Solving optimal control problems under interval uncertainty using robust model predictive control

Allahdadi, Mehdi; Soradi-Zeid, Samaneh; Torabi, Hamed; Hosseini, Elnaz

doi:10.22111/ijfs.2025.52170.9204

Document Type : Research Paper

Authors

¹ Mathematics Faculty, University of Sistan and Baluchestan, Zahedan, Iran

² Faculty of Industry and Mining (Khash), University of Sistan and Baluchestan, Zahedan, Iran

³ Faculty of Electrical and Computer Engineering, University of Sistan and Baluchestan, Zahedan, Iran

⁴ Faculty of Mathematics, University of Sistan and Baluchestan, Zahedan, Iran

https://doi.org/10.22111/ijfs.2025.52170.9204

Abstract

In this paper, as a novel control strategy that is implemented online, we use the model predictive control. In natural phenomena, some parameters may have uncertainty in which their exact amount is unknown, but we know that they are in a certain range. In this case, the predictive control formulation can be changed so that the control system becomes resistant to these parameters. In this article, two methods for solving optimal control problems with uncertainty will be proposed using model predictive control. In first one, we introduce the robust control, in which discretize the continuous time dynamic model and apply the predictive control algorithm of the resilient model to a typical system. In the second one, when the range of parameter has a certain uncertainty, it is appropriate that the solutions are in the form of interval values. So, we introduce the interval model predictive control and solve two sub-models for optimal control problems. According to the simulation results, it can be conclude that even though the dynamic of the system has severe uncertainty and the behavior of the system changes randomly between stable and unstable mode, but the predictive controller of the interval optimal control problems is well able to converge the state variables.

Keywords

Main Subjects

Optimization

References

[1] M. Athans, P. L. Falb, Optimal control: An introduction to the theory and its applications, New York, McGraw-Hill,
Inc, (1966). https://doi.org/10.1093/oso/9780198596752.001.0001

[2] M. Behrendt, A basic working principle of model predictive control, Wikipedia Commons, (2009). ISBN: 978-1-
80355-988-9

[3] J C. Butcher, A history of Runge-Kutta methods, Applied Numerical Mathematics, 20(3) (1996), 247-260. https:
//doi.org/10.1016/0168-9274(95)00108-5

[4] S. S. L. Chang, L. A. Zadeh, On fuzzy mappings and control, IEEE Transactions on Systems, Man, and Cybernetics,
1 (1972), 30-34. https://doi.org/10.1109/TSMC.1972.5408553

[5] B. Farhadinia, Necessary optimality conditions for fuzzy variational problems, Information Sciences, 181 (2011),
1348-1357. https://doi.org/10.1016/j.ins.2010.11.027

[6] B. Farhadinia, Pontryagin’s minimum principle for fuzzy optimal control problems, Iranian Journal of Fuzzy Systems,
2 (2014), 27-43. https://doi.org/10.22111/IJFS.2014.1489

[7] D. Filev, P. Angelov, Fuzzy optimal control, Fuzzy Sets and Systems, 47 (1992), 151-156. https://doi.org/10.
1016/0165-0114(92)90172-Z

[8] T. Geyer, Model predictive control of high power converters and industrial drives, John Wiley and Sons, (2016).
ISBN: 978-1-119-01090-6

[9] R. Goetsehel, W. Voxman, Elementary fuzzy calculus, Fuzzy Sets and Systems, 18(1) (1986), 13-43. https://doi.
org/10.1016/0165-0114(86)90026-6

[10] H. Guoqing, Y. Fengqi, Model predictive control for greenhouse condition adjustment and crop production prediction, Computer Aided Chemical Engineering, 51 (2022), 1051-1056. https://doi.org/10.1016/b978-0-323-95879-0.
50176-4

[11] D. Jardon, I. Sanchez, M. Sanchis, Fuzzy sets on uniform spaces, Iranian Journal of Fuzzy Systems, 20(6) (2023),
123-135. https://doi.org/10.22111/IJFS.2023.43774.7700

[12] O. Kaleva, Fuzzy differential equations, Fuzzy Sets and Systems, 24 (1987), 301-317. https://doi.org/10.1016/
0165-0114(87)90029-7

[13] G. Klir, B. Yuan, Fuzzy sets and fuzzy logic, intelligence, control and information, Prentice Hall Upper Saddle
River, NJ, (1999). ISBN 9780131011717

