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Volume 8 Issue 8
Aug.  2021

IEEE/CAA Journal of Automatica Sinica

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H. Mirinejad, T. Inanc, and Jacek M. Zurada, "Radial Basis Function Interpolation and Galerkin Projection for Direct Trajectory Optimization and Costate Estimation," IEEE/CAA J. Autom. Sinica, vol. 8, no. 8, pp. 1380-1388, Aug. 2021. doi: 10.1109/JAS.2021.1004081
Citation: H. Mirinejad, T. Inanc, and Jacek M. Zurada, "Radial Basis Function Interpolation and Galerkin Projection for Direct Trajectory Optimization and Costate Estimation," IEEE/CAA J. Autom. Sinica, vol. 8, no. 8, pp. 1380-1388, Aug. 2021. doi: 10.1109/JAS.2021.1004081

Radial Basis Function Interpolation and Galerkin Projection for Direct Trajectory Optimization and Costate Estimation

doi: 10.1109/JAS.2021.1004081
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  • This work presents a novel approach combining radial basis function (RBF) interpolation with Galerkin projection to efficiently solve general optimal control problems. The goal is to develop a highly flexible solution to optimal control problems, especially nonsmooth problems involving discontinuities, while accounting for trajectory accuracy and computational efficiency simultaneously. The proposed solution, called the RBF-Galerkin method, offers a highly flexible framework for direct transcription by using any interpolant functions from the broad class of global RBFs and any arbitrary discretization points that do not necessarily need to be on a mesh of points. The RBF-Galerkin costate mapping theorem is developed that describes an exact equivalency between the Karush–Kuhn–Tucker (KKT) conditions of the nonlinear programming problem resulted from the RBF-Galerkin method and the discretized form of the first-order necessary conditions of the optimal control problem, if a set of discrete conditions holds. The efficacy of the proposed method along with the accuracy of the RBF-Galerkin costate mapping theorem is confirmed against an analytical solution for a bang-bang optimal control problem. In addition, the proposed approach is compared against both local and global polynomial methods for a robot motion planning problem to verify its accuracy and computational efficiency.

     

