A journal of IEEE and CAA , publishes high-quality papers in English on original theoretical/experimental research and development in all areas of automation
Advanced Search
Volume 5 Issue 1
Jan.  2018

IEEE/CAA Journal of Automatica Sinica

  • JCR Impact Factor: 15.3, Top 1 (SCI Q1)
    CiteScore: 23.5, Top 2% (Q1)
    Google Scholar h5-index: 77， TOP 5
Turn off MathJax
Article Contents
Article Navigation >  IEEE/CAA Journal of Automatica Sinica  > 2018  >  5(1): 360-366
M. D. S. Aliyu, "An Iterative Relaxation Approach to the Solution of the Hamilton-Jacobi-Bellman-Isaacs Equation in Nonlinear Optimal Control," IEEE/CAA J. Autom. Sinica, vol. 5, no. 1, pp. 360-366, Jan. 2018. doi: 10.1109/JAS.2017.7510682
Citation: M. D. S. Aliyu, "An Iterative Relaxation Approach to the Solution of the Hamilton-Jacobi-Bellman-Isaacs Equation in Nonlinear Optimal Control," IEEE/CAA J. Autom. Sinica, vol. 5, no. 1, pp. 360-366, Jan. 2018. doi: 10.1109/JAS.2017.7510682
shu

An Iterative Relaxation Approach to the Solution of the Hamilton-Jacobi-Bellman-Isaacs Equation in Nonlinear Optimal Control

doi: 10.1109/JAS.2017.7510682

  • Department of Electrical Engineering, King Faisal University, Al-Ahsa 31982, Saudi Arabia

More Information
    Abstract
  • In this paper, we propose an iterative relaxation method for solving the Hamilton-Jacobi-Bellman-Isaacs equation (HJBIE) arising in deterministic optimal control of affine nonlinear systems. Local convergence of the method is established under fairly mild assumptions, and examples are solved to demonstrate the effectiveness of the method. An extension of the approach to Lyapunov equations is also discussed. The preliminary results presented are promising, and it is hoped that the approach will ultimately develop into an efficient computational tool for solving the HJBIEs.

     

