A journal of IEEE and CAA , publishes high-quality papers in English on original theoretical/experimental research and development in all areas of automation
Volume 6 Issue 2
Mar.  2019

IEEE/CAA Journal of Automatica Sinica

  • JCR Impact Factor: 11.8, Top 4% (SCI Q1)
    CiteScore: 17.6, Top 3% (Q1)
    Google Scholar h5-index: 77, TOP 5
Turn off MathJax
Article Contents
K. D. Do and A. D. Lucey, "Inverse Optimal Control of Evolution Systems and Its Application to Extensible and Shearable Slender Beams," IEEE/CAA J. Autom. Sinica, vol. 6, no. 2, pp. 395-409, Mar. 2019. doi: 10.1109/JAS.2019.1911381
Citation: K. D. Do and A. D. Lucey, "Inverse Optimal Control of Evolution Systems and Its Application to Extensible and Shearable Slender Beams," IEEE/CAA J. Autom. Sinica, vol. 6, no. 2, pp. 395-409, Mar. 2019. doi: 10.1109/JAS.2019.1911381

Inverse Optimal Control of Evolution Systems and Its Application to Extensible and Shearable Slender Beams

doi: 10.1109/JAS.2019.1911381
More Information
  • An optimal (practical) stabilization problem is formulated in an inverse approach and solved for nonlinear evolution systems in Hilbert spaces. The optimal control design ensures global well-posedness and global practical $\mathcal{K}_{\infty}$-exponential stability of the closed-loop system, minimizes a cost functional, which appropriately penalizes both state and control in the sense that it is positive definite (and radially unbounded) in the state and control, without having to solve a Hamilton-Jacobi-Belman equation (HJBE). The Lyapunov functional used in the control design explicitly solves a family of HJBEs. The results are applied to design inverse optimal boundary stabilization control laws for extensible and shearable slender beams governed by fully nonlinear partial differential equations.


