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Stability analysis of discrete-time switched systems with all unstable subsystems

  • *Corresponding author: Zhouchao Wei

    *Corresponding author: Zhouchao Wei

This work was supported by the National Natural Science Foundation of China (NNSFC) (Nos. 12172340), the Fundamental Research Funds for the Central Universities, China University of Geosciences (Wuhan) (Nos. G1323523061 and G1323523041).

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  • In this manuscript, it is investigated that the stability property of a discrete-time switched system consisting of all unstable subsystems. Under a switching signal with certain conditions, the sufficient constraints for asymptotic stability of a discrete-time switched system composed of all unstable subsystems are obtained via Lyapunov functions and the defined divergence time. Furthermore, based on this result, linear matrix inequalities are obtained for asymptotic stability of a linear discrete-time switched system composed of all unstable subsystems. The efficiency of the acquired theorems is exhibited by three numerical examples.

    Mathematics Subject Classification: Primary: 34D05, 37B25; Secondary: 93C10, 93C55, 93D20.

    Citation:

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  • Figure 1.  The trajectory of the state $ x_{1} $ of system (26) with a given switching signal (29) and an initial condition $ x_{0} = (1, 1)^{\top} $

    Figure 2.  The trajectory of the state $ x_{2} $ of system (26) with a given switching signal (29) and an initial condition $ x_{0} = (1, 1)^{\top} $

    Figure 3.  For system (26), the red stars are for $ \sigma(k) = A_{1} $ and the blue stars are for $ \sigma(k) = A_{2} $

    Figure 4.  The trajectory of the state $ x_{1} $ of system (30) with a given switching signal (33) and an initial condition $ x_{0} = (3, 4, 1)^{\top} $

    Figure 5.  The trajectory of the state $ x_{2} $ of system (30) with a given switching signal (33) and an initial condition $ x_{0} = (3, 4, 1)^{\top} $

    Figure 6.  The trajectory of the state $ x_{3} $ of system (30) with a given switching signal (33) and an initial condition $ x_{0} = (3, 4, 1)^{\top} $

    Figure 7.  For system (30), the red stars are for $ \sigma(k) = A_{1} $ and the blue stars are for $ \sigma(k) = A_{2} $

    Figure 8.  The trajectory of the state $ x_{1} $ of system (34) with a given switching signal (37) and an initial condition $ x_{0} = (8, 8, 6)^{\top} $

    Figure 9.  The trajectory of the state $ x_{2} $ of system (34) with a given switching signal (37) and an initial condition $ x_{0} = (8, 8, 6)^{\top} $

    Figure 10.  The trajectory of the state $ x_{3} $ of system (34) with a given switching signal (37) and an initial condition $ x_{0} = (8, 8, 6)^{\top} $

    Figure 11.  For system (34), the red stars are for $ \sigma(k) = A_{1} $ and the blue stars are for $ \sigma(k) = A_{2} $

