[go: up one dir, main page]
More Web Proxy on the site http://driver.im/ Skip to main content
Springer Nature Link
Log in

Finite-time Stabilization for Singular Markov Jump Systems with Generally Uncertain Transition Rates

  • Published:
Circuits, Systems, and Signal Processing Aims and scope Submit manuscript

Abstract

This paper studies the problem of finite-time stabilization for singular Markov jump systems (SMJSs) with time-varying delays and generally uncertain transition rates. First, a suitable Lyapunov–Krasovskii functional is constructed. And the criterion of the finite-time stability for singular Markov jump systems is analyzed. We adopt a better design approach for the matrix \({\bar{P}}_i\) satisfying the equality constraints, which is much less conservative. Then, a state feedback controller design method based on linear matrix inequalities (LMIs) is presented to ensure that the closed-loop system is finite-time stable. Finally, simulation examples are given to illustrate the results’ correctness and validity of this paper.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
£29.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (United Kingdom)

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

Data Availability

The data supporting the conclusions of this article are included within the article.

Abbreviations

\(R^n\) :

Euclidean space

The superscripts T :

The matrix transpose

The superscripts \(-1\) :

The matrix inverse

He(A):

\(A+A^T\)

\(*\) :

Symmetric term of symmetric matrix

\((\Omega ,{\mathcal {F}},{\mathcal {P}})\) :

A probability space

\(\Omega \) :

The sample space

\({\mathcal {F}}\) :

The \(\sigma -\) algebra of subsets of the sample space

\({\mathcal {P}}\) :

The probability measure on \({\mathcal {F}}\)

\({\mathbb {E}}\{\cdot \}\) :

The mathematical expectation operator

References

  1. A. Adnene, Controllability of delayed discrete Fornasini–Marchesini model via quantization and random packet dropouts. Math. Modell. Nat. Phenom. 17, 38 (2022). https://doi.org/10.1051/mmnp/2022040

    Article  MathSciNet  Google Scholar 

  2. A. Adnene, C. Jinde, A. Ahmed, Improved synchronization analysis of competitive neural networks with time-varying delays. Nonlinear Anal. Model. 23(1), 82–102 (2018). https://doi.org/10.15388/NA.2018.1.7

  3. A. Adnene, C. Jinde, A. Ahmed, E. Mohssine, Z. Mohammed, Z. Mohamed, Dynamics of delayed cellular neural networks in the Stepanov pseudo almost automorphic space. Discrete Contin. Dyn. Syst. 15(11), 3097–3109 (2022). https://doi.org/10.3934/dcdss.2022136

    Article  MathSciNet  Google Scholar 

  4. A. Adnene, T. Najeh, Stability analysis of inertial neural networks: a case of almost anti-periodic environment. Math. Methods Appl. Sci. 45(16), 10476–10490 (2022). https://doi.org/10.1002/mma.8379

    Article  MathSciNet  Google Scholar 

  5. F. Amato, M. Ariol, P. Dorato, Finite-time control of linear systems subject to parametric uncertainties and disturbances. Automatica 37(9), 1459–1463 (2001). https://doi.org/10.1016/S0005-1098(01)00087-5

    Article  Google Scholar 

  6. A. Barbata, M. Zasadzinski, H.S. Ali, Functional reduced order \(H_\infty \) decentralized observer based control for large scale interconnected nonlinear stochastic systems. IFAC-PapersOnLine. 53(2), 4991–4997 (2020). https://doi.org/10.1016/j.ifacol.2020.12.1093

    Article  Google Scholar 

  7. P. Baskar, S. Padmanabhan, M.S. Ali, Finite-time \(H_\infty \) control for a class of Markovian jumping neural networks with distributed time varying delays-LMI approach. Acta Math. Sci. 38(2), 561–579 (2018). https://doi.org/10.1016/S0252-9602(18)30766-5

    Article  MathSciNet  Google Scholar 

  8. E.K. Boukas, Z.K. Liu, Delay-dependent stabilization of singularly perturbed jump linear systems. Int. J. Control 77(3), 310–319 (2004). https://doi.org/10.1080/00207170310001657298

    Article  MathSciNet  Google Scholar 

  9. W.H. Chen, Z.H. Guan, X.M. Lu, Delay-dependent exponential stability of uncertain stochastic systems with multiple delays: an LMI approach. Syst. Control Lett. 54(6), 547–555 (2005). https://doi.org/10.1016/j.sysconle.2004.10.005

    Article  MathSciNet  Google Scholar 

  10. Y.C. Ding, H. Zhu, S.M. Zhong, Y. Zeng, Exponential mean-square stability of time-delay singular systems with Markovian switching and nonlinear perturbations. Appl. Math. Comput. 219(4), 2350–2359 (2012). https://doi.org/10.1016/j.amc.2012.08.067

