Skip to main content
SpringerLink
Log in

Dynamic output feedback stabilization for a class of nonsmooth stochastic nonlinear systems perturbed by multiple time-varying delays

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

In this paper, the output feedback stabilization problem is investigated for a class of low-order stochastic nonlinear time-delay systems with the lower-triangular form, where the powers of chained integrators are arbitrary real numbers between 0 and 1, and the multiple time-vary delays act on each system state. Because of the existence of low-order nonlinear terms, the system is not feedback linearizable and differentiable. Based on an extended adding a power integrator approach and a stability theory of stochastic continuous systems, an output feedback controller is systematically designed to ensure the global strong asymptotic stability of the closed-loop system. In the controller design, the negative effect of the multiple time-varying delays is counteracted by skillfully constructing a novel Lyapunov–Krasovskii functional, and the observer gains are determined by developing a recursive selection procedure. Finally, two numerical examples are provided to verify the effectiveness of the proposed method.

Graphic abstract

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
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

Data availability

The data that support the findings of this study are available from the corresponding author, Jinping Jia, upon reasonable request.

Abbreviations

LKF:

Lyapunov–Krasovskii functional

SNTDS:

Stochastic nonlinear time-delay system

HOS:

High-order system

AAPI:

Adding a power integrator

LOS:

Low-order systems

GSS:

Global strong stability

NGC:

Nonlinear growth condition

SWP:

Standard Wiener process

GSSP:

Globally strongly stable in probability

GSASP:

Globally strongly asymptotically stable in probability

References

  1. Gu, K., Niculescu, S.I.: Survey on recent results in the stability and control of time-delay systems. J. Dyn. Syst. Meas. Control 125(2), 158–165 (2003)

    Google Scholar 

  2. Krasovskii, N.N.: Stability of motion. Moscow (1959)

  3. Razumikhin, B.S.: Application of Lyapunov’s method to problems in the stability of systems with a delay. Avtom. Telemekhanika 21(6), 740–749 (1960)

    MathSciNet  Google Scholar 

  4. Olgac, N., Sipahi, R.: An exact method for the stability analysis of time-delayed linear time-invariant LTI systems. IEEE Trans. Autom. Control 47(5), 793–797 (2002)

    MathSciNet  Google Scholar 

  5. Richard, J.P.: Time-delay systems: an overview of some recent advances and open problems. Automatica 39(10), 1667–1694 (2003)

    MathSciNet  Google Scholar 

  6. Wang, Z., Zhuang, G., Xie, X., Wang, Y.: Timer-dependent LK functional-based \({H}_{\infty }\) impulsive filtering for delayed implicit hybrid systems. Nonlinear Dyn. 111, 15091–15107 (2023)

    Google Scholar 

  7. Ghaffari, V., Mobayen, S., ud Din, S., Rojsiraphisal, T., Vu, M.T.: Robust tracking composite nonlinear feedback controller design for time-delay uncertain systems in the presence of input saturation. ISA Trans. 129, 88–99 (2022)

    Google Scholar 

  8. Li, Z., Cao, G., Xie, W., Gao, R., Zhang, W.: Switched-observer-based adaptive neural networks tracking control for switched nonlinear time-delay systems with actuator saturation. Inf. Sci. 621, 36–57 (2023)

    Google Scholar 

  9. Mao, X.: A note on the LaSalle-type theorems for stochastic differential delay equations. J. Math. Anal. Appl. 268(1), 125–142 (2002)

    MathSciNet  Google Scholar 

  10. Mathiyalagan, K., Balachandran, K.: Finite-time stability of fractional-order stochastic singular systems with time delay and white noise. Complexity 21(S2), 370–379 (2016)

    MathSciNet  Google Scholar 

  11. Zhou, B., Luo, W.: Improved Razumikhin and Krasovskii stability criteria for time-varying stochastic time-delay systems. Automatica 89, 382–391 (2018)

    MathSciNet  Google Scholar 

  12. Cacace, F., Germani, A., Manes, C., Papi, M.: Predictor-based control of stochastic systems with nonlinear diffusions and input delay. Automatica 107, 43–51 (2019)

