Skip to main content
SpringerLink
Log in

General Stabilization for Stochastic System With Input Delay and Multiplicative Noise: Continuous-time Case

  • Regular Papers
  • Control Theory and Applications
  • Published:
International Journal of Control, Automation and Systems Aims and scope Submit manuscript

Abstract

This paper investigates the general stabilization issues for continuous-time stochastic dynamics whose input delay and multiplicative noise in control variable exist simultaneously. On the one hand, we present a set of necessary and sufficient conditions for stabilizing the considered stochastic dynamics in mean-square sense. Different from many previous works, one significant innovation is that our control policy is designed as the feedback of an extended state that contains the current available state and some past control information. On the other hand, another important innovation is that we for the first time generalize the notions of critical stabilization and essential destabilization to stochastic time-delay model in terms of spectral analysis technique, while the related necessary and sufficient stabilization conditions are derived respectively.

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

Similar content being viewed by others

References

  1. M. M. Gao, J. S. Zhao, Z. Y. Sun, and J. W. Xia, “Output feedback stabilization of stochastic nonlinear time-varying delay systems with unknown output function,” International Journal of Control, Automation, and Systems, vol. 20, no. 9, pp. 2839–2848, 2022.

    Article  Google Scholar 

  2. W. H. Zhang and B. S. Chen, “On stabilizability and exact observability of stochastic systems with their applications,” Autometica, vol. 40, no. 1, pp. 87–94, 2004.

    Article  MathSciNet  Google Scholar 

  3. W. H. Zhang, H. S. Zhang, and B. S. Chen, “Generalized Lyapunov equation approach to state-dependent stochastic stabilization detectability criterion,” IEEE Transactions on Automatic Control, vol. 53, no. 7, pp. 1630–1642, 2008.

    Article  MathSciNet  Google Scholar 

  4. Q. Q. Fang, Z. Y. Li, and L. S. Li, “Stability and stabilization of stochastic Neutral-type Markovian jump time-delay systems with two delays,” International Journal of Control, Automation, and Systems, vol. 20, no. 2, pp. 365–374, 2022.

    Article  Google Scholar 

  5. D. F. Zhang, Y. F. Gao, and S. L. Du, “Asymptotic stability analysis for switched stochastic nonlinear systems using mode-dependent uniformly stable functions,” International Journal of Control, Automation, and Systems, vol. 18, no. 9, pp. 272–279, 2021.

    Google Scholar 

  6. E. Gershon, U. Shaked, and I. Yaesh, “H∞ control and filtering of discrete-time stochastic systems with multiplicative noise,” Automatica, vol. 37, no. 3, pp. 409–417, 2001.

    Article  MathSciNet  Google Scholar 

  7. L. Zou, Z. D. Wang, and H. J. Gao, “Observer-based H∞ control of networked systems with stochastic communication protocol: The finite-horizon case,” Automatica, vol. 63, pp. 366–373, 2016.

    Article  MathSciNet  Google Scholar 

  8. M. A. Rami and X. Y. Zhou, “Linear matrix inequalities, Riccati equations, and indefinite stochastic linear quadratic controls,” IEEE Transactions on Automatic Control, vol. 45, no. 6, pp. 1131–1143, 2000.

    Article  MathSciNet  Google Scholar 

  9. T. Song and B. Liu, “Discrete-time mean-field stochastic linear-quadratic optimal control problem with finite horizon,” Asian Journal of Control, vol. 23, no. 2, pp. 979–989, 2021.

    Article  MathSciNet  Google Scholar 

  10. C. Hafizoglu, I. Lasiecka, T. Levajkovic, H. Mena, and A. Tuffaha, “The stochastic linear quadratic control problem with singular estimates,” SIAM Journal on Control and Optimization, vol. 55, no. 2, pp. 595–626, 2017.

    Article  MathSciNet  Google Scholar 

  11. W. H. Zhang, B. S. Chen, L. Sheng, and M. Gao, “Robust H2/H∞ filter design for a class of nonlinear stochastic systems with state-dependent noise,” Mathematical Problems in Engineering, vol. 2012, pp. 1–16, 2012.

    Google Scholar 

  12. X. F. Zong, T. Li, and J. F. Zhang, “Consensus conditions of continuous-time multi-agent systems with time-delays and measurement noises,” Automatica, vol. 99, pp. 412–419, 2019.

    Article  MathSciNet  Google Scholar 

  13. G. L. Chen, J. W. Xia, J. H. Park, H. Shen, and G. M. Zhuang, “Sampled data synchronization of stochastic Markovian jump neural network with time-varying delay,” IEEE Transactions on Neural Networks and learning Systems, vol. 33, no. 8, pp. 3829–3401, 2021.

