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Stability and robust stability for systems with a time-varying delay. (English) Zbl 1120.93043
Summary: To concern the stability and robust stability criteria for systems with time-varying delays, this note uses not only the time-varying-delayed state \(x(t-h(t))\) but also the delay-upper-bounded state \(x(t-\bar h)\) to exploit all possible information for the relationship among a current state \(x(t)\), an exactly delayed state \(x(t-h(t))\), a marginally delayed state \(x(t-\bar h)\), and the derivative of the state \(\dot x(t)\), when constructing Lyapunov-Krasovskii functionals and some appropriate integral inequalities, originally suggested by P. Park [IEEE Trans. Autom. Control 44, No. 4, 876–877 (1999; Zbl 0957.34069)]. Two fundamental criteria are provided for the cases where no bound of delay derivative is assumed and where an upper bound of delay derivative is assumed. Examples show the resulting criteria outperform all existing ones in the literature.

MSC:
93D09 Robust stability
93C05 Linear systems in control theory
93C15 Control/observation systems governed by ordinary differential equations
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