, these connectives are de ned in a unique way due to the two-valued logical operations. However, when the degree of membership to a set is no longer a value from the set {0; 1} but from the interval [0; 1], as it is the case in fuzzy sets, then the interpretation of logical connectives is neither unique nor so obvious. By introducing suitable axioms, newer interpretations of logical connectives are introduced which cover a wide range of suggested expressions [18]. In this paper, in the spirit of these new interpretations, we investigate a parametric class of t-norm and t-conorm operators. Furthermore, through the steps of reasoning we will present a fully parametric structure for reasoning mechanism as a uni ed framework. In what follows, we use computational level of expressions to interpret properties and relations of the connective operators. At this level, what we deal with are membership functions such as a; b; c; etc., which are de ned as For each x ∈ X : a(x) : X → [0; 1]; where X is the universe of discourse: 2.1. Fuzzy complement Bellman and Giertz [1] suggest the following as natural axioms for a negation operator c: C1: c(0) = 1; c(1) = 0, C2: c is strictly decreasing and continuous mapping, C3: c is involutive, i.e., c(c(a)) = a for a ∈ [0; 1]. A negation function is called strong if it satis es all three axioms. A suitable parametric strong negation is suggested by Yager [23]: c(a) = (1 − a q )1=q ;

q¿0

(2)

for q = 1, the negation function is the standard complement of crisp set theory which was also initially adopted for fuzzy set theory. Our further analysis is con ned to the standard complement operator. In classical set theory, complement operator relates union and intersection operators according to De Morgan laws c(∩ (a; b)) = ∪(c(a); c(b))

and

c(∪(a; b)) = ∩(c(a); c(b)):

In fuzzy set theory, we would also like to maintain the above relationships. Therefore, fuzzy set intersection and union operators are de ned mutually in order to satisfy De Morgan laws.

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2.2. Fuzzy set intersection and union The properties that are generally expected to be satis ed by commutative, associative class of intersection– union operators are identi ed by the following axioms [14]: F1: Boundary. ∩(a; 1) = a;

∩(a; 0) = 0

for all a in [0; 1];

∪(a; 0) = a;

∪(a; 1) = 1

for all a in [0; 1]:

F2: Commutativity. ∩(a; b) = ∩(b; a)

for all a; b in [0; 1];

∪(a; b) = ∪(b; a)

for all a; b in [0; 1]:

F3: Associativity. ∩(∩(a; b); c) = ∩(a; ∩(b; c))

for all a; b in [0; 1];

∪(∪(a; b); c) = ∪(a; ∪(b; c))

for all a; b in [0; 1]:

F4: Monotonicity. If a¿a0 and b¿b0 then: ∩(a; b)¿∩(a0 ; b);

∩(a; b)¿∩(a; b0 )

for all a; a0 ; b; b0 in [0; 1];

∪(a; b)¿∪(a0 ; b);

∪(a; b)¿∪(a; b0 )

for all a; a0 ; b; b0 in [0; 1]:

The above four properties have been used to characterize the functions t-norms, T : [0; 1] × [0; 1] → [0; 1], and t-conorms, S: [0; 1] × [0; 1] → [0; 1], which in turn de ne the commutative and associative class of conjuction– disjunction operators. Such functions are known as triangular norms and triangular conorms, respectively, introduced independently from the fuzzy set theory by Schweizer and Sklar [13] in the context of statistical metric spaces. The only distinction between T and S operators is in condition F1 in which the unit is 1 for t-norm and 0 for t-conorm. In order to cover various types, the parameterized families of t-norms and t-conorms have been suggested among which the Schweizer and Sklar’s operators [12] are adopted for further investigation T (a; b) = 1 − [(1 − a)p + (1 − b)p − (1 − a)p (1 − b)p ]1=p ; S(a; b) = [ap + bp − ap bp ]1=p ;

p¿0:

The extreme cases are when p tends to 0, 1, and in nity ( T (a; b) = Tmin (a; b) = min(a; b) = a ∧ b; Zadeh’s Operators p → ∞: S(a; b) = Smax (a; b) = max(a; b) = a ∨ b; ( T (a; b) = Tprod (a; b) = ab = a ⊗ b; Algebraic Operators p → 1: S(a; b) = Ssum (a; b) = a + b − ab = a ⊕ b;    a ∧ b if a ∨ b = 1   = a ∗ b;  T (a; b) = Tw (a; b) =  0 otherwise Drastic Operators p → 0:    a ∨ b if a ∧ b = 0   = a  b:  S(a; b) = Sw (a; b) = 1 otherwise

(3)

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By using properties F1 and F4, it can be easily proved that for any arbitrary t-norm T and t-conorm S and for all a; b ∈ [0; 1]: Tw (a; b)6T (a; b)6Tmin (a; b);

Sw (a; b)¿S(a; b)¿Smax (a; b):

(4)

The main features of the introduced parameterized form are its analytical simplicity and symmetry. These are basic advantages for our further implementation, as it will be demonstrated in sequel. Another advantage is that, as expressed by Eq. (4), as p changes continuously (p¿0), this parametric form does cover all t-norms and t-conorms from the special class of Zadeh’s operators to drastic operators. Analogous to classical set theory, De Morgan laws establish a link between union and intersection via complementation. If a t-norm T and a t-conorm S and a strong negation c satisfy De Morgan laws as ( T (a; b) = c(S(c(a); c(b))); for all a; b on [0; 1]: (5) S(a; b) = c(T (c(a); c(b))); then the triple (T; S; c) is called a De Morgan triple, and T and S are called n-duals (cojoints) of each other. All cojoint t-norms and t-conorms introduced above are duals when considered with the standard negation c(a) = 1 − a. 2.3. Extension of triangular norm and conorm functions Although t-norm and t-conorm functions are de ned as binary operators on [0, 1], their associativity property allows them to be extended to the n-ary operations as Tn : [0; 1]n → [0; 1]; ( (a1 ; a2 ; : : : ; an ) →

