Advertisement

Local exponential stabilization for a class of uncertain nonlinear impulsive periodic switched systems with norm-bounded input

  • Mohsen Ghalehnoie
  • Mohammad-R. Akbarzadeh-T.Email author
  • Naser Pariz
Original Research
  • 141 Downloads

Abstract

This paper investigates stabilization for a class of uncertain nonlinear impulsive periodic switched systems under a norm-bounded control input. The proposed approach studies stabilization criteria locally where the nonlinear dynamics satisfy the Lipschitz condition only on a subspace containing the origin, not on \(\mathbb {R}^{n}\). This makes the proposed approach applicable in most practical cases where the region of validity is limited due to physical issues. In presence of different resources of non-vanishing uncertainties, the main objective is to find a stabilizing control signal such that not only trajectories exponentially converge to a sufficient small ultimate bound, but also have the largest region of attraction. To this, for a more general model, we first propose several sufficient conditions using the common Lyapunov function approach. The proposed strategy allows the Lyapunov function to increase in some intervals, which is suitable when some of the subsystems are unstable and uncontrollable. We then apply these conditions to the targeted system, and the sufficient criteria are extracted in the forms of linear and bilinear matrix inequalities. To achieve the main goal, an optimization problem is also formulated which is solvable using augmented Lagrangian methods. Finally, some illustrative examples are presented to demonstrate the proposed approach.

Keywords

Exponential stability Impulsive switched systems Non-vanishing uncertainty Norm-bounded input Periodic switching scheme 

