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Stability Analysis of Discrete-Time Periodic Positive Systems with Delays

  • Tiantong PuEmail author
  • Qinzhen Huang
Chapter
Part of the Lecture Notes in Control and Information Sciences book series (LNCIS, volume 480)

Abstract

Addressed in this paper is the stability issue of discrete-time periodic positive systems with constant delay. The positivity condition of the periodic system with constant delay is given, and three sufficient and necessary conditions for the asymptotic stability of the considered system are established. Note that the positivity condition in this paper is different from that in the Ref. (Bougatef et al, On the stabilization of a class of periodic positive discrete time systems, 2010. [3]), which gives the sufficient condition of the system without delay. The sufficient and necessary condition of the positivity of the system with delay is produced in this paper. Finally, a numerical example is given to demonstrate the effectiveness.

Keywords

Positive system Periodic system Delay Stability 

Notes

Acknowledgements

This work was partially supported by National Nature Science Foundation (61673016), Innovative Research Team of the Education Department of Sichuan Province (15TD0050), Sichuan Youth Science and Technology Innovation Research Team (2017TD0028).

References

  1. 1.
    Almér, S., Jönsson, U.: Dynamic phasor analysis of periodic systems. IEEE Trans. Autom. Control 54(8), 2007–2012 (2009)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Benvenuti, L., Farina, L.: Positive and compartmental systems. IEEE Trans. Autom. Control 47(2), 370–373 (2002)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bougatef, N., Chaabane, M., Bachelier, O., Mehdi, D., Cauet, S.: On the stabilization of a class of periodic positive discrete time systems. In: the 49th IEEE Conference on Decision and Control, pp. 4311–4316. Atlanta, GA, USA (2010)Google Scholar
  4. 4.
    Farina, L., Rinaldi, S.: Positive Linear Systems: Theory and Applications. Wiley, New York (2000)CrossRefGoogle Scholar
  5. 5.
    Gao, X., Liu, X.: Chaos control for periodically forced complex Duffing’s system based on fuzzy model. Math. Methods Phys. Methods Simul. Sci. Technol. 1(1), 205–215 (2008)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Hayakawa, T.: Compartmental modeling and adaptive control of stochastic nonnegative systems. In: Positive Systems, pp. 351–358 (2006)Google Scholar
  7. 7.
    Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press, Cambridge (1985)CrossRefGoogle Scholar
  8. 8.
    Jacquez, J.A.: Compartmental Analysis in Biology and Medicine. The. University of Michigan Press, Ann Arbor (1985)zbMATHGoogle Scholar
  9. 9.
    Liu, X.: Constrained control of positive systems with delays. IEEE Trans. Autom. Control 54(7), 1596–1600 (2009)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Liu, X.: Stability analysis of switched positive systems: a switched linear copositive Lyapunov function method. IEEE Trans. Circuits Syst. II: Expr. Briefs 56(5), 414–418 (2009)CrossRefGoogle Scholar
  11. 11.
    Liu, X.: Stability analysis of discrete-time \(p\)-periodic positive systems (in Chinese). J. Southwest Univ. Natl. (Nat. Sci. Ed.), 37(3), 342–347 (2011)Google Scholar
  12. 12.
    Liu, X., Yu, W., Wang, L.: Stability analysis of positive systems with bounded time-varying delays. IEEE Trans. Circuits Syst. II: Expr. Briefs 56(7), 600–604 (2009)CrossRefGoogle Scholar
  13. 13.
    Liu, X., Yu, W., Wang, L.: Stability analysis for continuous-time positive systems with time-varying delays. IEEE Trans. Autom. Control 55(4), 1024–1028 (2010)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Liu, X., Zhong, S., Zhao, Q.: Dynamics of delayed switched nonlinear systems with applications to cascade systems. Automatica 87, 251–257 (2018)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Longhi, S., ùA.M.: Fault detection for linear periodic systems using a geometric approach. IEEE Trans. Autom. Control 54(7), 1637–1643 (2009)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Van den Hof, J.M.: Positive linear observers for linear compartmental systems. SIAM. J. Control Optim. 36(2), 590–608 (1998)Google Scholar
  17. 17.
    Yang, Z., Xu, D.: Existence and exponential stability of periodic solution for impulsive delay differential equations and applications. Nonlinear Anal. 64(1), 130–145 (2006)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.School of Electrical and Information EngineeringSouthwest Minzu UniversityChengduPeople’s Republic of China

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