Sn : [0; 1]n → [0; 1]; T (Tn−1 (a1 ; a2 ; : : : ; an−1 ); an ); S(Sn−1 (a1 ; a2 ; : : : ; an−1 ); an );

where, T (: : :) and S(: : :) are the binary operators. It is proved that the n-ary operators Tn and Sn satisfy similar properties as the original binary T and S. Theorem 1 (Ruan and Kerre [11]). The n-ary operators Tn and Sn on [0; 1] satisfy the following properties: FE1: Tn (a1 ; a2 ; : : : ; ai−1 ; 1; ai+1 ; : : : ; an ) = Tn−1 (a1 ; a2 ; : : : ; an−1 ; an+1 ; : : : ; an ); Tn (a1 ; a2 ; : : : ; ai−1 ; 0; ai+1 ; : : : ; an ) = 0: Sn (a1 ; a2 ; : : : ; ai−1 ; 0; ai+1 ; : : : ; an ) = Sn−1 (a1 ; a2 ; : : : ; an−1 ; an+1 ; : : : ; an ); Sn (a1 ; a2 ; : : : ; ai−1 ; 1; ai+1 ; : : : ; an ) = 1: FE2: Tn (a1 ; a2 ; : : : ; an ) = Tn (a(1) ; a(2) ; : : : ; a(n) ); Sn (a1 ; a2 ; : : : ; an ) = Sn (a(1) ; a(2) ; : : : ; a(n) ); where  is a permutation of {1; 2; : : : ; n}. FE3: Tn (a1 ; a2 ; : : : ; an ) = Ti+1 (a1 ; : : : ; ai ; Tn−i (ai+1 ; : : : ; aj ; : : : ; an )) = Tn−j+1 (Tj (a1 ; : : : ; aj ); aj+1 ; : : : ; an ); Sn (a1 ; a2 ; : : : ; an ) = Si+1 (a1 ; : : : ; ai ; Sn−i (ai+1 ; : : : ; aj ; : : : ; an )) = Sn−j+1 (Sj (a1 ; : : : ; aj ); aj+1 ; : : : ; an ):

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FE4:

( (∀i ∈ {1; 2; : : : ; n}:

ai 6a0i ) ⇒

Tn (a1 ; a2 ; : : : ; an )6Tn (a01 ; a02 ; : : : ; a0n ); Sn (a1 ; a2 ; : : : ; an )6Sn (a01 ; a02 ; : : : ; a0n ):

Moreover, we need to justify that De Morgan laws hold for n-ary triangular operators, as well. Theorem 2. If the binary operators T and S are duals with standard negation; then their corresponding n-ary extensions Tn and Sn are also duals. Proof. By using mathematical induction, De Morgan law is true for n = 2. Suppose that it is also true for n = k; then, ( Tk (a1 ; a2 ; : : : ; ak ) = 1 − Sk ((1 − a1 ); (1 − a2 ); : : : ; (1 − ak )); Sk (a1 ; a2 ; : : : ; ak ) = 1 − Tk ((1 − a1 ); (1 − a2 ); : : : ; (1 − ak )): Consider n = k + 1, then, Tk+1 (a1 ; a2 ; : : : ; ak+1 ) = T (Tk (a1 ; a2 ; : : : ; ak ); ak+1 ) = 1 − S[(1 − Tk (a1 ; a2 ; : : : ; ak )); (1 − ak+1 )] = 1 − S[Sk ((1 − a1 ); (1 − a2 ); : : : ; (1 − ak )); (1 − ak+1 )] = 1 − Sk+1 ((1 − a1 ); (1 − a2 ); : : : ; (1 − ak+1 )): The same proof can be given for Sk+1 (a1 ; a2 ; : : : ; ak+1 ). We may call Tn and Sn , the extension of triangular norm and conorms to n arguments, and we will omit the subscript n and simply write T and S for the class of mapping generated by the triangular norms and conorms. Now, we have to extend the parameterized form of t-norm and t-conorm (Eq. (3)) to n arguments. The extension of t-conorm will be derived here and in order to obtain extended parameterized t-norm we simply use De Morgan law T (a1 ; a2 ; : : : ; an ) = 1 − S((1 − a1 ); (1 − a2 ); : : : ; (1 − an )):

(6)

Extending the binary formulation of the t-conorm to three arguments, by using the associativity property, yields S(a1 ; a2 ; a3 ) = S(S(a1 ; a2 ); a3 ) = {[(ap1 + ap2 − ap1 ap2 )1=p ]p + ap3 − [(ap1 + ap2 − ap1 ap2 )1=p ]p ap3 }1=p = (ap1 + ap2 + ap3 − ap1 ap2 − ap1 ap3 − ap2 ap3 + ap1 ap2 ap3 )1=p and, in the same way, the formulation for n arguments becomes  1=p n n n X n n X n X n X X X Y api − api apj + api apj apk ± · · · ± api  : S(a1 ; a2 ; : : : ; an ) =  i=1

i=1 j¿i

i=1 j¿i k¿j

(7)

i=1

From Eq. (7), the computational complexity increases exponentially when the number of arguments (n) becomes large. With a rather good algorithm we need (2n − 2) additions and (2n − n − 1) multiplications and (n) power operations to compute the above formulation. Hence, the computational complexity is of O(2n ). This problem is a bottleneck for further applications of fuzzy reasoning in fuzzy systems and control.

