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An efficient Legendre pseudospectral method for solving nonlinear quasi bang-bang optimal control problems. (English) Zbl 1277.65104

Summary: In this paper, an efficient Legendre pseudospectral approach for the accurate solution of nonlinear quasi bang-bang optimal control problems (OCPs) is investigated. In this approach, after linearizing the dynamical system, control and state functions are considered as piecewise constant and piecewise continuous polynomials, respectively, and the switching points are also taken as decision variables. Furthermore, for simplicity in discretization, a integral formulation of the dynamical equations is considered. Thereby, the problem is converted into a mathematical programming problem which can be solved by well-developed parameter optimization algorithms. Through a numerical implementation we show the efficiency of the proposed method via comparing with a classical pseudospectral method and other discretization approaches.

MSC:

65N35 Spectral, collocation and related methods for boundary value problems involving PDEs
49J15 Existence theories for optimal control problems involving ordinary differential equations
90C30 Nonlinear programming
93C05 Linear systems in control theory
49M25 Discrete approximations in optimal control
90C05 Linear programming

Software:

Matlab
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Full Text: DOI

References:

[1] K. P. Badakhshan and A. V. Kamyad, Numerical Solution of Nonlinear Optimal Control Problems Using Non-linear Programming, Appl. Math. Comput, 187 (2007), 1511-1519. · Zbl 1118.65067 · doi:10.1016/j.amc.2006.09.074
[2] R. E. Bellman, Dynamic Programming, Princeton University Press, Princeton, NJ, 1957. · Zbl 0077.13605
[3] J. T. Betts, Survey of numerical methods for trajectory optimization. Journal of Guidance, Control, and Dynamics, 21 (1998) 193-207. · Zbl 1158.49303 · doi:10.2514/2.4231
[4] J. T. Betts, Practical methods for optimal control using nonlinear programming, Advances in Design and Control 3, SIAM, Philadelphia, PA, 2001. · Zbl 0995.49017
[5] A. E. Bryson, and Y. C. Ho, Applied optimal Control, Hemisphere, New York, 1975.
[6] C. Canuto, M. Hussaini, A. Quarteroni, T. A. Zang, Spectral Methods in Fluid Dynamics. Springer Series in Computational Physics. Springer: Berlin, 1991. · Zbl 0717.76004
[7] M. Dehghan, M. Shamsi, Numerical solution of two-dimensional parabolic equation subject to nonstandard boundary specifications using the pseudospectral Legendre method. Numerical Methods for Partial Differential Equations, 22(2006), 1255-266. · Zbl 1108.65101 · doi:10.1002/num.20150
[8] G. Elnagar, M. A. Kazemi, M. Razzaghi, The pseudospectral Legendre method for discretiz- ing optimal control problems. IEEE Transactions on Automatic Control, 40 (1995), 1793-1796. · Zbl 0863.49016 · doi:10.1109/9.467672
[9] F. Fahroo, I. M. Ross, Direct trajectory optimization by a Chebyshev pseudospectral method, J. Guidance. Control. Dynamics, 25(2002) 160-166.
[10] R. Fletcher, Practical Methods of Optimization, Wiley, Chichester, 1987. · Zbl 0905.65002
[11] B. Fornberg, A Practical Guide to Pseudospectral Methods. Cambridge University Press: Cambridge, 1998. · Zbl 0912.65091
[12] W. Kang, Q. Gong, I. M. Ross, Convergence of pseudospectral methods for nonlinear optimal control problems with discontinuous controller, 44th IEEE Conference on Decision and Control and European Control Conference (CDC-ECC’05), Sevil, Spain, 2005, 2799-2804.
[13] H. Ma, T. Qin, and W. zhang, An efficient Chebyshev algorithm for the solution of optimal control problems, IEEE trans. Autom. Control, 56 (2011) 675-680. · Zbl 1368.49036
[14] J. Nocedal, S. J. Wright, Numerical Optimization, Springer Series in Operations Research. Springer, New York, NY, 1999.
[15] M. H. Noori Skandari, E. Tohidi, Numerical solution of a class of nonlinear optimal control problems using linearization and discretization, Applied Mathematics, 2 (2011), 271-277.
[16] M, Razzaghi, G. Elnagar, Numerical solution of the controlled Duffing oscillator by the pseudospectral method, J. Comput. Appl. Math, 56 (1994) 253-261. · Zbl 0827.65074 · doi:10.1016/0377-0427(94)90081-7
[17] M, Razzaghi, S. Yousefi, Legendre wavelets method for constrained optimal control prob- lems, Math. Meth. Appl. Sci, 25 (2002), 529-539. · Zbl 1001.49033 · doi:10.1002/mma.299
[18] I. M. Ross, J. Rea, F. Fahroo, Exploiting higher-order derivatives in computational optimal control. Proceedings of the 2002 IEEE Mediterranean Conference, Lisbon, Portugal, July 2002.
[19] M. Shamsi, M. Dehghan, Recovering a time-dependent coefficient in a parabolic equa- tion from overspeciffied boundary data using the pseudospectral Legendre method, Numerical Methods for Partial Differential Equations, 23 (2007), 196-210. · Zbl 1107.65085 · doi:10.1002/num.20174
[20] M. Shamsi, A modified pseudospectral scheme for accurate solution of Bang-Bang optimal control problems, Optimal, Control, Appl, Methods, DOI: 10.1002/oca.967. · Zbl 1272.49065 · doi:10.1002/oca.967
[21] O. V. Stryk, Numerical Solution of Optimal Control Problems by Direct Collocation, In: R. Bulrisch, A. Miele, J. Stoer and K. H. Well, Eds., Optional Control of Variations, Opti- mal Control Theory and Numerical Methods, International Series of Numerical Mathematics, Birkhuser Verlag, Basel, 1993, 129-143. · Zbl 0790.49024
[22] E. Tohidi, O. R. N. Samadi, M. H. Farahi, Legendre approximation for solving a class of nonlinear optimal control problems, Journal of Mathematical Finance, 1 (2011), 8-13.
[23] E. Tohidi, M. H. Noori Skandari, A New Approach for a Class of Nonlinear Optimal Control Problems Using Linear Combination Property of Intervals, Journal of Computations and Modelling, 1 (2011), 145-156. · Zbl 1238.49047
[24] E. Tohidi, O. R. N. Samadi, Optimal control of nonlinear Volterra integral equations via Legendre Polynomials, IMA Journal of Mathematical Control and Information, DOI: 10.1093/imamci/DNS014. · Zbl 1275.49056 · doi:10.1093/imamci/dns014
[25] L. N. Trefethen, Spectral methods in Matlab. Software-Environments-Tools 10. IAM, So- ciety for Industrial and Applied Mathematics: Philadelphia, PA, 2000.
[26] V. Yen, M. Nagurka, Linear quadratic optimal control via Fourier-based state parameteri- zation. Journal of Dynamic Systems, Measurement and Control, 113 (1991), 206-215. · Zbl 0765.49022 · doi:10.1115/1.2896367
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