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

Performance Output Tracking for an ODE Cascaded with Schrödinger Equation Subject to Disturbances

  • Published:
Journal of Dynamical and Control Systems Aims and scope Submit manuscript

Abstract

In this paper, we are concerned with the performance output tracking for a Schrödinger PDE-ODE cascaded system with external disturbances enter in all possible channels. The main challenge of the problem is the fact that the disturbances are non-collocated to the controller. By proper trajectory planning approach, this difficulty can be overcome by converting non-collocated configurations into the collocated ones. Then a state observer is designed in terms of the tracking errors. Finally, the feedback control is proposed by applying the backstepping technique. The stability of the closed-loop system and the exponential convergence of the regulation error are proved. Some numerical simulations illustrate that the proposed approach is very effective.

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

Similar content being viewed by others

References

  1. Byrnes CI, Lauko IG, Gilliam DS, Shubov VI. Output regulation for linear distributed parameter systems. IEEE Trans Automat Cont 2000;45 (12):2236–2252.

    Article  MathSciNet  MATH  Google Scholar 

  2. Chen WH. Disturbance observer based control for nonlinear systems. IEEE/ASME Trans Mechatron 2004;9(4):706–710.

    Article  Google Scholar 

  3. Cui HY, Chen YN, Xu GQ. 2021. Stabilization for schrödinge equation with internal damping and boundary disturbance. J Dyn Control Syst, https://doi.org/10.1007/s10883-021-09564-z.

  4. Davison EJ. The robust control of a servomechanism problem for linear time-invariant multivariable systems. IEEE Trans Automat Control 1976; 21(1):25–34.

    Article  MathSciNet  MATH  Google Scholar 

  5. Deutscher J. Output regulation for linear distributed-parameter systems using finite-dimensional dual observers. Automatica 2011;47(11):2468–2473.

    Article  MathSciNet  MATH  Google Scholar 

  6. Deutscher J. Backstepping approach to the output regulation of boundary controlled parabolic PDEs. Automatica 2015;57:56–64.

    Article  MathSciNet  MATH  Google Scholar 

  7. Deutscher J. Finite-time output regulation for linear 2×2 hyperbolic systems using backstepping. Automatica 2017;75:54–62.

    Article  MathSciNet  MATH  Google Scholar 

  8. Desoer CA, Lin CA. Tracking and disturbance rejection of MIMO nonlinear systems with PI controller. IEEE Trans Automat Control 1985;30(9):861–867.

    Article  MathSciNet  MATH  Google Scholar 

  9. Evans L. 1997. Partial differential equations. Providence, Island: American math soc.

  10. Feng H, Guo BZ. A new active disturbance rejection control to output feedback stabilization for a one-dimensional anti-stable wave equation with disturbance. IEEE Trans Automat Control 2017;62(62):3774–3787.

    Article  MathSciNet  MATH  Google Scholar 

  11. Feng H, Guo BZ. On stability equivalence between dynamic output feedback and static output feedback for a class of second order infinite-dimensional systems. SIAM J Control Optim 2015;53(4):1934–1955.

    Article  MathSciNet  MATH  Google Scholar 

  12. Francis BA, Wonham WM. The internal model principle of control theory. Automatica 1976;12:457–465.

    Article  MathSciNet  MATH  Google Scholar 

  13. Guo BZ, Liu JJ. Sliding mode control and active disturbance rejection control to the stabilization of one-dimensional Schródinger equation subject to boundary control matched disturbance. Internat J Robust Nonlinear Control 2014;24: 2194–2212.

    Article  MathSciNet  MATH  Google Scholar 

  14. Guo W, Shao ZC, Krstic M. Adaptive rejection of harmonic disturbance anticollocated with control in 1D wave equation. Automatica 2017;79:17–26.

    Article  MathSciNet  MATH  Google Scholar 

  15. Guo W, Zhou HC, Krstic M. Adaptive error feedback regulation problem for 1D wave equation. Int J Robust Nonlinear Control 2018;28:4309–4329.

    Article  MathSciNet  MATH  Google Scholar 

  16. Guo W, Guo BZ. Performance output tracking for a wave equation subject to unmatched general boundary harmonic disturbance. Automatica 2016; 68:194–202.

    Article  MathSciNet  MATH  Google Scholar 

  17. Haroche S, Raimond J-M. Exploring the quantum: atoms, cavities, and photons. USA: Oxford University Press; 2013.

    MATH  Google Scholar 

  18. Immonen E, Pohjolainen S. Feedback and feedforward output regulation of bounded uniformly continuous signals for infinite-dimensional systems. SIAM J Control Optim 2006;45(5):1714–1735.

    Article  MathSciNet  MATH  Google Scholar 

  19. Jia YN, Liu JJ. Output feedback stabilization of an ODE-schrödinge cascade system subject to boundary control matched unknown disturbance. J Dyn Control Syst 2020;26(2):393–405.

