208
Views
6
CrossRef citations to date
0
Altmetric
Original Articles

A robust pseudospectral method for numerical solution of nonlinear optimal control problems

ORCID Icon & ORCID Icon
Pages 1146-1165 | Received 06 Sep 2019, Accepted 28 Jul 2020, Published online: 24 Aug 2020

References

  • F.S. Acton, Numerical Methods that Work, MAA spectrum, Mathematical Association of America 1990.
  • U. Ali and Y. Wardi, Multiple shooting technique for optimal control problems with application to power aware networks, IFAC-PapersOnLine 48 (2015), pp. 286–290. doi: 10.1016/j.ifacol.2015.11.189
  • M.S. Aronna, J.F. Bonnans, and P. Martinon, A shooting algorithm for optimal control problems with singular arcs, J. Optim. Theory. Appl. 158 (2013), pp. 419–459. doi: 10.1007/s10957-012-0254-8
  • E. Ashpazzadeh, B. Han, M. Lakestani, and M. Razzaghi, Derivative-orthogonal wavelets for discretizing constrained optimal control problems, Int. J. Syst. Sci. 51 (2020), pp. 786–810. doi: 10.1080/00207721.2020.1739356
  • J. Berrut and L. Trefethen, Barycentric Lagrange Interpolation, SIAM Rev. 46 (2004), pp. 501–517. doi: 10.1137/S0036144502417715
  • R. Bertrand and R. Epenoy, New smoothing techniques for solving bang-bang optimal control problems -- numerical results and statistical interpretation, Opt. Control Appl. Methods 23 (2002), pp. 171–197. doi: 10.1002/oca.709
  • J. Betts, Survey of numerical methods for trajectory optimization, J. Guid. Control. Dyn. 21 (1998), pp. 193–207. doi: 10.2514/2.4231
  • J.T. Betts, Practical Methods for Optimal Control and Estimation Using Nonlinear Programming, 2nd ed. Cambridge University Press, New York, NY, USA, 2009,
  • J.F. Bonnans, The shooting approach to optimal control problems, IFAC Proc. Vol. 46 (2013), pp. 281–292. doi: 10.3182/20130703-3-FR-4038.00158
  • A.E. Bryson Jr. and Y.C. Ho, Applied Optimal Control: Optimization, Estimation, and Control, Hemisphere Publishing Corp., Washington, DC, 1975.
  • C. Canuto, M. Hussaini, A. Quarteroni, and T. Zang, Spectral Methods in Fluid Dynamics, Springer, Berlin, 1991. Springer Series in Computational Physics,
  • J. Dennis and R. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations, Classics in Applied Mathematics, Society for Industrial and Applied Mathematics, 1996.
  • M. Diehl, H. Bock, H. Diedam, and P.B. Wieber, Fast direct multiple shooting algorithms for optimal robot control, Lecture Notes Control Inform. Sci. 340 (2006), pp. 65–93. doi: 10.1007/978-3-540-36119-0_4
  • G. Elnagar, M. Kazemi, and M. Razzaghi, Pseudospectral legendre method for discretizing optimal control problems, IEEE. Trans. Automat. Contr. 40 (1995), pp. 1793–1796. doi: 10.1109/9.467672
  • B.C. Fabien, Indirect solution of inequality constrained and singular optimal control problems via a simple continuation method, J. Dyn. Syst. Meas. Control. 136 (2014), pp. 1–14. doi: 10.1115/1.4025596
  • B.C. Fabien, Parallel indirect solution of optimal control problems, Optim. Control Appl. Methods 35 (2014), pp. 204–230. doi: 10.1002/oca.2064
  • F. Fahroo and I. Ross, Direct trajectory optimization by a Chebyshev pseudospectral method, J. Guid. Control. Dyn. 25 (2002), pp. 160–166. doi: 10.2514/2.4862
  • B. Fornberg, A Practical Guide to Pseudospectral Methods, Cambridge University Press, Cambridge, 1998.