Mathematics Subject Classification

93C30 93C10 

References

  1. 1.
    Haddad, W.M., Chellaboina, V., Nersesov, S.G.: Impulsive and Hybrid Dynamical Systems: Stability, Dissipativity, and Control. Princeton University Press, Princeton (2006)CrossRefzbMATHGoogle Scholar
  2. 2.
    Shi, R., Jiang, X., Chen, L.: The effect of impulsive vaccination on an SIR epidemic model. Appl. Math. Comput. 212, 305–311 (2009).  https://doi.org/10.1016/j.amc.2009.02.017 MathSciNetzbMATHGoogle Scholar
  3. 3.
    Sun, X.-M., Wang, W.: Integral input-to-state stability for hybrid delayed systems with unstable continuous dynamics. Automatica 48, 2359–2364 (2012).  https://doi.org/10.1016/j.automatica.2012.06.056 MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Pang, G., Liang, Z., Xu, W., Li, L., Fu, G.: A pest management model with stage structure and impulsive state feedback control. Discrete Dyn. Nat. Soc. 2015, 1–12 (2015).  https://doi.org/10.1155/2015/617379 MathSciNetCrossRefGoogle Scholar
  5. 5.
    Jiao, J., Cai, S., Chen, L.: Dynamics of a plankton-nutrient chemostat model with hibernation and it described by impulsive switched systems. J. Appl. Math. Comput. 53, 583–598 (2017).  https://doi.org/10.1007/s12190-015-0983-6 MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Hamed, K.A., Grizzle, J.W.: Event-based stabilization of periodic orbits for underactuated 3-D bipedal robots with left-right symmetry. IEEE Trans. Robot. 30, 365–381 (2014).  https://doi.org/10.1109/TRO.2013.2287831 CrossRefGoogle Scholar
  7. 7.
    Posa, M., Tobenkin, M., Tedrake, R.: Stability analysis and control of rigid-body systems with impacts and friction. IEEE Trans. Autom. Control (2015).  https://doi.org/10.1109/TAC.2015.2459151 zbMATHGoogle Scholar
  8. 8.
    Yang, X., Yang, Z., Nie, X.: Exponential synchronization of discontinuous chaotic systems via delayed impulsive control and its application to secure communication. Commun. Nonlinear Sci. Numer. Simul. 19, 1529–1543 (2014).  https://doi.org/10.1016/j.cnsns.2013.09.012 MathSciNetCrossRefGoogle Scholar
  9. 9.
    Fang, T., Sun, J.: Stability of complex-valued impulsive and switching system and application to the Lü system. Nonlinear Anal. Hybrid Syst. 14, 38–46 (2014).  https://doi.org/10.1016/j.nahs.2014.04.004 MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Li, J., Ma, R., Dimirovski, G.M.: Adaptive impulsive observers for a class of switched nonlinear systems with unknown parameter. Asian J. Control 19, 1153–1163 (2017).  https://doi.org/10.1002/asjc.1464 MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Zhao, X., Shi, P., Yin, Y., Nguang, S.K.: New results on stability of slowly switched systems: a multiple discontinuous Lyapunov function approach. IEEE Trans. Autom. Control 62, 3502–3509 (2017).  https://doi.org/10.1109/TAC.2016.2614911 MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Xiang, W., Xiao, J.: Stabilization of switched continuous-time systems with all modes unstable via dwell time switching. Automatica 50, 940–945 (2014).  https://doi.org/10.1016/j.automatica.2013.12.028 MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Li, L., Liu, L., Yin, Y.: Stability analysis for discrete-time switched nonlinear system under MDADT switching. IEEE Access 5, 18646–18653 (2017).  https://doi.org/10.1109/ACCESS.2017.2751584 CrossRefGoogle Scholar
  14. 14.
    Cai, C., Teel, A.R., Goebel, R.: Smooth Lyapunov functions for hybrid systems-part I: existence is equivalent to robustness. IEEE Trans. Autom. Control 52, 1264–1277 (2007).  https://doi.org/10.1109/TAC.2007.900829 CrossRefzbMATHGoogle Scholar
  15. 15.
    Heemels, W.P.M.H., De Schutter, B., Lunze, J., Lazar, M.: Stability analysis and controller synthesis for hybrid dynamical systems. Philos. Trans. R. Soc. A Math. Phys. Eng. Sci. 368, 4937–4960 (2010).  https://doi.org/10.1098/rsta.2010.0187 MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Lin, H., Antsaklis, P.J.: Stability and stabilizability of switched linear systems: a survey of recent results. IEEE Trans. Autom. Control 54, 308–322 (2009).  https://doi.org/10.1109/TAC.2008.2012009 MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Xu, F., Dong, L., Wang, D., Li, X., Rakkiyappan, R.: Globally exponential stability of nonlinear impulsive switched systems. Math. Notes 97, 803–810 (2015).  https://doi.org/10.1134/S0001434615050156 MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Xu, H., Teo, K.L.: Exponential stability with \(L_{2}\)-gain condition of nonlinear impulsive switched systems. IEEE Trans. Autom. Control 55, 2429–2433 (2010).  https://doi.org/10.1109/TAC.2010.2060173 CrossRefzbMATHGoogle Scholar
  19. 19.
    Chen, Y., Fei, S., Zhang, K.: Stabilization of impulsive switched linear systems with saturated control input. Nonlinear Dyn. 69, 793–804 (2012).  https://doi.org/10.1007/s11071-011-0305-y MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Liu, C., Xiang, Z.: Robust L \(\infty \) reliable control for impulsive switched nonlinear systems with state delay. J. Appl. Math. Comput. 42, 139–157 (2013).  https://doi.org/10.1007/s12190-012-0625-1 MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    She, Z., Lu, J., Liang, Q., Ge, S.S.: Dwell time based stabilisability criteria for discrete-time switched systems. Int. J. Syst. Sci. 48, 3087–3097 (2017).  https://doi.org/10.1080/00207721.2017.1367430 MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Tian, Y., Cai, Y., Sun, Y., Gao, H.: Finite-time stability for impulsive switched delay systems with nonlinear disturbances. J. Frankl. Inst. 353, 3578–3594 (2016).  https://doi.org/10.1016/j.jfranklin.2016.06.021 MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Feng, G., Cao, J.: Stability analysis of impulsive switched singular systems. IET Control Theory Appl. 9, 863–870 (2015).  https://doi.org/10.1049/iet-cta.2013.1142 MathSciNetCrossRefGoogle Scholar
  24. 24.
