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Volume 6 Issue 4
Jul.  2019

IEEE/CAA Journal of Automatica Sinica

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Aye Aye Than and Junmin Wang, "Stabilization of the Cascaded ODE-Schrödinger Equations Subject to Observation With Time Delay," IEEE/CAA J. Autom. Sinica, vol. 6, no. 4, pp. 1027-1035, June 2019. doi: 10.1109/JAS.2019.1911588
Citation: Aye Aye Than and Junmin Wang, "Stabilization of the Cascaded ODE-Schrödinger Equations Subject to Observation With Time Delay," IEEE/CAA J. Autom. Sinica, vol. 6, no. 4, pp. 1027-1035, June 2019. doi: 10.1109/JAS.2019.1911588

Stabilization of the Cascaded ODE-Schrödinger Equations Subject to Observation With Time Delay

doi: 10.1109/JAS.2019.1911588
Funds:  Manuscript received September 9, 2018; revised October 12, 2018; accepted November 8, 2018. This work was supported by the National Natural Science Foundation of China (61673061)
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  • This paper focuses on the stabilization of the cascaded Schrödinger-ODE equations subject to the observation with time delay. Both observer and predictor systems are designed to estimate the state variable on the time interval $[0, t-\tau]$ when the observation is available, and to predict the state variable on the time interval $[t-\tau, t]$ when the observation is not available, respectively. Based on the estimated state variable and the output feedback stabilizing controller using the backstepping method, it is shown that the closed-loop system is exponentially stable.

     

