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Stability analysis and synthesis of nonlinear time-delay systems via linear Takagi-Sugeno fuzzy models. (English) Zbl 1002.93051
Consider the time-delay system $\dot x(t)= \sum^r_{i=1} h_i(z(t))[A_{1i} x(t)+ A_{2i}x(t- \tau_i(t))],\tag{1}$ $$h_i(z(t))\geq 0$$, for $$i= 1,2,\dots, r$$ and $$\sum^r_{j=1} h_j(z(t))= 1$$ for all $$t$$. Having used a quadratic Lyapunov function $$V(x(t))= x^T(t) Px(t)$$ and the Razumikhin conditions we have
Theorem 1. The equilibrium of the continuous-time fuzzy system with time delay described by (1) is asymptotically stable in the large if there exist a common matrix $$P> 0$$ and $$r$$ matrices $$S_i> 0$$ such that $A^T_{1i}P+ PA_{1i}+ P+ PA_{2i} S_iA^T_{2i}P< 0,\quad P\geq S^{-1}_i,\quad\text{for }i= 1,2,\dots,r.$ Later, more complex time-delay systems are considered.

##### MSC:
 93D20 Asymptotic stability in control theory 93C42 Fuzzy control/observation systems 93C23 Control/observation systems governed by functional-differential equations 93D30 Lyapunov and storage functions
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