    MathSciNet  Google Scholar 

  13. Yao, L., Zhang, W., Xie, X.: Stability analysis of random nonlinear systems with time-varying delay and its application. Automatica 117, 108994 (2020)

    MathSciNet  Google Scholar 

  14. Zhang, Y., Zhang, J., Liu, X.: Bifurcation analysis and \({H}_\infty \) control of a stochastic competition model with time delay and harvesting. Nonlinear Dyn. 109(2), 1217–1232 (2022)

    Google Scholar 

  15. Astolfi, A., Karagiannis, D., Ortega, R.: Nonlinear and Adaptive Control with Applications. Springer Verlag, London (2008)

    Google Scholar 

  16. Kanellakopoulos, I., Kokotovic, P.V., Stephen Morse, A.: Systematic design of adaptive controllers for feedback linearizable systems. IEEE Trans. Autom. Control 36(11), 1241–1253 (1991)

    MathSciNet  Google Scholar 

  17. Krstic, M., Kanellakopoulos, I., Kokotovic, P.: Nonlinear and Adaptive Control Design. John Wiley & Sons, New York (1995)

    Google Scholar 

  18. Liu, L., Kong, M., Yin, S.: Memoryless parameter-dependent control strategy of stochastic strict-feedback time delay systems. J. Frankl. Inst. 360(4), 2436–2456 (2023)

    MathSciNet  Google Scholar 

  19. Yusun, F., Tian, Z., Shi, S.: State feedback stabilization for a class of stochastic time-delay nonlinear systems. IEEE Trans. Autom. Control 48(2), 282–286 (2003)

    MathSciNet  Google Scholar 

  20. Meng, Q., Ma, Q., Zhou, G.: Adaptive output feedback control for stochastic uncertain nonlinear time-delay systems. IEEE Trans. Circuits Syst. II Express Br. 69(7), 3289–3293 (2022)

    Google Scholar 

  21. Min, H., Shengyuan, X., Zhang, B., Ma, Q.: Output-feedback control for stochastic nonlinear systems subject to input saturation and time-varying delay. IEEE Trans. Autom. Control 64(1), 359–364 (2018)

    MathSciNet  Google Scholar 

  22. Wang, H., Liu, P.X., Shi, P.: Observer-based fuzzy adaptive output-feedback control of stochastic nonlinear multiple time-delay systems. IEEE Trans. Cybern. 47(9), 2568–2578 (2017)

    Google Scholar 

  23. Liu, Y., Shaojie, X., Ma, H.: Switched-observer-based adaptive DSC design of nonstrict-feedback switched stochastic nonlinear time-delay systems. Nonlinear Anal. Hybrid Syst. 38, 100917 (2020)

    MathSciNet  Google Scholar 

  24. Chen, C., Sun, Z.: A unified approach to finite-time stabilization of high-order nonlinear systems with an asymmetric output constraint. Automatica 111, 108581 (2020)

    MathSciNet  Google Scholar 

  25. Jia, J., Chen, W., Dai, H., Li, J.: Global stabilization of high-order nonlinear systems under multi-rate sampled-data control. Nonlinear Dyn. 94, 2441–2453 (2018)

    Google Scholar 

  26. Qian, C., Lin, W.: Output feedback control of a class of nonlinear systems: a nonseparation principle paradigm. IEEE Trans. Autom. Control 47(10), 1710–1715 (2002)

    MathSciNet  Google Scholar 

  27. Lin, W., Qian, C.: Adding one power integrator: a tool for global stabilization of high-order lower-triangular systems. Syst. Control Lett. 39(5), 339–351 (2000)

    MathSciNet  Google Scholar 

  28. Xu, K., Wang, H., Liu, P.X.: Adaptive fixed-time control for high-order stochastic nonlinear time-delay systems: an improved Lyapunov–Krasovskii function. IEEE Trans. Cybern. 54, 776–786 (2023)

    Google Scholar 

  29. Xue, L., Liu, Z., Zhang, W.: Decentralized tracking control for a class of stochastic high-order time-delay nonlinear systems under arbitrary switchings. J. Frankl. Inst. 357(2), 887–905 (2020)

