access icon free Geometric approach for observability and accessibility of discrete-time non-linear switched impulsive systems

This study is concerned with observability and accessibility of discrete-time non-linear switched impulsive systems, for which the problem is more complicated than that for general discrete systems and has novel features. A new method from the combination of differential geometric theory and Lie group analysis is developed. Specifically, under the geometric framework of the system, infinitesimal invariance for the observability and accessibility is investigated, respectively. Based on the invariance investigation, explicit criteria for local observability and local accessibility are derived. Numerical examples are discussed to show the effectiveness of the proposed methods, and the features of observability and accessibility for discrete switched impulsive systems.

Inspec keywords: invariance; Lie groups; observability; differential geometry; nonlinear control systems; discrete time systems

Other keywords: discrete-time nonlinear switched impulsive systems; accessibility; differential geometric theory; observability; Lie group analysis; invariance investigation

Subjects: Control system analysis and synthesis methods; Nonlinear control systems; Discrete control systems

References

    1. 1)
      • 16. Sun, Z., Ge, S.: ‘Analysis and synthesis of switched linear control systems’, Automatica, 2005, 41, (2), pp. 181195 (doi: 10.1016/j.automatica.2004.09.015).
    2. 2)
      • 20. Isidori, A.: ‘Nonlinear control systems’ (Springer-Verlag, 1995).
    3. 3)
      • 2. Ma, S., Zhang, C., Zhu, S.: ‘Robust stability for discrete-time uncertain singular Markov jump systems with actuator saturation’, IET Control Theory Appl., 2011, 5, (2), pp. 255262 (doi: 10.1049/iet-cta.2010.0057).
    4. 4)
      • 10. Ge, S., Sun, Z., Lee, T.: ‘Reachability and controllability of switched linear discrete-time systems’, IEEE Trans. Autom. Control, 2001, 46, (9), pp. 14371441 (doi: 10.1109/9.948473).
    5. 5)
      • 18. Liu, B., Hill, D.J.: ‘Comparison principle and stability of discrete-time impulsive hybrid systems’, IEEE Trans Circuits Syst., Regul. Pap., 2009, 56, (1), pp. 233245 (doi: 10.1109/TCSI.2008.924897).
    6. 6)
      • 14. Lee, J.W., Khargonekar, P.P.: ‘Detectability and stabilizability of discrete-time switched linear systems’, IEEE Trans. Autom. Control, 2009, 54, (3), pp. 424437 (doi: 10.1109/TAC.2009.2012966).
    7. 7)
      • 24. Albertini, F., Sontag, E.D.: ‘Discrete-time transitivity and accessibility: analytic systems’, SIAM J. Control Optim., 1993, 31, pp. 15991622 (doi: 10.1137/0331075).
    8. 8)
      • 26. Jakubczyk, B., Sontag, E.D.: ‘Controllability of nonlinear discrete-time systems: a Lie-algebraic approach’, SIAM J. Control Optim., 1990, 28, pp. 133 (doi: 10.1137/0328001).
    9. 9)
      • 17. Li, C.X., Sun, J.T., Sun, R.Y.: ‘Stability analysis of a class of stochastic differential delay equations with nonlinear impulsive effects’, J. Franklin Inst. B, 2010, 347, (7), pp. 11861198 (doi: 10.1016/j.jfranklin.2010.04.017).
    10. 10)
      • 12. Wu, A.-G., Feng, G., Duan, G.-R., Gao, H.: ‘Stabilising slow-switching laws for switched discrete-time linear systems’, IET Control Theory Appl., 2011, 5, (16), pp. 18431858 (doi: 10.1049/iet-cta.2010.0643).
    11. 11)