  • Z. Foroozandeh, M. Shamsi, V. Azhmyakov, and M. Shafiee, A modified pseudospectral method for solving trajectory optimization problems with singular arc, Math. Methods. Appl. Sci. 40 (2017), pp. 1783–1793. doi: 10.1002/mma.4097
  • Z. Foroozandeh, M. Shamsi, and M.D.R. De Pinho, A hybrid direct–indirect approach for solving the singular optimal control problems of finite and infinite order, Iran. J. Sci. Technol., Trans. A: Sci. 42 (2018), pp. 1545–1554. doi: 10.1007/s40995-017-0176-2
  • Z. Foroozandeh, M. Shamsi, and M.D.R. De Pinho, A mixed-binary non-linear programming approach for the numerical solution of a family of singular optimal control problems, Int. J. Control. 92 (2019), pp. 1551–1566. doi: 10.1080/00207179.2017.1399216
  • D. Garg, Advances in global pseudospectral methods for optimal control, Ph.D. diss., University of Florida, 2011.
  • W. Gautschi, Numerical Analysis: An Introduction, Birkhäuser, Boston, 1997.
  • W. Gautschi, Orthogonal Polynomials: Computation and Approximation, Oxford University Press, Oxford, 2004.
  • A. Hermant, Optimal control of the atmospheric reentry of a space shuttle by an homotopy method, Optim. Control Appl. Methods 32 (2011), pp. 627–646. doi: 10.1002/oca.961
  • N. Higham, The numerical stability of barycentric Lagrange interpolation, IMA J. Numer. Anal. 24l (2004), pp. 547–556. doi: 10.1093/imanum/24.4.547
  • H. Keller, Numerical Solution of Two Point Boundary Value Problems, SIAM, Philadelphia, PA, 1976. CBMS-NSF Regional Conference Series in Applied Mathematics.
  • C. Kelley, Solving Nonlinear Equations with Newton's Method, Fundamentals of Algorithms, Society for Industrial and Applied Mathematics, 2003 .
  • M. Khaksar-e Oshagh and M. Shamsi, Direct pseudo-spectral method for optimal control of obstacle problem -- an optimal control problem governed by elliptic variational inequality, Math. Methods. Appl. Sci. 40 (2017), pp. 4993–5004.
  • M. Khaksar-e Oshagh and M. Shamsi, An adaptive wavelet collocation method for solving optimal control of elliptic variational inequalities of the obstacle type, Comput. Math. Appl. 75 (2018), pp. 470–485. doi: 10.1016/j.camwa.2017.09.026
  • D. Kirk, Optimal Control Theory, Prentice-Hall, Englewood Cliffs, NJ, 1970. 
  • S. Lenhart and J. Workman, Optimal Control Applied to Biological Models, Chapman & Hall/CRC Mathematical and Computational Biology, CRC Press, 2007.
  • H. Maurer, Numerical solution of singular control problems using multiple shooting techniques, J. Optim. Theory. Appl. 18 (1976), pp. 235–257. doi: 10.1007/BF00935706
  • M. Mehrpouya and M. Shamsi, Gauss pseudospectral and continuation methods for solving two-point boundary value problems in optimal control theory, Appl. Math. Model. 39 (2015), pp. 5047–5057. doi: 10.1016/j.apm.2015.04.009
  • M. Mehrpouya, M. Shamsi and V. Azhmyakov, An efficient solution of hamiltonian boundary value problems by combined gauss pseudospectral method with differential continuation approach, J. Franklin. Inst. 351 (2014), pp. 4765–4785. doi: 10.1016/j.jfranklin.2014.07.005
  • M. Mehrpouya, M. Shamsi and M. Razzaghi, A combined adaptive control parametrization and homotopy continuation technique for the numerical solution of bang-bang optimal control problems, ANZIAM J. 56 (2014), pp. 48–65.
  • H. Nosratipour, F. Sarani, O. Solaymani Fard and A. Hashemi Borzabadi, An adaptive nonmonotone truncated Newton method for optimal control of a class of parabolic distributed parameter systems, Eng. Comput. 36 (2020), pp. 689–702. doi: 10.1007/s00366-019-00724-1
