Читать книгу Artificial Intelligence and Quantum Computing for Advanced Wireless Networks - Savo G. Glisic - Страница 69

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 used. A set of rules in the conjunctive antecedent form divides the input domain into a lattice of fuzzy hyper‐boxes, parallel with the axes. Each of the hyper‐boxes is a Cartesian product‐space intersection of the corresponding univariate fuzzy sets. The number of rules in the conjunctive form needed to cover the entire domain is given by where p is the dimension of the input space, and N_i is the number of linguistic terms of the i‐th antecedent variable.

By combining conjunctions, disjunctions‚ and negations, various partitions of the antecedent space can be obtained; the boundaries are, however, restricted to the rectangular grid defined by the fuzzy sets of the individual variables. As an example, consider the rule “If x₁ is not A₁₃ and x₂ is A₂₁ then …”

The degree of fulfillment of this rule is computed using the complement and intersection operators:

(4.42)

The antecedent form with multivariate membership functions, Eq. (4.31), is the most general one, as there is no restriction on the shape of the fuzzy regions. The boundaries between these regions can be arbitrarily curved and opaque to the axes. Also, the number of fuzzy sets needed to cover the antecedent space may be much smaller than in the previous cases. Hence, for complex multivariable systems, this partition may provide the most effective representation.

Defuzzification: In many applications, a crisp output y is desired. To obtain a crisp value, the output fuzzy set must be defuzzified. With the Mamdani inference scheme, the center of gravity (COG) defuzzification method is used. This method computes the y coordinate of the COG of the area under the fuzzy set B′:

(4.43)

where F is the number of elements y_j in Y. The continuous domain Y thus must be discretized to be able to compute the COG.