Article
Full entry |
PDF
(1.3 MB)
Feedback
PDF
(1.3 MB)
Feedback
Keywords:
Least squares support vector machines; Optimal control problems; Legendre orthogonal polynomials; Regression; Artificial intelligence
Least squares support vector machines; Optimal control problems; Legendre orthogonal polynomials; Regression; Artificial intelligence
Summary:
In this paper, a new application of the Least Squares Support Vector Regression (LS-SVR) with Legendre basis functions as mapping functions to a higher dimensional future space is considered for solving optimal control problems. At the final stage of LS-SVR, an optimization problem is formulated and solved using Maple optimization packages. The accuracy of the method are illustrated through numerical examples, including nonlinear optimal control problems. The results demonstrate that the proposed method is capable of solving optimal control problems with high accuracy.
References:
[1] Amman, H. M., Kendrick, D. A.: Computing the 58 (1998), 2, 185-191. DOI
[2] Betts, J. T.: Survey of numerical methods for trajectory optimization. J. Guidance Control Dynamics (1998), 98124-2207. Zbl 1158.49303
[3] Bock, H. G., Plitt, K. J.: A multiple shooting algorithm for direct solution of optimal control problems. In: IFAC 9th congress 1984, pp. 1603-1608. DOI
[4] Cortes, C., Vapnik, V.: Support-vector networks. Mach. Learn 20 (1995), 3, 273-297. DOI
[5] Elnagar, G. N., Razzaghi, M.: A collocation-type method for linear quadratic optimal control problems. Opt. Control Appl. Methods 8 (1997), 3, 227-235. DOI 10.1002/(SICI)1099-1514(199705/06)18:3<227::AID-OCA598>3.0.CO;2-A | MR 1456947
[6] Kafash, B., Delavarkhalafi, A., Karbassi, S. M.: Application of Chebyshev polynomials to derive efficient algorithms for the solution of optimal control problems. Scientia Iranica 19 (2012), 3, 795-805. DOI
[7] Kang, S., Wang, J., Li, C., Shan, J.: Nonlinear optimal control with disturbance rejection for asteroid landing. J. Franklin Inst. 355 (2018), 16, 8027-8048. DOI | MR 3864034
[8] Kirk, D. E.: Optimal Control Theory. Prentice-Hall, Englewood Cliffs 1970.
[9] Latifi, S., Parand, K., Delkhosh, M.: Generalized Lagrange-Jacobi-Gauss-Radau collocation method for solving a nonlinear optimal control problem with the classical diffusion equation. European Phys. J. Plus, 2020.
[10] Loxton, R. C., Teo, K. L., Rehbock, V., Ling, W. K.: Optimal switching instants for a switched-capacitor DC/DC power converter. Automatica J. IFAC 45 (2009), 4, 973-980. DOI | MR 2535357
[11] Marzban, H. R., Razzaghi, M.: Optimal control of linear systems via hybrid of block-pulse functions and Legendre polynomials. J. Franklin Inst. 341 (2004), 279-293. DOI | MR 2054477
[12] Mohammadi, K. M., Jani, M., Parand, K.: A least squares support vector regression for anisotropic diffusion filtering. arXiv preprint arXiv, 2022.
[13] Naevdal, E.: Solving continuous-time optimal-control problems with a spreadsheet. J. Econom. Educ. 34 (2003), 2, 99-122. DOI
[14] Naraigh, L. O., Byrne, A.: Piecewise-constant optimal control strategies for controlling the outbreak of COVID-19 in the Irish population. Math. Biosci 330 (2020), 108496. DOI | MR 4167193
[15] Parand, K., Latifi, S., Delkhosh, M., Moayeri, M. M.: Generalized Lagrangian Jacobi-Gauss-Radau collocation method for solving a nonlinear 2-D optimal control problem with the classical diffusion equation. arXiv preprint arXiv, 2018.
[16] Parand, K., Razzaghi, M., Sahleh, R., Jani, M.: Least squares support vector regression for solving Volterra integral equations.
[17] Pakniyat, A., Parand, K., Jani, M.: Least squares support vector regression for differential equations on unbounded domains. Chaos Solitons Fractals, 2021. MR 4299700
[18] Pontryagin, L. S., Boltyanskii, V., Gamkrelidze, R., Mischenko, E.: The Mathematical Theory of Optimal Processes. Wiley Interscience, 1962. MR 0166037
[19] Rabiei, K., Parand, K.: Collocation method to solve inequality-constrained optimal control problems of arbitrary order. Engrg. Comput. 36 (2020), 1, 115-125. DOI 10.1007/s00366-018-0688-1
[20] Razzaghi, M., Nazarzadeh, J., Nikravesh, K. Y.: A collocation method for optimal control of linear systems with inequality constraints. Math.Problems Engrg. 3 (1998), 6, 503-515. DOI 10.1155/S1024123X97000653
[21] Rifkin, R., Yeo, G., Poggio, T.: Regularized least-squares classification. Nato Sci. Ser. Sub Ser. III Comput. Syst. Sci. 190 (2003), 131-154.
