Fuzzy inference is a method that interprets the values in the input vector and, based on some sets of rules, assigns values to the output vector. In fuzzy logic, the truth of any statement becomes a matter of a degree.

Fuzzy inference is the process of formulating the mapping from a given input to an output using fuzzy logic. The mapping then provides a basis from which decisions can be made or patterns discerned. The process of fuzzy inference involves all of the pieces described so far, i.e., membership functions, fuzzy logic operators, and if-then rules. Two main types of fuzzy inference systems can be implemented: Mamdani-type (1977) and Sugeno-type (1985). These two types of inference systems vary somewhat in the way outputs are determined.

Mamdani-type inference expects the output membership functions to be fuzzy sets. After the aggregation process, there is a fuzzy set for each output variable, which needs defuzzification. It is possible, and sometimes more efficient, to use a single spike as the output membership function rather than a distributed fuzzy set. This, sometimes called a singleton output membership function, can be considered a pre-defuzzified fuzzy set. It enhances the efficiency of the defuzzifi-cation process because it greatly simplifies the computation required by the more general Mamdani method, which finds the centroid of a two-dimensional function. Instead of integrating across the two-dimensional function to find the cen-troid, the weighted average of a few data points can be used.

The Sugeno method of fuzzy inference is similar to the Mamdani method in many respects. The first two parts of the fuzzy inference process, fuzzifying the inputs and applying the fuzzy operator, are exactly the same. The main difference between Mamdani-type and Sugeno-type fuzzy inference is that the output membership functions are only linear or constant for the Sugeno-type fuzzy inference. A typical fuzzy rule in a first-order Sugeno fuzzy model has the form

If x is A and y is B, then z = px + qy + r (11.119)

where A and B are fuzzy sets in the antecedent, while p, q, and r are all constants. Higher-order Sugeno fuzzy models are possible, but they introduce significant complexity with little obvious merit. Because of the linear dependence of each rule on the system's input variables, the Sugeno method is ideal for acting as an interpolating supervisor of multiple linear controllers that are to be applied, respectively, to different operating conditions of a dynamic nonlinear system. A Sugeno fuzzy inference system is extremely well suited to the task of smoothly interpolating the linear gains that would be applied across the input space, i.e., it is a natural and efficient gain scheduler. Similarly, a Sugeno system is suitable for modeling nonlinear systems by interpolating multiple linear models.

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