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  • [1]
    B. B. Ren, J. M. Wang, and M. Krstic, " Stabilization of an ODESchrödinger cascade,” Syst. Control Lett., vol. 62, no. 6, pp. 503–510, Jun. 2013.
    [2]
    S. Tang and C. Xie, " State and output feedback boundary control for a coupled PDE-ODE system,” Syst. Control Lett., vol. 60, no. 8, pp. 540–545, Aug. 2011.
    [3]
    J. M. Wang, J. J. Liu, B. Ren, and J. Chen, " Sliding mode control to stabilization of cascaded heat PDE-ODE systems subject to boundary control matched disturbance,” Automatica, vol. 52, pp. 23–34, Feb. 2015. doi: 10.1016/j.automatica.2014.10.117
    [4]
    G. A. Susto and M. Krstic, " Control of PDE-ODE cascades with Neumann interconnections,” J. Franklin Institute, vol. 347, no. 1, pp. 284–314, Feb. 2010. doi: 10.1016/j.jfranklin.2009.09.005
    [5]
    J. M. Wang, B. Z. Guo, and M. Krstic, " Wave equation stabilization by delays equal to even multiples of the wave propagation time,” SIAM J. Control and Optimization, vol. 49, no. 2, pp. 517–554, Mar. 2011. doi: 10.1137/100796261
    [6]
    J. M. Wang, X. W. Lv, and D. X. Zhao, " Exponential stability and spectral analysis of the pendulum system under position and delayed position feedbacks,” Internat. J. Control, vol. 84, no. 5, pp. 904–915, May 2011. doi: 10.1080/00207179.2011.582886
    [7]
    M. Krstic, " Compensating a string PDE in the actuation or sensing path of an unstable ODE,” IEEE Trans. Automatic Control, vol. 54, no. 6, pp. 1362–1368, May 2009. doi: 10.1109/TAC.2009.2015557
    [8]
    B. Z. Guo, C. Z. Xu, and H. Hammouri, " Output feedback stabilization of a one-dimensional wave equation with an arbitrary time delay in boundary observation,” ESAIM:Control,Opti. Cal. Variations, vol. 18, no. 1, pp. 22–35, Jan. 2012. doi: 10.1051/cocv/2010044
    [9]
    K. Y. Yang, X. Ren, and J. Zhang, " Output feedback stabilization of an unstable wave equation with observations subject to time delay,” J. Syst. Sci. Complex, vol. 29, no. 1, pp. 99–118, Feb. 2016. doi: 10.1007/s11424-016-5169-2
    [10]
    H. Y. Zhu, H. N. Wu, and J. W. Wang, " H disturbance attenuation for nonlinear coupled parabolic PDE-ODE systems via fuzzy-model-based control approach,” IEEE Trans. Syst.,Man,Cybernetics:Systems, vol. 47, no. 8, pp. 1814–1825, Aug. 2017. doi: 10.1109/TSMC.2016.2531701
    [11]
    H. N. Wu, H. Y. Zhu, and J. W. Wang, " Observer-based H sampleddata fuzzy control for a class of nonlinear parabolic PDE systems,” IEEE Trans. Fuzzy Systems, vol. 26, no. 2, pp. 454–473, Apr. 2018. doi: 10.1109/TFUZZ.2017.2686337
    [12]
    H. N. Wu, H. Y. Zhu, and J. W. Wang, " H2 fuzzy control for a class of nonlinear coupled ODE-PDE with input constraint,” IEEE Trans. Fuzzy Systems, vol. 23, no. 3, pp. 593–604, Jun. 2015. doi: 10.1109/TFUZZ.2014.2318180
    [13]
    J. H. Su, " Further result on the robust stability of linear systems with a single time delay,” Syst. Control Letters, vol. 23, no. 5, pp. 375–379, Nov. 1994. doi: 10.1016/0167-6911(94)90071-X
    [14]
    W. Michiels and S. I. Niculescu, " On the delay sensitivity of Smith predictors. Time delay systems: theory and control,” Internat. J. Systems Sci, vol. 34, pp. 8–9, Jul. 2003.
    [15]
    I. Gumowski and C. Mira, Optimization in the Control Theory and Practice, Cambridge University Press: 1968.
    [16]
    K. Y. Yang and J. M. Wang, " Pointwise stabilisation of a string with time delay in the observation,” Internat. J. Control, vol. 90, no. 11, pp. 2394–2405, Sept. 2017. doi: 10.1080/00207179.2016.1250159
    [17]
    R. Datko, J. Lagnese, and P. M. Polis, " An example on the effect of time delays in boundary feedback stabilization of wave equations,” SIAM J. Control and Optimization, vol. 24, no. 1, pp. 152–156, Jan. 1986. doi: 10.1137/0324007
    [18]
    R. Datko, " Two examples of ill-posedness with respect to time delays revisited,” IEEE Trans. Automatic Control, vol. 42, no. 4, pp. 511–515, Apr. 1997. doi: 10.1109/9.566660
    [19]
    M. Krstic, " Control of an unstable reaction-diffusion PDE with long input delay,” Syst. Control Lett, vol. 58, no. 10–11, pp. 773–782, Sept. 2009. doi: 10.1016/j.sysconle.2009.08.006
    [20]
    B. Z. Guo and K. Y. Yang, " Dynamic stabilization of an Euler-Bernoulli beam equation with time delay in boundary observation,” Automatica, vol. 45, no. 6, pp. 1468–1475, Jun. 2009. doi: 10.1016/j.automatica.2009.02.004
    [21]
    I. Karafyllis and M. Krstic, " Predictor feedback for delay systems: Implementations and approximations”, Mathematics, Systems & Control: Foundations & Applications, Birkhäuser: 2017.
    [22]
    W. H. Fleming, Future Directions in Control Theory: A Mathematical Perspective, SIAM Reports Issues Math. Sci., SIAM, Philadelphia: 1988.
    [23]
    K. Y. Yang and J. M. Wang, " Pointwise feedback stabilization of an Euler-Bernoulli beam in observations with time delay”, ESAIM: Control, Optimisation and Calculus of Variations, vol. 25, Article no. 4, 23 pages, Mar. 2019.
    [24]
    A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, New York: 1983.
    [25]
    B. Z. Guo and J. M. Wang, Control of Wave and Beam PDEs: The Riesz Basis Approach, Springer, Switzerland: 2019.
    [26]
    F. R. Curtain, " The Salamon-Weiss class of wellposed infinite dimensional linear systems: A survey,” IMA J. Mathematical Control and Information, vol. 14, no. 2, pp. 207–223, Jan. 1997. doi: 10.1093/imamci/14.2.207
    [27]
    I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Continuous and Approximation Theories Ⅱ, Abstract Hyperbolic-Link Systems Over a Finite Time Horizon, Cambridge, Cambridge Univ. Press: 2000.
    [28]
    B. Z. Guo, " Riesz basis approach to the stabilization of a flexible beam with a tip mass,” SIAM J. Control and Optimization, vol. 39, pp. 1736–1747, 2001. doi: 10.1137/S0363012999354880
    [29]
    B. Z. Guo, " Riesz basis property and exponential stability of controlled Euler-Bernoulli beam equations with variable coefficients,” SIAM J. Control and Optimization, vol. 40, pp. 1905–1923, 2002. doi: 10.1137/S0363012900372519

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    Highlights

    • We present a unified method to compensate the Schrödinger equation cascaded by the ODE equations, where the observation of the system has the time delay.
    • We prove the well-posedness of the open-loop system in the sense of the Salamon-Weiss well-posed infinite dimensional system theory.
    • We design the observer and predictor systems, respectively, at the time interval when the observation is available, and when the observation is not available, so that we can construct a control law with the estimated state by the observer and predictor to stabilize the cascaded PDE-ODE system.
    • We prove the closed-loop system is exponentially stable for the smooth initial values.

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