Variable weighted synthesis inference method for fuzzy reasoning and fuzzy systems

Keywords-Fuzzy reasoning, Fuzzy system, Variable weighted synthesis inference method, CRI method, Fuzzy control. 1. I N T R O D U C T I O N It is well known that fuzzy inference is a theoretical foundation of fuzzy control systems and fuzzy expert systems. At present, general fuzzy inference methods include the CRI (compositional rules of inference) method, (+,-)-centroid method, simple inference method, function inference method, characteristic expansion inference method, full implication triple I method, and so on [17]. Among them, the CRI method is widely adopted in applications. The CRI method is realized by fuzzy implication operators and fuzzy relation composition, where the selection of appropriate fuzzy implication operators appears critical. The Mamdani implication operator '%" is one of the most commonly used implication operators in practice. In the CRI method built upon Mamdani implication, inference relations are defined as fuzzy relations between antecedents and consequents of inference rules by the Mamdani implication operator "A", and then fuzzy relation composition operators are used to composite an input fuzzy set and the total inference relation. Through analyzing its mathematical expression, we find out that the expression has the form of a variable weighted synthesis function, in which the variable weights are determined by some fuzzy relations between input fuzzy sets and rule antecedents and vary with input fuzzy sets, and the *Author to whom all correspondence should be addressed. Supported by National Natural Science Foundation of China (60474023), Research Fund for Doctoral Program of Higher Education (20020027013), Science and Technology Key Project Fund of Ministry of Education (03184), and Major State Basic Research Development Program of China (2002CB312200). 0898-1221/06/$ see front matter (~) 2006 Elsevier Ltd. All rights reserved. Typeset by .AA~-TEX doi:10.1016/j.camwa.2006.08.021 306 Y.-Z. ZHANG AND H.-X. L1 synthesis is realized by composing rule consequents with variable weights. This idea conforms to the principle of variable weighted synthesis in multifactorial decision-making. The notion of variable weights was introduced for the synthetic decision-making analysis in [8]. Reference [9] gave the axiomatic definitions of variable weight vectors and state variable weight vectors in accordance with varying regularity of weights, and presented the principle of variable weights saying that "variable weight vectors are made by normalized Hadamard products of a constant weight vector and state variable weight vectors." The concept of synthesis functions, also named multifactorial functions, was introduced for reducing dimensions in [10]. If the parameters in synthesis functions are variable, the synthesis functions are called variable weighted synthesis functions and have the ability of merging information. In this paper, we propose the variable weighted synthesis inference (VWSI) method by applying the principle of variable weighted synthesis to fuzzy inference. In Section 2, we give some preliminary knowledge. In Section 3, we introduce the notions of states of rules, state variable weights of rules and variable weights of rules, and also present the VWSI method and some algorithmic models. In Section 4, we discuss the construction of variable weights of rules. In Section 5, we analyze the interpolation mechanism of fuzzy systems constructed by several VWSI algorithmic models. In Section 6, we discuss the relationship between the fuzzy systems based on VWSI algorithms and the fuzzy systems based on commonly used fuzzy inference algorithms. In Section 7, we give a method for determining constant weights of rules and a simulation experiment. DEFINITION 2.1. to [0, 1] m, where 2. P R E L I M I N A R I E S An m-dimensional variable weight is a mapping W = (Wl, . . . , Win) fYom [0, 1] m wi : [0, 1]'~--~ [0, 1]; ( x l , . . . , x m ) H w , ( x l , . . . , x m ) , l < / < m . A variable weight with reward is an m-dimensional variable weight W satisfying the following three conditions: (w.1) normality, i.e., E , ~ w,(x~, . . . , x~) = 1, (w.2) continuity, i.e., every mapping wi is continuous with respect to any variable x j (1 < i, j <_ m), (w.3) reward law, i.e., every mapping wi is monotonically increasing with respect to the variable xi (1 < i < m). A variable weight with penalty is an m-dimensional variable weight W satisfying (w.1), (w.2) and the following condition: (w.3 ~) penalty law, i.e., every mapping wi is monotonically decreasing with respect to the variable xi (1 < i < m). A synthesis function Mm, also named a multifactorial function, is essentially a projection from m-dimensional space to one-dimensional space [10]. In many cases, the spaces may be transformed into the unit closed intervals. Then Mm : [0, 1] m ~ [0, 1] and it is called a standard synthesis function. Standard synthesis functions can be divided into additive ones and nonadditive ones. DEFINITION 2.2. An additive m-ary standard synthesis function is a mapping Mm : [0, 1] "~ ---* [0, 1]; ( X l , . . . , Zm) H M m ( X l , . . . , Xm) satisfying the following three conditions: (m.1) xi <_ yi (1 < i < m) ~ M m ( X l , . . . , X m ) <_ M m ( y l , . . . , y m ) , (m.2) Mm is continuous with respect to any variable xi (1 < i < m), (re.a) h :lx < < V :I iA nonadditive m-ary standard synthesis function is a mapping [0, 1] [0, 1]; ( x l , . . . , Variable Weighted Synthesis 307 satisfying (m.1), (m.2) and the following condition: (m.3') Mm(Xl , . . . ,Xm) :> vim_lXi or Mm(Xl , . . . ,Xm) < himlXi. EXAMPLE 2.1. The following mappings from [0, 1] m to [0, 1] are additive m-ary standard synthesis functions: A(xi,..., x~) A A x~; i=1 V ( X l . . . . ,Xm) A--V x i ; i=1 m E (x l . . . . 'gCm) ~--" E W i X i ' i=1 where w, • [0, 11 and Eim=, w, = 1; Mm(xl, • . . , Xm) =" V wixi, i=1 where w~ E [0, 1] and Vi~l wi = 1; m M ~ ( ~ , . . . , ~ ) ~ V(~ , ^x~), i= l where wi E [0, 1] and Viral wi = 1; \ i = l / m where p > 0, wi e [0, 1] and }--]~=1 wi = 1. EXAMPLE 2.2. The mapping I-I : [0, 1] m ~ [0, 1] given by m H( )°H X l ~ . . . ~ Z m = 2C i i=1 is a nonadditive m-ary standard synthesis function. (1)