[14] B. Kouvaritakis, M. Cannon, Model predictive control, Switzerland: Springer International Publishing, 38(13-56)
(2016), 7. https://doi.org/10.1007/978-3-319-24853-0

[15] X. Liu, L. Qiu, J. Rodriguez, W. Wu, J. Ma, Z. Peng, Y. Fang, Data-driven neural predictors-based robust MPC
for power converters, IEEE Transactions on Power Electronics, 37(10) (2022), 11650-11661. https://doi.org/10.
1109/TPEL.2022.3171100

[16] X. Liu, L. Qiu, W. Wu, J. Ma, Y. Fang, Z. Peng, J. Rodr´ıguez, Event-triggered ESO-based robust MPC for power
converters, IEEE Trnsactions on Industrial Electronics, 70(2) (2023), 2144-2152. https://doi.org/10.1109/TIE.
2022.3167135

[17] X. Luo, T. Tang, X. Lu, J. Lv, H. Liu, Robust constraint satisfaction and stability of virtually coupled train set with
uncertain dynamics: A dual-mode robust mpc approach, Transportation Research Part C: Emerging Technologies,
156 (2023), 104356. https://doi.org/10.1016/j.trc.2023.104356

[18] J. A. M´endez, A. Marrero, J. A. Reboso, A. Le´on, Adaptive fuzzy predictive controller for anesthesia delivery,
Control Engineering Practice, 46 (2016), 1-9. https://doi.org/10.1016/j.conengprac.2015.09.009

[19] F. Micheli, T. Summers, J. Lygeros, Data-driven distributionally robust MPC for systems with uncertain dynamics,
In 2022 IEEE 61st Conference on Decision and Control (CDC), (2022), 4788-4793. https://doi.org/10.1109/
CDC51059.2022.9992469

[20] M. Najariyan, M. H. Farahi, Optimal control of fuzzy linear controled system with fuzzy initial conditions, Iranian
Journal of Fuzzy Systems, 10 (2013). https://doi.org/10.22111/IJFS.2013.807

[21] A. Pinto, Design and analysis of a mass-spring-damper system for active stabilization of a dynamic platform,
Mechanical Engineering for Sustainable Development: ISME-2024, 49 (2025), 308. https://doi.org/10.21741/
9781644903438-31

[22] L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, E. F. Mishchenko, The mathematical theory of optimal
processes, Translated from the Russian by Trirogoff KN, edited by Neustadt LW, Interscience Publishers John Wiley
and Sons, Inc New York, (1962). https://doi.org/10.1017/S0008439500032112

[23] M. Schwenzer, M. Ay, T. Bergs, D. Abel, Review on model predictive control: An engineering perspective, The
International Journal of Advanced Manufacturing Technology, 117(5) (2021), 1327-1349. https://doi.org/10.
1007/s00170-021-07682-3

[24] D. E. Seborg, T. F. Edgar, D. A. Mellichamp, F. J. Doyle III, Process dynamics and control, John Wiley and Sons,
(2016). ISBN 978-0470128671

[25] W. M. Tulczyjew, The Euler-Lagrange resolution in differential geometrical methods in mathematical physics,
Springer, Berlin, Heidelberg, (1980), 22-48. https://doi.org/10.1007/BFb0089725

[26] X. Wu, J. Shen, Y. Li, K. Y. Lee, Fuzzy modeling and predictive control of superheater steam temperature for power
plant, ISA Transactions, 56 (2015), 241-251. https://doi.org/10.1016/j.isatra.2014.11.018

[27] Z. Xie, S. Li, P. K.Wong, W. Li, J. Zhao, An improved gain-scheduling robust MPC for path following of autonomous
independent-drive electric vehicles with time-varying and uncertainties, Vehicle System Dynamics, 63(4) (2025),
695-721. https://doi.org/10.1080/00423114.2024.2351574

[28] J. Yen, R. Langari, Fuzzy logic, intelligence, control, and information, Prentice Hall Upper Saddle River, NJ,
(1999). ISBN: 978-0135258170

[29] L. A. Zade, Fuzzy sets, Informations and Control, 8 (1965), 338-353. https://doi.org/10.1016/S0019-9958(65)
90241-X

Solving optimal control problems under interval uncertainty using robust model predictive control

References

References

Volume 22, Issue 6November and December 2025Pages 147-165

Volume 22, Issue 6
November and December 2025
Pages 147-165