  • H. Oberle and W. Grimm, BNDSCO – A program for the numerical solution of optimal control problems, Tech. Rep. 515-89/22, Institute for Flight Systems Dynamics, DLR, Oberpfaffenhofen, Germany, 1989.
  • M. Osborne, On shooting methods for boundary value problems, J. Math. Anal. Appl. 27 (1969), pp. 417–433. doi: 10.1016/0022-247X(69)90059-6
  • B. Pan, P. Lu, X. Pan, and Y. Ma, Double-homotopy method for solving optimal control problems, J. Guid. Control. Dyn. 39 (2016), pp. 1706–1720. doi: 10.2514/1.G001553
  • H. Peng, Q. Gao, Z. Wu, and W. Zhong, Symplectic approaches for solving two-point boundary-value problems, J. Guid. Control. Dyn. 35 (2012), pp. 653–659. doi: 10.2514/1.55795
  • H. Peng, Q. Gao, Z. Wu, and W. Zhong, Symplectic algorithms with mesh refinement for a hypersensitive optimal control problem, Int. J. Comput. Math. 92 (2015), pp. 2273–2289. doi: 10.1080/00207160.2014.979810
  • H. Peng, X. Wang, M. Li, and B. Chen, An hp symplectic pseudospectral method for nonlinear optimal control, Commun. Nonlinear Sci. Numer. Simul. 42 (2017), pp. 623–644. doi: 10.1016/j.cnsns.2016.06.023
  • H. Peng, F. Li, J. Liu, and Z. Ju, A symplectic instantaneous optimal control for robot trajectory tracking with differential–algebraic equation models, IEEE Trans. Ind. Electron. 67 (2020), pp. 3819–3829. doi: 10.1109/TIE.2019.2916390
  • K. Rabiei and K. Parand, Collocation method to solve inequality-constrained optimal control problems of arbitrary order, Eng. Comput. 36 (2020), pp. 115–125. doi: 10.1007/s00366-018-0688-1
  • I. Ross, A Primer on Pontryagin's Principle in Optimal Control, 2nd ed., Collegiate Publishers, San Francisco, 2015.
  • Z. Sabeh, M. Shamsi and M. Dehghan, Distributed optimal control of the viscous burgers equation via a legendre pseudo-spectral approach, Math. Methods. Appl. Sci. 39 (2016), pp. 3350–3360. doi: 10.1002/mma.3779
  • S. Sethi and G. Thompson, Optimal Control Theory: Applications to Management Science and Economics, Kluwer Academic Publishers, Boston, MA, 2000.
  • L. Shampine, I. Gladwell and S. Thompson, Solving ODEs with MATLAB, Cambridge University Press, Cambridge, 2003.
  • J. Stoer and R. Bulirsch, Introduction to numerical analysis, 3rd ed., Texts in Applied Mathematics Vol. 12, Springer-Verlag, New York, 2002. MR 1923481 (2003d:65001).
  • L.N. Trefethen, Spectral Methods in MATLAB, SIAM, Philadelphia, 2000.
  • E. Trélat, Optimal control and applications to aerospace: some results and challenges, J. Optim. Theory. Appl. 154 (2012), pp. 713–758. doi: 10.1007/s10957-012-0050-5
  • S. Upreti, Optimal Control for Chemical Engineers, CRC Press, Boca Raton, 2016.
  • H. Wang and S. Xiang, On the convergence rates of Legendre approximation, Math. Comput. 81 (2012), pp. 861–877. doi: 10.1090/S0025-5718-2011-02549-4
  • X. Wang, H. Peng, S. Zhang, B. Chen and W. Zhong, A symplectic pseudospectral method for nonlinear optimal control problems with inequality constraints, ISA Trans. 68 (2017), pp. 335–352. doi: 10.1016/j.isatra.2017.02.018
  • X. Wang, H. Peng, S. Zhang, B. Chen and W. Zhong, A symplectic local pseudospectral method for solving nonlinear state-delayed optimal control problems with inequality constraints, Int. J. Robust Nonlin. Control 28 (2018), pp. 2097–2120. doi: 10.1002/rnc.4003

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.