    MathSciNet  Google Scholar 

  30. Xie, X., Liu, L.: A homogeneous domination approach to state feedback of stochastic high-order nonlinear systems with time-varying delay. IEEE Trans. Autom. Control 58(2), 494–499 (2013)

    MathSciNet  Google Scholar 

  31. Cao, Y., Zhao, J., Sun, Z.: State feedback stabilization problem of stochastic high-order and low-order nonlinear systems with time-delay. AIMS Mathematics 8(2), 3185–3203 (2023)

    MathSciNet  Google Scholar 

  32. Wang, H., Zhu, Q.: Output-feedback stabilization of a class of stochastic high-order nonlinear systems with stochastic inverse dynamics and multidelay. Int. J. Robust Nonlinear Control 31(12), 5580–5601 (2021)

    MathSciNet  Google Scholar 

  33. Xue, L., Zhang, W., Lin, Y.: Global output tracking control for high-order stochastic nonlinear systems with SISS inverse dynamics and time-varying delays. J. Frankl. Inst. 353(13), 3249–3270 (2016)

    MathSciNet  Google Scholar 

  34. Song, Z., Zhai, J.: Decentralized output feedback stabilization for switched stochastic high-order nonlinear systems with time-varying state/input delays. ISA Transactions 90, 64–73 (2019)

    Google Scholar 

  35. Liu, L., Xie, X.: Output-feedback stabilization for stochastic high-order nonlinear systems with time-varying delay. Automatica 47(12), 2772–2779 (2011)

    MathSciNet  Google Scholar 

  36. Ogata, K.: Modern Control Engineering, 5th edn. Prentice Hall, New Jersey (2010)

    Google Scholar 

  37. Gu, K., Chen, J., Kharitonov, V.L.: Stability of Time-Delay Systems. Springer Science & Business Media, Cham (2003)

  38. Hua, C., Liu, P.X., Guan, X.: Backstepping control for nonlinear systems with time delays and applications to chemical reactor systems. IEEE Trans. Ind. Electron. 56(9), 3723–3732 (2009)

    Google Scholar 

  39. Qian, C., Lin, W.: A continuous feedback approach to global strong stabilization of nonlinear systems. IEEE Trans. Autom. Control 46(7), 1061–1079 (2001)

    MathSciNet  Google Scholar 

  40. Li, J., Qian, C.: Global finite-time stabilization by dynamic output feedback for a class of continuous nonlinear systems. IEEE Trans. Autom. Control 51(5), 879–884 (2006)

    MathSciNet  Google Scholar 

  41. Jia, J., Chen, W., Dai, H.: Multirate sampled-data stabilization for a class of low-order lower-triangular nonlinear systems. Int. J. Robust Nonlinear Control 28(6), 2121–2130 (2018)

    MathSciNet  Google Scholar 

  42. Wang, X., Huang, S., Xiang, Z.: Output feedback finite-time stabilization of a class of nonlinear time-delay systems in the p-normal form. Int. J. Robust Nonlinear Control 30(11), 4418–4432 (2020)

    MathSciNet  Google Scholar 

  43. Liu, L., Shengyuan, X., Zhang, Y.: Finite-time output feedback control for a class of stochastic low-order nonlinear systems. Int. J. Control 90(7), 1457–1465 (2017)

    MathSciNet  Google Scholar 

  44. Cui, R., Xie, X.: Finite-time stabilization of stochastic low-order nonlinear systems with time-varying orders and FT-SISS inverse dynamics. Automatica 125, 109418 (2021)

    MathSciNet  Google Scholar 

  45. Huang, S., Xiang, Z.: Finite-time stabilization of a class of switched stochastic nonlinear systems under arbitrary switching. Int. J. Robust Nonlinear Control 26(10), 2136–2152 (2016)

    MathSciNet  Google Scholar 

  46. Chen, C.C., Chen, G.: A new approach to stabilization of high-order nonlinear systems with an asymmetric output constraint. Int. J. Robust Nonlinear Control 30(2), 756–775 (2020)