    Article  MathSciNet  Google Scholar 

  11. P. Dorato, Short time stability in linear time-varying systems. Proc. IRE Int. Conv. Rec. 4, 83–87 (1961)

    Google Scholar 

  12. D. Hernandez, L. Fridman, M. Golkani, Continuous sliding-mode control for singular systems: CSM for Singular systems. Int. J. Robust Nonlinear 28(10), 3454–3474 (2018). https://doi.org/10.1002/rnc.4094

    Article  Google Scholar 

  13. L.V. Hien, H.M. Trinh, Stability analysis of two-dimensional Markovian jump state-delayed systems in the Roesser model with uncertain transition probabilities. Inf. Sci. 367–368, 403–417 (2016). https://doi.org/10.1016/j.ins.2016.06.011

    Article  Google Scholar 

  14. Z. Jin, Q. Zhang, J. Ren, The approximation of the T-S fuzzy model for a class of nonlinear singular systems with impulses. Neural Comput. Appl. 32(14), 10387–10401 (2020). https://doi.org/10.1007/s00521-019-04576-0

    Article  Google Scholar 

  15. T. Kaczorek, Realization problem for singular positive continuous-time systems with delays. Control Cybern. 36(1), 47–57 (2007)

    MathSciNet  Google Scholar 

  16. N. Krasovskii, E. Lidskii, Analytical design of controllers in systems with random attributes. I. Statement of the problem, method of solving. Avtomat i Telemeh. 22(1), 1145–1150 (1961)

  17. G.Y. Kulikov, M.V. Kulikova, Square-root high-degree cubature Kalman filters for state estimation in nonlinear continuous-discrete stochastic systems. Eur. J. Control. 59, 58–68 (2021). https://doi.org/10.1016/j.ejcon.2021.02.002

    Article  MathSciNet  Google Scholar 

  18. J. Lee, W.M. Haddad, On finite-time stability and stabilization of nonlinear hybrid dynamical systems. ATMS Math. 6(6), 5535–5562 (2021). https://doi.org/10.3934/math.2021328

    Article  MathSciNet  Google Scholar 

  19. F.L. Lewis, A survey of linear singular systems. Circ. Syst. Signal Process. 5(1), 3–36 (1986). https://doi.org/10.1007/BF01600184

    Article  MathSciNet  Google Scholar 

  20. J. Li, X.W. Mu, K. Li, Event-triggered finite-time bounded and finite-time stability for Networked Control Systems under DoS attacks. Int. J. Syst. Sci. 51(15), 2820–2836 (2020). https://doi.org/10.1080/00207721.2020.1802625

    Article  MathSciNet  Google Scholar 

  21. S.Y. Li, Y.C. Ma, Finite-time dissipative control for singular Markovian jump systems via quantizing approach. Nonlinear Anal. Hybrid 27, 323–340 (2018). https://doi.org/10.1016/j.nahs.2017.10.007

    Article  MathSciNet  Google Scholar 

  22. X.Z. Lin, S.H. Li, Z. Yun, Finite-time stability of switched linear systems with subsystems which are not finite-time stable. IET Control Theory Appl. 8(12), 1137–1146 (2014). https://doi.org/10.1049/iet-cta.2013.0648

    Article  MathSciNet  Google Scholar 

  23. J.Z. Liu, Y.G. Niu, Z.W. Lin, Passivity, feedback equivalence and global stabilization of nonlinear Markovian jump systems. Asian J. Control 17(3), 806–820 (2014). https://doi.org/10.1002/asjc.929

    Article  MathSciNet  Google Scholar 

  24. M. Ogura, C.F. Martin, Stability analysis of positive semi-Markovian jump linear systems with state resets. SIAM J. Control. Optim. 52(3), 1809–1831 (2014). https://doi.org/10.1137/130925177

    Article  MathSciNet  Google Scholar 

  25. A. Parivallal, R. Sakthivel, C. Wang, Guaranteed cost leaderless consensus for uncertain Markov jumping multi-agent systems. J. Exp. Theor. Artif. Intell. 35(2), 257–273 (2023). https://doi.org/10.1080/0952813X.2021.1960631

    Article  Google Scholar 

  26. C. Park, N.K. Kwon, P.G. Park, Dynamic output feedback control for continuous time singular Markovian jump systems. Int. J. Robust Nonlinear 28(11), 3521–3531 (2018). https://doi.org/10.1002/rnc.4093

    Article  MathSciNet  Google Scholar 

  27. H.L. Ren, G.D. Zong, Robust input-output finite-time filtering for uncertain Markovian jump nonlinear systems with partially known transition probabilities. Int. J. Adapt. Control 31(10), 1437–1455 (2017). https://doi.org/10.1002/acs.2777

    Article  MathSciNet  Google Scholar 

  28. R. Sakthivel, H. Divya, A. Parivallal, Quantized fault detection filter design for networked control system with Markov Jump parameters. Circuits Syst. Signal Process. 40, 4741–4758 (2021). https://doi.org/10.1007/s00034-021-01693-x

