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By Jacek Kluska

This publication is targeted on mathematical research and rigorous layout tools for fuzzy keep an eye on platforms according to Takagi-Sugeno fuzzy versions, often referred to as Takagi-Sugeno-Kang versions. the writer offers a slightly common analytical concept of tangible fuzzy modeling and keep an eye on of constant and discrete-time dynamical structures. major awareness is paid to usability of the consequences for the keep watch over and computing device engineering neighborhood and for this reason uncomplicated and straightforward knowledge-bases for linguistic interpretation were used. The technique relies at the author’s theorems pertaining to equivalence among regular Takagi-Sugeno structures and a few classification of multivariate polynomials. It combines some great benefits of fuzzy procedure conception and classical regulate concept. Classical keep watch over concept should be utilized to modeling of dynamical crops and the controllers. they're all corresponding to the set of Takagi-Sugeno style fuzzy principles. The strategy combines the simplest of fuzzy and standard keep watch over concept. It allows linguistic interpretability (also referred to as transparency) of either the plant version and the controller. with regards to linear structures and a few category of nonlinear structures, engineers can in lots of instances without delay follow famous classical instruments from the regulate conception either for research, and the layout of closed-loop fuzzy keep watch over structures. accordingly the most target of the publication is to set up finished and unified analytical foundations for fuzzy modeling utilizing the Takagi-Sugeno rule scheme and their purposes for fuzzy keep an eye on, id of a few type of nonlinear dynamical methods and type challenge solver design.

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Additional info for Analytical Methods in Fuzzy Modeling and Control

Sample text

Ms ⎥ ⎥ ⎢ ⎥ ⎢ = ⎢ ... ⎥ , ⎥ ⎢ ⎢ mr ⎥ ⎥ ⎢ ⎢ . ⎥ ⎣ .. ⎦ ⎡ M[2] ⎤ q1 ⎢ .. ⎥ ⎢ . ⎥ ⎥ ⎢ ⎢ qs ⎥ ⎥ ⎢ ⎥ ⎢ = ⎢ ... ⎥ , ⎥ ⎢ ⎢ qr ⎥ ⎥ ⎢ ⎢ . ⎥ ⎣ .. ⎦ ⎡ Q[2] m2n q2n where m1 , . . , m2n ∈ {N1 , P1 } × · · · × {Nn , Pn }, and q1 , . . , q2n ∈ R. 31)), we immediately obtain that the outputs of such systems are the same. Of course, the above systems have different generators ⎤ ⎤ ⎡ ⎡ g0 g0 ⎥ ⎥ ⎢ ⎢ .. ⎥ ⎥ ⎢ ⎢ ⎢ r1 . r ⎥ ⎢ s1 . s ⎥ ⎢ z1 · · · znn ⎥ ⎢ z1 · · · znn ⎥ ⎥ ⎥ ⎢ ⎢ ⎥ ⎥ ⎢ ⎢ .. (z , . . , z ) = g[1] (z1 , .

Zn−1 ) 1 aT βn + bT αn , bT − aT Ω−1 n−1 z g Ln n n−1 (z1 , . . , zn−1 ) (βn − zn ) aT Ω−1 n−1 gn−1 (z1 , . . , zn−1 ) αn + βn (αn + zn ) bT Ω−1 n−1 gn−1 (z1 , . . , zn−1 ) + . αn + βn = But aT Ω−1 n−1 gn−1 (z1 , . . , zn−1 ) = Sn−1 (z1 , . . , zn−1 | q1 , . . , q2n−1 ) , and bT Ω−1 n−1 gn−1 (z1 , . . , zn−1 ) = Sn−1 (z1 , . . , zn−1 | q2n−1 +1 , . . , q2n ) . Thus, βn − z n Sn−1 (z1 , . . , zn−1 | q1 , . . , q2n−1 ) αn + βn αn + zn + Sn−1 (z1 , . . , zn−1 | q2n−1 +1 , . . , q2n ) αn + βn = Nn (zn ) Sn−1 (z1 , .

Thus, both systems of rules describe exactly the same system, indeed. 10 independently in−1 in zn ” of the order of “If-then” rules provided that every element “z1i1 · · · zn−1 of the generator corresponds both to the appropriate sequence of labels Ai1 , . . , Ain−1 , Ain ∈ {N1 , P1 } × . . × {Nn−1 , Pn−1 } × {Nn , Pn } , which occurs in the antecedent of the rule, and to the appropriate element qv in the consequents vector q. An extension of the above result for MIMO systems is straightforward and will be omitted.

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