Skip to main content
SpringerLink
Log in

Finite-Time Spatial Sampled-Data Control for Reaction–Diffusion Systems

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

Abstract

The finite-time control is considered for reaction–diffusion systems (RDSs) under a designed spatial sampled-data controller (SSDC). Firstly, an SSDC is designed and the finite-time stability is investigated for RDSs, based on the designed controller. In terms on Lyapunov functional method and inequality techniques, a sufficient condition is obtained to ensure the finite-time stability. Then, uncertain RDSs are studied and the robust finite-time stability is considered. Under the designed SSDC, a criterion is obtained to ensure the uncertain RDSs to achieve robust finite-time stability. When there exist external disturbances, the \(H_\infty \) performance of RDSs is investigated. Both distributed disturbances and boundary disturbances are considered, and sufficient conditions are obtained to guarantee the finite-time \(H_\infty \) performance with the designed SSDC. Finally, examples are given to verify the effectiveness of our theoretical results.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

Data availability

All data generated or analyzed during this study are included in this article

References

  1. M. Bahreini, J. Zarei, Robust finite-time stabilization for networked control systems via static output-feedback control: Markovian jump systems approach. Circuits Syst. Signal Process. 37(4), 1523–1541 (2018)

    Article  MathSciNet  Google Scholar 

  2. W.Z. Chen, Y.L. Huang, S.Y. Ren, Passivity and robust passivity of delayed Cohen–Grossberg neural networks with and without reaction–diffusion terms. Circuits Syst. Signal Process. 37(7), 2772–2804 (2018)

    Article  MathSciNet  Google Scholar 

  3. Y. Chen, H.B. Zou, R.Q. Lu, A.K. Xue, Finite-time stability and dynamic output feedback stabilization of stochastic systems. Circuits Syst. Signal Process. 33(2), 53–69 (2014)

    Article  MathSciNet  Google Scholar 

  4. J.C. Doyle, K. Glover, P.P. Khargonekar, B.A. Francis, State-space solutions to standard \(H_2\) and \(H_\infty \) control problems. IEEE Trans. Autom. Control 34(8), 831–847 (1989)

    Article  Google Scholar 

  5. J. Feng, G.X. Wen, Adaptive NN consensus tracking control of a class of nonlinear multi-agent systems. Neurocomputing 151(1), 288–295 (2015)

    Article  Google Scholar 

  6. E. Fridman, Sampled-data distributed \(H_\infty \) control of transport reaction systems. SIAM J. Control Optim. 51(2), 1500–1527 (2013)

    Article  MathSciNet  Google Scholar 

  7. E. Fridman, A. Blighovsky, Robust sampled-data control of a class of semilinear parabolic systems. Automatica 48(5), 826–836 (2012)

    Article  MathSciNet  Google Scholar 

  8. L.N. Guin, S. Acharya, Dynamic behaviour of a reaction–diffusion predator-prey model with both refuge and harvesting. Nonlinear Dyn. 88(2), 1501–1533 (2017)

    Article  MathSciNet  Google Scholar 

  9. G.H. Hardy, J.E. Littlewood, G. Polya, Inequalities (Cambridge University Press, Cambridge, 1988)

    MATH  Google Scholar 

  10. Y.R. Hong, J. Huang, Y.S. Xu, On an output feedback finite-time stabilization problem. IEEE Trans. Autom. Control 46(2), 305–309 (2001)

    Article  MathSciNet  Google Scholar 

  11. J. Huang, Y. Shi, \(H_\infty \) state-feedback control for semi-Markov jump linear systems with time-varying delays. J. Dyn. Syst. Meas. Control 135(4), 1004–1012 (2013)

    Article  Google Scholar 

  12. S.P. Huang, Z.R. Xiang, Finite-time stabilization of switched stochastic nonlinear systems with mixed odd and even powers. Automatica 73, 130–137 (2016)

    Article  MathSciNet  Google Scholar 

  13. W. Kang, E. Fridman, Distributed sampled-data control of Kuramoto–Sivashinsky equation. Automatica 95, 514–524 (2018)

    Article  MathSciNet  Google Scholar 

  14. H.J. Liang, G.L. Liu, H.G. Zhang, T.W. Huang, Neural-network-based event-triggered adaptive control of nonaffine nonlinear multiagent systems with dynamic uncertainties. IEEE Trans. Neural Netw. Learn. Syst. (2020). https://doi.org/10.1109/TNNLS.2020.3003950

    Article  Google Scholar 

  15. H. Ma, H.Y. Li, R.Q. Lu, T.W. Huang, Adaptive event-triggered control for a class of nonlinear systems with periodic disturbances. Sci. China Inf. Sci. 63(5), 1–5 (2020). https://doi.org/10.1007/s11432-019-2680-1

    Article  MathSciNet  Google Scholar 

  16. X.R. Mao, Stochastic Differential Equations and Applications (Horwood Publishing Limited, Sawston, 2007)

    MATH  Google Scholar 

  17. K. Mathiyalagan, J.H. Park, R. Sakthivel, Finite-time boundedness and dissipativity analysis of networked cascade control systems. Nonlinear Dyn. 84(4), 2149–2160 (2016)

    Article  MathSciNet  Google Scholar 

  18. S. Matoba, N. Nakamura, H. Nakamura, H. Nishitani. Robust finite-time control of robot manipulators, in: Proceedings of the 18th World Congress the International Federation of Automatic Control Milano (Italy, 2011)

  19. M.M. Moore, Real-world applications for brain-computer interface technology. IEEE Trans. Neural Syst. Rehabil. Eng. 11(2), 162–165 (2003)

