access icon free Dwell-time-dependent stability results for impulsive systems

This study is concerned with the dwell-time stability for impulsive systems under periodic or aperiodic impulses. A Lyapunov-like functional approach is established to study the stability for impulsive systems. The Lyapunov-like functional is time-varying and decreasing but is not imposed definite positive nor continuous. A specific Lyapunov-like functional is constructed by introducing the integral of state together with the cross-terms of the integral and impulsive states. A tight bounding is obtained for the derivative of this functional with the help of the improved Jensen inequality reported recently and the integral equation of the impulsive system. On the basis of the Lyapunov-like functional approach, new dwell-time-dependent stability results with ranged dwell-time, maximal dwell-time and minimal dwell-time are derived for periodic or aperiodic impulsive systems. The stability results have less conservatism than some existing ones, which are illustrated by numerical examples.

Inspec keywords: stability; functional equations; sampled data systems; integral equations; Lyapunov methods; time-varying systems

Other keywords: tight bounds; cross-terms; impulsive states; impulsive system stability; improved Jensen inequality; time-varying Lyapunov-like functional approach; integral states; integral equation; maximal dwell-time; periodic impulsive systems; dwell-time-dependent stability; aperiodic impulsive systems; periodic impulses; aperiodic impulses; minimal dwell-time

Subjects: Integral equations (numerical analysis); Nonlinear and functional equations (numerical analysis); Discrete control systems; Stability in control theory; Time-varying control systems