    • FullText(HTML)
  • loading
    • References(36)
  • [1]
    M. Athans and P. L. Falb, Optimal Control: An Introduction to the Theory and its Applications. New York: Dover Publishers, 2006. http://ci.nii.ac.jp/ncid/BA03468181
    [2]
    D. E. Kirk, Optimal Control Theory. Englewood Cliffs: Prentice Hall, 1970. doi: 10.1007/978-3-540-29678-2_4256
    [3]
    R. Abraham and J. E. Marsden, Foundations of Mechanics. Reading, MA, USA: Addison Wesley, 1978. doi: 10.1088/0951-7715/15/4/316/meta
    [4]
    T. Basar and P. Bernhard, H∞ Optimal Control and Related Minimax Design, Boston: Birkhauser, 1991.
    [5]
    M. D. S. Aliyu, Nonlinear H∞ Control, Hamiltonian Systems and Hamilton-Jacobi Equations. Boca Raton, FL. USA: CRC Press, Taylor and Francis, 2011.
    [6]
    V. Barbu and G. Da Prata, Hamilton-Jacobi Equations in Hilbert Spaces. London: Pitman Advanced Publishing Program, 1983.
    [7]
    Y. Feng, B. D. O. Anderson, and M. Rotkowitz, "A game theoretic algorithm to compute local stabilizing solutions of HJBI equations in nonlinear H∞ control, " Automatica, vol. 45, no. 4, pp. 881-888, Apr. 2009. http://www.ams.org/mathscinet-getitem?mr=2535347
    [8]
    M. D. S. Aliyu, "An Approach for solving the Hamilton-Jacobi-Isaacs equation (HJIE) in nonlinear H∞ control, " Automatica, vol. 39, no. 5, pp. 877-884, May 2003. http://dl.acm.org/citation.cfm?id=2239328
    [9]
    M. D. S. Aliyu, "A transformation approach for solving the hamiltonjacobi-bellman equation in H2 deterministic and stochastic optimal control of affine nonlinear Systems, " Automatica, vol. 39, no. 7, pp. 1243-1249, Jul. 2003. http://www.sciencedirect.com/science/article/pii/S0005109803000803
    [10]
    M. D. S. Aliyu and L. Smolinsky, "A parametrization approach for solving the Hamilton-Jacobi equation and application to the A2 Toda lattice, " Nonlinear Dyn. Syst. Theory, vol. 5, no. 4, pp. 323-344. 2005. https://www.sciencedirect.com/science/article/pii/S1570865998800094
    [11]
    M. D. S. Aliyu, "Adaptive solution of Hamilton-Jacobi-Isaacs equation and Practical H∞ stabilization of nonlinear systems, " in Proc. IEEE Int. Conf. Control Applications, Anchorage, Alaska, USA, 2000, pp. 343-348. http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=897448
    [12]
    A. Al-Tamimi, F. L. Lewis, and M. Abu-Khalaf, "Discrete-time nonlinear HJB solution using approximate dynamic programming: Convergence proof, " IEEE Trans. Syst., Man Cybern., vol. 38, no. 4, pp. 943-949, Aug. 2008. http://www.ncbi.nlm.nih.gov/pubmed/18632382
    [13]
    S. H. Jr. Benton, The Hamilton-Jacobi Equation: A Global Approach. New York: Academic Press, 1977. https://catalogue.nla.gov.au/Record/656384
    [14]
    S. T. Glad, "Robustness of nonlinear state feedback-a survey, " Automatica, vol. 23, no. 4, pp. 425-435, Jul. 1987. http://dl.acm.org/citation.cfm?id=29286
    [15]
    D. L. Lukes, "Optimal regulation of nonlinear dynamical systems, " SIAM J. Control, vol. 7, no. 1, pp. 75-100, Feb. 1969. doi: 10.1137%2F0307007
    [16]
    P. Tsiotras, M. Corless, and M. A. Rotea, "An L2 disturbance attenuation solution to the nonlinear benchmark problem, " Int. J. Robust Nonlinear Control, vol. 8, no. 4-5, pp. 311-330, Dec. 1998. http://www.ams.org/mathscinet-getitem?mr=1611574
    [17]
    M. J. Yazdanpanah, K. Khorasani, and R. V. Patel, "Uncertainty compensation for a flexible-link manipulator using nonlinear H∞ control, " Int. J. Control, vol. 69, no. 6, pp. 753-771, Apr. 1998. http://www.ams.org/mathscinet-getitem?mr=1691654
    [18]
    R. W. Beard, G. N. Saridis, and J. T. Wen, "Galerkin approximations of the generalized Hamilton-Jacobi-Bellman equation, " Automatica, vol. 33, No. 12, pp. 2159-2177, Dec. 1997. http://dl.acm.org/citation.cfm?id=280068&preflayout=flat
    [19]
