Artificial Intelligence and Quantum Computing for Advanced Wireless Networks. Savo G. Glisic

the max and min operations are taken over different domains, their order can be changed as follows:

      (4.38)

      Denote as the degree of fulfillment of the i‐th rule’s antecedent. The output fuzzy set of the linguistic model is thus

      (4.39)

The entire algorithm, called the max‐min or Mamdani inference, is summarized in Algorithm 4.1 and visualized in Figure 4.5.

Figure 4.5 A schematic representation of the Mamdani inference algorithm.

Algorithm 4.1 Mamdani (max‐min) inference

       1. Compute the degree of fulfillment by . Note that for a singleton fuzzy set ( otherwise) the equation for βi simplifies to . 2. Derive the output fuzzy sets . 3. Aggregate the output fuzzy sets .

      Design Example 4.4

      Let us take the input fuzzy set A′ = [1,0.6,0.3,0] from the previous example and compute the corresponding output fuzzy set by the Mamdani inference method. Step 1 yields the following degrees of fulfillment:

      In step 2, the individual consequent fuzzy sets are computed:

      Finally, step 3 gives the overall output fuzzy set:

      which is identical to the result from the previous example.

      Multivariable systems: So far, the linguistic model was presented in a general manner covering both the single‐input and single‐output (SISO) and multiple‐input and multiple‐output (MIMO) cases. In the MIMO case, all fuzzy sets in the model are defined on vector domains by multivariate membership functions. It is, however, usually more convenient to write the antecedent and consequent propositions as logical combinations of fuzzy propositions with univariate membership functions. Fuzzy logic operators, such as the conjunction, disjunction, and negation (complement), can be used to combine the propositions. Furthermore, a MIMO model can be written as a set of multiple‐input and single‐output (MISO) models, which is also convenient for the ease of notation. Most common is the conjunctive form of the antecedent, which is given by

      (4.40)

      Note that the above model is a special case of Eq. (4.31), as the fuzzy set A_i in Eq. (4.31) is obtained as the Cartesian product of fuzzy sets A_ij : A_i = A_i1 × A_i2 × · · · × A_ip . Hence, the degree of fulfillment (step 1 of Algorithm 4.1) is given by

(4.41)

      Other conjunction operators, such as the product, can be