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In order to signi cantly reduce the number of arithmetic operations, we change Eq. (7) to the following form, which can be con rmed by inspection S(a1 ; a2 ; : : : ; an ) = [ap1 + (1 − ap1 )[ap2 + (1 − ap2 )[ap3 + (1 − ap3 )[· · · [apn−2 + (1 − apn−2 )(apn−1 + [1 − apn−1 ]apn ] · · ·]]]]1=p :

(8)

Then, for a set of (a1 ; a2 ; : : : ; an ) (n¿2), the formulation can be computed by the following algorithm called Algorithm 1. Step 1: Compute ap1 ; ap2 ; : : : ; apn as A1 ; A2 ; : : : ; An , respectively. Step 2: S = An Step 3: LOOP i FROM n − 1 TO 1 STEP −1; S = Ai + (1 − Ai ) × S END Step 4: S = S 1=p . The number of arithmetic operations in the above algorithm is (2n−2) additions, and (n−1) multiplications and (n + 1) power operations, reducing the computational complexity to O(n). Therefore, computational complexity is a linear function of the number of arguments, which sounds sucient for further implementation. According to Eq. (6), for a computation of a t-norm, two other steps as an initial step which is to complement all arguments a1 ; a2 ; : : : ; an , and a nal step which is to subtract S from 1, must be added to Algorithm 1 as the following Algorithm 2. Step 1: Compute (1 − a1 ); (1 − a2 ); : : : ; (1 − an ) as a01 ; a02 ; : : : ; a0n , respectively. 0p 0p Step 2: Compute a0p 1 ; a2 ; : : : ; an as A1 ; A2 ; : : : ; An , respectively. Step 3: T = An Step 4: LOOP i FROM n − 1 TO 1 STEP −1; T = Ai + (1 − Ai ) × T END Step 5: T = 1 − T 1=p .

3. Implication of individual rules According to the theory of approximate reasoning [4,5], each fuzzy rule of the form IF U1 is B1 AND U2 is B2 AND : : : AND Ur is Br THEN V is D; can be translated into a canonical proposition of the form (U1 ; U2 ; : : : ; Ur ; V ) is R; where R is a fuzzy relation de ned on the Cartesian product universe X1 × X2 × · · · × Xr × Y . According to the analysis presented in Section 2, it is suitable to use t-norm operators to de ne conjunctions in the antecedent of the multi-input rule. Furthermore, modeling of implication relation based on the use of fuzzy logic is not unique. In membership domain, each entry of the implication relation (B1 ∩ B2 ∩ · · · ∩ Br )

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→ D is denoted as R(x1 ; x2 ; : : : ; xr ; y) = I (T 0 (B1 (x1 ); B2 (x2 ); : : : ; Br (xr )); D(y)):

(9)

Two extreme paradigms for forming the implication relation are conjunctive method and disjunctive method. Under conjunctive implication, the fuzzy relation R is simply the conjunction of antecedent and consequent spaces. Therefore, Rc (x1 ; x2 ; : : : ; xr ; y) = T (T 0 (B1 (x1 ); B2 (x2 ); : : : ; Br (xr )); D(y));

(10)

where T is t-norm operator (with parameter p) for rule implication, and T 0 is t-norm operator (with parameter q) for rule antecedent aggregation. This approach, which is based on heuristics, is actually an approximation of implication functions. On the other side, disjunctive approach is obtained directly by generalizing the material implication de ned in classical set theory as: B → D ≡ B ∪ D. Therefore, we have Rd (x1 ; x2 ; : : : ; xr ; y) = S(c(T 0 (B1 (x1 ); B2 (x2 ); : : : ; Br (xr ))); D(y)) = S(S 0 ((1 − B1 (x1 )); (1 − B2 (x2 )); : : : ; (1 − Br (xr ))); D(y));

(11)

where S and S 0 are t-conorm operators with parameters p and q, respectively. 4. Aggregation of the rules Selection of an operator for aggregation of the rules depends on the selection of implication operator for the individual rules. Suppose that the total knowledge of a system is expressed by the rule set “(U1 ; U2 ; : : : ; Ur ; V ) is R”, or, in a more expanded form (U1 ; U2 ; : : : ; Ur ; V ) is R1 ALSO (U1 ; U2 ; : : : ; Ur ; V ) is R2 ALSO : : : ALSO (U1 ; U2 ; : : : ; Ur ; V ) is Rn :

(12)

Further, suppose that each basic proposition (individual rule) is a conjoin of the constituents, i.e., “Uj is Bij ”, and “V is Di ”. Then each individual rule is stated as a conjunctive implication (Eq. (10)). Furthermore, the combination of all the rules must be stated as a disjunction (union) operation. In other words, the ALSO connective should be an OR operator; a t-conorm RM (x1 ; x2 ; : : : ; xr ; y) = S(Rc1 (x1 ; x2 ; : : : ; xr ; y); Rc2 (x1 ; x2 ; : : : ; xr ; y); : : : ; Rcn (x1 ; x2 ; : : : ; xr ; y)):

(13)

The reason for such selection is that, since we have adopted t-norm operation for the implication of individual rules, there is always a possibility of having a zero output from at least one of the rules, i.e., when the antecedent membership value becomes zero (property FE1 in Section 2.2). In order to eliminate the eect of such a rule in the rule set, it is required to implement t-conorm operation for the aggregation of the rules according to the property FE1 in Section 2.2. This method of aggregation, which is a heuristic approximation of reasoning, coincides with Mamdani’s approximation approach introduced originally by Mamdani [8] and was applied successfully by Mamdani and Assilian [9] to control dynamic systems. On the other hand, if each basic proposition, i.e., each individual rule, is regarded as “[c(Uj is Bij )] ∪ [V is Di ]”, which is the disjunctive implication approach, then the knowledge “(U1 ; U2 ; : : : ; Ur ; V ) is R” should be considered as a conjunction (intersection) of the rules. In other words, the ALSO connective is AND operator; a t-norm RL (x1 ; x2 ; : : : ; xr ; y) = T (Rd1 (x1 ; x2 ; : : : ; xr ; y); Rd2 (x1 ; x2 ; : : : ; xr ; y); : : : ; Rdn (x1 ; x2 ; : : : ; xr ; y)):

(14)

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A similar argument is put forth for this selection. Since there is a t-conorm operation for the implication of each individual rule, there is always a chance of getting 1 from at least one of the rules. This happens when the antecedent membership value becomes zero (property FE1 in Section 2.2). In this case, the eect of such a rule should be cancelled by the rule aggregation operator; a condition which is satis ed by t-norm operator, according to the property FE1 in Section 2.2. This method based on formal logic is called formal logical approach. We refer to [22] for more details.