    R. W. Beard and T. W. Mclain, "Successive Galerkin approximation algorithms for nonlinear optimal and robust control, " Int. J. Control, vol. 71, no. 5, pp. 717-743, Nov. 1998. doi: 10.1080/002071798221542
    [20]
    A. Isidori and W. Kang, " H∞ control via measurement feedback for general nonlinear systems, " IEEE Trans. Automat. Control, vol. 40, no. 3, pp. 466-472, Mar. 1995. http://www.ams.org/mathscinet-getitem?mr=1319245
    [21]
    A. Isidori and W. Lin, "Global L2-Gain design for a class of nonlinear systems, " Syst. Control Lett. , vol. 34, no. 5, pp. 295-302, Jul. 1998. http://dl.acm.org/citation.cfm?id=294951.294960
    [22]
    J. Huang, "An algorithm to solve the discrete HJI equation arising in the L2 gain optimization problem, " Int. J. Control, vol. 72, No. 1, pp. 49-57, Jan. 1999. http://www.ams.org/mathscinet-getitem?mr=1670722
    [23]
    H. Guillard, S. Monaco, and D. Normand-Cyrot, "Approximated solutions to nonlinear discrete-time H∞ Control, " IEEE Trans. Automat. Control, vol. 40, no. 12, pp. 2143-2148, Dec. 1995. doi: 10.1109/9.478342
    [24]
    M. Abu-Khalaf, F. L. Lewis, and L. Huang, "Policy iterations on the Hamilton-Jacobi-Isaacs equation for H∞ state feedback control with input saturation, " IEEE Trans. Automat. Control, vol. 51, no. 12, pp. 1989-1993, Dec. 2006 http://www.ams.org/mathscinet-getitem?mr=2289992
    [25]
    M. Abu-Khalaf and F. L. Lewis, "Nearly optimal control laws for nonlinear systems with saturating actuators using a neural network HJB approach, " Automatica, vol. 41, no. 5, pp. 779-791, May 2005. http://dl.acm.org/citation.cfm?id=2239385
    [26]
    T. Ohtsuka, "Solutions to the Hamilton-Jacobi equation with algebraic gradients, " IEEE Trans. Automat. Control, vol. 56, no. 8, pp. 1874-1885, Aug. 2011. doi: 10.1109/TAC.2010.2097130
    [27]
    E. Zeidler, Nonlinear Functional Analysis and Its Applications: Ⅰ: FixedPoint Theorems. New York: Springer-Verlag, 1986. http://www.springer.com/us/book/9780387909141
    [28]
    J. M. Ortega and W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables. London: Academic Press, 1970. http://www.researchgate.net/publication/220690833_Iterative_Solution_of_Non-Linear_Equation_in_Several_Variables
    [29]
    H. K. Khalil, Nonlinear Systems, 3 rd Edition, Prentice Hall, Upper Saddle River, NJ, USA, 2002.
    [30]
    S. Bittanti, A. J. Laub, and J. Willems, The Riccati Equation. Germany: Springer-Verlag, 1991.
    [31]
    K. M. Zhou, J. C. Doyle, and K. Glover, Robust and Optimal Control. New Jersey: Prentice Hall, 1996. http://www.sciencedirect.com/science/article/pii/S2405896316307522
    [32]
    G. G. L. Meyer and H. J. Payne, "An iterative method of solution of the algebraic Riccati equation, " IEEE Trans. Automat. Control, vol. 17, no. 4, pp. 550-551, Aug. 1972. doi: 10.1109/TAC.1972.1100059
    [33]
    D. L. Kleinman, "On an Iterative technique for Riccati equation Computations, " IEEE Trans. Automat. Control, vol. 13, no. 1, pp. 114-115, Feb. 1968. doi: 10.1109/tac.1968.1098829
    [34]
    K. Vit, "Iterative solution of the Riccati equation, " IEEE Trans. Automat. Control, vol. 17, no. 2, pp. 258-259, Apr. 1972. doi: 10.1109/tac.1972.1099945
    [35]
    J. M. Saniuk and I. B. Rhodes, "A matrix inequality associated with bounds on Solutions of algebraic Riccati and Lyapunov equations, " IEEE Trans. Automat. Control, vol. 32, no. 8, pp. 739-740, Aug. 1987. doi: 10.1109/TAC.1987.1104700
    [36]
    J. M. A. Scherpen and A. J. Van der Schaft, "Normalized coprime factorizations and balancing for unstable nonlinear systems, " Int. J. Control, vol. 60, no. 6, pp. 1193-1222, 1994. doi: 10.1080/00207179408921517
    • Related Articles
    • Supplements(0)

Catalog

    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

    Figures(3)

    Get Citation
    PDF
    XML

    Article Metrics

    Article views (852) PDF downloads(50) Cited by()

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return