    Article  MathSciNet  Google Scholar 

  14. Y. Y. Li, S. Liu, M. Y. Zhong, and S. X. Ding, “State estimation for stochastic discrete-time systems with multiplicative noises and unknown inputs over fading channels,” Applied Mathematics and Computation, vol. 320, no. 1, pp. 116–130, 2018.

    Article  MathSciNet  Google Scholar 

  15. C. H. Shan, W. D. Zhou, H. Y. Shan, and L. Liu, “A new variational Bayesian-based Kalman filter with random measurement delay and Non-Gaussian noises,” International Journal of Control, Automation, and Systems, vol. 20, no. 8, pp. 2594–2605, 2022.

    Article  Google Scholar 

  16. A. R. Abbasi, “Probabilistic load flow based on holomorphic embedding, kernel density estimator and saddle point approximation including correlated uncertainty variables,” Electric Power Systems Research, vol. 4, pp. 106178, 2020.

    Article  Google Scholar 

  17. K. Abdollah, K. Reza, R. Mohammad, and A. Alireza, “An smart stochastic approach to model plug-in hybrid electric vehicles charging effect in the optimal operation of microgrids,” Journal of Intelligent and Fuzzy Systems, vol. 28, no. 2, pp. 835–842, 2015.

    Article  Google Scholar 

  18. K. Abdollah, A. Somayeh, A. Alireza, and T. Sajad, “Optimal probabilistic reconfiguration of smart distribution grids considering penetration of plug-in hybrid electric vehicles,” Journal of Intelligent and Fuzzy Systems, vol. 29, no. 5, pp. 1847–1855, 2015.

    Article  Google Scholar 

  19. K. Rahmani, F. Kavousifard, and A. Abbasi, “Consideration effect of wind farms on the network reconfiguration in the distribution systems in an uncertain environment,” Journal of Experimental and Theoretical Artificial Intelligence, vol. 29, no. 5, pp. 955–1009, 2017.

    Article  Google Scholar 

  20. W. J. Ye and Z. Y. Yu, “Exact controllability of linear mean-field stochastic systems and observability inequality for mean-field backward stochastic differential equations,” IEEE Transactions on Industrial Electronics, vol. 24, no. 1, pp. 237–248, 2020.

    MathSciNet  Google Scholar 

  21. H. Ting, W. H. Zhang, and H. J. Ma, “Essential instability and essential destabilisation of linear stochastic systems,” IET Control Theory and Applications, vol. 5, no. 2, pp. 334–340, 2011.

    Article  MathSciNet  Google Scholar 

  22. E. Fridman, A. Seuret, and J. P. Richard, “Robust sampleddata stabilization of linear systems: An input delay approach,” Automatica, vol. 40, no. 8, pp. 1441–1446, 2004.

    Article  MathSciNet  Google Scholar 

  23. H. C. Yan, X. H. Huang, H. Zhang, and M. Wang, “Delay-dependent robust stability criteria of uncertain stochastic systems with time-varying delay,” Chaos, Solitons and Fractals, vol. 40, no. 4, pp. 1668–1679, 2009.

    Article  MathSciNet  Google Scholar 

  24. W. H. Chen, Z. H. Guan, and X. M. Lu, “Delay-dependent exponential stability of uncertain stochastic systems with multiple delays: an LMI approachr,” Systems and Control Letters, vol. 54, no. 6, pp. 547–555, 2005.

    Article  MathSciNet  Google Scholar 

  25. L. R. Huang and X. R. Mao, “Robust delayed-state-feedback stabilization of uncertain stochastic systems,” Automatica, vol. 45, no. 5, pp. 1332–1339, 2009.

    Article  MathSciNet  Google Scholar 

  26. B. Li and G. Yang, “Robust stabilization and H∞ control of uncertain stochastic time-delay systems with nonlinear perturbation,” International Journal of Robust and Nonlinear Control, vol. 26, no. 15, pp. 3274–3291, 2016.

    Article  MathSciNet  Google Scholar 

  27. H. S. Zhang and J. J. Xu, “Control for Itô stochastic systems with input delay,” IEEE Transactions on Automatic Control, vol. 62, no. 1, pp. 350–365, 2016.

    Article  Google Scholar 

  28. M. J. Zhou and Y. M. Fu, “Stability and stabilization for discrete-time Markovian jump stochastic systems with Piecewise Homogeneous transition probabilities,” International Journal of Control, Automation, and Systems, vol. 17, no. 9, pp. 2165–2173, 2021.

    Article  Google Scholar 

  29. H. S. Zhang, J. W. Xia, J. H. Park, W. Sun, and G. M. Zhuang, “Interval stability and interval stabilization of linear stochastic systems with time-varying delay,” International Journal of Robust and Nonlinear Control, vol. 31, no. 6, pp. 2334–2347, 2021.