[22] Schwartz, A.: Theory and Implementation of Numerical Mathod Based on Runge-Kutta Integration for Solving Optimal Control Problems. Ph.D. Thesis, University of California, 1996. MR 1395832
[23] Shen, J., Tang, T., Wang, L-L.: Spectral Methods: Algorithms, Analysis and Applications. Springer Science and Business Media, 2011. MR 2867779
[24] Suykens, J. A., Vandewalle, J.: Least squares support vector machine classifiers. Neural Process. Lett. 9 (1999), 3, 293-300. DOI
[25] Suykens, J. A. K., Gestel, T. Van, Brabanter, J. De, Moor, B. De, Vandewalle, J.: Least Squares Support Vector Machines. World Scientific, 2002.
[26] Teo, K. L., Goh, C. J., Wong, K. H.: Unified Computational Approach to Optimal Control Problems. Longmann Scientific and Technical, 1991. MR 1153024
[27] Sabermahani, S., Ordokhani, Y., Rabiei, K., Razzaghi, M.: Solution of optimal control problems governed by Volterra integral and fractional integro-differential equations. J. Vibration Control 29 (2023), 15.-16, 3796-3808. DOI | MR 4617773
[28] Rabiei, K., Razzaghi, M.: An approach to solve fractional optimal control problems via fractional-order Boubaker wavelets. J. Vibration Control 29 (2023), 7-8, 1806-1819. DOI | MR 4560887
[29] Heydari, M. H., Razzaghi, M., Avazzadeh, Z.: Orthonormal piecewise Bernoulli functions: Application for optimal control problems generated. Vibration Control 29 (2023), 5-6, 1164-1175. DOI | MR 4548331
[30] Heydari, M. H., Tavakoli, R., Razzagh, M.: Application of the extended Chebyshev cardinal wavelets in solving fractional optimal control problems with ABC fractional derivative. Int. J. Systems Sci. 53 (2022), 12, 2694-2708. DOI | MR 4496198
[31] Ghanbari, G., Razzaghi, M.: Numerical solutions for fractional optimal control problems by using generalised fractional-order Chebyshev wavelets. Int. J. Systems Sci. 53 (2022), 4, 778-792. DOI | MR 4385669
[32] Heydari, M. H., Razzaghi, M.: A new class of orthonormal basis functions: application for fractional optimal control problems. Int. J. Systems Sci. 53 (2022), 2, 240-252. DOI 10.1080/00207721.2021.1947411 | MR 4369004
[33] Lakestani, M., Edrisi-Tabrizi, Y., Razzaghi, M.: Study of B-spline collocation method for solving fractional optimal control problems. Trans. Inst. Measurement Control 43 (2021), 11, 2425-2437. DOI
[34] Dehestani, H., Ordokhani, Y., Razzaghi, M.: Fractional-order Bessel wavelet functions for solving variable order fractional optimal control problems with estimation error. Int. J. Systems Sci. 51 (2020), 6, 1032-1052. DOI | MR 4095675
[35] Mashayekhi, S., Razzaghi, M.: An approximate method for solving fractional optimal control problems by hybrid functions. J. Vibration Control 24 (2018), 9, 1621-1631. DOI 10.1177/1077546316665956 | MR 3785609
[36] Mohammed, J. K., Khudair, A. R.: A novel numerical method for solving optimal control problems using fourth-degree hat functions. J. Partial Differential Equations Appl. Math. 7 (2023), 9, 100507. DOI
[37] Hassani, H., Machado, J. A. Tenreiro, Avazzadeh, Z., Naraghirad, E., Dahaghin, M. Sh.: Generalized Bernoulli polynomials: Solving nonlinear 2D fractional optimal control problems. J. Scient. Comput. 83 (2020), 2, 1-21. DOI 10.1007/s10915-020-01213-0 | MR 4091557
[38] Suykens, J. A. K., Vandewalle, J., Moor, B. D.: Optimal control by least squares support vector machines. J. Partial Differential Equations Appl. Math. 14 (2001), 1, 23-35.
Search
Partner of