  • loading
  • [1]
    B. D. O. Anderson and J. B. Moore, Linear Optimal Control. New Jersey: Prentice-Hall, 1971.
    M. Hinze, R. Pinnau, M. Ulbrich, and S. Ulbrich, Optimization with PDE Constraints, Mathematical Modeling: Theory and Applications, vol. 23, New York: Springer, 2009.
    J. L. Lions, Optimal Control of Systems Governed by Partial Differential Equations. New York-Berlin: Springer-Verlag, 1971.
    V. Barbu, E. N. Barron, and R. Jensen, "The necessary conditions for optimal control in Hilbert spaces, " Journal of Mathematical Analysis and Applications, vol. 133, pp. 151-162, 1988. doi: 10.1016/0022-247X(88)90372-1
    F. Tröltzsch, Optimal Control of Partial Differential Equations: Theory, Methods and Applications, Graduate Studies in Mathematics, vol. 112, American Mathematical Society, 2010.
    I. Lasiecka, Mathematical Control Theory of Coupled PDEs. Philadelphia: SIAM, 2002.
    M. D. Chekroun, A. Kröner, and H. Liu, "Galerkin approximations of nonlinear optimal control problems in Hilbert spaces, " Electronic Journal of Differential Equations, vol. 2017, no. 189, pp. 1-40, 2017. http://arxiv.org/abs/1704.00427
    R. Sepulchre, M. Jankovic, and P. Kokotovíc, Constructive Nonlinear Control. New York: Springer, 1997.
    M. Krstíc and H. Deng, Stabilization of Nonlinear Uncertain Systems. London: Springer, 1998.
    M. Krstíc and Z. H. Li, "Inverse optimal design of input-to-state stabilizing nonlinear controllers, " IEEE Transactions on Automatic Control, vol. 43, no. 3, pp. 336-350, 1998. doi: 10.1109/9.661589
    M. Krstíc, "On global stabilization of Burgers equation by boundary control, " Systems & Control Letters, vol. 37, pp. 123-141, 1999. http://www.researchgate.net/publication/224276844_On_global_stabilization_of_Burgers'_equation_by_boundary_control
    K. D. Do, "Inverse optimal gain assignment control of evolution systems and its application to boundary control of marine risers, " Automatica, Submitted, 2018.
    K. D. Do, "Stability of nonlinear stochastic distributed parameter systems and its applications, " Journal of Dynamic Systems, Measurement, and Control, vol. 138, pp. 1-12, 2016.
    K. D. Do, "Modeling and boundary control of translational and rotational motions of nonlinear slender beams in three-dimensional space, " Journal of Sound and Vibration, vol. 389, pp. 1-23, 2017. doi: 10.1016/j.jsv.2016.10.044
    A. L. Zuyev, Partial Stabilization and Control of Distributed Parameter Systems with Elastic Elements. Switzerland: Springer International Publishing, 2015.
    A. Cavallo, G. de Maria, and C. N. S. Pirozzi, Active Control of Flexible Structures. London: Springer, 2010.
    T. Meurer, D. Thull, and A. Kugi, "Flatness-based tracking control of a piezoactuated Euler-Bernoulli beam with non-collocated output feedback: theory and experiments, " International Journal of Control, vol. 81, pp. 475-493, 2008. doi: 10.1080/00207170701579429
    F. F. Jin and B. Z. Guo, "Lyapunov approach to output feedback stabilization for the Euler-Bernoulli beam equation with boundary input disturbance, " Automatica, vol. 52, pp. 95-102, 2015. doi: 10.1016/j.automatica.2014.10.123
    K. D. Do, "Stochastic boundary control design for extensible marine risers in three dimensional space, " Automatica, vol. 77, pp. 184-197, 2017. doi: 10.1016/j.automatica.2016.11.032
    W. He, S. Nie, and T. Meng, "Modeling and vibration control for a moving beam with application in a drilling riser, " IEEE Transactions on Control Systems Technology, vol. 25, pp. 1036-1043, 2017. doi: 10.1109/TCST.2016.2577001
    W. He, T. M. D. Huang, and X. Li, "Adaptive boundary iterative learning control for an Euler-Bernoulli beam system with input constraint, " IEEE Transactions on Neural Networks and Learning Systems, In press, 2017. http://www.ncbi.nlm.nih.gov/pubmed/28320681
    K. D. Do and A. D. Lucey, "Boundary stabilization of extensible and unshearable marine risers with large in-plane deflection, " Automatica, vol. 77, pp. 279-292, 2017. doi: 10.1016/j.automatica.2016.11.044
    K. D. Do and A. D. Lucey, "Stochastic stabilization of slender beams in space: Modeling and boundary control, " Automatica, vol. 91, pp. 279-293, 2018. doi: 10.1016/j.automatica.2018.01.017