5. Inference of the rule set We now consider the problem of nding the output value in its fuzzy environment. At the moment, consider single-input–single-output (SISO) system. Given the relationship “(U; V ) is R” and the information that U equals to a fuzzy set A, the problem of interest becomes that of nding the fuzzy value for V . From the mathematical point of view, this can be seen as solving the equation. From a logical point of view, this can be seen as a generalized form of Modus Ponens (GMP). It is known as Zadeh’s compositional rule of inference (CRI) [26], the process of nding a solution, making a deduction in the theory of approximate reasoning, consists of two steps: (1) Combine the system input proposition and the rule-set relation via the conjunction operation. (2) Project onto the variable of interest. Therefore, at the propositional level of expression, having “(U; V ) is R” and “U is A”, one obtains the relationship “(U; V ) is G” where G = A ∩ R is a fuzzy set de ned on the Cartesian product universe X × Y with membership function: G(x; y) = T 00 (A(x); R(x; y)), where, T 00 is a t-norm for CRI conjunction operation. Now, we are interested in obtaining the output that is generated by the rule, considering that the input U is fuzzy set A. Applying the projection principle, we get a value F for V as “V is F”, where F is a fuzzy subset of Y such that F = Projx G:

(15)

The membership function of the projection of fuzzy set G onto the output space Y is _ _ (G(x; y)) = [T 00 (A(x); R(x; y))] = Max[T 00 (A(x); R(x; y))]; F(y) = x

x

x

(16)

W where x means the maximum for all values of x. Expression 16 can be written in a more compact form: F = A ◦R, where “◦” is known as the composition W operator which represents a combination of “ x ” and “T 00 ” operators. Two dierent approaches de ning the relation R, Mamdani’s approximation and formal logical approach, lead to two formulations for obtaining the reasoning solution [22]. 5.1. Reasoning based on Mamdani’s approximation In conjunctive rule implication approach, we recall that individual rules are combined via disjunction operation R=

n [

Ri ;

i=1

where each fuzzy rule Ri is interpreted as a fuzzy intersection of the fuzzy sets Bi and Di Ri = Bi ∩ Di ;

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Ri is de ned on the Cartesian product space X × Y and has membership function Ri (x; y) = T (Bi (x); Di (y)): For a given input fuzzy set U = A, the fuzzy output, obtained by this method is " n # [ (Bi ∩ Di ) FM = A ◦

(17)

i=1

and in computational level, its membership function is _ _ [T 00 [A(x); S(R1 (x; y); R2 (x; y); : : : ; Rn (x; y))]] FM (y) = [T 00 [A(x); R(x; y)]] = x

=

_

x 00

[T [A(x); S[T (B1 (x); D1 (y)); T (B2 (x); D2 (y)); : : : ; T (Bn (x); Dn (y))]]]:

(18)

x

In Eq. (18) several conditions should be satis ed for the selection of t-norm and t-conorm operators. First of all, according to two similar approaches for the selection of the CRI conjunction operator by Trillas and Valverde [16] and Dubois and Prade [3], T 00 should be the same as the t-norm T assigned for rule implication in Section 3 to possess certain desired properties. This is reviewed in more detail in [19]. However, if the observation of the reasoning (the input A) is a crisp set, which is true in most cases of fuzzy modeling and control application, T 00 vanishes in the nal reasoning formulation because of the boundary condition implemented on the CRI conjunction t-norm operator, as we will see in the foregoing derivation. Secondly, as investigated in [21], selection of t-norm T and t-conorm S operators in Eq. (18) should satisfy the basic requirement of fuzzy reasoning, i.e., if we have a system input which is the same as the antecedent of a rule in the rule base, then the reasoning result should be the same as the consequent of the rule. Moreover, there are some constraints on membership functions depending on composition and implication operators as discussed in [25]. Again, for a crisp observation, it can be seen that the basic requirement for fuzzy reasoning and the membership function constraints are already satis ed. Another concern with the above reasoning formulation (Eq. (18)) is that, in general, t-norm and t-conorm classes do not have the property of distributivity. Therefore, for a fuzzy input A(x), we cannot distribute the t-norm operation among the t-conorms in Eq. (18). This means that for the case of fuzzy input, we have to rst combine all rules to get the rule set relation R, and then compose it with the fuzzy input. This is called ÿrst-aggregate-then-infer (FATI), and regarding the computation and memory considerations, this method is very time and memory consuming and has practical limitations. Fortunately, there is a more ecient method if the input of the reasoning is crisp. If the input is x then the input fuzzy set A is interpreted as a fuzzy singleton with membership function ( 1 if x = x∗ ; ∗ (19) A (x) = 0 if x 6= x∗ : In this case, we can implement the distributivity property for any type of t-norm and t-conorm family. We will prove this for two arguments below, and by using the associativity property, this proof can easily be extended to the n-ary operation. Theorem 3. If A∗ (x) is a singleton; and M and N are fuzzy relations deÿned on X × Y; then the following distributivity relation holds for t-norm T 00 and t-conorm S T 00 [A∗ (x); S[M (x; y); N (x; y)]] = S[T 00 [A∗ (x); M (x; y)]; T 00 [A∗ (x); N (x; y)]]:

