Membership functions

A membership function is a curve that defines how each point in the input space is mapped to a membership value, or degree of membership, between 0 and 1. In the literature, the input space is sometimes referred to as the universe of discourse. The only condition a membership function must really satisfy is that it must vary between 0 and 1. Additionally, it is possible, in a fuzzy set, to have a partial membership, such as "the weather is rather hot." The function itself can be an arbitrary curve whose shape can be defined as a function that suits the particular problem from the point of view of simplicity, convenience, speed, and efficiency.

Based on signals usually obtained from sensors and common knowledge, membership functions for the input and output variables need to be defined. The inputs are described in terms of linguistic variables as, for example, very high, high, okay, low, and very low, as shown in Figure 11.24. It should be noted that, depending on the problem, different sensors could be used showing different parameters such as distance, angle, resistance, slope, etc.

The output can be adjusted in a similar way, according to some membership functions—for example, the ones presented in Figure 11.25. In both cases,

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FiGURE 11.25 Membership functions for linguistic variables describing motor operation.

membership curves other than the triangular can be used, such as trapezoidal, quadratic, Gaussian (exponential), cos-function, and many others.

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