    Article  MathSciNet  MATH  Google Scholar 

  20. Krstic M, Smyshlyaev A. 2008. Boundary control of PDEs: a course on backstepping designs. Philadelphia PA: SIAM.

  21. Liu JJ, Guo BZ. Robust tracking error feedback control for a one-dimensional schrödinger equation. IEEE Trans Automat Control 2022;67:1120–1134.

    Article  MathSciNet  MATH  Google Scholar 

  22. Liu XF, Xu GQ. Output-based stabilization of Timoshenko beam with the boundary control and input distributed delay. J Dyn Control Syst 2016;22 (2):347–67.

    Article  MathSciNet  MATH  Google Scholar 

  23. Li S, Yang J, Chen WH, Chen X. Disturbance observer-based control methods and applications. Boca Raton: CRC press; 2014.

    Google Scholar 

  24. Logemann H, Ilchmann A. An adaptive servomechanism for a class of infinite-dimensional systems. SIAM J Control Optim 1994;32(4):917–936.

    Article  MathSciNet  MATH  Google Scholar 

  25. Natarajan V, Gilliam DS, Weiss G. The state feedback regulator problem for regular linear systems. IEEE Trans Automat Control 2014;59:2708–2723.

    Article  MathSciNet  MATH  Google Scholar 

  26. Paunonen L. Controller design for robust output regulation of regular linear systems. IEEE Trans Automat Control 2016;61:2974–2986.

    Article  MathSciNet  MATH  Google Scholar 

  27. Paunonen L, Pohjolainen S. The internal model principle for systems with unbounded control and observation, SIAM. J. Control Optim 2014;52: 3967–4000.

    Article  MathSciNet  MATH  Google Scholar 

  28. Paunonen L. Robust controllers for regular linear systems with infinite-dimensional exosystems. SIAM J Control Optim 2017;55:1567–1597.

    Article  MathSciNet  MATH  Google Scholar 

  29. Pisano A, Orlov Y, Usai E. Tracking control of the uncertain heat and wave equation via power-fractional and sliding-mode techniques. SIAM J Control Optim 2011;49(2):363–382.

    Article  MathSciNet  MATH  Google Scholar 

  30. Rebarber R, Weiss G. Internal model based tracking and disturbance rejection for stable well-posed systems. Automatica 2003;39(9):1555–1569.

    Article  MathSciNet  MATH  Google Scholar 

  31. Reed M. 1981. Simon B Methods of modern mathematical physics I: functional analysis, Academic Press.

  32. Ren BB, Wang JM, Krstic M. Stabilization of an ODE-schrödinger Cascade. Syst Control Lett 2013;62:503–510.

    Article  MATH  Google Scholar 

  33. Tucsnak M, Weiss G. Observation and control for operator semigroups. Switzerland: Birkhäuser: Basel; 2009.

    Book  MATH  Google Scholar 

  34. Wen RL, Feng H. 2021. Performance output tracking for cascaded heat partial differential equation-ordinary differential equation systems subject to unmatched disturbance. Int J Robust Nonlinear Control:1–22.

  35. Yang KY. Stabilization of one-dimensional schrödinge equation under joint feedback control with delayed observation. J Dyn Control Syst 2019;25:275–288.

    Article  MathSciNet  MATH  Google Scholar 

  36. Yao X, Guo L. Composite anti-disturbance control for Markovian jump nonlinear systems via disturbance observer. Automatica 2013;49(8):2538–2545.

    Article  MathSciNet  MATH  Google Scholar 

  37. Zhang X, Feng H, Chai SG. Performance output exponential tracking for a wave equation with a general boundary disturbance. Syst Control Lett 2016;98:79–85.

    Article  MathSciNet  MATH  Google Scholar 

  38. Zhao DX, Wang JM. Exponential stability and spectral analysis of the inverted pendulum system under two delayed position feedbacks. J Dyn Control Syst 2012;18(2):269–95.

    Article  MathSciNet  MATH  Google Scholar 

  39. Zhou HC, Guo BZ. Performance output tracking for one-dimensional wave equation subject to unmatched general disturbance and non-collocated control. Eur J Control 2018;39:39–52.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Funding

This work was supported by the Basic Research Program of Shanxi Province (Free Exploration) Project under Grant: 20210302123181, 20210302124688, the Scientific and Technological Innovation Programs of Higher Education Institutions in Shanxi under Grant STIP2021L416, and the Youth fund of Shanxi University of Finance and Economics (Grant No. QN-2019024).

Author information

Authors and Affiliations

  1. College of Mathematics, Taiyuan University of Technology, Taiyuan, 030024, Shanxi, China

    Yi-Jing Li & Jun-Jun Liu

Authors
  1. Yi-Jing Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jun-Jun Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jun-Jun Liu.

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

Li, YJ., Liu, JJ. Performance Output Tracking for an ODE Cascaded with Schrödinger Equation Subject to Disturbances. J Dyn Control Syst 29, 901–917 (2023). https://doi.org/10.1007/s10883-022-09631-z

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10883-022-09631-z

Keywords

Mathematics Subject Classification (2010)

Search

Navigation