References

    1. 1)
      • 1. Bainov, D., Simeonov, P.: ‘Systems with impulse effects: stability, theory and application’ (Halsted Press, New York, 1989).
    2. 2)
      • 7. Hetel, L., Daafouz, J., Tarbouriech, S., et al: ‘Stabilization of linear impulsive systems through a nearly-periodic reset’, Nonlinear Anal. Hybrid, 2013, 7, (1), pp. 415.
    3. 3)
      • 12. Antunes, D., Hespanha, J., Silvestre, C.: ‘Stability of networked control systems with asynchronous renewal links: an impulsive systems approach’, Automatica, 2013, 49, (2), pp. 402413.
    4. 4)
      • 2. Yang, T.: ‘Impulsive control theory’ (Springer-Verlag, Berlin, 2001).
    5. 5)
      • 6. Sivashankar, N., Khargonekar, P.P.: ‘Characterization of the L2at-induced norm for linear systems with jumps with applications to sampled- data systems’, SIAM J. Control Optim., 1994, 32, (2), pp. 11281150.
    6. 6)
      • 9. Verriest, E.I., Pepe, P.: ‘Time optimal and optimal impulsive control for coupled differential difference point delay systems with an application in forestry’, in Loiseau, J.J., Michiels, W., Niculescu, S.I. (Eds.): ‘Topics in time delay systems’ (Springer, Berlin, Heidelberg, 2009), pp. 255265.
    7. 7)
      • 26. Gronwall, T.H.: ‘Note on the derivatives with respect to a parameter of the solutions of a system of differential equations’, Ann. Math., 1991, 20, (4), pp. 292296.
    8. 8)
      • 19. Briat, C., Seuret, A.: ‘Convex dwell-time characterizations for uncertain linear impulsive systems’, IEEE Trans. Autom. Control, 2012, 57, (12), pp. 32413246.
    9. 9)
      • 17. Wu, X.T., Tang, Y., Zhang, W.B.: ‘Input-to-state stability of impulsive stochastic delayed systems under linear assumptions’, Automatica, 2016, 66, pp. 195204.
    10. 10)
      • 29. Liu, X.: ‘Stability results for impulsive differential systems with applications to population growth models’, Dyn. Stab. Syst., 1994, 9, pp. 163174.
    11. 11)
      • 24. Zeng, H.B., He, Y., Wu, M., et al: ‘Free-matrix-based integral inequality for stability analysis of systems with time-varying delay’, IEEE Trans. Autom. Control, 2015, 60, (10), pp. 27682772.
    12. 12)
      • 18. Wang, H.M., Duan, S.K., Li, C.D., et al: ‘Stability of impulsive delayed linear differential systems with delayed impulses’, J. Franklin Inst., 2015, 352, (8), pp. 30443068.
    13. 13)
      • 16. Naghshtabrizi, P., Hespanha, J.P., Teel, A.R.: ‘Exponential stability of impulsive systems with application to uncertain sampled-data systems’, Syst. Control Lett., 2008, 57, (5), pp. 378385.
    14. 14)
      • 5. Michel, A.N., Hou, L., Liu, D.: ‘Stability of dynamical systems-continuous, discontinuous and discrete systems’ (Birkhäuser, Boston, 2008).
    15. 15)
      • 25. Shao, H.Y., Lam, J., Feng, Z.G.: ‘Sampling-interval-dependent stability for linear sampled-data systems with non-uniform sampling’, Int. J. Syst. Sci., 2016, 47, (12), pp. 28932900.
    16. 16)
      • 3. Cai, C., Teel, A.R., Goebel, R.: ‘Converse Lyapunov theorems and robust asymptotic stability for hybrid systems’. Proc. American Control Conf., Portland, Oregon, USA, June 2005, pp. 1217.
    17. 17)
      • 8. Stone, L., Shulgin, B., Agur, Z.: ‘Theoretical examination of the pulse vaccination policy in the SIR epidemic model’, Math. Comput. Model., 2000, 31, (4-5), pp. 201215.
    18. 18)
      • 10. Loxton, R.C., Teo, K.L., Rehbock, V., et al: ‘Optimal switching instants for a switched-capacitor DC/DC power converter’, Automatica, 2009, 45, (4), pp. 973980.
    19. 19)
      • 13. Oishi, Y., Fujiok, H.: ‘Stability and stabilization of aperiodic sampled-data control systems using robust linear matrix inequalities’, Automatica, 2010, 46, (8), pp. 13271333.
    20. 20)
      • 22. Briat, C., Seuret, A.: ‘Robust stability of impulsive systems: a functional-based approach’. Proc. Fourth IFAC conf. Analysis Design Hybrid Systems, June 2012, pp. 611.
    21. 21)
      • 28. Gyurkovics, É., Takács, T.: ‘Multiple integral inequalities and stability analysis of time delay systems’, Syst. Control Lett., 2016, 96, pp. 7280.
    22. 22)
      • 20. Briat, C.: ‘Convex conditions for robust stability analysis and stabilization of linear aperiodic impulsive and sampled-data systems under dwell-time constraints’, Automatica, 2013, 49, (11), pp. 34493457.
    23. 23)
      • 15. Seuret, A., Gouaisbaut, F.: ‘Wirtinger-based integral inequality: application to time-delay systems’, Automatica, 2013, 49, (9), pp. 28602866.
    24. 24)
      • 14. Shao, H., Han, Q.-L., Zhu, X., et al: ‘Sampling-interval-dependent stability for sampled-data systems with state quantization’, Int. J. Robust Nonlinear Control, 2014, 24, (17), pp. 29953008.
    25. 25)
      • 27. Seuret, A., Gouaisbaut, F.: ‘Hierarchy of LMI conditions for the stability analysis of time-delay’, Syst. Control Lett., 2015, 81, pp. 17.
    26. 26)
      • 21. Briat, C., Seuret, A.: ‘A looped-functional approach for robust stability analysis of linear impulsive systems’, Syst. Control Lett., 2012, 61, (10), pp. 980988.
    27. 27)
      • 23. Gu, K.: ‘An integral inequality in the stability problem of time-delay systems’. Proc. 39th IEEE Conf. Decision Control, December 2000, pp. 28052810.
    28. 28)
      • 4. Hespanha, J.P., Liberzon, D., Teel, A.R.: ‘Lyapunov conditions for input-to-state stability of impulsive systems’, Automatica, 2008, 44, (11), pp. 27352744.
    29. 29)
      • 11. Shao, H., Han, Q.-L., Zhao, J., et al: ‘A separation method of transmission delays and data packet dropouts from a lumped input delay in the stability problem of networked control systems’, Int. J. Robust Nonlinear Control, 2016, Published online in Wiley Online Library (wileyonlinelibrary.com), doi: 10.1002/rnc.3647.
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