    Article  Google Scholar 

  20. E. Moulay, M. Dambrine, N. Yeganefar, W. Perruquetti, Finite-time stability and stabilization of time-delay systems. Syst. Control Lett. 57(7), 561–566 (2008)

    Article  MathSciNet  Google Scholar 

  21. Y. Orlov, Finite time stability and robust control synthesis of uncertain switched systems. SIAM J. Control Optim. 43(4), 1253–1271 (2004)

    Article  MathSciNet  Google Scholar 

  22. Y.N. Pan, P.H. Du, H. Xue, H.K. Lam, Singularity-free fixed-time fuzzy control for robotic systems with user-defined performance. IEEE Trans. Fuzzy Syst. (2020). https://doi.org/10.1109/TFUZZ.2020.2999746

    Article  Google Scholar 

  23. R. Rakkiyappan, S. Dharani, Q. Zhu, Synchronization of reaction–diffusion neural networks with time-varying delays via stochastic sampled-data controller. Nonlinear Dyn. 79(1), 485–500 (2015)

    Article  MathSciNet  Google Scholar 

  24. R. Rakkiyappan, S. Dharani, Sampled-data synchronization of randomly coupled reaction–diffusion neural networks with Markovian jumping and mixed delays using multiple integral approach. Neural Comput. Appl. 28(3), 449–462 (2017)

    Article  Google Scholar 

  25. B. Wang, Q.X. Zhu, Stability analysis of Markov switched stochastic differential equations with both stable and unstable subsystems. Syst. Control Lett. 105, 55–61 (2017)

    Article  MathSciNet  Google Scholar 

  26. C. Wu, L. Wu, J. Liu, Z. Jiang, Active defense-based resilient sliding mode control under denial-of-service attacks. IEEE Trans. Inf. Forensics Secur. 15, 237–249 (2020)

    Article  Google Scholar 

  27. C.W. Wu, J.X. Liu, Y.Y. Xiong, L.G. Wu, Observer-based adaptive fault-tolerant tracking control of nonlinear nonstrict-feedback systems. IEEE Trans. Neural Netw. Learn. Syst. 29(7), 3022–3033 (2018)

    MathSciNet  Google Scholar 

  28. K.N. Wu, X.Z. Liu, P. Shi, C.C. Lim, Boundary control of linear stochastic reaction–diffusion systems. Int. J. Robust Nonlinear Control 29(1), 268–282 (2019)

    Article  MathSciNet  Google Scholar 

  29. Z.R. Xiang, C.H. Qiao, M.S. Mahmoud, Finite-time analysis and \(H_\infty \) control for switched stochastic systems. J. Frankl. Inst. 349(3), 915–927 (2012)

    Article  MathSciNet  Google Scholar 

  30. W.B. Xiao, L. Cao, H.Y. Li, R.Q. Lu, Observer-based adaptive consensus control for nonlinear multi-agent systems with time-delay. Sci. China Inf. Sci. 63(3), 132202:1–132202:17 (2020)

    Article  MathSciNet  Google Scholar 

  31. Z.G. Yan, W.H. Zhang, G.S. Zhang, Finite-time stability and stabilization of Ito stochastic systems with Markovian switching: mode-dependent parameter approach. IEEE Trans. Autom. Control 60(9), 2428–2433 (2014)

    Article  MathSciNet  Google Scholar 

  32. J.L. Yin, S.Y. Khoo, Z.H. Man, X.H. Yu, Finite-time stability and instability of stochastic nonlinear systems. Automatica 47(12), 2671–2677 (2011)

    Article  MathSciNet  Google Scholar 

  33. G. Zames, Feedback and optimal sensitivity: model reference transformations, multiplicative seminorms, and approximate inverses. IEEE Trans. Autom. Control 26(2), 301–320 (1981)

    Article  MathSciNet  Google Scholar 

  34. J.Y. Zhai, Finite-time output feedback stabilization for stochastic high-order nonlinear systems. Circuits Syst. Signal Process. 33(12), 3809–3837 (2014)

    Article  MathSciNet  Google Scholar 

  35. Y. Zhang, H.H. Liu, F. Yan, J. Zhou, Oscillatory behaviors in genetic regulatory networks mediated by microRNA with time delays and reaction–diffusion terms. IEEE Trans. Nanobiosci. 16(3), 166–176 (2017)

    Article  Google Scholar 

  36. Q.X. Zhu, Razumikhin-type theorem for stochastic functional differential equations with Levy noise and Markov switching. Int. J. Control 90(8), 1703–1712 (2017)

    Article  Google Scholar 

  37. Z.Q. Zuo, H.C. Li, Y. Liu, Y.J. Wang, On finite-time stochastic stability and stabilization of Markovian jump systems subject to partial information on transition probabilities. Circuits Syst. Signal Process. 31(6), 1973–1983 (2012)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Harbin Institute of Technology, Weihai, 264209, China

    Kai-Ning Wu, Zhen Wang & Yun-Zhu Wang

  2. School of Automotive Engineering, Harbin Institute of Technology, Weihai, 264209, China

    Zhiquan Cui

Authors
  1. Kai-Ning Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zhen Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yun-Zhu Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Zhiquan Cui
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zhiquan Cui.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

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

This work is supported by Natural Science Foundations of Shandong Province under Grant ZR2018MF018.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wu, KN., Wang, Z., Wang, YZ. et al. Finite-Time Spatial Sampled-Data Control for Reaction–Diffusion Systems. Circuits Syst Signal Process 40, 4833–4849 (2021). https://doi.org/10.1007/s00034-021-01716-7

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00034-021-01716-7

Keywords

Search

Navigation