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Volume 6 Issue 1
Jan.  2019

IEEE/CAA Journal of Automatica Sinica

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Article Contents
Shaoxia Feng, Juan Wang and Jun Zhao, "Stability and Robust Stability of Switched Positive Linear Systems With All Modes Unstable," IEEE/CAA J. Autom. Sinica, vol. 6, no. 1, pp. 167-176, Jan. 2019. doi: 10.1109/JAS.2017.7510718
 Citation: Shaoxia Feng, Juan Wang and Jun Zhao, "Stability and Robust Stability of Switched Positive Linear Systems With All Modes Unstable," IEEE/CAA J. Autom. Sinica, vol. 6, no. 1, pp. 167-176, Jan. 2019.

# Stability and Robust Stability of Switched Positive Linear Systems With All Modes Unstable

##### doi: 10.1109/JAS.2017.7510718
Funds:

National Natural Science Foundation of China 61703288

National Natural Science Foundation of China 61603079

National Natural Science Foundation of China 61873174

• This paper is concerned with the stability and robust stability of switched positive linear systems (SPLSs) whose subsystems are all unstable. By means of the mode-dependent dwell time approach and a class of discretized co-positive Lyapunov functions, some stability conditions of switched positive linear systems with all modes unstable are derived in both the continuous-time and the discrete-time cases, respectively. The copositive Lyapunov functions constructed in this paper are timevarying during the dwell time and time-invariant afterwards. In addition, the above approach is extended to the switched interval positive systems. A numerical example is proposed to illustrate our approach.

