An efficient hp spectral collocation method for nonsmooth optimal control problems

Hedayati, Mehrnoosh ; Ahsani Tehrani, Hojjat ; Fakharzadeh Jahromi, Alireza ; Noori Skandari, Mohammad Hadi ; Baleanu, Dumitru

Kybernetika, Tome 58 (2022) no. 6, pp. 843-862

Abstract (VO)

One of the most challenging problems in the optimal control theory consists of solving the nonsmooth optimal control problems where several discontinuities may be present in the control variable and derivative of the state variable. Recently some extended spectral collocation methods have been introduced for solving such problems, and a matrix of differentiation is usually used to discretize and to approximate the derivative of the state variable in the particular collocation points. In such methods, there is typically no condition for the continuity of the state variable at the switching points. In this article, we propose an efficient hp spectral collocation method for the general form of nonsmooth optimal control problems based on the operational integration matrix. The time interval of the problem is first partitioned into several variable subintervals, and the problem is then discretized by considering the Legendre-Gauss-Lobatto collocation points. Here, the switching points are unknown parameters, and having solved the final discretized problem, we achieve some approximations for the optimal solutions and the switching points. We solve some comparative numerical test problems to support of the performance of the suggested approach.

Détail

DOI : 10.14736/kyb-2022-6-0843

Classification : 4, 9, 65M70, MSC
Keywords: nonsmooth optimal control; hp-method; Lagrange interpolating polynomials; Legendre-Gauss-Lobatto points

Bibliographie

[1] Agamawi, Y., Hager, W., Rao, A. V.: Mesh refinement method for optimal control problems with discontinuous control profiles. In: AIAA Guidance, Navigation, and Control Conference 2017, pp. 1506.

[2] Aly, G. M., Chan, W. C.: Application of a modified quasilinearization technique to totally singular optimal control problems. Int. J. Control 17 (1973), 809-815. | DOI

[3] Aronna, M. S., Bonnans, J. F., Martinon, P.: A shooting algorithm for optimal control problems with singular arcs. J. Optim. Theory Appl. 158 (2013), 419-459. | DOI | MR

[4] Betts, J. T.: Practical methods for optimal control using nonlinear programming, ser. In: Advances in Design and Control, SIAM Press, Philadelphia 2001, 3. | MR

[5] Betts, J. T., Huffman, W. P.: Mesh refinement in direct transcription methods for optimal control. J. Optimal Control Appl. Methods 19 (1998), 1-21. | DOI | MR

[6] Berkmann, P., Pesch, H. J.: Abort landing in windshear: optimal control problem with third-order state constraint and varied switching structure. J. Optim. Theory Appl.85 (1995), 21-57. | DOI | MR

[7] Canuto, C., Hussaini, M., Quarteroni, A., Zang, T. A.: Spectral Methods in Fluid Dynamics. Springer Series Comput. Physics, Springer, Berlin 1991. | MR

[8] Cuthreli, J. E., Biegler, L. T.: On the optimization of differential-algebraic processes. J. Amer. Inst. Chemical Engineers 33 (1987), 1257-1270. | DOI | MR

[9] Cuthrell, J. E., Biegler, L. T.: Simultaneous optimization and solution methods for batch reactor control profiles. J. Computers Chemical Engrg. 13 (1989), 49-62. | DOI

[10] Dadebo, S. A., McAuley, K. B.: On the computation of optimal singular controls. In: Proc. International Conference on Control Applications. IEEE (1995), pp. 150-155.

[11] Dadebo, S. A., McAuley, K. B., McLellan, P. J.: On the computation of optimal singular and bang-bang controls. J. Optim. Control Appl. Methods 19 (1998), 287-297. | DOI | MR

[12] Darby, C. L., Hager, W., Rao, A. V.: An hp-adaptive pseudospectral method for solving optimal control problems. J. Optimal Control Appl. Methods 32 (2011), 476-502. | DOI | MR

[13] Dolan, E., More, J. J., Munson, T. S.: Benchmarking optimization software with COPS 3.0, ANL/ MCS-273. ANL/MCS-TM-273. Argonne National Lab., Argonne, IL (US), 2004.

[14] Foroozandeh, Z., Shamsi, M., Azhmyakov, V., Shafiee, M.: A modified pseudospectral method for solving trajectory optimization problems with singular arc. Math. Methods Appl. Sci. 40 (2016), 1783-1793. | DOI | MR

[15] Fu, W., Lu, Q.: Multiobjective optimal control of FOPID controller for hydraulic turbine governing systems based on reinforced multiobjective harris hawks optimization coupling with hybrid strategies. Complexity (2020), 1-15. | DOI

[16] Gao, W., Jiang, Y., Jiang, Z. P., Chai, T.: Output-feedback adaptive optimal control of interconnected systems based on robust adaptive dynamic programming. J. Automatica 72 (2016), 37-45. | DOI | MR

[17] Goddard, R. H.: A method of reaching extreme altitudes. Nature 105 (1920), 809-811. | DOI

[18] Graichen, K., Petit, N.: Solving the Goddard problem with thrust and dynamic pressure constraints using saturation functions, Proceedings of the 17th World Congress. IFAC Proceed. 41 (2008), 14301-14306. | DOI

[19] Hager, W., Liu, J., Mohapatra, S., Rao, V., Wang, X-SH.: Convergence rate for a Gauss collocation method applied to constrained optimal control. SIAM J. Control Optim. 56 (2017), 1386-1411. | DOI | MR

[20] Henriques, J. C. C., Lemos, J. M., Gato, L. M C., Falcao, A. F. O.: A high-order discontinuous Galerkin method with mesh refinement for optimal control. J. Automatica 85 (2017), 70-82. | DOI | MR

