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Stability analysis and systematic design of Takagi–Sugeno fuzzy control systems. (English) Zbl 1142.93373

Discussed are Takagi-Sugeno models of the form \[ \begin{gathered} R_i:\text{ If }x_1(t)\text{ is }F_{1i}\text{ and }x_2(t)\text{ is }F_{2i}\text{ and}\dots x_n(t)\text{ is }F_{ni}\text{ then }dx(t)/dt= A_ix(k)+ B_i u(t),\\ i= 1,2,\dots, 1,\end{gathered} \] where \(x(t)= [x_1x_2 x_n]^T\) denotes a state vector and \((A_i, B_i)\) stands for the matrices of the corresponding local model. The extended Lyapunov stability criterion applied to the rule-based system presented above is concerned with the structural information about rules “activated” within some region and this helps relax the stability conditions (in which the \(n\times n\) positive definite symmetric matrix \(P\) involves only a subset of matrices \(P_1,P_2,\dots, P_r\) pertaining to the individual rules). The mechanism of stability verification is then presented. Numeric examples are also included in this study.

MSC:

93C42 Fuzzy control/observation systems
93D05 Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory
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