An efficient pseudospectral method for solving a class of nonlinear optimal control problems. (English) Zbl 1291.49020

Tohidi, Emran; Pasban, Atena; Kilicman, A.; Lotfi Noghabi, S.

Abstr. Appl. Anal. 2013, Article ID 357931, 7 p. (2013).

Summary: This paper gives a robust pseudospectral scheme for solving a class of nonlinear optimal control problems (OCPs) governed by differential inclusions. The basic idea includes two major stages. At the first stage, we linearize the nonlinear dynamical system by an interesting technique which is called linear combination property of intervals. After this stage, the linearized dynamical system is transformed into a multi domain dynamical system via computational interval partitioning. Moreover, the integral form of this multidomain dynamical system is considered. Collocating these constraints at the Legendre Gauss Lobatto (LGL) points together with using the Legendre Gauss Lobatto quadrature rule for approximating the involved integrals enables us to transform the basic OCPs into the associated nonlinear programming problems (NLPs). In all parts of this procedure, the associated control and state functions are approximated by piecewise constants and piecewise polynomials, respectively. An illustrative example is provided for confirming the accuracy and applicability of the proposed idea.

MSC:

49M25

Discrete approximations in optimal control

References:

[1]	Betts, J. T., Survey of numerical methods for trajectory optimization, Journal of Guidance, Control, and Dynamics, 21, 2, 193-207 (1998) · Zbl 1158.49303
[2]	Tohidi, E.; Soleymani, F.; Kilicman, A., Robustness of operational matrices of differentiation for solving state-space analysis and optimal control problems, Abstract and Applied Analysis, 2013 (2013) · Zbl 1272.49066
[3]	Betts, J. T., Practical Methods for Optimal Control Using Nonlinear Programming. Practical Methods for Optimal Control Using Nonlinear Programming, Advances in Design and Control, 3 (2001), Philadelphia, Pa, USA: Society for Industrial and Applied Mathematics, Philadelphia, Pa, USA · Zbl 0995.49017
[4]	Elnagar, G.; Kazemi, M. A.; Razzaghi, M., The pseudospectral Legendre method for discretizing optimal control problems, IEEE Transactions on Automatic Control, 40, 10, 1793-1796 (1995) · Zbl 0863.49016
[5]	Fahroo, F.; Ross, I. M., Direct trajectory optimization by a Chebyshev pseudospectral method, Journal of Guidance, Control, and Dynamics, 25, 1, 160-166 (2002)
[6]	Tohidi, E.; Samadi, O. R. N.; Farahi, M. H., Legendre approximation for solving a class of nonlinear optimal control problems, Journal of Mathematical Finance, 1, 8-13 (2011)
[7]	Toutounian, F.; Tohidi, E.; Kilicman, A., Fourier operational matrices of differentiation and transmission: Introduction and Applications, Abstract and Applied Analysis, 2013 (2013) · Zbl 1275.65036
[8]	Tohidi, E.; Samadi, O. R. N., Optimal control of nonlinear Volterra integral equations via Legendre Polynomials, IMA Journal of Mathematical Control and Information, 30, 1, 67-83 · Zbl 1275.49056
[9]	Bryson,, A. E.; Ho, Y. C., Applied Optimal Control (1975), New York, NY, USA: Hemisphere, New York, NY, USA
[10]	Noori Skandari, M. H.; Tohidi, E., Numerical solution of a class of nonlinear optimal control problems using linearization and discretization, Applied Mathematics, 2, 5, 646-652 (2011)
[11]	Tohidi, E.; Noori Skandari, M. H., A new approach for a class of nonlinear optimal control problems using linear combination property of intervals, Journal of Computations and Modelling, 1, 145-156 (2011) · Zbl 1238.49047
[12]	Shamsi, M., A modified pseudospectral scheme for accurate solution of Bang-Bang optimal control problems, Optimal Control Applications and Methods, 32, 6, 668-680 (2011) · Zbl 1272.49065
[13]	von Stryk, O.; Bulrisch, R.; Miele, A.; Stoer, J., Numerical solution of optimal control problems by direct collocation, Optional Control of Variations, Optimal Control Theory and Numerical Methods. Optional Control of Variations, Optimal Control Theory and Numerical Methods, International Series of Numerical Mathematics, 129-143 (1993), Basel, Switzerland: Birkhäuser, Basel, Switzerland