Non-fragile Control for Positive Markov Jump Systems with Actuator Saturation

  • Shicheng Li
  • Junfeng ZhangEmail author
  • Yun Chen
  • Xianglei Jia
Part of the Lecture Notes in Control and Information Sciences book series (LNCIS, volume 480)


This paper is concerned with the non-fragile control for a class of positive Markov jump systems in both continuous-time and discrete-time contexts. The systems under consideration are subject to actuator saturation and partially known transition probabilities. First, a stochastic co-positive Lyapunov function is constructed. Using the Lyapunov function, a set of mode-dependent state feedback control laws and attraction domain gains are designed based on a gain matrix decomposition technique. Furthermore, a cone is chosen as the attraction domain. The free weighting vectors are used to tackle incomplete transition probabilities. Sufficient conditions for non-fragile stochastic stability of the resulting closed-loop systems are developed by solving a linear programming problem. Compared with existing methods in the literature, the presented approach designs not only controller gains but also attraction domain gains with less conservativeness. Finally, two numerical examples are given to show the validity of the proposed approach.


Positive Markov jump systems Actuator saturation Non-fragile controller Linear programming 



The authors thank the anonymous reviewers and associate editor for their valuable suggestions and comments which have helped to improve the quality of the paper. This work was supported in part by the National Nature Science Foundation of China (61873314, 61503107, and U1509205), the Zhejiang Provincial Natural Science Foundation of China (S18F030001), and the Foundation of Key Laboratory of System Control and Information Processing, Ministry of Education, P.R. China.


  1. 1.
    Bunks, C., McCarthy, D., Al-Ani, T.: Condition-based maintenance of machines using hidden Markov models. Mech. Syst. Signal Process. 14(4), 597–612 (2000)CrossRefGoogle Scholar
  2. 2.
    Berman A., Plemmons R.J.: Nonnegative matrices in the mathematical sciences. Society for Industrial and Applied Mathematics (1994)Google Scholar
  3. 3.
    Farina L., Rinaldi S.: Positive Linear Systems: Theory and Applications. Wiley (2011)Google Scholar
  4. 4.
    Kaczorek T.: Stability of positive continuous-time linear systems with delays. In: European IEEE Control Conference (ECC), pp. 1610–1613 (2009)Google Scholar
  5. 5.
    Liu, X.: Constrained control of positive systems with delays. IEEE Trans. Autom. Control 54(7), 1596–1600 (2009)MathSciNetCrossRefGoogle Scholar
  6. 6.
    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)Google Scholar
  7. 7.
    Liu, H., Boukas, E.K.B., Sun, F., et al.: Controller design for Markov jumping systems subject to actuator saturation. Automatica 42(3), 459–465 (2006)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Lien, C.H.: \(H_{\infty }\) non-fragile observer-based controls of dynamical systems via LMI optimization approach. Chaos, Solitons Fractals 34(2), 428–436 (2007)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Ma, S., Zhang, C.: \(H_{\infty }\) control for discrete-time singular Markov jump systems subject to actuator saturation. J. Franklin Inst. 349(3), 1011–1029 (2012)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Olfati-Saber, R., Fax, J.A., Murray, R.M.: Consensus and cooperation in networked multi-agent systems. Proc. IEEE 95(1), 215–233 (2007)CrossRefGoogle Scholar
  11. 11.
    Park, I.S., Kwon, N.K., Park, P.G.: A linear programming approach for stabilization of positive Markovian jump systems with a saturated single input. Nonlinear Anal. Hybrid Syst. 29, 322–332 (2018)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Qi, W., Gao, X.: \(L_{1}\) Control for positive Markovian jump systems with time-varying delays and partly known transition rates. Circuits. Syst. Signal Process. 34(8), 2711–2726 (2015)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Shu, Z., Lam, J., Xiong, J.: Non-fragile exponential stability assignment of discrete-time linear systems with missing data in actuators. IEEE Trans. Autom. Control 54(3), 625–630 (2009)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Wang, J., Zhao, J.: Stabilisation of switched positive systems with actuator saturation. IET Control Theory Appl. 10(6), 717–723 (2016)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Xiao L., Hassibi A., How J.P.: Control with random communication delays via a discrete-time jump system approach. In: Proceedings of the 2000 IEEE American Control Conference, vol. 3, pp. 2199–2204 (2000)Google Scholar
  16. 16.
    Xu, S., Lam, J., Wang, J., et al.: Non-fragile positive real control for uncertain linear neutral delay systems. Syst. Control Lett. 52(1), 59–74 (2004)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Yang, G.H., Wang, J.L.: Non-fragile \(H_{\infty }\) control for linear systems with multiplicative controller gain variations. Automatica 37(5), 727–737 (2001)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Zhang, Y., He, Y., Wu, M., et al.: Stabilization for Markovian jump systems with partial information on transition probability based on free-connection weighting matrices. Automatica 47(1), 79–84 (2011)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Zhang, J., Zhao, X., Zhu, F., et al.: \(L_{1}/l_{1}\)-Gain analysis and synthesis of Markovian jump positive systems with time delay. ISA Trans. 63, 93–102 (2016)CrossRefGoogle Scholar
  20. 20.
    Zhu, S., Han, Q.L., Zhang, C.: \(l_{1}\)-gain performance analysis and positive filter design for positive discrete-time Markov jump linear systems: A linear programming approach. Automatica 50(8), 2098–2107 (2014)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Zhang, D., Zhang, Q., Du, B.: \(L_{1}\) fuzzy observer design for nonlinear positive Markovian jump system. Nonlinear Anal. Hybrid Syst. 27, 271–288 (2018)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Shicheng Li
    • 1
  • Junfeng Zhang
    • 1
    • 2
    Email author
  • Yun Chen
    • 1
  • Xianglei Jia
    • 1
  1. 1.School of AutomationHangzhou Dianzi UniversityHangzhouChina
  2. 2.Key Laboratory of System Control and Information ProcessingMinistry of Education of ChinaShanghaiChina

Personalised recommendations