  • [1] T. AdrianU. IoanE. Daniela and T. George, Towards nonconservative conditions for equilibrium stability, applications to switching systems with control delay, Communications in Nonlinear Science and Numerical Simulation, 121 (2023), 107188. 
    [2] L. Allerhand and U. Shaked, Robust stability and stabilization of linear switched systems with dwell time, IEEE Transactions on Automatic Control, 56 (2011), 381-386.  doi: 10.1109/TAC.2010.2097351.
    [3] M. BoccadoroY. WardiM. Egerstedt and E. Verriesst, Optimal control of switching surfaces in hybrid dynamical systems., Discrete Event Dynamic Systems: Theory and Applications, 15 (2005), 433-448.  doi: 10.1007/s10626-005-4060-4.
    [4] L. Cheng, X. Xu, Y. Xue and H. Zhang, Stability analysis of switched systems with all subsystems unstable: A matrix polynomial approach, ISA Transactions, 114 2021, 99-105. doi: 10.1016/j.isatra.2020.12.031.
    [5] P. ChengS. HeX. Luan and F. Liu, Finite-region asynchronous ${H}_{\infty}$ control for 2D Markov jump systems, Automatica, 129 (2021), 109590.  doi: 10.1016/j.automatica.2021.109590.
    [6] G. ChesiP. ColaneriJ. GeromelR. Middleton and R. Shorten, A nonconservative LMI condition for stability of switched systems with guaranteed dwell time, IEEE Transactions on Automatic Control, 57 (2012), 1297-1302.  doi: 10.1109/TAC.2011.2174665.
    [7] R. DecarloM. BranickyS. Pettersson and B. Lennartson, Perspectives and results on the stability and stabilizability of hybrid systems, Proceedings of the IEEE, 88 (2000), 1069-1082.  doi: 10.1109/5.871309.
    [8] K. Ding and Q. Zhu, Intermittent static output feedback control for stochastic delayed-switched positive systems with only partially measurable information, IEEE Transactions on Automatic Control, 68 (2023), 8150-8157.  doi: 10.1109/TAC.2023.3293012.
    [9] L. FanQ. Zhu and W. Zheng, Stability analysis of switched stochastic nonlinear systems with state-dependent delay, IEEE Transactions on Automatic Control, 69 (2024), 2567-2574.  doi: 10.1109/TAC.2023.3315672.
    [10] C. Y. F. HoB. LingY. LiuP. Tam and K. Teo, Optimal pwm control of switched-capacitor dc-dc power converters via model transformation and enhancing control techniques, IEEE Transactions on Circuits and Systems I: Regular Papers, 55 (2008), 1382-1391.  doi: 10.1109/TCSI.2008.916442.
    [11] C.-L. JinQ.-G. WangR. Wang and D. Wu, Stabilization of switched systems with unstable modes via mode-partition-dependent adt, Journal of the Franklin Institute, 360 (2023), 1308-1326.  doi: 10.1016/j.jfranklin.2022.10.015.
    [12] J. JinJ. RamirezS. WeeD. LeeY. Kim and G. Gans, A switched-system approach to formation control and heading consensus for multi-robot systems, Intelligent Service Robotics, 11 (2018), 207-224.  doi: 10.1007/s11370-018-0246-0.
    [13] H. Li, Stability analysis of time-varying switched systems via indefinite difference Lyapunov functions, Nonlinear Analysis: Hybrid Systems, 48 (2023), 101329.  doi: 10.1016/j.nahs.2022.101329.
    [14] Z. LiC. Wen and Y. Soh, Stabilization of a class of switched systems via designing switching laws, IEEE Transactions on Automatic Control, 46 (2001), 665-670.  doi: 10.1109/9.917674.
    [15] H. Lin and P. J. Antsaklis, Stability and stabilizability of switched linear systems: A survey of recent results, IEEE Transactions on Automatic Control, 54 (2009), 308-322.  doi: 10.1109/TAC.2008.2012009.
    [16] J. Lu and L. Brown, A multiple Lyapunov functions approach for stability of switched systems, In Proceedings of the 2010 American Control Conference, (2010), 3253-3256.
    [17] S. LuoF. Deng and W. Chen, Unified dwell time-based stability and stabilization criteria for switched linear stochastic systems and their application to intermittent control, International Journal of Robust and Nonlinear Control, 28 (2018), 2014-2030.  doi: 10.1002/rnc.3997.
    [18] X. MaoH. ZhuW. Chen and H. Zhang, New results on stability of switched continuous-time systems with all subsystems unstable, ISA Transactions, 87 (2019), 28-33. 
    [19] A. Morse, Supervisory control of families of linear set-point controllers - part I. exact matching, IEEE Transactions on Automatic Control, 41 (1996), 1413-1431.  doi: 10.1109/9.539424.
    [20] S. Pettersson and B. Lennartson, Stabilization of hybrid systems using a min-projection strategy, Proceedings of the 2001 American Control Conference, (Cat. No.01CH37148), 1 (2001), 223-228.  doi: 10.1109/ACC.2001.945546.
