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Stability of time-delay systems via Wirtinger-based double integral inequality. (English) Zbl 1377.93123
Summary: Based on the Wirtinger-based integral inequality, a double integral form of the Wirtinger-based integral inequality (hereafter called as Wirtinger-based double integral inequality) is introduced in this paper. To show the effectiveness of the proposed inequality, two stability criteria for systems with discrete and distributed delays are derived within the framework of Linear Matrix Inequalities (LMIs). The advantage of employing the proposed inequalities is illustrated via two numerical examples.

##### MSC:
 93D05 Lyapunov and other classical stabilities (Lagrange, Poisson, $$L^p, l^p$$, etc.) in control theory 93C15 Control/observation systems governed by ordinary differential equations 93C05 Linear systems in control theory
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##### References:
 [1] Ariba, Y., Gouaisbaut, F., & Johansson, K. H. (2010). Stability interval for time-varying delay systems. In Proceedings of the 49th IEEE conference on decision and control. · Zbl 1303.93135 [2] Chen, W.-H.; Zheng, W. X., Delay-dependent robust stabilization for uncertain neutral systems with distributed delays, Automatica, 43, 1, 95-104, (2007) · Zbl 1140.93466 [3] Graham, A., Kronecker products and matrix calculus: with applications, (1982), John Wiley & Sons [4] Gu, K., A further refinement of discretized Lyapunov functional method for the stability of time-delay systems, International Journal of Control, 74, 10, 967-976, (2001) · Zbl 1015.93053 [5] Gu, K.; Kharitonov, V. L.; Chen, J., Stability of time-delay systems, (2003), Birkhäuser · Zbl 1039.34067 [6] He, Y.; Wang, Q. G.; Lin, C.; Wu, M., Delay-range-dependent stability for systems with time-varying delay, Automatica, 43, 2, 371-376, (2007) · Zbl 1111.93073 [7] Kim, J. H., Note on stability of linear systems with time-varying delay, Automatica, 47, 9, 2118-2121, (2011) · Zbl 1227.93089 [8] Moon, Y. S.; Park, P.; Kwon, W. H., Delay-dependent robust stabilisation 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] Park, P. G.; Ko, J. W.; Jeong, C., Reciprocally convex approach to stability of systems with time-varying delays, Automatica, 47, 1, 235-238, (2011) · Zbl 1209.93076 [11] Seuret, A., & Gouaisbaut, F. (2012). On the use of the Wirtinger inequalities for time-delay systems. In 10th IFAC workshop on time delay systems. Boston: États-Unis. · Zbl 1364.93740 [12] Seuret, A.; Gouaisbaut, F., Wirtinger-based integral inequality: application to time-delay systems, Automatica, 49, 9, 2860-2866, (2013) · Zbl 1364.93740 [13] Seuret, A., & Gouaisbaut, F. (2014). Complete quadratic Lyapunov functionals using Bessel-Legendre inequality. In Proc. 13th European control conference, Strasborug; France. · Zbl 1276.93070 [14] Skelton, R. E.; Iwasaki, T.; Grigoradis, K. M., A unified algebraic approach to linear control design, (1997), Taylor & Francis [15] Sun, J.; Liu, G. P.; Chen, J., Delay-dependent stability and stabilization of neutral time-delay systems, International Journal of Robust and Nonlinear Control, 19, 1364-1375, (2009) · Zbl 1169.93399 [16] Zheng, M.; Li, K.; Fei, M., Comments on Wirtinger-based integral inequality: application to time-delay systems [automatica 49 (2013) 2860-2866], Automatica, 50, 1, 300-301, (2014)
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