•  [1] R. Shorten, F. Wirth, and D. Leith, "A positive systems model of TCP-like congestion control: asymptotic results, " IEEE/ACM Trans. Netw., vol. 14, no. 3, pp. 616-629, Jun. 2006. http://www.researchgate.net/publication/2938477_A_positive_systems_model_of_TCP-like_congestion [2] A. Jadbabaie, J. Lin, and A. S. Morse, "Coordination of groups of mobile autonomous agents using nearest neighbor rules, " IEEE Trans. Autom. Control, vol. 48, no. 6, pp. 988-1001, Jun. 2003. https://ieeexplore.ieee.org/document/1205192 [3] M. Bolajraf, F. Tadeo, T. Alvarez, and M. A. Rami, "State-feedback with memory for controlled positivity with application to congestion control, " IET Control Theory Appl., vol. 4, no. 10, pp. 2041-2048, Oct. 2010. http://www.researchgate.net/publication/224185829_State-feedback_with_memory_for_controlled_positivity_with_application_to_congestion_control [4] J. Zhao and D. J. Hill, "On stability, L$_2$-gain and H$_{\infty}$ control for switched systems, " Automatica, vol. 44, no. 5, pp. 1220-1232, May 2008. [5] J. Wang and J. Zhao, "Stability analysis and control synthesis for a class of cascade switched nonlinear systems with actuator saturation, " Circ. Syst. Signal Process., vol. 33, no. 9, pp. 2961-2970, Sep. 2014. [6] X. M. Sun, J. Zhao, and D. J. Hill, "Stability and L$_2$-gain analysis for switched delay systems: A delay-dependent method, " Automatica, vol. 42, no. 10, pp. 1769-1774, Oct. 2006. [7] E. Fornasini and M. E. Valcher, "Asymptotic stability and stabilizability of special classes of discrete-time positive switched systems, " Linear Algebra Appl., vol. 438, no. 4, pp. 1814-1831, Feb. 2013. https://www.sciencedirect.com/science/article/pii/S0024379511006112 [8] F. Blanchini, P. Colaneri, and M. E. Valcher, "Co-positive Lyapunov functions for the stabilization of positive switched systems, " IEEE Trans. Automat. Control, vol. 57, no. 12, pp. 3038-3050, Dec. 2012. http://www.researchgate.net/publication/260661819_Co-Positive_Lyapunov_Functions_for_the_Stabilization_of_Positive_Switched_Systems [9] E. Fornasini and M. E. Valcher, "On the stability of continuous-time positive switched systems, " in Proc. American Control Conference (ACC), Baltimore, MD, USA, 2010, pp. 6225-6230. http://www.researchgate.net/publication/224162584_On_the_stability_of_continuous-time_positive_switched_systems [10] L. Gurvits, R. Shorten, and O. Mason, "On the stability of switched positive linear systems, " IEEE Trans. Automat. Control, vol. 52, no. 6, pp. 1099-1103, Jun. 2007. https://www.researchgate.net/publication/3032757_On_the_Stability_of_Switched_Positive_Linear_Systems [11] Z. Liu, Y. Z. Wang, and G. D. Zhao, "Exponential stability for positive switched linear systems with delays and impulses, " in Proc. 33rd Chinese Control Conference (CCC), Nanjing, China, 2014, pp. 2469-2474. https://www.researchgate.net/publication/286595317_Exponential_stability_for_positive_switched_linear_systems_with_delays_and_impulses [12] X. Y. Ding, L. Shu, and Z. H. Wang, "On stability for switched linear positive systems, " Math. Comput. Modell., vol. 53, no. 5-6, pp. 1044- 1055, Mar. 2011. https://www.sciencedirect.com/science/article/pii/S0895717710005534 [13] X. D. Zhao, L. X. Zhang, and P. Shi, "Stability of a class of switched positive linear time-delay systems, " Int. J. Robust Nonlinear Control, vol. 23, no. 5, pp. 578-589, Mar. 2013. https://www.researchgate.net/publication/259717122_Stability_of_a_Class_of_Switched_Positive_Linear_Time-Delay_Systems [14] X. D. Zhao, L. X. Zhang, P. Shi, and H. R. Karimi, "Stabilization of a class of slowly switched positive linear systems: state-feedback control, " in Proc. American Control Conference (ACC), Montreal, QC, Canada, 2012, pp. 5282-5286. Stabilization of a class of slowly switched positive linear systems: state-feedback control [15] O. Mason and R. Shorten, "On linear copositive Lyapunov functions and the stability of switched positive linear systems, " IEEE Trans. Automat. Control, vol. 52, no. 7, pp. 1346-1349, Jul. 2007. http://www.researchgate.net/publication/3032814_On_Linear_Copositive_Lyapunov_Functions_and_the_Stability_of_Switched_Positive_Linear_Systems?ev=sim_pub [16] X. W. Liu, "Stability analysis of switched positive systems: a switched linear copositive Lyapunov function method, " IEEE Trans. Circ. Syst. Ⅱ: Express Briefs, vol. 56, no. 5, pp. 414-418, May 2009. http://www.researchgate.net/publication/224441998_Stability_Analysis_of_Switched_Positive_Systems_A_Switched_Linear_Copositive_Lyapunov_Function_Method [17] X. D. Zhao, L. X. Zhang, P. Shi, and M. Liu, "Stability of switched positive linear systems with average dwell time switching, " Automatica, vol. 48, no. 6, pp. 1132-1137, Jun. 2012. https://www.sciencedirect.com/science/article/pii/S0005109812001045 [18] H. Yang, V. Cocquempot, and B. Jiang, "On stabilization of switched nonlinear systems with unstable modes, " Syst. Control Lett., vol. 58, no. 10-11, pp. 703-708, Oct.-Nov. 2009. [19] J. Lian and J. Liu, "New results on stability of switched positive systems: an average dwell-time approach, " IET Control Theory Appl., vol. 7, no. 12, pp. 1651-1658, Aug. 2013. http://www.researchgate.net/publication/260586544_New_results_on_stability_of_switched_positive_systems_an_average_dwell-time_approach [20] R. Q. Shi, X. M. Tan, X. D. Zhao, and X. L. Zheng, "Stability and l$_1$-gain analysis for switched delay positive systems with stable and unstable subsystems, " Circ. Syst. Signal Process., vol. 34, no. 5, pp. 1683-1696, May 2015. [21] W. M. Xiang and J. Xiao, "Stabilization of switched continuous-time systems with all modes unstable via dwell time switching, " Automatica, vol. 50, no. 3, pp. 940-945, Mar. 2014. https://www.sciencedirect.com/science/article/pii/S0005109813005876 [22] K. Q. Gu, V. L. Kharitonov, and J. Chen, Stability of Time-Delay Systems. Boston, USA: Birkhäuser, 2003. [23] L. Allerhand and U. Shaked, "Robust stability and stabilization of linear switched systems with dwell time, " IEEE Trans. Automat. Control, vol. 56, no. 2, pp. 381-386, Feb. 2011. http://www.researchgate.net/publication/224208117_Robust_stability_and_stabilization_of_linear_switched_systems_with_dwell_time [24] L. G. Wu, X. J. Su, and P. Shi, "Sliding mode control with bounded $\ell_2$ gain performance of Markovian jump singular time-delay systems, " Automatica, vol. 48, no. 8, pp. 1929-1933, Aug. 2012. [25] L. G. Wu, X. J. Su, and P. Shi, "Output feedback control of Markovian jump repeated scalar nonlinear systems, " IEEE Trans. Automat. Control, vol. 59, no. 1, pp. 199-204, Jan. 2014. [26] L. X. Zhang and B. Jiang, "Stability of a class of switched linear systems with uncertainties and average dwell time switching, " Int. J. Innov. Comput. Inform. Control, vol. 6, no. 2, pp. 667-676, Feb. 2010. https://www.researchgate.net/publication/279546251_Stability_of_a_class_of_switched_linear_systems_with_uncertainties_and_average_dwell_time_switching [27] J. J. Liu, K. J. Zhang, and H. K. Wei, "Robust stability of positive switched systems with dwell time, " Int. J. Syst. Sci., vol. 47, no. 11, pp. 2553-2562, Aug. 2016. http://www.researchgate.net/publication/273168026_robust_stability_of_positive_switched_systems_with_dwell_time [28] J. F. Zhang, Z. Z. Han, H. Wu, and J. Huang, "Robust stabilization of discrete-time positive switched systems with uncertainties and average dwell time switching, " Circ. Syst. Signal Process., vol. 33, no. 1, pp. 71 -95, Jan. 2014. [29] J. F. Zhang, W. Zhang, X. S. Cai, and Z. Z. Han, "Stability and stabilization of positive switched systems under asynchronous switching, " in Proc. International Conference on Mechatronics and Control (ICMC), Jinzhou, China, 2014, pp. 347-352. https://www.researchgate.net/publication/283302183_Stability_and_stabilization_of_positive_switched_systems_under_asynchronous_switching [30] Y. Li, H. B. Zhang, and Q. X. Zheng, "Robust stability and $L_1$-gain analysis of interval positive switched T-S fuzzy systems with mode-dependent dwell time, " Neurocomputing, vol. 235, pp. 90-97, Apr. 2017.

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