    Article  Google Scholar 

  29. R. Sakthivel, M. Joby, K. Mathiyalagan, S. Santra, Mixed \(H_\infty \) and passive control for singular Markovian jump systems with time delays. J. Frankl. Inst. 352(10), 4446–4466 (2015). https://doi.org/10.1016/j.jfranklin.2015.06.017

    Article  Google Scholar 

  30. S. Saravanan, M.S. Ali, H. Alsulami, M.S. Alhodaly, Robust \(H_\infty \) filtering for finite-time boundedness of Markovian jump system with distributed time-varying delays. Int. J. Syst. Sci. 51(2), 368–380 (2020). https://doi.org/10.1080/00207721.2020.1716097

    Article  MathSciNet  Google Scholar 

  31. A. Seuret, F. Gouaisbaut, Wirtinger-based integral inequality: application to time-delay systems. Automatica 49(9), 2860–2866 (2013). https://doi.org/10.1016/j.automatica.2013.05.030

    Article  MathSciNet  Google Scholar 

  32. E. Uezato, M. Ikeda, Strict LMI conditions for stability, robust stabilization, and \(H_\infty \) control of descriptor systems. IEEE Conf. Decis. Control 4(38), 4092–4097 (1999). https://doi.org/10.1109/CDC.1999.828001

    Article  Google Scholar 

  33. G. Wang, Z. Li, Q. Zhang, C. Yang, Robust finite-time stability and stabilization of uncertain Markovian jump systems with time-varying delay. Appl. Math. Comput. 293, 377–393 (2017). https://doi.org/10.1016/j.amc.2016.08.044

    Article  MathSciNet  Google Scholar 

  34. Z.G. Wu, H.Y. Su, J. Chu, Delay-dependent \(H_\infty \) filtering for singular Markovian jump time-delay systems. Signal Process. 90(6), 1815–1824 (2010). https://doi.org/10.1016/j.sigpro.2009.11.029

    Article  Google Scholar 

  35. W. Xia, W.X. Zheng, S. Xu, Event-triggered filter design for Markovian jump delay systems with nonlinear perturbation using quantized measurement. Int. J. Robust Nonlinear 29, 4644–4664 (2019). https://doi.org/10.1002/rnc.4645

    Article  MathSciNet  Google Scholar 

  36. X. Xing, D.Y. Yao, Q. Lu, Finite-time stability of Markovian jump neural networks with partly unknown transition probabilities. Neurocomputing 159(2), 282–287 (2015). https://doi.org/10.1016/j.neucom.2015.01.033

    Article  Google Scholar 

  37. X.M. Yao, L.G. Wu, W.X. Zheng, Fault detection filter design for Markovian jump singular systems with intermittent measurements. IEEE Trans. Signal Process. 59(7), 3099–3109 (2011). https://doi.org/10.1109/TSP.2011.2141666

    Article  MathSciNet  Google Scholar 

  38. Q. Zhang, J. Lu, \(H_\infty \) control for singular fractional-order interval systems: The \(0<\alpha <1\) case. ISA T. 110, 105–116 (2021). https://doi.org/10.1016/j.isatra.2020.10.003

  39. Y.Q. Zhang, C.X. Liu, X.W. Mu, Robust finite-time stabilization of uncertain singular Markovian jump systems. Appl. Math. Model. 36(10), 5109–5121 (2012). https://doi.org/10.1016/j.apm.2011.12.052

    Article  MathSciNet  Google Scholar 

  40. P. Zhao, D.H. Zhai, Y.G. Sun, Adaptive finite-time control of a class of Markovian jump nonlinear systems with parametric and dynamic uncertainties. Nonlinear Anal. Hybrid 29, 234–246 (2018). https://doi.org/10.1016/j.nahs.2018.02.003

    Article  MathSciNet  Google Scholar 

  41. G.M. Zhuang, J.W. Xia, J.S. Zhao, Nonfragile \(H_\infty \) output tracking control for uncertain singular Markovian jump delay systems with network-induced delays and data packet dropouts. Complexity 21(6), 396–411 (2016). https://doi.org/10.1002/cplx.21699

    Article  MathSciNet  Google Scholar 

Download references

Funding

No funding was received.

Author information

Authors and Affiliations

  1. College of Science, Northeastern University, Nanhu Street, Shenyang, 110004, Liaoning, China

    Xinru Ai, Juan Zhou & Guoyi Liu

Authors
  1. Xinru Ai
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Juan Zhou
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Guoyi Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Juan Zhou.

Ethics declarations

Conflict of interest

The authors declared no competing interests with respect to the research, authorship and/or publication of this article.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ai, X., Zhou, J. & Liu, G. Finite-time Stabilization for Singular Markov Jump Systems with Generally Uncertain Transition Rates. Circuits Syst Signal Process 43, 3410–3439 (2024). https://doi.org/10.1007/s00034-023-02554-5

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00034-023-02554-5

Keywords

Search

Navigation