    M. S. D. Queiroz, M. Dawson, S. Nagarkatti, and F. Zhang, Lyapunov- Based Control of Mechanical Systems. Boston: Birkhauser, 2000.
    J. U. Kim and Y. Renardy, "Boundary control of the Timoshenko beam, " SIAM Journal of Control and Optimization, vol. 25, pp. 1417-1429, 1987. doi: 10.1137/0325078
    O. Morgul, "Dynamic boundary control of the Timoshenko beam, " Automatica, vol. 28, pp. 1255-1260, 1992. doi: 10.1016/0005-1098(92)90070-V
    C. Mei, "Hybrid wave/mode active control of bending vibrations in beams based on the advanced Timoshenko theory, " Journal of Sound and Vibration, vol. 322, pp. 29-38, 2009. doi: 10.1016/j.jsv.2008.11.003
    G. Xu and H. Wang, "Stabilisation of Timoshenko beam system with delay in the boundary control, " International Journal of Control, vol. 86, pp. 1165-1178, 2013. doi: 10.1080/00207179.2013.787494
    W. He, T. Meng, J. K. Liu, and H. Qin, "Boundary control of a Timoshenko beam system with input dead-zone, " International Journal of Control, vol. 88, pp. 1257-1270, 2015. doi: 10.1080/00207179.2014.1003098
    W. He, S. S. Ge, and C. Liu, "Adaptive boundary control for a class of inhomogeneous Timoshenko beam equations with constraints, " IET Control Theory & Applications, vol. 8, pp. 1285-1292, 2014. http://www.researchgate.net/publication/265416616_Adaptive_boundary_control_for_a_class_of_inhomogeneous_Timoshenko_beam_equations_with_constraints
    T. Endo, F. Matsuno, and Y. Jia, "Boundary cooperative control by flexible Timoshenko arms, " Automatica, vol. 81, pp. 377-389, 2017. doi: 10.1016/j.automatica.2017.04.017
    M. Krstíc, A. A. Siranosian, and A. A. Smyshlyaev, "Backstepping boundary controllers and observers for the slender Timoshenko beam: Part Ⅰ control design, " in Proc. 45th IEEE Conference on Decision and Control, San Diego, CA, pp. 3938-3943, 2006. https://ieeexplore.ieee.org/document/1656581/
    M. Krstíc, A. A. Siranosian, A. A. Smyshlyaev, and M. Bement, "Backstepping boundary controllers and observers for the slender Timoshenko beam: Part Ⅱ stability and simulations, " in Proc. American Control Conference, Minneapolis, MN, pp. 2412-2417, 2006. http://ieeexplore.ieee.org/xpl/articleDetails.jsp?tp=&arnumber=4177226
    M. Krstíc and A. Smyshlyaev, Boundary Control of PDEs: A Course on Backstepping Designs. Philadelphia, PA: Society for Industrial and Applied Mathematics, 2008.
    K. D. Do, "Stabilization of exact nonlinear Timoshenko beams in space by boundary feedback, " Journal of Sound and Vibration, vol. 422, pp. 278-299, 2018. doi: 10.1016/j.jsv.2018.02.005
    L. Gawarecki and V. Mandrekar, Stochastic Differential Equations in Infinite Dimensions with Applications to Stochastic Partial Differential Equations. Berlin: Springer, 2011.
    P. L. Chow, Stochastic Partial Differential Equations. Boca Raton: Chapman & Hall/CRC, 2007.
    G. Hardy, J. E. Littlewood, and G. Polya, Inequalities. Cambridge: Cambridge University Press, 2nd ed., 1989.
    H. Khalil, Nonlinear Systems. Prentice Hall, 2002.
    K. D. Do, "Global inverse optimal stabilization of stochastic nonholonomic systems, " Systems & Control Letters, vol. 75, pp. 41-55, 2015. http://www.sciencedirect.com/science/article/pii/S0167691114002369
    M. Krstíc, I. Kanellakopoulos, and P. Kokotovíc, Nonlinear and Adaptive Control Design. New York: Wiley, 1995.
    A. Love, A Treatise on the Mathematical Theory of Elasticity. Cambridge University Press, 3rd ed., 1920.
    R. A. Adams and J. J. F. Fournier, Sobolev Spaces. Oxford, UK: Academic Press, 2nd ed., 2003.
    M. M. Bernitsas, J. E. Kokarakis, and A. Imron, "Large deformation three-dimensional static analysis of deep water marine risers, " Applied Ocean Research, vol. 7, pp. 178-187, 1985. doi: 10.1016/0141-1187(85)90024-0
    J. M. Niedzwecki and P. Y. F. Liagre, "System identification of distributed-parameter marine riser models, " Ocean Engineering, vol. 30, no. 11, pp. 1387-1415, 2003. doi: 10.1016/S0029-8018(02)00110-5


    通讯作者: 陈斌, bchen63@163.com
    • 1. 

      沈阳化工大学材料科学与工程学院 沈阳 110142

    1. 本站搜索
    2. 百度学术搜索
    3. 万方数据库搜索
    4. CNKI搜索


    Article Metrics

    Article views (1691) PDF downloads(66) Cited by()


    DownLoad:  Full-Size Img  PowerPoint