[21] Hu, G. S., Ong, C. J., Teo, C. L.: Minimum-time control of a crane with simultaneous traverse and hoisting motions. J. Optim. Theory Appl. 120 (2004), 395-416. | DOI | MR

[22] Huang, H. P., McClamroch, N. H.: Time-optimal control for a robotic contour following problem. IEEE J, Robotics Automat. 4 (1998), 140-149. | DOI

[23] Jain, D., Tsiotras, P.: Trajectory optimization using multiresolution techniques. J. Guidance Control Dynamics 31 (2008), 1424-1436. | DOI

[24] Kim, J. H. R., Maurer, H., Astrov, Y. A., Bode, M., Purwins, H. G.: High-speed switch-on of a semiconductor gas discharge image converter using optimal control methods. J. Comput. Phys. 170 (2001), 395-414. | DOI

[25] Ledzewicz, U., Schattler, H.: Optimal bang-bang controls for a two-compartment model in cancer chemotherapy. J. Optim. Theory Appl. 114 (2002), 609-637. | DOI | MR

[26] Luus, R.: On the application of iterative dynamic programming to singular optimal control problems. IEEE Trans. Automat. Control 37 (1992), 1802-1806. | DOI | MR

[27] Martinon, P., Bonnans, F., Laurent-Varin, J., Trelat, E.: Numerical study of optimal trajectories with singular arcs for an Ariane 5 launcher. J. Guidance Control Dynamics 32 (2009), 51-55. | DOI

[28] Marzban, H. R., Hoseini, S. M.: A composite Chebyshev finite difference method for nonlinear optimal control problems. Commun. Nonlinear Sci. Numerical Simul- 18 (2012), 1347-1361. | DOI | MR | Zbl

[29] Maurer, H.: Numerical solution of singular control problems using multiple shooting techniques. J. Optim. Theory Appl. 18 (1973), 235-259. | DOI | MR

[30] Mehra, R. K., Davis, R. E.: A generalized gradient method for optimal control problems with inequalitv constraints and singular arcs. IEEE Trans. Automat. Control 17 (1972), 69-79. | DOI | MR

[31] Olsder, G. J.: On open- and closed-loop bang-bang control in nonzero-sum differential games. SIAM J. Control Optim. 40 (2001), 1087-1106. | DOI | MR

[32] Ross, I. M., Gong, Q., Kang, W.: A pseudospectral method for the optimal control of constrained feedback linearizable systems. Inst. Electrical and Electronic Engineers Transactions on Automatic Control 51 (2006), 1115-1129. | DOI | MR

[33] Savku, E., Weber, G. W.: A stochastic maximum principle for a markov regime-switching jump-diffusion model with delay and an application to finance. J. Optim. Theory Appl. 179 (2018), 696-721. | DOI | MR

[34] Seywald, H., Cliff, E. M.: Goddard problem in presence of a dynamic pressure limit. J. Guidance Control Dynamics 16 (1993), 776-781. | DOI

[35] Shamsi, M.: A modified pseudospectral scheme for accurate solution of Bang-Bang optimal control problems. J. Optim. Control Appl. Methods 32 (2011), 668-680. | DOI | MR

[36] Shen, J., Tang, T., Wang, L.: Spectral Methods Algorithms, Analysis and Applications. Springer Series in Computational Mathematics 2011. | MR

[37] Shirin, A., Klickstein, I. S., Feng, S., Lin, Y. T., Hlavacek, W. S., Sorrentino, F.: Prediction of optimal drug schedules for controlling autophagy. Nature 9 (2019), 1-15.

[38] Signori, A.: Vanishing parameter for an optimal control problem modeling tumor growth. Asymptotic Analysis 117 (2020), 4-66. | DOI | MR

[39] Skandari, M. H. N., Tohidi, E.: Numerical solution of a class of nonlinear optimal control problems using linearization and discretization. Appl. Math. 2 (2011), 646-652. | DOI | MR

[40] Speyer, J. L.: Periodic optimal flight. J. Guidance Control Dynamics 19 (1996), 745-755. | DOI

[41] Sun, D. Y.: The solution of singular optimal control problems using the modified line-up competition algorithm with region-relaxing strategy. ISA Trans. 49 (2010), 106-113. | DOI

[42] Tabrizidooz, H. R., Marzban, H. R., Pourbabaee, M., Hedayati, M.: A composite pseudospectral method for optimal control problems with piecewise smooth solutions. J. Franklin Inst. 35 (2017), 2393-2414. | DOI | MR

[43] Trelat, E.: Optimal control and applications to aerospace: some results and challenges. J. Optim. Theory Appl. 154 (2012), 713-758. | DOI | MR

[44] Tsiotras, P., Kelley, H. J.: Goddard problem with constrained time of flight. J. Guidance Control Dynamics 15 (1992), 289-296. | DOI

[45] Williams, P.: A Gauss-Lobatto quadrature method for solving optimal control problems. In: Proc. Seventh Biennial Engineering Mathematics and Applications Conference 2005, The Anzim Journal 47 (2006), C101-C115. | DOI | MR

[46] Zhang, X. Y.: Convergence analysis of the multistep Legendre pseudo-spectral method for Volterra integral equations with vanishing delays. J. Computat. Appl. Math. 321 (2017), 284-301. | DOI | MR

[47] Zhao, Y., Tsiotras, P.: A density-function based mesh refinement algorithm for solving optimal control problems. In: Infotech and Aerospace Conference 2009, 2009-2019.