Example of Fuzzy Modeling

Let us assume that we like to approximate the function z = shown in Fig. 28.3, in the regions = [-2, 2] by a Takagi-Sugeno fuzzy model. 

Approximating this relationship by three linear functions might be a somewhat crude approximation, we therefore choose to develop five linear functions for the following values of x X = [-1.75, -1.1, 0, 1.1, 1.75]. From the slope to the curve in any of these points the following five linear models can be derived  

For this illustration Gaussian membership functions of the following form are chosen  

It should be pointed out that each membership curve has two degrees of freedom, x and (7, although (7 has been chosen the same for all the membership curves. 

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