(20)

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Proof. (a) If x = x∗ then A∗ (x) = 1, and therefore, recalling the property F1 of the triangular norms, we get T 00 [A∗ (x); S[M (x; y); N (x; y)]] = T 00 [1; S[M (x; y); N (x; y)]] = S[M (x; y); N (x; y)]; S[T 00 [A∗ (x); M (x; y)]; T 00 [A∗ (x); N (x; y)]] = S[T 00 [1; M (x; y)]; T 00 [1; N (x; y)]] = S[M (x; y); N (x; y)]: Hence, the left- and the right-hand side of Eq. (20) are equal. (b) If x 6= x∗ then A∗ (x) = 0, and therefore, recalling the property F1 of the triangular norms, we have T 00 [A∗ (x); S[M (x; y); N (x; y)]] = T 00 [0; S[M (x; y); N (x; y)]] = 0; S[T 00 [A∗ (x); M (x; y)]; T 00 [A∗ (x); N (x; y)]] = S[T 00 [0; M (x; y)]; T 00 [0; N (x; y)]] = S[0; 0] = 0: Hence, again the left- and the right-hand side of Eq. (20) are equal. Returning to Eq. (18) and applying the distributivity property for singleton input A∗ , we have _ [S[T 00 [A∗ (x); T (B1 (x); D1 (y))]; : : : ; T 00 [A∗ (x); T (Bn (x); Dn (y))]]]: FM (y) =

(21)

x

Again, we consider two sets of values for x: (a) if x = x∗ then A∗ (x) = 1; and therefore, we have FM (y) = S[T (B1 (x∗ ); D1 (y)); T (B2 (x∗ ); D2 (y)); : : : ; T (Bn (x∗ ); Dn (y))]; (b) if x 6= x∗ then A∗ (x) = 0; and therefore, we have FM (y) = 0. Thus, for a crisp input x∗ : FM (y) = S[T (B1 (x∗ ); D1 (y)); T (B2 (x∗ ); D2 (y)); : : : ; T (Bn (x∗ ); Dn (y))]:

(22)

This means that we can re each single rule rst, i.e., to compute Bi (x∗ ), and then, for each single rule calculate individual fuzzy output Ri and nally, aggregate all fuzzy output Ri of all rules to obtain the inferred fuzzy output FM (y). This method is called ÿrst-infer-then-aggregate (FITA); and it is computationally faster than FATI. Thus, it is proved here that for a crisp input, the two approaches always give the same fuzzy output. This is stated in the following theorem. Theorem 4. For crisp input, the reasoning result based on Mamdani’s approximation approach is identical for FATI and FITA methods. From Eq. (22), it is concluded that for the case that the input is a crisp set, for multi-input singleoutput systems, the antecedent of each rule i is a conjunction of r fuzzy sets Bi1 ; Bi2 ; : : : ; Bir . For crisp input x∗ = (x1∗ ; x2∗ ; : : : ; xr∗ ), the rule ring step for the rule i is to compute as i (x∗ ) = T 0 (Bi1 (x1∗ ); Bi2 (x2∗ ); : : : ; Bir (xr∗ ));

(23)

where i is called the degree of ring (DOF) of rule i [7]. The fuzzy output membership function is FM (y) = S[T (1 (x∗ ); D1 (y)); T (2 (x∗ ); D2 (y)); : : : ; T (n (x∗ ); Dn (y))]:

(24)

It is observed that generally there are two t-norm and one t-conorm operators in the fuzzy output membership function that must be selected among dierent classes of triangular norms. In MISO systems, the number of input variables (r) and rules (n) are usually more than two, which makes the computations more complicated unless some simple triangular operators such as Zadeh’s or drastic or so on are selected. That is the main reason why people usually tend to use these norms in practical applications. We do not restrict ourselves to simple and extreme triangular operators but give more exibility to the system representation by adjusting the

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operators among the in nite continuous variation of parameters in the parameterized form. This is applicable by taking the bene t of fast algorithms suggested in Section 2, for calculating parameterized triangular norms and conorms, and also by some further simpli cations which will be implemented in Section 7. Through this approach, the most appropriate set of operators can be selected for the system under investigation. 5.2. Reasoning based on formal logical approach This method is based on the disjunctive approach of rule implication, in which the relation of the rule set is conjunction of individual rules R=

n \

Ri :

i=1

For each rule, we derive the fuzzy relation membership function (Eq. (11)) for SISO systems as Ri (x; y) = S[(1 − Bi (x)); Di (y)]: Therefore, for a fuzzy input A, the fuzzy output is " n # \ [c(Bi ) ∪ Di ] : FL = A ◦

(25)

i=1

At computational level, the membership function of the fuzzy output is FL (y) =

_

[T 00 [A(x); R(x; y)]] =

x

=

_

_

[T 00 [A(x); T [R1 (x; y); R2 (x; y); : : : ; Rn (x; y)]]]

x 00

[T [A(x); T [S[(1 − B1 (x)); D1 (y)]; S[(1 − B2 (x)); D2 (y)]; : : : ; S[(1 − Bn (x)); Dn (y)]]]]:

(26)

x

If the input is crisp, then A∗ (x) is a fuzzy singleton (Eq. (19)), then, T 00 [A∗ (x); T [S[(1 − B1 (x)); D1 (y)]; : : : ; S[(1 − Bn (x)); Dn (y)]]] ( T [S[(1 − B1 (x)); D1 (y)]; : : : ; S[(1 − Bn (x)); Dn (y)]] if x = x∗ ; = 0 if x = 6 x∗ :

(27)

Finally, we can write FL (y) = T [S[(1 − B1 (x∗ )); D1 (y)]; S[(1 − B2 (x∗ )); D2 (y)]; : : : ; S[(1 − Bn (x)); Dn (y)]]:

(28)

If we choose dual t-norm and t-conorm operators, by using De Morgan law, we can derive an alternative form for logical reasoning FL (y) = 1 − S[T [B1 (x∗ ); (1 − D1 (y))]; T [B2 (x∗ ); (1 − D2 (y))]; : : : ; T [Bn (x∗ ); (1 − Dn (y))]]; FL (y) = 1 − S[T [B1 (x∗ ); D1 (y)]; T [B2 (x∗ ); D2 (y)]; : : : ; T [Bn (x∗ ); Dn (y)]]; where Di (y) = 1 − Di (y) is the complement of Di (y).

