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.


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