## 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

Zbl 0957.34069
Full Text:

### References:

 [1] Boyd, S.; Ghaoui, L.E.; Feron, E.; Balakrishnan, V., Linear matrix inequalities in system and control theory, (1994), SIAM Philadelphia · Zbl 0816.93004 [2] Han, Q.-L., On robust stability of neutral systems with time-varying discrete delay and norm-bounded uncertainty, Automatica, 40, 6, 1087-1092, (2004) · Zbl 1073.93043 [3] He, Y.; Wu, M.; She, J.-H.; Liu, G.-P., Delay-dependent robust stability criteria for uncertain neutral systems with mixed delays, Systems & control letters, 51, 1, 57-65, (2004) · Zbl 1157.93467 [4] He, Y.; Wu, M.; She, J.-H.; Liu, G.-P., Parameter-dependent Lyapunov functional for stability of time-delay systems with polytopic-type uncertainties, IEEE transactions on automatic control, 49, 5, 828-832, (2004) · Zbl 1365.93368 [5] Jiang, X.; Han, Q.-L., On $$\mathcal{H}_\infty$$ control for linear systems with interval time-varying delay, Automatica, 41, 12, 2099-2106, (2005) · Zbl 1100.93017 [6] Jing, X.-J.; Tan, D.-L.; Wang, Y.-C., An LMI approach to stability of systems with severe time-delay, IEEE transactions on automatic control, 49, 7, 1192-1195, (2004) · Zbl 1365.93226 [7] Lien, C.-H., Delay-dependent stability criteria for uncertain neutral systems with multiple time-varying delays via LMI approach, IEE Proceedings: control theory and applications, 152, 6, 707-714, (2005) [8] Moon, Y.S.; Park, P.; Kwon, W.H.; Lee, Y.S., Delay-dependent robust stabilization of uncertain state-delayed systems, International journal of control, 74, 14, 1447-1455, (2001) · Zbl 1023.93055 [9] Park, P., A delay-dependent stability criterion for systems with uncertain time-invariant delays, IEEE transactions on automatic control, 44, 4, 876-877, (1999) · Zbl 0957.34069 [10] Wu, M.; He, Y.; She, J.-H.; Liu, G.-P., Delay-dependent criteria for robust stability of time-varying delay systems, Automatica, 40, 8, 1435-1439, (2004) · Zbl 1059.93108 [11] Yue, D.; Han, Q.-L., A delay-dependent stability criterion of neutral systems and its application to a partial element equivalent circuit model, IEEE transactions on circuits and systems-II: express briefs, 51, 12, 685-689, (2004)
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