## 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

### Keywords:

fuzzy models; stability; fuzzy partition; Takagi-Sugeno model
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### References:

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