N.M. Dmitruk. Multiply closed control strategy in a linear terminal problem of optimal guaranteed control ... P. 66-82

This paper deals with an optimal control problem for a linear discrete system with disturbances. It is required to steer the system robustly to a given terminal set in a finite time while minimizing the guaranteed value of a terminal cost function. A multiply closed control strategy is introduced; it takes into account the assumption that, at several future times, the state of the system will be measured exactly and the control input will be corrected. An efficient numerical method for constructing a suboptimal multiply closed strategy is proposed. The results of numerical experiments show an improvement in the performance under the optimal control strategy when the number of closing instants increases as well as in comparison to the optimal open-loop worst-case control while maintaining comparable computation times.

Keywords: linear system, disturbances, robust optimal control, control strategy, algorithm

Received June 1, 2022

Revised July 15, 2022

Accepted July 18, 2022

Funding Agency: This work was supported by the National Program for Scientific Research of the Republic of Belarus “Convergence 2025” (project no.

Natalia Mikhailovna Dmitruk, Cand. Sci. ( Phys.-Math.), Belarusian State University, Minsk, 220030 Belarus, e-mail: dmitrukn@bsu.by


1.   Witsenhausen H. A minimax control problem for sampled linear systems. IEEE Transactions on Automatic Control, 1968, vol. 13, no. 1, pp. 5–21. doi: 10.1109/TAC.1968.1098788 

2.   Kurzhanskii A.B. Upravlenie i nablyudenie v usloviyakh neopredelennosti [Control and observation under conditions of uncertainty]. Moscow: Nauka Publ., 1977, 392 p.

3.   Krasovskii N.N. Upravlenie dinamicheskoi sistemoi [Control of a dynamical system]. Moscow: Nauka Publ., 1985, 520 p.

4.   Lee J.H., Yu Z. Worst-case formulations of model predictive control for systems with bounded parameters. Automatica, 1997, vol. 33, no. 5, pp. 763–781. doi: 10.1016/S0005-1098(96)00255-5 

5.   Bemporad A., Borrelli F., Morari M. Min-max control of constrained uncertain discrete-time linear systems. IEEE Transactions on Automatic Control, 2003, vol. 48, no. 9, pp. 1600–1606. doi: 10.1109/TAC.2003.816984 

6.   Goulart P.J., Kerrigan E.C., Maciejowski J.M. Optimization over state feedback policies for robust control with constraints. Automatica, 2006, vol. 42, no. 4, pp. 523–533. doi: 10.1016/j.automatica.2005.08.023 

7.   Gabasov R., Kirillova F.M., Kostina E.A. Closed state feedback for optimization of uncertain control systems. Part 1. Single loop. Autom. Remote Control, 1996, vol. 57, no. 7, pp. 1008–1015.

8.   Gabasov R., Kirillova F.M., Kostina E.A. Closed-loop state feedback for optimization of uncertain control systems. II: Multiply closed feedback. Autom. Remote Control, 1996, vol. 57, no. 8, pp. 1137–1145.

9.   Balashevich N.V., Gabasov R., Kirillova F.M. The construction of optimal feedback from mathematical models with uncertainty. Comput. Math. Math. Phys., 2004, vol. 44, no. 2, pp. 247–267.

10.   Kostyukova O., Kostina E. Robust optimal feedback for terminal linear-quadratic control problems under disturbances. Mathematical Programming, 2006, vol. 107, no. 1, pp. 131–153. doi: 10.1007/s10107-005-0682-4 

11.   Kostina E., Kostyukova O. Worst-case control policies for (terminal) linear-quadratic control problems under disturbances. Int. J. Robust Nonlinear Control, 2009, vol. 19, no. 17, pp. 1940–1958. doi: 10.1002/rnc.1417 

12.   Chong K., Kostyukova O., Kurdina M. Guaranteed control policy with arbitrary set of correction points for linear-quadratic system with delay. Control and Cybernetics, 2010, vol. 39, no. 3, pp. 739–768.

13.   Dmitruk N.M. Optimal strategy with one closing instant for a linear optimal guaranteed control problem. Comput. Math. Math. Phys., 2018, vol. 58, no. 5, pp. 642–658. doi: 10.1134/S096554251805007X 

14.   Kastsiukevich D.A., Dmitruk N.M. A method for constructing an optimal control strategy in a linear terminal problem. Journal of the Belarusian State University. Mathematics and Informatics, 2021, no. 2, pp. 38–50. doi: 10.33581/2520-6508-2021-2-38-50 

15.   Boyd S., Vandenberghe L. Convex optimization. Cambridge: Cambridge Univ. Press, 2004, 716 p. ISBN: 9780521833783 .

16.   Gal T. Postoptimal analyses, parametric programming, and related topics. Berlin; NY: De Gruyter, 1994, 437 p. doi: 10.1515/9783110871203 .

17.   Borrelli F. Constrained optimal control for hybrid systems. Berlin; Heidelberg: Springer, 2003. 206 p.

Cite this article as: N.M. Dmitruk. Multiply closed control strategy in a linear terminal problem of optimal guaranteed control. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2022, vol. 28, no. 3, pp. 66–82; Proceedings of the Steklov Institute of Mathematics (Suppl.), 2022, Vol. 319, Suppl. 1, pp. S112–S128.