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

By Jacek Kluska

This e-book is concentrated on mathematical research and rigorous layout equipment for fuzzy regulate structures in line with Takagi-Sugeno fuzzy types, also known as Takagi-Sugeno-Kang models.

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75 ... ....... ..... ... .... ..... ... ... ... ... ... ... ...... ... ... .. ... ... .. ... ... ... ... ... 5 ... ... ... ... ... ..... ... ... ... .. .. ... ... ... ... . . .. . . .. ... 25 ... ... ... ...... . ... ..... ...... . 8 1 z Assumptions and Linguistic Interpretation of Linear Membership Functions We will mainly use linear membership functions for input variables. They are conceptually the simplest, have a clear interpretation and play a crucial role in many applications in the fuzzy modeling and control.

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. 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.

The notion of the generator and the fundamental matrix of the rule-based system belong to the most important ones, both for the theory and applications. 26). It is worth adding that every Boolean (or switching) funcn tion {0, 1} → {0, 1} has a unique representation as a multi-linear polynomial. Such representation has been originally introduced by Zhegalkin [219] and was called canonical polynomial form of a Boolean function and plays an important role in many applications [6], [46], [48], [118].

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