(29)

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For MISO system we have FL (y) = T [S[(1 − 1 (x∗ )); D1 (y)]; S[(1 − 2 (x∗ )); D2 (y)]; : : : ; S[(1 − n (x∗ )); Dn (y)]];

(30)

FL (y) = 1 − S[T [1 (x∗ ); D1 (y)]; T [2 (x∗ ); D2 (y)]; : : : ; T [n (x∗ ); Dn (y)]];

(31)

or

where i is the degree of ring of the ith rule. Eq. (31) looks similar to Eq. (25) for Mamdani’s reasoning. In other words, it is concluded that in the logical approach, we can implement the same reasoning mechanism as Mamdani’s approach with a modi cation that the consequents of each rule is considered as the complement of the original rule set with the nal result being complemented. This similarity of the reasoning mechanism for both approaches provides a more convenient ground to compare the theoretical features of each approach. As we can observe, a kind of complementation-related behavior exists between Mamdani’s and logical approaches for the singleton input case. In the logical method, we consider the complement of the consequents and then we complement the whole function to obtain the result, when the input is crisp. However, the nal output is totally dierent for two methods [20], as we will see in Example 1 at the end of this section. However, there is another advantage with representing the logical method by Eq. (31), which will become more clear in the following discussion. 5.3. Uniÿed parameterized fuzzy reasoning method Yager [24] analyzed the advantages and disadvantages of both approaches of rule aggregation without any conclusion about the superiority of either of them. He also considered the situation where the output of the two methods is combined to give a new individual output as [25] E(y) = ÿFL (y) + (1 − ÿ)FM (y);

06ÿ61:

(32)

We will adopt this approach for constructing the parameterized frame for reasoning process. However, it should be mentioned that for the case of crisp input (which is mostly the case in fuzzy modeling and control), we introduced a uni ed reasoning mechanism for both Mamdani’s and logical approaches (Eqs. (24) and (31)), so, we are not concerned with the problem of aggregating the inferred output fuzzy set by such an operator that would be a compromise between the aggregating operators used in two methods. In other words, in the proposed uni ed fuzzy reasoning method, reasoning operators are exactly the same. This is the second advantage of representing the logical reasoning as Eq. (31). As a result, the proposed parameterized reasoning method is suggested as E(y) = ÿ(1 − S[T [1 (x∗ ); D1 (y)]; T [2 (x∗ ); D2 (y)]; : : : ; T [n (x∗ ); Dn (y)]]) + (1 − ÿ)S[T [1 (x∗ ); D1 (y)]; T [2 (x∗ ); D2 (y)]; : : : ; T [n (x∗ ); Dn (y)]]:

(33)

Example 1. Let us consider the fuzzy model of a nonlinear system with 3 inputs x1 ; x2 and x3 and single output y as illustrated in Fig. 1. For a set of inputs x = {1:25; 3:0; 4:2}, Fig. 2 shows dierent fuzzy outputs due to dierent values of reasoning parameters: p (the parameter of t-norm and t-conorm operators for rule implication and rule aggregation, T and S); q (the parameter of t-norm operator for rule antecedent connection, T 0 ), and ÿ (the combination parameter of two inference approaches). It is observed that the fuzzy output has a considerable variation with

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Fig. 1. Fuzzy model of a nonlinear system.

respect to various combinations of reasoning parameters; a matter that should be considered seriously in fuzzy modeling and control.

6. Defuzziÿcation of the output One important issue of the reasoning process, specially in fuzzy modeling and control is the problem of selecting a crisp value y∗ based on the output fuzzy set E(y). Two commonly used defuzzi cation methods are the center of area (COA), and mean of maxima (MOM). In the COA method, one calculates the output of the defuzzi er yC∗ as follows: R y1

yC∗

yE(y) dy y = R 0y1 ; E(y) dy y0

where the real interval Y = [y0 ; y1 ] is the universe of discourse of the output.

(34)

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Fig. 2. Fuzzy output of the nonlinear system for input set [1:25; 3:0; 4:2].

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In the MOM method, the output of the defuzzi er is calculated as ∗ = yM

1 X yi ; m

(35)

yi ∈G

where G is the set of elements in Y which attain the maximum value of E(y) and m is the cardinality of G. An eort to make a generalized defuzzi cation method is the method of heights [10] in which the intervals of the universe that correspond to membership grades lower than a certain given level  are completely discounted and the defuzzi cation value is calculated by the application of the COA in the interval of Y that has membership grade not less than : R y10 yE(y) dy y0 ; (36) yH∗ = R 0y0 1 y0 E(y) dy 0