    MathSciNet  Google Scholar 

  47. Ikeda, N., Watanabe, S.: Stochastic Differential Equations and Diffusion Processes, 2nd edn. North-Holland, Amsterdam (1981)

    Google Scholar 

  48. Liu, S., Ge, S.S., Zhang, J.: Adaptive output-feedback control for a class of uncertain stochastic non-linear systems with time delays. Int. J. Control 81(8), 1210–1220 (2008)

    MathSciNet  Google Scholar 

  49. Mao, X.: Stochastic Differential Equations and Applications, 2nd edn. Horwood Publishing Limited, Chichester (2007)

    Google Scholar 

  50. Kurzweil, J.: On the inversion of Lyapunov’s second theorem on stability of motion. Am. Math. Soc. Transl. 2(24), 19–77 (1956)

    Google Scholar 

  51. Qian, C., Lin, W.: Non-Lipschitz continuous stabilizers for nonlinear systems with uncontrollable unstable linearization. Syst. Control Lett. 42(3), 185–200 (2001)

    MathSciNet  Google Scholar 

  52. Zhang, J., Liu, Y.: Nonsmooth adaptive control design for a large class of uncertain high-order stochastic nonlinear systems. Math. Probl. Eng. (2012). https://doi.org/10.1155/2012/808035

  53. Li, F., Liu, Y.: Global stability and stabilization of more general stochastic nonlinear systems. J. Math. Anal. Appl. 413(2), 841–855 (2014)

    MathSciNet  Google Scholar 

  54. Jia, J., Dai, H., Zhang, F., Huang, J.: Global stabilization of low-order stochastic nonlinear systems with multiple time-varying delays by a continuous feedback control. Appl. Math. Comput. 429, 127234 (2022)

  55. Hardy, G., Littlewood, J.E., P\(\acute{o}\)lya, G.: Inequalities, 2nd edn. Cambridge University Press, Cambridge (1992)

  56. Sun, Z.Y., Zhang, X.H., Xie, X.J.: Global continuous output-feedback stabilization for a class of high-order nonlinear systems with multiple time delays. J. Frankl. Inst. 351(8), 4334–4356 (2014)

    MathSciNet  Google Scholar 

  57. Yang, B., Lin, W.: Homogeneous observers, iterative design, and global stabilization of high-order nonlinear systems by smooth output feedback. IEEE Trans. Autom. Control 49(7), 1069–1080 (2004)

    MathSciNet  Google Scholar 

  58. Qian, C., Li, J.: Global finite-time stabilization by output feedback for planar systems without observable linearization. IEEE Trans. Autom. Control 50(6), 885–890 (2005)

  59. Higham, D.J.: An algorithmic introduction to numerical simulation of stochastic differential equations. SIAM Rev. 43(3), 525–546 (2001)

    MathSciNet  Google Scholar 

Download references

Funding

This work is supported by the National Natural Science Foundation of China (Grant Nos. 62063031, 12101454), the College Teachers Innovation Foundation of Gansu Province of China (Grant No. 2023B-132), the Research Project of Tianshui Normal University (Grant No. TDJ2023-02) and the Foundation Project of Chongqing Normal University (Grant No. 23XLB013).

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Tianshui Normal University, Tianshui, 741001, People’s Republic of China

    Jinping Jia, Fandi Zhang & Jianwen Huang

  2. School of Aerospace Science and Technology, Xidian University, Xi’an, 710071, People’s Republic of China

    Hao Dai

  3. School of Mathematical Sciences, Chongqing Normal University, Chongqing, 401331, People’s Republic of China

    Jianwen Huang

Authors
  1. Jinping Jia
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hao Dai
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Fandi Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Jianwen Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to Jinping Jia or Fandi Zhang.

Ethics declarations

Conflict of interest

We declare that we do not have any commercial or associative interest that represents a conflict of interest in connection with the work submitted.

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

Jia, J., Dai, H., Zhang, F. et al. Dynamic output feedback stabilization for a class of nonsmooth stochastic nonlinear systems perturbed by multiple time-varying delays. Nonlinear Dyn 112, 7093–7111 (2024). https://doi.org/10.1007/s11071-024-09377-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-024-09377-2

Keywords

Search

Navigation