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

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

      (4.38)mu Subscript upper B Sub Superscript prime Subscript Baseline left-parenthesis normal y right-parenthesis equals max left-brace max left-bracket mu Subscript upper A Sub Superscript prime Subscript Baseline left-parenthesis normal x right-parenthesis logical-and mu Subscript upper A Sub Subscript i Subscript Baseline left-parenthesis normal x right-parenthesis right-bracket logical-and mu Subscript upper B Sub Subscript i Subscript Baseline left-parenthesis normal y right-parenthesis right-brace period

      Denote beta Subscript i Baseline equals max Underscript upper X Endscripts left-bracket mu Subscript upper A prime Baseline left-parenthesis normal x right-parenthesis logical-and mu Subscript upper A Sub Subscript i Subscript Baseline left-parenthesis normal x right-parenthesis right-bracket as the degree of fulfillment of the i‐th rule’s antecedent. The output fuzzy set of the linguistic model is thus

      (4.39)mu Subscript upper B Sub Superscript prime Subscript Baseline left-parenthesis normal y right-parenthesis equals max left-bracket beta Subscript i Baseline logical-and mu Subscript upper B Sub Subscript i Subscript Baseline left-parenthesis normal y right-parenthesis right-bracket comma normal y element-of upper Y period

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

      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:

StartLayout 1st Row beta 1 equals max left-bracket mu Subscript upper A Sub Superscript prime Subscript Baseline left-parenthesis x right-parenthesis logical-and mu Subscript upper A 1 Baseline left-parenthesis x right-parenthesis right-bracket equals max left-parenthesis left-bracket 1 comma 0.6 comma 0.3 comma 0 right-bracket logical-and left-bracket 1 comma 0.6 comma 0 comma 0 right-bracket right-parenthesis equals 1 comma 2nd Row beta 2 equals max left-bracket mu Subscript upper A Sub Superscript prime Subscript Baseline left-parenthesis x right-parenthesis logical-and mu Subscript upper A 2 Baseline left-parenthesis x right-parenthesis right-bracket equals max left-parenthesis left-bracket 1 comma 0.6 comma 0.3 comma 0 right-bracket logical-and left-bracket 0 comma 0.4 comma 1 comma 0.4 right-bracket right-parenthesis equals 0.4 3rd Row beta 3 equals max left-bracket mu Subscript upper A Sub Superscript prime Subscript Baseline left-parenthesis x right-parenthesis logical-and mu Subscript upper A 3 Baseline left-parenthesis x right-parenthesis right-bracket equals max left-parenthesis left-bracket 1 comma 0.6 comma 0.3 comma 0 right-bracket logical-and left-bracket 0 comma 0 comma 0.1 comma 1 right-bracket right-parenthesis equals 0.1 period EndLayout comma

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

StartLayout 1st Row upper B prime 1 equals beta 1 logical-and upper B 1 equals 1 logical-and left-bracket 1 comma 1 comma 0.6 comma 0 comma 0 right-bracket equals left-bracket 1 comma 1 comma 0.6 comma 0 comma 0 right-bracket 2nd Row upper B prime 2 equals beta 2 logical-and upper B 2 equals 0.4 logical-and left-bracket 0 comma 0 comma 0.3 comma 0.9 comma 1 right-bracket equals left-bracket 0 comma 0 comma 0.3 comma 0.4 comma 0.4 right-bracket comma 3rd Row upper B prime 3 equals beta 3 logical-and upper B 3 equals 0.1 logical-and left-bracket 1 comma 1 comma 0.6 comma 0 comma 0 right-bracket equals left-bracket 0.1 comma 0.1 comma 0.1 comma 0 comma 0 right-bracket period EndLayout

      Finally, step 3 gives the overall output fuzzy set:

upper B equals max mu Subscript upper B prime Sub Subscript i Baseline equals left-bracket 1 comma 1 comma 0.6 comma 0.4 comma 0.4 right-bracket comma

      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)script upper R Subscript i Baseline colon If x 1 is upper A Subscript i Baseline 1 Baseline and x 2 is upper A Subscript i Baseline 2 Baseline and ellipsis and x Subscript p Baseline is upper A Subscript italic i p Baseline then y is upper B Subscript i Baseline comma i equals 1 comma 2 comma ellipsis comma upper K period

      Note that the above model is a special case of Eq. (4.31), as the fuzzy set Ai in Eq. (4.31) is obtained as the Cartesian product of fuzzy sets Aij : Ai = Ai1 × Ai2 × · · · × Aip . Hence, the degree of fulfillment (step 1 of Algorithm 4.1) is given by

      Other conjunction operators, such as the product, can be

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