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FUZZY INFERENCE Siti Zaiton Mohd Hashim, PhD Fuzzy Inference Introduction Mamdani Mamdani--style inference Sugeno--style inference Sugeno Building a fuzzy expert system 9/29/2011 2 Introduction Fuzzy inference is the process of formulating the mapping from a given input to an output using fuzzy theory of fuzzy sets. The process of fuzzy inference: membership functions, fuzzy logic operators, and fuzzy if- if-then rules. Two known types of fuzzy inference systems (in the Fuzzy Logic Toolbox): Mamdani Mamdani--type Sugeno Sugeno--type. 9/29/2011 3 Introduction Fuzzy inference systems have been successfully applied in fields: automatic control, data classification, decision analysis, expert systems, and computer vision. 9/29/2011 4 Introduction Because of its multidisciplinary nature, fuzzy inference systems are associated with a number of names: fuzzy fuzzy--rule rule--based systems, fuzzy expert systems, fuzzy modeling, fuzzy associative memory, fuzzy logic controllers, and simply (and ambiguously) fuzzy systems. 9/29/2011 5 Mamdani’s Fuzzy Inference Mamdani's fuzzy inference method is the most commonly seen fuzzy methodology and was among the first control systems built using fuzzy set theory. It was proposed in 1975 by Ebrahim Mamdani [Mam75] as an attempt to control a steam engine and boiler combination by synthesizing a set of linguistic control rules obtained from experienced human operators. Mamdani's effort was based on Lotfi Zadeh's 1973 paper on fuzzy algorithms for complex systems and decision processes [Zad73]. 9/29/2011 6 Mamdani’s Mamdani ’s Fuzzy Inference Mamdani-type inference process is performed in four Mamdani- steps: 1. Fuzzification of the input and output variables 2. Fuzzy logical operations 3. Implication method 4. Aggregation 5. Defuzzification Mamdani--type inference expects the output Mamdani membership functions to be fuzzy sets. sets. 9/29/2011 7 Sample problem We examine a simple two- two-input one- one-output problem that includes three rules: Rule:: 1 Rule Rule:: 1 Rule IF x is A3 IF project_funding is adequate OR y is B1 OR project_staffing is small THEN z is C1 THEN risk is low Rule: 2 Rule: Rule: 2 Rule: IF x is A2 IF project_funding is marginal AND y is B2 AND project_staffing is large THEN z is C2 THEN risk is normal Rule: 3 Rule: Rule: 3 Rule: IF x is A1 IF project_funding is inadequate THEN z is C3 THEN risk is high 9/29/2011 8 Fuzzification The first step is to take the crisp inputs, x1 and y1 (project funding and project staffing), staffing), and determine the degree to which these inputs belong to each of the appropriate fuzzy sets. Crisp Input Crisp Input x1 y1 1 1 B1 B2 A1 A2 A3 0.7 0.5 0.2 0.1 0 0 x1 X y1 Y (x = A1) = 0.5 (y = B1) = 0.1 (x = A2) = 0.2 (y = B2) = 0.7 9/29/2011 9 Fuzzy logical operation / Rule evaluation After fuzzification, we know the degree to which each part of the antecedent has been satisfied for each rule. If the antecedent of a given rule has more than one part, the fuzzy operator is applied to obtain one number that represents the result of the antecedent for that rule. This number will then be applied to the output function. The input to the fuzzy operator is two or more membership values from fuzzified input variables. The output is a single truth value.