Exponential stabilization of the class of the switched systems with mixed time varying delays in state and control
DOI:
https://doi.org/10.56764/hpu2.jos.2024.3.1.47-56Abstract
This paper presents the problem of exponential stabilization of switched systems with mixed time-varying delays in state and control. Based on the partitioning of the stability state regions into convex cones, a constructive geometric design for switching laws is put forward. By using an improved Lyapunov–Krasovskii functional in combination with matrix knowledge, we design a state feedback controller that guarantees the closed-loop system to be exponentially stable. The obtained conditions are given in terms of linear matrix inequalities (LMIs), which can be effectively decoded in polynomial time by various computational tools such as the LMI tool in MATLAB software. A numerical example is proposed to illustrate the effectiveness of the obtained results.
References
[1] Y. Zhang, X. Liu, and X. Shen, “Stability of switched systems with time delay,” Nonlinear Anal. Hybrid Syst., vol. 1, no. 1, pp. 44–58, Mar. 2007, doi: 10.1016/j.nahs.2006.03.001.
[2] Z. Sun, S.S. Ge, Switched Linear Systems: Control and Design, Springer, London, 2005, doi: 10.1007/1-84628-131-8.
[3] A.V. Savkin, R.J. Evans, Hybrid dynamical systems: Controller and sensor switching problems, Boston, MA: Birkhäuser Boston, 2002. doi: 10.1007/978-1-4612-0107-6.
[4] V. N. Phat, T. Botmart, and P. Niamsup, “Switching design for exponential stability of a class of nonlinear hybrid time-delay systems,” Nonlinear Anal. Hybrid Syst., vol. 3, no. 1, pp. 1–10, Feb. 2009, doi: 10.1016/j.nahs.2008.10.001.
[5] F. Gao, S. Zhong, and X. Gao, “Delay-dependent stability of a type of linear switching systems with discrete and distributed time delays,” Appl. Math. Comput., vol. 196, no. 1, pp. 24–39, Feb. 2008, doi: 10.1016/j.amc.2007.05.053.
[6] P. T. Nam and V. N. Phat, “Robust stabilization of linear systems with delayed state and control,” J. Optim. Theory Appl., vol. 140, no. 2, pp. 287–299, Sep. 2008, doi: 10.1007/s10957-008-9453-8.
[7] F. Uhlig, “A recurring theorem about pairs of quadratic forms and extensions: a survey,” Linear Algebra Appl., vol. 25, no. 2, pp. 219–237, Jun. 1979, doi: 10.1016/0024-3795(79)90020-x.
[8] Y.-E. Wang, Hamid Reza Karimi, and D. Wu, “Conditions for the stability of switched systems containing unstable subsystems,” IEEE Trans. Circuits Syst. II Express Briefs, vol. 66, no. 4, pp. 617–621, Apr. 2019, doi: 10.1109/tcsii.2018.2852766.
[9] Z. Wang, J. Sun, J. Chen, and Y. Bai, “Finite‐time stability of switched nonlinear time‐delay systems,” Int. J. Robust Nonlinear Control, vol. 30, no. 7, pp. 2906–2919, May. 2020, doi: 10.1002/rnc.4928.
[10] J. Wei, H. Zhi, K. Liu, and X. Mu, “Stability of mode-dependent linear switched singular systems with stable and unstable subsystems,” J. Franklin Inst., vol. 356, no. 5, pp. 3102–3114, Mar. 2019, doi: 10.1016/j.jfranklin.2019.02.014.
[11] X. Zhao, L. Zhang, P. Shi, and M. Li, “Stability of switched positive linear systems with average dwell time switching,” Automatica, vol. 48, no. 6, pp. 1132–1137, Jun. 2012, doi: 10.1016/j.automatica.2012.03.008.
[12] W. Zhang, J. Fang, and W. Cui, “Exponential stability of switched genetic regulatory networks with both stable and unstable subsystems,” J. Franklin Inst., vol. 350, no. 8, pp. 2322–2333, Oct. 2013, doi: 10.1016/j.jfranklin.2013.06.007.
[13] J. Zhang, Z. Han, F. Zhu, and J. Huang, “Stability and stabilization of positive switched systems with mode-dependent average dwell time,” Nonlinear Anal. Hybrid Syst., vol. 9, no. 8, pp. 42–55, Aug. 2013, doi: 10.1016/j.nahs.2013.01.005.
[14] D. Zhang and L. Yu, “Exponential stability analysis for neutral switched systems with interval time-varying mixed delays and nonlinear perturbations,” Nonlinear Anal. Hybrid Syst., vol. 6, no. 2, pp. 775–786, May 2012, doi: 10.1016/j.nahs.2011.10.002.
[15] L. Zhou, Daniel, and G. Zhai, “Stability analysis of switched linear singular systems,” Automatica, vol. 49, no. 5, pp. 1481–1487, May 2013, doi: 10.1016/j.automatica.2013.02.002.
[16] W. Xiang, “Stabilization for continuous-time switched linear systems: A mixed switching scheme,” Nonlinear Anal. Hybrid Syst., vol. 36, no. 5, p. 100872, May 2020, doi: 10.1016/j.nahs.2020.100872.
[17] R. Yang, S. Liu, X. Li, and T. Huang, “Stability analysis of delayed fractional-order switched systems,” Trans. Inst. Meas. Control, vol. 45, no. 3, pp. 502–511, Sep. 2022, doi: 10.1177/01423312221116713.
[18] L. V. Hien and V. N. Phat, “Exponential stabilization for a class of hybrid systems with mixed delays in state and control,” Nonlinear Anal. Hybrid Syst., vol. 3, no. 3, pp. 259–265, Aug. 2009, doi: 10.1016/j.nahs.2009.01.009.
[19] S. Boyd, L. E. Ghaoui, E. Feron and V. Balakrishnan, Linear matrix inqualities in systems and control theory, Philadenphia, SIAM, 1994, doi: 10.1137/1.9781611970777.
[20] K. Gu, “An integral inequality in the stability problem of time delay systems,” in Proceedings of the 39th IEEE conference on decision and control, vol. 3, no. 3, pp. 2805–2810, Aug. 2009, doi: 10.1109/CDC.2000.914233.
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