An efficient design for solving discrete optimal control problem with time-varying multi-delays

Document Type : Research Article

Authors

• S.M. Abdolkhaleghzade
• S. Effati
• S.A. Rakhshan

Department of Applied Mathematics, Faculty of Mathematical Sciences, Ferdowsi University of Mashhad, Mashhad, Iran.

10.22067/ijnao.2022.78220.1168

Abstract

The focus of this article is on the study of discrete optimal control problems (DOCPs) governed by time-varying systems, including time-varying delays in control and state variables. DOCPs arise naturally in many multi-stage control and inventory problems where time enters discretely in a natural fashion. Here, the Euler--Lagrange formulation (which are two-point boundary values with time-varying multi-delays) is employed as an efficient technique to solve DOCPs with time-varying multi-delays. The main feature of the procedure is converting the complex version of the discrete-time optimal control problem into a simple form of differential equations. Since the main problem is in discrete form, then the Euler--Lagrange equation changes to an algebraic system with initial and final conditions. The graphic representation of numerical simulation results shows that the proposed method can effectively and reliably solve DOCPs with time-varying multi-delays.

Keywords

• Discrete-time optimal control problem with time-varying delay
• Euler–Lagrange equations
• Pontryagin maximum principle

Main Subjects

• Optimal Control

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References

1. Al-Tamimi, A., Lewis, F.L., and Abu-Khalaf, M. Discrete-time nonlinear HJB solution using approximate dynamic programming: Convergence proof. IEEE Trans. Syst. Man Cybern. Part B (Cybernetics), 38(4), (2008) 943–949.
2. Essounaini, A., Labzai, A., Laarabi, H., and Rachik, M. Mathematical modeling and optimal control strategy for a discrete time model of COVID-19 variants. Commun. Math. Biol. Neurosci. (2022) Article-ID 2022.
3. Ferreira, R.A. Discrete Calculus. In Discrete Fractional Calculus and Fractional Difference Equations (pp. 1–14). Springer, Cham, 2022.
4. Fisher, M.E., and Jennings, L.S. Discrete-time optimal control problems with general constraints. ACM Trans. Math. Softw. (TOMS), 18(4), (1992) 401–413.
5. Haddad, W.M., Lee, J., and Bhat, S.P. Asymptotic and finite time semistability for nonlinear discrete-time systems with application to network consensus. IEEE Trans. Automat. Contr. (2022).
6. Hale, J. K., and Lunel, S.V. Stability and control of feedback systems with time delays. Int. J. Syst. Sci. 34(8-9), (2003) 497–504.
7. Hasegawa, Y. Control problems of discrete-time dynamical systems (Vol. 447) Springer, 2013.
8. Kot, M. A first course in the calculus of variations (Vol. 72). American Mathematical Society, 2014.
9. Lewis, F. L., Vrabie, D., and Syrmos, V.L. Optimal control. John Wiley and Sons, 2012.

10. Li, B., Teo, K. L., and Duan, G.R. Optimal control computation for discrete time time-delayed optimal control problem with all-time-step inequality constraints. Int. J. Innov. Comput. Inf. Control. 6(3), (2010) 521–532.
11. Li, X., Wang, R., Du, S., and Li, T. An improved exponential stability analysis method for discrete-time systems with a time-varying delay. Int. J. Robust Nonlinear Control. 32(2), (2022) 669–681.
12. Lu, J., Wei, Q., Wang, Z., Zhou, T., and Wang, F.Y. Event-triggered optimal control for discrete-time multi-player non-zero-sum games using parallel control. Inf. Sci. 584, (2022) 519–535.
13. Mariconda, C., and Tonolo, A. Discrete calculus. Methods for counting Springer, 2016.
14. Mehraeen, S., Dierks, T., Jagannathan, S., and Crow, M.L. Zero-sum two-player game theoretic formulation of affine nonlinear discrete-time systems using neural networks. IEEE Trans. Cybern. 43(6), (2012) 1641–1655.
15. Miller, F.P., Vandome, A.F., and McBrewster, J. Gâteaux Derivative: Directional Derivative, Differential Calculus, World War I, Locally Con-vex Topological Vector Space, Topological Vector Space, Banach Space, Fréchet Derivative, Functional Derivative, Alphascript Publishing, 2010.
16. Naz, R. A current-value Hamiltonian approach to discrete-time optimal control problems in economic growth theory. J. Differ. Equ. Appl. (2022) 1–11.
17. Park, J. H., Lee, T.H., Liu, Y., and Chen, J. Dynamic systems with time delays: Stability and control. Singapore: Springer, 2019.
18. Rovelli, C., and Zatloukal, V. Natural discrete differential calculus in physics. Found. Phys. 49(7), (2019) 693–699.
19. Tang, G., Sun, H., and Pang, H. Approximately optimal tracking control for discrete time-delay systems with disturbances. Prog. Nat. Sci. 18(2), (2008) 225–231.
20. Wei, Q., Liu, D., and Lin, H. Value iteration adaptive dynamic programming for optimal control of discrete-time nonlinear systems. IEEE Trans. Cybern. 46(3), (2015) 840–853.
21. Xu, J., Wang, J., Rao, J., Zhong, Y., and Wang, H. Adaptive dynamic programming for optimal control of discrete-time nonlinear system with state constraints based on control barrier function. Int. J. Robust Nonlinear Control. 32(6), (2022) 3408–3424.

22. Zanma, T., Yamamoto, N., Koiwa, K. and Liu, K.Z. Optimal control input for discrete‐time networked control systems with data dropout. IET Cyber-Phys. Syst.: Theory Appl. 2022.
23. Zhao, H., Chen, D., and Hou, L. The optimal control of delay discretetime linear system with control constraint. IEEE ICCA 2010 (pp. 1699–1704). IEEE.