where all elements in Y0 = [y00 ; y10 ] have membership grades more than . It is obvious that in this method, the case  = 0 implies the COA method and  = Maxy∈Y [E(y)] implies the MOM method. As a better attempt for generalization, Yager and Filev [25] suggest a general defuzzi cation method, based on the probabilistic nature of the selection process among the values of a fuzzy set, called basic defuzzi cation distribution (BADD) method R y1  yE (y) dy y ∗ ; ¿0: (37) yB = R 0y1  E (y) dy y0 As we can see, BADD method is essentially a family of defuzzi cation methods parameterized by parameter . For  = 1, BADD method implies the COA defuzzi cation method and for  → ∞, it implies the MOM defuzzi cation method. For  = 0, the BADD defuzzi cation method coincides with arithmetic mean of the universal set Y . By varying  continuously in the real interval, it is possible to have more appropriate mappings from the fuzzy set to the crisp value depending on the system behavior. Example 2. Following the fuzzy model inference process for the nonlinear system in Example 1, we would like to have the defuzzi ed values of y for input variables x = {1:25; 3:0; 4:2} with respect to dierent values of p; q; , and ÿ. Fig. 3 shows the results. It is obvious from the gure that for one input set, a wide range of crisp output values can be derived from the fuzzy model of the nonlinear system due to dierent values of reasoning and defuzzi cation parameters. By adjusting these parameters in the uni ed parameterized formulation, we are able to modify the fuzzy model in order to obtain a closer behavior to the real system. 7. Simpliÿed parameterized reasoning approach Until now, we have constructed a general parameterized frame for the reasoning process in fuzzy modeling and control. It was shown that we can obtain an analytical relation between input and output variables, using parameterized triangular functions, when fast algorithm for calculating triangular operators are used. Furthermore, it was shown that the computation eort is low enough for most applications. We are now interested in seeing if it is possible to derive an even simpler formulation for reasoning process in fuzzy modeling and control. The following simpli cation is based on the Schweizer and Sklar’s operators, Eq. (3).

M.R. Emami et al. / Fuzzy Sets and Systems 108 (1999) 59–81

Fig. 3. Defuzzi ed output of the nonlinear system for input set [1:25; 3:0; 4:2].

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76

We start from the t-conorm parameterized formulation and derive the rst term of its Tailor expansion around the middle point of the domain [0; 1] × [0; 1] × · · · × [0; 1], i.e., (0:5; 0:5; : : : ; 0:5): {z } | {z } | n times

n times

S(a1 ; a2 ; : : : ; an ) = [ap1 + (1 − ap1 )[ap2 + (1 − ap2 )[ap3 + (1 − ap3 )[· · · [apn−2 + (1 − apn−2 )[apn−1 + (1 − apn−1 )apn ]]· · ·]]]]1=p ; n X @S ˜ 1 ; a2 ; : : : ; an ) = S(0:5; 0:5; : : : ; 0:5) + S(a1 ; a2 ; : : : ; an ) ∼ = S(a {z } | @ai i=1

n times

(ai − 0:5):

(38)

(0:5; 0:5;:::;0:5)

Now, let us rewrite some of the terms in Eq. (38) as follows: " n #1=p n! 1 X i−1 (n−i)p (−1) ; (2) Jn = S(0:5; 0:5; : : : ; 0:5) = n {z } | 2 i!(n − i)!

(39)

i=1

n times

@S Kn = @ai (0:5; 0:5;:::;0:5)  =

1 2(n−1)



n−1 X j=0

 1=p−1 n X (n − 1)! n! (2)(n−1−j)p   (2)(n−j)p  (−1) j (−1) j−1 ; j!(n − 1 − j)! j!(n − j)!

(40)

j=1

  ˜ 1 ; a2 ; : : : ; an ) = Jn + Kn a1 + a2 + · · · + an − n S(a 2  n  = Kn (a1 + a2 + · · · + an ) + Jn − Kn = Kn (a1 + a2 + · · · + an ) + Hn ; 2

(41)

where Hn = Jn − (n=2)Kn . Instead of expanding the parameterized t-norm function, it is more convenient to obtain the approximate form, T˜ , by using De Morgan law. One can verify that the result is the same as the Tailor expansion directly derived about the middle point for the t-norm function ˜ − a1 ); (1 − a2 ); : : : ; (1 − an )) T˜ (a1 ; a2 ; : : : ; an ) = 1 − S((1  n  = Kn (a1 ; a2 ; : : : ; an ) + 1 − Jn − Kn = Kn (a1 ; a2 ; : : : ; an ) + Ln ; 2

(42)

where Ln = 1 − Jn − (n=2)Kn . Therefore, the only dierence between simpli ed t-norm and t-conorm functions is due to the constant part of the linear function, which in turn is a function of parameter p. In this way, all families of parameterized triangular functions are approximated by lines with dierent slopes and constant values. It is worth noting that choosing parameter p, for each fuzzy model, Hn and Ln are computed once and are constant during the reasoning process. Thus, the calculation of approximate triangular norms for n arguments simply needs n addition and one multiplication.

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The linearized form of Schweizer and Sklar triangular functions lets us have a simpler analytical relation between input and output, while we still keep the exibility of changing the nature of the reasoning by changing parameter p. Let us derive the simpli ed parameterized fuzzy reasoning formulation E(y) = ÿ(1 − S[T [1 (x∗ ); D1 (y)]; T [2 (x∗ ); D2 (y)]; : : : ; T [n (x∗ ); Dn (y)]]) + (1 − ÿ)S[T [1 (x∗ ); D1 (y)]; T [2 (x∗ ); D2 (y)]; : : : ; T [n (x∗ ); Dn (y)]]; "

"

˜ E(y) = ÿ 1 − Kn K2

n X

∗

i (x ) + K2

i=1

" + (1 − ÿ) Kn K2

˜ E(y) = [K2 Kn ]

n X

∗

i (x ) + K2

n X

Di (y) + nL2 + Hn

n X

!

