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Stability analysis of the discrete Takagi-Sugeno fuzzy model with time-varying consequent uncertainties. (English) Zbl 0993.93018

The study is concerned with the fuzzy dynamic model governed by “\(m\)” Takagi-Sugeno rules where the \(p\)th rule reads as follows \[ \begin{split} \text{- if }x(k)\text{ is }A_{p1}\text{ and }\dots\text{ and }x(k- n+1)\text{ is }A_{pn}\\ \text{then }x_p(k+ 1)= a_{p1} x(k)+\cdots+ a_{pm}x(k- n+1).\end{split} \] The linear part (subsystem) standing in the consequent part of the rule can be written in a matrix form \(A_px(k)\), and this in turn leads to the output of the system governed by the expression \[ x(k+1)= \sum^m_{p=1} w_p(k) A_px(k)/ \sum^m_{p=1} w_p(k) \] (here the \(w_p(k)\)’s are the degrees of activation of the individual rules). It is assumed that the matrix \(A_p\) comes with time-varying uncertainty (say \(A_p+\Delta A_p(k)\)) where the uncertainty matrices \(\Delta A_p(k)\) are not known and their values are constrained to lie within known compact bounding sets. For this class of TS models, conditions for their global asymptotic stability are derived.

MSC:

93C42 Fuzzy control/observation systems
93D20 Asymptotic stability in control theory
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