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  • [1]
    X. Wang J. Liu, and H. Peng, “Computational techniques for nonlinear optimal control,” in Symplectic Pseudospectral Methods for Optimal Control. Intelligent Systems, Control and Automation: Science and Engineering, vol. 97, Springer, Singapore, 2021.
    [2]
    B. A. Conway, “A survey of methods available for the numerical optimization of continuous dynamic systems,” J. Optim. Theory Appl., vol. 152, no. 2, pp. 271–306, Feb. 2012.
    [3]
    A. V. Rao, “Trajectory Optimization: A Survey,” in: Optimization and Optimal Control in Automotive Systems, Lecture Notes in Control and Information Sciences vol. 455, Springer, Cham, pp. 3–21, 2014.
    [4]
    W. W. Hager, “Runge-Kutta methods in optimal control and the transformed adjoint system,” Numer. Math., vol. 87, pp. 247–282, 2000. doi: 10.1007/s002110000178
    [5]
    C. Jiang, K. Xie, C. Yu, et al., “A sequential computational approach to optimal control problems for differential-algebraic systems based on efficient implicit Runge–Kutta integration,” Appl. Math. Modelling, vol. 58 pp. 313–330, Jun. 2018.
    [6]
    J. Mei, F. Zhang, J. Zang, Y. Zhao, and H. Yan, “Trajectory optimization of the 6-degrees-of-freedom high-speed parallel robot based on B-spline curve,” Scientific Progress, vol. 103, no. 1, pp. 1–26, 2020.
    [7]
    Y. E. Tabriz and A. Heydari, “Generalized B-spline functions method for solving optimal control problems,” Comput. Methods Differential Equations, vol. 2, no. 4, pp. 243–255, 2014.
    [8]
    X. Tang and J. Chen, “Direct trajectory optimization and costate estimation of infinite-horizon optimal control problems using collocation at the flipped legendre-gauss-radau points,” IEEE/CAA J. Autom. Sinica, vol. 3, pp. 174–183, 2016. doi: 10.1109/JAS.2016.7451105
    [9]
    I. M. Ross and M. Karpenko, “A review of pseudospectral optimal control: From theory to flight,” Annual Reviews in Control, vol. 36, no. 2, pp. 182–197, 2012. doi: 10.1016/j.arcontrol.2012.09.002
    [10]
    H. Ma, T. Qin, and W. Zhang, “An efficient Chebyshev algorithm for the solution of optimal control problems,” IEEE Trans. Autom. Control, vol. 56, no. 3, pp. 675–680, 2011. doi: 10.1109/TAC.2010.2096570
    [11]
    D. A. Benson, G. T. Huntington, T. P. Thorvaldsen, and A. V. Rao, “Direct trajectory optimization and costate estimation via an orthogonal collocation method,” J. Guidance Control Dynamics, vol. 29, no. 6, pp. 1435–1440, 2006. doi: 10.2514/1.20478
    [12]
    G. Elnagar, M.A. Kazemi, and M. Razzaghi, “The pseudospectral Legendre method for discretizing optimal control problems,” IEEE Trans. Autom. Control, vol. 40, no. 10, pp. 1793–1796, 1995. doi: 10.1109/9.467672
    [13]
    Q. Gong, I. M. Ross, and F. Fahroo, “Spectral and pseudospectral optimal control over arbitrary grids,” J. Optim. Theory Appl., vol. 169, no. 3, pp. 759–783, 2016. doi: 10.1007/s10957-016-0909-y
    [14]
    M. A. Mehrpouya and H. Peng, “A robust pseudospectral method for numerical solution of nonlinear optimal control problems,” Intern. J. Comput. Math., 2020. DOI: 10.1080/00207160.2020.1807521.
    [15]
    M. Shamsi, “A modified pseudospectral scheme for accurate solution of bang-bang optimal control problems,” Optim. Control Appl. Meth., vol. 32, pp. 668–680, 2011. doi: 10.1002/oca.967
    [16]
    E. Tohidi and S. L. Noghabi, “An efficient Legendre pseudospectral method for solving nonlinear quasi bang-bang optimal control problems,” J. Applied Math. Statistics Inform., vol. 8, no. 2, pp. 73–85, 2012. doi: 10.2478/v10294-012-0016-0
    [17]
    J. Wei, X Tang, and J. Yan, “Costate estimation for a multiple-interval pseudospectral method using collocation at the flipped Legendre-Gauss-Radau Points,” IEEE/CAA J. Autom. Sinica, pp. 1–15, Nov. 2016. DOI: 10.1109/JAS.2016.7510028.
    [18]
    H. Ghassemi, M. Maleki, and M. Allame, “On the modification and convergence of unconstrained optimal control using pseudospectral methods,” Optimal Control Applications &Methods, vol. 42, no. 3, pp. 717–743, 2021.
    [19]
    H. Mirinejad, “A radial basis function method for solving optimal control problems,” Ph.D. dissertation, University of Louisville, May 2016.
    [20]
    J. A. Rad, S. Kazem, and K. Parand, “Optimal control of a parabolic distributed parameter system via radial basis functions,” Commun. Nonlinear Sci. Numer. Simul., vol. 19, no. 8, pp. 2559–2567, 2014. doi: 10.1016/j.cnsns.2013.01.007
    [21]
    H. Mirinejad and T. Inanc, “A radial basis function method for direct trajectory optimization,” in Proc. American Control Conf., Chicago, IL. 2015, pp. 4923–4928.
    [22]
    H. Mirinejad and T. Inanc, “An RBF collocation method for solving optimal control problems,” Robotics Autonomous Syst., vol. 87, pp. 219–225, 2017. doi: 10.1016/j.robot.2016.10.015
    [23]
    H. Mirinejad and T. Inanc, “RBF method for optimal control of drug administration in the anemia of hemodialysis patients,” in Proc. 41st Annu. Northeast Biomed. Eng. Conf., Troy, NY, 2015.
    [24]
    H. Mirinejad and T. Inanc, “Individualized anemia management using a radial basis function method,” in Proc. IEEE Great Lakes Biomed. Conf., Milwaukee, WI, 2015.
    [25]
    H. Mirinejad, T. Inanc, M. Brier, and A. Gaweda, “RBF-based receding horizon control approach to personalized anemia treatment,” in Proc. 41st Annu. Northeast Biomed. Eng. Conf., Troy, NY, 2015.
    [26]
    S. Hubbert, Q. Thong, L. Gia, and T. M. Morton, Spherical Radial Basis Functions, Theory and Applications. 1st ed., Springer International Publishing, 2015.
    [27]
    K. E. Atkinson and W. Han, Theoretical Numerical Analysis: A Functional Analysis Framework. 3rd ed., Springer-Verlag, New York, 2009.
    [28]
    I. M. Ross, “A historical introduction to the covector mapping principle,” in Proc. AAS/AIAA Astrodynamics Specialist Conf., Lake Tahoe, CA, 2005.
    [29]
    Q. Gong, I. M. Ross, W. Kang, and F. Fahroo, “Connections between the covector mapping theorem and convergence of pseudospectral methods for optimal control,” Computat. Optim. and Appl., vol. 41, pp. 307–335, 2008. doi: 10.1007/s10589-007-9102-4
    [30]
    M. Ross and F. Fahroo, User’s manual For DIDO 2002: A MATLAB application package for dynamic optimization, Dept. Aeronautics and Astronautics, Naval Postgraduate School, AA-02-002, Monterey, CA, 2002.
    [31]
    R. Bhattacharya, “OPTRAGEN, a MATLAB toolbox for optimal trajectory generation,” in Proc. 45th IEEE Conf. Decision and Control, San Diego, CA, Dec. 2006.
    [32]
    P. E. Gill, W. Murray, and M. A. Saunders, “SNOPT: An SQP algorithm for large-scale constrained optimization,” SIAM Review, vol. 47, no. 1, pp. 99–131, 2005. doi: 10.1137/S0036144504446096

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    Highlights

    • An accurate, direct method proposed for solving optimal control problems (especially nonsmooth)
    • RBF-Galerkin, the proposed method, offers a highly flexible framework for direct transcription
    • Any type of global RBFs (as interpolants) and any arbitrary discretization points can be used
    • RBF-Galerkin uses global RBF interpolation and Galerkin projection for trajectory optimization
    • Costate estimation provided via RBF-Galerkin Costate Mapping Theorem

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