Analytical Methods in Fuzzy Modeling and Control (Studies in by Jacek Kluska

By Jacek Kluska

This publication is concentrated on mathematical research and rigorous layout tools for fuzzy keep watch over structures in line with Takagi-Sugeno fuzzy types, often referred to as Takagi-Sugeno-Kang models.

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M. 53) Every column θj is assigned to a single system output Sj , (j = 1, . . , m). 49). The fundamental matrix Ω of the system is a concatenation of 2n columns which are values of the generator g for the vertices of the hypercuboid Dn , where every vertex corresponds to the appropriate antecedent of the rule. Proof. 4. 30). 7 Equivalence Problem in the Rule-Based Systems The problem of equivalence between the systems of fuzzy “If-then” rules is important especially when one compares the outcomes obtained by various experts or designers, and the number of inputs is greater than two.

Let us consider a MISO TS system with the inputs z1 , . . , zn and the output S (see Fig. 9). This system is defined by 2n rules in the form of implications Fig. 13) n where (i1 , . . ,in ) = “z1 is Ai1 and ... 14) 10 2 MISO Takagi-Sugeno Fuzzy System with Linear Membership Functions and Aik = Nk , for ik = 0 , Pk , for ik = 1 k = 1, . . , n. e. we will consider a zero-order Takagi-Sugeno model [180]. In more general TS systems, the consequents are polynomials of the first or higher order or more complicated functions of input variables.

10. Suppose the MIMO P1-TS system with the inputs constituting the vector [z1 , . . , zn ]T = z ∈ Dn and the outputs S1 , . . 50). The row vector of crisp outputs S (z) = [S1 , . . 52) and n Θ = [θ1 , . . 1 ] ∈ R2 , j = 1, . . , m. 53) Every column θj is assigned to a single system output Sj , (j = 1, . . , m). 49). The fundamental matrix Ω of the system is a concatenation of 2n columns which are values of the generator g for the vertices of the hypercuboid Dn , where every vertex corresponds to the appropriate antecedent of the rule.

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