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Journal of Systems Science and Complexity

, Volume 30, Issue 6, pp 1258–1269 | Cite as

The stability of Schrödinger equation with boundary control matched disturbance

  • Xiaoying Zhang
  • Shugen ChaiEmail author
Article

Abstract

This paper studies the stability of Schrödinger equation with boundary control matched disturbance. The time-varying gain extended state observer is utilized to estimate disturbance and state. Meanwhile, the authors get a continuous controller by the active disturbance rejection control strategy, which shows that the closed-loop system of Schrödinger equation is asymptotically stable. These results are illustrated by simulation examples.

Keywords

Active disturbance rejection control extended state observer Schrödinger equation stability 

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References

  1. [1]
    Guo B Z and Jin F F, Output feedback stabilization for one-dimensional wave equation subject to boundary disturbance, IEEE Transactions on Automatic Control, 2015, 60: 824–830.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    Guo B Z, Zhou H C, AL-Fhaid A S, et al., Stabilization of Euler-Bernoulli beam equation with boundary moment control and disturbance by active disturbance rejection control and sliding mode control approaches, Journal of Dynamic and Control Systems, 2014, 20: 539–558.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    Guo B Z and Zhou H C, Active disturbance rejection control for rejecting boundary disturbance from multi-dimensional Kirchhoff plate via boundary control, SIAM Journal on Control and Optimization, 2014, 52: 2800–2830.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    Guo B Z and Liu J J, Sliding mode control and active disturbance rejection control to the stabilization of one-dimensional Schrödinger equation subject to boundary control matched disturbance, International Robust Nonlinear Control, 2014, 24: 2194–2212.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    Guo B Z and Yang K Y, Output feedback stabilization of one-dimensional Schrödinger equation by boundary observation with time delay, IEEE Transactions on Automatic Control, 2010, 55: 1226–1232.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    Feng H and Guo B Z, Output feedback stabilization of an unstable wave equation with general corrupted boundary observation, Automatica, 2014, 50: 3164–3172.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    Guo W and Guo B Z, Parameter estimation and non-collocated adaptive stabilization for a wave equation subject to general boundary harmonic disturbance, IEEE Transactions on Automatic Control, 2013, 58: 1631–1643.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    Guo W and Guo B Z, Stabilization and regulator design for a one-dimensional unstable wave equation with input harmonic disturbance, International Journal of Robust and Nonlinear Control, 2013, 23: 514–533.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    Han J Q, From PID to active distubancee rejection control, IEEE Transactions on Industrial Electronics, 2009, 56: 900–906.CrossRefGoogle Scholar
  10. [10]
    Zhao Z L and Guo B Z, On active disturbance rejection control for nonlinear systems using time-varying again, European Journal of Control, 2015, 23: 62–70.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    Guo B Z and Zhao Z L, On the convergence of an extended state observer for nonlinear systems with uncertainty, Systems Control Letters, 2011, 60: 420–430.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    Krstic M, Guo B Z, and Smyshlyaev A, Boundary controllers and observers for the linear Schrödinger equation, SIAM Journal on Control and Optimization, 2011, 49: 1479–1497.MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    Machtyngier E and Zuazua E, Stablization of the Schrödinger equation, Portugaliae Mathematica, 1994, 51: 243–256.MathSciNetzbMATHGoogle Scholar
  14. [14]
    Chen G and Coleman M P, Improving low order eigenfrequency estimates derived from the wave propagation method for an Euler-Bernoulli beam, Journal of Sound and Vibration, 1997, 204(4): 696–704.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    Tucsnak M and Weiss G, Observation and Control for Operator Semigroups, Springer, Berlin, 2009.CrossRefzbMATHGoogle Scholar
  16. [16]
    Guo B Z and Chai S G, Control Theory of Infinite Dimensional Linear Systems, Science Press, Beijing, 2012.Google Scholar
  17. [17]
    Gohberg I C and Krěin M G, Introduction to the theory of linear nonselfadjoint operators, Trans of Math Monographs, AMS Providence, Rhode Island, 1969.Google Scholar
  18. [18]
    Komornik V and Loreti P, Fourier Series in Control Theory, Springer-Verlag, New York, 2005.zbMATHGoogle Scholar
  19. [19]
    Weiss G, Admissible observation operators for linear semigroups, Israel Journal of Mathematics, 1989, 65: 17–43.MathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    Weiss G, Admissibility of unbounded control operators, SIAM Journal on Control and Optimization, 1989, 27: 527–545.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Institute of Systems Science, Academy of Mathematics and Systems Science, CAS and Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.School of Mathematical SciencesShanxi UniversityTaiyuanChina
  2. 2.Department of MathematicsShanxi Agriculture UniversityTaiguChina

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