    Article  MathSciNet  Google Scholar 

  30. L. Schenato, B. Sinopoli, M. Franceschetti, K. Poolla, and S. S. Sastry, “Foundations of control and estimation over lossy networks,” Proceedings of the IEEE, vol. 95, no. 1, pp. 163–187, 2007.

    Article  Google Scholar 

  31. X. M. Zhang, Q. L. Han, X. H. Ge, D. R. Ding, L. Ding, D. Yue, and C. Peng, “Networked control systems: A survey of trends and techniques,” IEEE/CAA Journal of Automatica Sinica, vol. 7, no. 1, pp. 1–17, 2020.

    MathSciNet  Google Scholar 

  32. W. Chen and L. Qiu, “Linear quadratic optimal control of continuous time LTI systems with random input gains,” IEEE Transactions on Automatic Control, vol. 61, no. 7, pp. 2008–2013, 2015.

    Article  MathSciNet  Google Scholar 

  33. C. Tan, H. S. Zhang, and W. S. Wong, “Delay-dependent algebraic riccati equation to stabilization of networked control systems: continuous-time case,” IEEE Transactions on Cybernetics, vol. 48, no. 10, pp. 2783–2794, 2017.

    Article  Google Scholar 

  34. W. H. Zhang, “General D-stability and D-stabilization for linear stochastic systems: Continuous-time case,” IEEE International Conference on Control and Automation, 2010.

  35. B. Chentouf and Z. J. Han, “On the stabilization of an overhead crane system with dynamic and delayed boundary conditions,” IEEE Transactions on Automatic Control, vol. 65, no. 10, pp. 4273–4280, 2019.

    Article  MathSciNet  Google Scholar 

  36. T. Hou, W. H. Zhang, and B. S. Chen, “Study on general stability and stabilizability of linear discrete-time stochastic systems,” Asian Journal of Control, vol. 13, no. 6, pp. 977–987, 2011.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Engineering, Qufu Normal University, Rizhao, Shandong, 276826, China

    Cheng Tan, Jianying Di & Zhengqiang Zhang

  2. Department of Information Engineering, The Chinese University of Hong Kong, Shatin, N. T., Hong Kong

    Wing Shing Wong

Authors
  1. Cheng Tan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jianying Di
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Zhengqiang Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Wing Shing Wong
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Cheng Tan.

Additional information

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

This work was supported in part by the National Key R&D Program of China under Grant 2021YFE0193900; the National Natural Science Foundation of China under Grants 62173206; China Postdoctoral Science Foundation under Grant 2021M691849; the Natural Science Foundation of Shandong Province under Grant ZR2021ZD13.

Cheng Tan received his B.S. and M.S. degrees from School of Information Science and Engineering, Shandong University of Science and Technology, Qingdao, China, in 2010 and 2012, respectively, and a Ph.D. degree from the School of Control Science and Engineering, Shandong University, Jinan, China in 2016. He is now an associate professor in College of Engineering, Qufu Normal University. His research interests include networked control system, stochastic control, time-delay system, and optimization control.

Jianying Di received her B.S. degree from the School of Mathematical Sciences from Qufu Normal University, Qufu, Shandong, China, in 2020. She is currently pursuing an M.S. degree in Qufu Normal University, Rizhao, Shandong. Her research interests include linear stochastic control and stochastic stability.

Zhengqiang Zhang received his B.Sc. degree in mathematics and an M.Sc. degree in control theory from Qufu Normal University, Qufu, China, in 2000 and 2003, respectively, and a Ph.D. degree in control theory from Nanjing University of Science and Technology, Nanjing, China, in 2011. He is currently a Professor in the School of Engineering, Qufu Normal University. His current research interests include adaptive control, control of nonlinear systems, fault tolerant control, and time-delay systems.

Wing Shing Wong received his combined master and bachelor’s degree from Yale University and his M.S. and Ph.D. degrees from Harvard University. He worked for the AT&T Bell Laboratories from 1982 until he joined the Chinese University of Hong Kong in 1992, where he is now Choh-Ming Li Research Professor of Information Engineering. He was the Chairman of the Department of Information Engineering from 1995 to 2003 and the Dean of the Graduate School from 2005 to 2014. He served as Science Advisor at the Innovation and Technology Commission of the HKSAR government from 2003 to 2005. He has participated in a variety of research projects on topics ranging from mobile communication, networked control to network control.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Tan, C., Di, J., Zhang, Z. et al. General Stabilization for Stochastic System With Input Delay and Multiplicative Noise: Continuous-time Case. Int. J. Control Autom. Syst. 22, 527–536 (2024). https://doi.org/10.1007/s12555-022-1184-3

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12555-022-1184-3

Keywords

Search

Navigation