      • 13. Shen, L., Sun, J.: ‘Approximate controllability of stochastic impulsive systems with control-dependent coefficients’, IET Control Theory Appl., 2011, 5, (16), pp. 18891894 (doi: 10.1049/iet-cta.2010.0422).
    12. 12)
      • 3. Ammar, S., Mabrouk, M., Vivalda, J.C.: ‘On the genericity of the differential observability of controlled discrete-time systems’, SIAM J. Control Optim., 2007, 46, (6), pp. 21822198 (doi: 10.1137/060677938).
    13. 13)
      • 23. Albertini, F., D’Alessandro, D.: ‘Observability and forward–backward observability of discrete-time nonlinear systems’, Math. Control Signals Syst., 2002, 15, (4), pp. 275290 (doi: 10.1007/s004980200011).
    14. 14)
      • 25. Aranda-Bricaire, E., Kotta, U., Moog, C.: ‘Linearization of discrete-time systems’, SIAM J. Control Optim., 1996, 34, pp. 19992023 (doi: 10.1137/S0363012994267315).
    15. 15)
      • 19. Zhao, S.: ‘A Lie algebraic condition for exponential stability of discrete hybrid systems and application to hybrid synchronization’, Chaos (Woodbury, NY), 2011, 21, (2), p. 023125 (doi: 10.1063/1.3594046).
    16. 16)
      • 15. Pan, S., Sun, J., Zhao, S.: ‘Stabilization of discrete-time Markovian jump linear systems via time-delayed and impulsive controllers’, Automatica, 2008, 44, (11), pp. 29542958 (doi: 10.1016/j.automatica.2008.04.004).
    17. 17)
      • 6. Rieger, K., Schlacher, K., Holl, J.: ‘On the observability of discrete-time dynamic systems-A geometric approach’, Automatica, 2008, 44, (8), pp. 20572062 (doi: 10.1016/j.automatica.2007.11.007).
    18. 18)
      • 11. Chen, L., Sun, J.: ‘Nonlinear boundary value problem of first order impulsive functional differential equations’, J. Math. Anal. Appl., 2006, 318, (2), pp. 726741 (doi: 10.1016/j.jmaa.2005.08.012).
    19. 19)
      • 7. Zhao, S., Sun, J.: ‘A geometric method for observability and accessibility of discrete impulsive nonlinear systems’, Int. J. Syst. Sci., 2012, DOI:10.1080/00207721.2012.659695.
    20. 20)
      • 1. Huang, H., Feng, G.: ‘Improved approach to delay-dependent stability analysis of discrete-time systems with time-varying delay’, IET Control Theory Appl., 2010, 4, (10), pp. 21522159 (doi: 10.1049/iet-cta.2009.0225).
    21. 21)
      • 9. Ji, Z., Lin, H., Lee, T.H.: ‘A new perspective on criteria and algorithms for reachability of discrete-time switched linear systems’, Automatica, 2009, 45, (6), pp. 15841587 (doi: 10.1016/j.automatica.2009.02.024).
    22. 22)
      • 22. Olver, P.J.: ‘Applications of Lie groups to differential equations’ (Springer-Verlag, 2000).
    23. 23)
      • 8. Zhang, B.: ‘Controllability and observability at infinity of linear time-varying descriptor systems’, IET Control Theory Appl., 2009, 3, (12), pp. 16411647 (doi: 10.1049/iet-cta.2008.0412).
    24. 24)
      • 4. Diblík, J., Khusainov, D.Y., Růžičková, M.: ‘Controllability of linear discrete systems with constant coefficients and pure delay’, SIAM J. Control Optim., 2008, 47, pp. 11401149 (doi: 10.1137/070689085).
    25. 25)
      • 5. Holl, J., Schlacher, K.: ‘Analysis and nonlinear control of implicit discrete-time dynamic systems’. Proc. 16th IFAC World Congress, 2005.
    26. 26)
      • 21. Aranda-Bricaire, E., Kotta, U.: ‘Generalized controlled invariance for discrete-time nonlinear systems with an application to the dynamic disturbance decoupling problem’, IEEE Trans. Autom. Control, 2001, 46, (1), pp. 165171 (doi: 10.1109/9.898712).
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