Stability Analysis of Switched Linear Systems with Uncertainty and Delays

  • Yiwei Feng
  • Yudan Cai


Combining the common quadratic Lyapunov functional approach and free-weighting matrix approach, this paper is devoted to the stability analysis of continuous-time switched linear systems (CTSLS) with uncertainty and time-delays. A particular class of matrix inequalities, the so-called Lyapunov–Metzler inequalities are introduced for the CTSLS to investigate the stability and performance in the presence of the uncertainty and delays. We provide sufficient conditions in terms of the Linear Matrix Inequality criterions to guarantee delay-dependent asymptotically stability under the uncertainty of the CTSLS. The combination of switching rule and switching output feedback controllers which will be designed to stabilize the CTSLS and satisfy a prespecified \({\mathcal{L}}_{2}\) gain performance. A example used jitterbug tools is provided to illustrate the effectiveness of the proposed method.


CTSLS Lyapunov–Metzler inequalities Stability Jitterbug 


  1. 1.
    Liberzon, D. (2003). Switching in systems and control. Boston: Birkhäuser.CrossRefMATHGoogle Scholar
  2. 2.
    Deaecto, G. S., Geromel, J. C., Garcia, F. S., et al. (2010). Switched affine systems control design with application to DC–DC converters. Control Theory & Applications Iet, 4(7), 1201–1210.CrossRefGoogle Scholar
  3. 3.
    Cristiano, R., Carvalho, T., Tonon, D. J., & Pagano, D. J. (2017). Hopf and Homoclinic bifurcations on the sliding vector field of switching systems in R3: A case study in power electronics. Physica D Nonlinear Phenomena, 347, 12–20.MathSciNetCrossRefGoogle Scholar
  4. 4.
    Baldi, S., Yuan, S., Endel, P., et al. (2016). Dual estimation: Constructing building energy models from data sampled at low rate. Applied Energy, 169, 81–92.CrossRefGoogle Scholar
  5. 5.
    Baldi, S., Michailidis, I., Ntampasi, V., & et al. (2017). A simulation-based traffic signal control for congested urban traffic networks. Transportation Science.Google Scholar
  6. 6.
    Aleksandrov, A., & Kosov, A. (2017). Stability analysis of hybird mechanical systems with switched nonlinear nonhomogeneous positional forces. In The conference in St. Petersburg, Russia (pp. 1–4).Google Scholar
  7. 7.
    Mancilla-Aguilar, J. L., Haimovich, H., & García, R. A. (2017). Global stability results for switched systems based on weak Lyapunov functions. IEEE Transactions on Automatic Control, 62(6), 2764–2777.MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Lu, J., She, Z., Ge, S. S., et al. (2018). Stability analysis of discrete-time switched systems via multi-step multiple Lyapunov-like functions nonlinear analysis hybrid systems, 27, 44–61.Google Scholar
  9. 9.
    Long, L. (2017). Multiple Lyapunov functions-based small-gain theorems for switched interconnected nonlinear systems. IEEE Transactions on Automatic Control, 62(8), 3943–3958.MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Zhai, D., Lu, A. Y., Dong, J., et al. (2017). Adaptive fuzzy tracking control for a class of switched uncertain nonlinear systems: An adaptive state-dependent switching law method. IEEE Transactions on Systems Man & Cybernetics Systems, 99, 1–10.CrossRefGoogle Scholar
  11. 11.
    Ding, X., & Liu, X. (2017). Stability analysis for switched positive linear systems under state-dependent switching. International Journal of Control, Automation and Systems, 307, 1–8.Google Scholar
  12. 12.
    Hespanha, J. P. & Morse, A. S. (1999). Stability of switched systems with average dwell-time. In Proceedings of the IEEE conference on decision and control, 2002 (Vol. 3, pp. 2655–2660).Google Scholar
  13. 13.
    Regaieg, M. A., Kchaou, M., Gassara, H., & et al. (2017). Average dwell-time approach to H∞ control of time-varying delay switched systems. In International conference on sciences and techniques of automatic control and computer engineering (pp. 741–746).Google Scholar
  14. 14.
    Yotha, N., Botmart, T., & Mouktonglang, T. (2012). Exponential stability for a class of switched nonlinear systems with mixed time-varying delays via an average dwell-time method. ISRN Mathematical Analysis, 2015, 2012.MathSciNetMATHGoogle Scholar
  15. 15.
    Yuan, S., & Baldi, S. (2017). Stabilization of switched linear systems using quantized output feedback via dwell-time switching. In IEEE international conference on control & automation (pp. 236–241).Google Scholar
  16. 16.
    Guo, R. W., & Wang, Y. Z. (2017). Region stability analysis for switched nonlinear systems with multiple equilibria. International Journal of Control, Automation and Systems, 15(2), 567–574.CrossRefGoogle Scholar
  17. 17.
    Li, L., Liu, L., & Yin, Y. (2017). Stability analysis for discrete-time switched nonlinear system under MDADT switching. IEEE Access, 99, 1–1.Google Scholar
  18. 18.
    Zhao, J., & Hill, D. J. (2008). On stability, L2-gain and H∞ control for switched system. Automatica, 44(5), 1220–1232.MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Wang, D., Wang, W., & Shi, P. (2009). Robust fault detection for switched linear systems with state delays. IEEE Transactions on Systems Man & Cybernetics Part B Cybernetics A Publication of the IEEE Systems Man & Cybernetics Society, 39(3), 800–805.CrossRefGoogle Scholar
  20. 20.
    Liu, J., Zhu, C., & Li, Z. (2017). Robust stabilization for constrained switched positive linear systems with uncertainties and delays. In Control and decision conference.Google Scholar
  21. 21.
    Ohta, Y., Wada, T., & Siljak, D. D. (2001). Stability analysis of discontinuous nonlinear systems via piecewise linear Lyapunov functions. In American control conference, 2001. Proceedings of the. IEEE (Vol. 6, pp. 4852–4857).Google Scholar
  22. 22.
    Thuan, M. V., Trinh, H., & Huong, D. C. (2017). Reachable sets bounding for switched systems with time-varying delay and bounded disturbances. International Journal of Systems Science, 48(3), 1–11.MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Yu, Y., Cheng, X., Xu, H., et al. (2017). Improved results on exponential stability of discrete-time switched delay systems. Discrete and Continuous Dynamical Systems-Series B (DCDS-B), 22(1), 199–208.MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Duan, C., & Wu, F. (2012). Switching control synthesis for discrete-time switched linear systems via modified Lyapunov–Metzler inequalities. In American control conference (pp. 3186–3191).Google Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.College of Electrical and Information EngineeringLanZhou University of TechnologyLanzhouChina

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