#

Di (y) + nL2 + Hn ;

i=1

" Di + (1 − 2ÿ)K2 Kn

i=1

=P

##

i=1

i=1

n X

!

n X

n X

# i (x∗ ) + (1 − 2ÿ)nL2 Kn + (1 − 2ÿ)Hn + ÿ(1 − nK2 Kn )

i=1

Di (y) + Q1

i=1

n X

i (x∗ ) + Q2 = P

i=1

n X

Di (y) + Q;

(43)

i=1

where P = [K2 Kn ];

Q1 = (1 − 2ÿ)K2 Kn ;

Q2 = (1 − 2ÿ)nL2 Kn + (1 − 2ÿ)Hn + ÿ(1 − nK2 Kn )

and " Q = (1 − 2ÿ)K2 Kn

n X

# i (x∗ ) + (1 − 2ÿ)nL2 Kn + (1 − 2ÿ)Hn + ÿ(1 − nK2 Kn ) :

i=1

˜ Now, with the above formulation, it is possible to compute the defuzzi cation step faster. Since E(y) is a ∗ summation of Di ’s and the input eect as i (x ) appears as an addition term in Q, the MOM method would be irrelevant for this form of fuzzy output. Therefore, we are restricted to choose the Center of Area method as the defuzzi cation method: Ry P Ry R y1 P R y1 ˜ yE(y) dy P y01 y Di (y) dy + Q y01 y dy P yDi (y) dy + 0:5Q(y12 − y02 ) y0 y ∗ R y1 P R y1 = = y˜ = R y1 P 0R y1 ˜ P dy P Di (y) dy + Q(y1 − y0 ) Di (y) dy + Q E(y) dy y0

y0

Pn P i=1 yi0 Ai + 0:5Q(y12 − y02 ) Pn ; = P i=1 Ai + Q(y1 − y0 )

y0

y0

(44)

where yi0 is the center of area of consequent of the ith rule Di and Ai is the area under the curve Di . For a trapezoidal membership function such as Fig. 4, the centroid and area can be simply calculated as A = 0:5[(yd − ya ) + (yc − yb )];

(45)

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Fig. 4. Trapezoidal membership function.

y0 =

1=6[(yc2 + yd2 + yc yd ) − (ya2 + yb2 + ya yb )] ; A

y0 = Min [ya ]i

and

i=1; n

y1 = Max [yd ]i : i=1; n

(46) (47)

From Eq. (44), for each input set x∗ = [x1 ; x2 ; : : : ; xn ], the defuzzi ed output can be immediately calculated by obtaining the centroid and areas of the individual consequent fuzzy sets Di ’s and parameters P and Q. Example 3. We return to the fuzzy model of the nonlinear system in Example 1. Fig. 5 shows the exact and approximate values of the defuzzi ed output for variation of reasoning parameters p; q; and ÿ. As we can see, the approximate values have the relative error of about 1% at most, which is a reasonable dierence between the exact and approximate results. A good evaluation of the proposed simpli ed function is to compare it with Yager’s [25], pp. 177, and Sugeno’s [15] simpli ed reasoning formulations. Both simpli cation functions are based on Mamdani’s reasoning approach, and the approximate crisp output is calculated as Pn y˜ ∗Y; S = Pi=1 n i=1

0 i yi

;

(48)

i

where, in Yager’s function: i

= min(Bi1 (x1∗ ); Bi2 (x2∗ ); : : : ; Bir (xr∗ ));

(49)

and, in Sugeno’s function: i

= Bi1 (x1∗ ) × Bi2 (x2∗ ) × · · · × Bir (xr∗ ):

(50)

In order to make the results comparable, we execute the parameterized reasoning function, for p = 0:0001 and q = 0:0001, to compare the result with Yager’s formulation in Fig. 6; and choose p = 0:0001 and q = 1, to compare the result with Sugeno’s formulation in Fig. 7. For the speci c system and inputs, it is observed that the proposed simpli cation is better than Yager’s function and close to Sugeno’s function. Unlike Yager’s and Sugeno’s simpli ed reasoning formulations, which are heuristic in nature, the proposed simpli ed approach is based on the mathematical analysis. This makes it more reliable to generalize the speci c results. Besides, the parameterized form of the proposed simpli ed function makes the reasoning process continuously to vary among dierent selections of p; q; and ÿ.

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Fig. 5. Approximate (y˜∗ ) and exact (y∗ ) defuzzi ed output for input set [1.25, 3.0, 4.2].

79

80

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Fig. 6. Yager’s (y˜∗Y ) and proposed simpli ed function (y˜∗ ) for Example 1.

Fig. 7. Sugeno’s (y˜∗S ) and proposed simpli ed function (y˜∗ ) for Example 1.

8. Conclusions Our goal in this paper was to construct a general and uni ed framework for the reasoning process in fuzzy modeling and control. We started from the basic elements of reasoning, the connective operators (AND, ALSO, and IF–THEN), and adopted a special parameterized family of triangular functions which, due to its simplicity and symmetry, is appropriate for our purpose. By extending the binary operation to n-ary operation, and by proving the validity of De Morgen law for n-ary operation, we were able to parameterize the reasoning formulation for multi-input single-output systems. Focusing on systems with crisp input, which is the case in most applications of fuzzy modeling and control, and by proving the property of distributivity for triangular functions in this case, it was shown that the two methods of inference from a rule set, rst-aggregate-then-infer (FATI) and rst-infer-then-aggregate (FITA), always give the same fuzzy output. Moreover, we presented a uni ed reasoning mechanism for Mamdani’s approximation and formal logical reasoning approaches that leads to a uni ed parameterized fuzzy reasoning method. As a result, four reasoning parameters p; q; , and ÿ, were introduced whose variation will cause a continuous range of variation for reasoning mechanism. Therefore, we are no longer restricted to stay at the extremes in any of the steps of reasoning process. In a fuzzy model, these parameters can be optimized based on the proximity of the model to the input–output data obtained from the real system. This has been investigated in detail in [6].

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