    Wang, Y.-E., Sun, X.-M., Wang, W., Zhao, J.: Stability properties of switched nonlinear delay systems with synchronous or asynchronous switching. Asian J. Control 17, 1187–1195 (2015).  https://doi.org/10.1002/asjc.964 MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Chiang, M.-L., Fu, L.-C.: Robust output feedback stabilization of switched nonlinear systems with average dwell time. Asian J. Control 16, 264–276 (2014).  https://doi.org/10.1002/asjc.699 MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Wang, B., Zhang, H., Wang, G., Dang, C., Zhong, S.: Asynchronous control of discrete-time impulsive switched systems with mode-dependent average dwell time. ISA Trans. 53, 367–372 (2014).  https://doi.org/10.1016/j.isatra.2013.11.019 CrossRefGoogle Scholar
  27. 27.
    Zhao, X., Zhang, L., Shi, P., Liu, M.: Stability and stabilization of switched linear systems with mode-dependent average dwell time. IEEE Trans. Autom. Control 57, 1809–1815 (2012).  https://doi.org/10.1109/TAC.2011.2178629 MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Branicky, M.S.: Multiple Lyapunov functions and other analysis tools for switched and hybrid systems. IEEE Trans. Autom. Control 43, 475–482 (1998).  https://doi.org/10.1109/9.664150 MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Hui, Ye, Michel, A.N., Ling, Hou: Stability theory for hybrid dynamical systems. IEEE Trans. Autom. Control 43, 461–474 (1998).  https://doi.org/10.1109/9.664149 MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Xie, X., Xu, H., Zhang, R.: Exponential stabilization of impulsive switched systems with time delays using guaranteed cost control. Abstr. Appl. Anal. 2014, 1–8 (2014).  https://doi.org/10.1155/2014/126836 MathSciNetzbMATHGoogle Scholar
  31. 31.
    Gao, L., Wang, D.: Input-to-state stability and integral input-to-state stability for impulsive switched systems with time-delay under asynchronous switching. Nonlinear Anal. Hybrid Syst. 20, 55–71 (2016).  https://doi.org/10.1016/j.nahs.2015.12.002 MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Li, P., Lam, J., Cheung, K.C.: Stability, stabilization and L2-gain analysis of periodic piecewise linear systems. Automatica 61, 218–226 (2015).  https://doi.org/10.1016/j.automatica.2015.08.024 CrossRefzbMATHGoogle Scholar
  33. 33.
    Hespanha, J.: Uniform stability of switched linear systems: extensions of LaSalle’s invariance principle. IEEE Trans. Autom. Control 49, 470–482 (2004).  https://doi.org/10.1109/TAC.2004.825641 MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Lu, L., Lin, Z.: Design of switched linear systems in the presence of actuator saturation. IEEE Trans. Autom. Control 53, 1536–1542 (2008).  https://doi.org/10.1109/TAC.2008.921021 MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Benzaouia, A., Akhrif, O., Saydy, L.: Stabilisation and control synthesis of switching systems subject to actuator saturation. Int. J. Syst. Sci. 41, 397–409 (2010).  https://doi.org/10.1080/00207720903045791 MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Ni, W., Cheng, D.: Control of switched linear systems with input saturation. Int. J. Syst. Sci. 41, 1057–1065 (2010).  https://doi.org/10.1080/00207720903201865 MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Poznyak, A., Polyakov, A., Azhmyakov, V.: Attractive Ellipsoids in Robust Control. Springer, Cham (2014)CrossRefzbMATHGoogle Scholar
  38. 38.
    Silva, L.F.P., Leite, V.J.S., Castelan, E.B., Klug, M.: Local stabilization of time-delay nonlinear discrete-time systems using Takagi–Sugeno models and convex optimization. Math. Probl. Eng. 2014, 1–10 (2014).  https://doi.org/10.1155/2014/587510 MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Derinkuyu, K., Pınar, M.Ç.: On the S-procedure and some variants. Math. Methods Oper. Res. 64, 55–77 (2006).  https://doi.org/10.1007/s00186-006-0070-8 MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Kocvara, M., Stingl, M.: PENNON: software for linear and nonlinear matrix inequalities. In: Anjos, M.F., Lasserre, J.B. (eds.) Handbook on Semidefinite, Conic and Polynomial Optimization, pp. 755–791. Springer US, New York (2012)CrossRefGoogle Scholar
  41. 41.
    Hien, L.V., Phat, V.N.: Exponential stabilization for a class of hybrid systems with mixed delays in state and control. Nonlinear Anal. Hybrid Syst. 3, 259–265 (2009).  https://doi.org/10.1016/j.nahs.2009.01.009 MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Lu, J., Brown, L.J.: A multiple Lyapunov functions approach for stability of switched systems. In: Proceedings of the 2010 American Control Conference. pp. 3253–3256. IEEE (2010)Google Scholar
  43. 43.
    Yang, H., Jiang, B., Zhao, J.: On finite-time stability of cyclic switched nonlinear systems. IEEE Trans. Autom. Control 60, 2201–2206 (2015).  https://doi.org/10.1109/TAC.2014.2366856 MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Barkhordari Yazdi, M., Jahed-Motlagh, M.R.: Stabilization of a CSTR with two arbitrarily switching modes using modal state feedback linearization. Chem. Eng. J. 155, 838–843 (2009).  https://doi.org/10.1016/j.cej.2009.09.008 CrossRefGoogle Scholar
  45. 45.
    Magni, L., Nicolao, G.D., Magnani, L., Scattolini, R.: A stabilizing model-based predictive control algorithm for nonlinear systems. Automatica 37, 1351–1362 (2001).  https://doi.org/10.1016/S0005-1098(01)00083-8 MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Bakošová, M., Puna, D., Dostál, P., Závacká, J.: Robust stabilization of a chemical reactor. Chem. Pap. (2009).  https://doi.org/10.2478/s11696-009-0046-2 Google Scholar

Copyright information

© Korean Society for Computational and Applied Mathematics 2018

Authors and Affiliations

  1. 1.Center of Excellence on Soft Computing and Intelligent Information Processing, Department of Electrical Engineering, Faculty of EngineeringFerdowsi University of MashhadMashhadIran
  2. 2.Department of EECSUniversity of California at BerkeleyBerkeleyUSA

Personalised recommendations