    [21] C. RenS. HeX. LuanF. Liu and H. Karimi, Finite-time L2-gain asynchronous control for continuous-time positive hidden Markov jump systems via T-S fuzzy model approach, IEEE Transactions on Cybernetics, 51 (2021), 77-87. 
    [22] J. RidenourJ. HuN. Pettis and Y. Lu, Low-power buffer management for streaming data, IEEE Transactions on Circuits and Systems for Video Technology, 17 (2007), 143-157.  doi: 10.1109/TCSVT.2006.888025.
    [23] Y. TianY. Cai and Y. Sun, Stability of switched nonlinear time-delay systems with stable and unstable subsystems, Nonlinear Analysis: Hybrid Systems, 24 (2017), 58-68.  doi: 10.1016/j.nahs.2016.11.003.
    [24] S. Vassilyev and A. Kosov, Common and multiple Lyapunov functions in stability analysis of nonlinear switched systems, AIP Conf. Proc., 1493 (2012), 1066-1073.  doi: 10.1063/1.4765620.
    [25] B. WangQ. Zhu and S. Li, Stability analysis of discrete-time semi-Markov jump linear systems with time delay, IEEE Transactions on Automatic Control, 68 (2023), 6758-6765.  doi: 10.1109/TAC.2023.3240926.
    [26] B. WangQ. Zhu and S. Li, Stabilization of discrete-time hidden semi-markov jump linear systems with partly unknown emission probability matrix, IEEE Transactions on Automatic Control, 69 (2024), 1952-1959.  doi: 10.1109/TAC.2023.3272190.
    [27] M. WicksP. Peleties and R. Decarlo, Switched controller synthesis for the quadratic stabilisation of a pair of unstable linear systems, European Journal of Control, 4 (1998), 140-147.  doi: 10.1016/S0947-3580(98)70108-6.
    [28] W. Xiang and J. Xiao, Stabilization of switched continuous-time systems with all modes unstable via dwell time switching, Automatica, 50 (2014), 940-945.  doi: 10.1016/j.automatica.2013.12.028.
    [29] Z. Xiang and W. Xiang, Stability analysis of switched systems under dynamical dwell time control approach, International Journal of Systems Science, 40 (2009), 347-355.  doi: 10.1080/00207720802436240.
    [30] H. XiaoQ. Zhu and H. Karimi, Stability of stochastic delay switched neural networks with all unstable subsystems: A multiple discretized Lyapunov-Krasovskii functionals method, Information Sciences, 582 (2022), 302-315.  doi: 10.1016/j.ins.2021.09.027.
    [31] X. XuX. Mao and H. Zhang, Stability analysis of switched system with all subsystems unstable under novel average dwell time criteria, IEEE Access, 7 (2019), 44959-44965. 
    [32] H. YangB. JiangV. Cocquempot and H. Zhang, Stabilization of switched nonlinear systems with all unstable modes: Application to multi-agent systems, IEEE Transactions on Automatic Control, 56 (2011), 2230-2235.  doi: 10.1109/TAC.2011.2157413.
    [33] H. YeA. Michel and L. Hou, Stability theory for hybrid dynamical systems, IEEE Transactions on Automatic Control, 43 (1998), 461-474.  doi: 10.1109/9.664149.
    [34] Z. YeD. ZhangC. DengH. Yan and G. Feng, Finite-time resilient sliding mode control of nonlinear UMV systems subject to Dos attacks, Automatica, 156 (2023), 111170.  doi: 10.1016/j.automatica.2023.111170.
    [35] Q. Yu and H. Lv, Stability analysis for discrete-time switched systems with stable and unstable modes based on a weighted average dwell time approach, Nonlinear Analysis: Hybrid Systems, 38 (2020), 100949.  doi: 10.1016/j.nahs.2020.100949.
    [36] Q. Yu and X. Yuan, Stability analysis for positive switched systems having stable and unstable subsystems based on a weighted average dwell time scheme, ISA Transactions, 136 (2023), 275-283. doi: 10.1016/j.isatra.2022.10.019.
    [37] Q. Yu and X. Zhao, Stability analysis of discrete-time switched linear systems with unstable subsystems, Applied Mathematics and Computation, 273 (2016), 718-725.  doi: 10.1016/j.amc.2015.10.039.
    [38] G. Zhai, B. Hu, K. Yasuda and A. Michel, Stability analysis of switched systems with stable and unstable subsystems: An average dwell time approach, in Internat. J. Systems Sci., 32 (2001), 1055-1061. doi: 10.1080/00207720116692.
    [39] G. Zhai, I. Matsune, J. Imae and T. Kobayashi, A note on multiple Lyapunov functions and stability condition for switched and hybrid systems, in 2007 IEEE International Conference on Control Applications, (2007), 226-231.
    [40] H. ZhangD. XieH. Zhang and G. Wang, Stability analysis for discrete-time switched systems with unstable subsystems by a mode-dependent average dwell time approach, disturbance estimation and mitigation, ISA Transactions, 53 (2014), 1081-1086. 
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