Document Type : Research Article
Author
Laboratoire de Conception et Conduite des Systèmes de Production, Université Mouloud Mammeri, 15 000 Tizi-Ouzou, Algérie
10.30473/coam.2026.76729.1376
Abstract
In this paper, a systematic design approach for the parametric optimization of uncertain dynamical systems with bounded parameter uncertainties is proposed. The methodology proceeds in two stages. In the first stage, the control input is expressed as a finite linear combination of polynomial basis functions, and an approximate analytical solution of the state equation is derived using the Variational Iteration Method, which is a semi-analytical iterative technique that avoids discretization and linearization. Substituting this approximate trajectory into the performance index yields a robust min–max optimization problem parameterized by the control coefficients. In the second stage, the robust optimization problem is converted into a tractable scenario optimization problem by drawing a finite number of independent and identically distributed samples from the uncertainty set. The resulting problem is solved using a genetic algorithm. The effectiveness of the proposed approach is demonstrated through two application examples. The first example concerns an uncertain linear-quadratic regulator, and the second addresses an uncertain nonlinear optimal control problem. Optimal control and state trajectories are provided for a set of samples. In addition, the optimal value of the performance index is reported, showing that this value does not exceed the threshold imposed by the proposed approach. The paper concludes by discussing limitations, including dependence on the accuracy of the Variational Iteration Method approximation and the assumption of a known probability distribution over the uncertainty set, and identifies directions for future research.
Highlights
- A two-stage parametric optimization framework is proposed for the optimal control of uncertain dynamical systems with bounded parameter uncertainties.
- The control input is represented as a finite linear combination of polynomial basis functions to enable a tractable parametric formulation.
- The Variational Iteration Method is employed to derive an approximate analytical solution of the state equation, avoiding discretization and linearization.
- Substituting the approximate trajectory into the performance index yields a robust min–max optimization problem parameterized by the control coefficients.
- The robust problem is converted into a scenario optimization problem via finite i.i.d. sampling from the uncertainty set, ensuring probabilistic feasibility guarantees.
- The resulting scenario optimization problem is solved using a genetic algorithm.
- The approach is validated on two application examples: an uncertain linear-quadratic regulator and an uncertain nonlinear optimal control problem.
- The framework is extensible to fractional-order systems, multi-state multi-input systems, and constrained optimal control problems.
Keywords
- Optimal control
- Parametric optimization
- Uncertain system
- Variational iteration method
- Scenario approach
Main Subjects
[1] Akkouche, A., Maidi, A., Aidene, M. (2014). “Optimal control of partial differential equations based on the Variational Iteration Method”. Computers & Mathematics with Applications, 68(5), 622–631. https://doi.org/10.1016/j.camwa.2014.07.007
[2] Allahdadi, M., Soradi-Zeid, S., Torabi, H., Hosseini, E. (2025). “Solving optimal control problems under interval uncertainty using robust model predictive control”. Iranian Journal of Fuzzy Systems, 22(6), 147–165. https://doi.org/10.22111/ijfs.2025. 52170.9204
[3] Berkani, S., Manseur, F., Maidi, A. (2012). “Optimal control based on the variational iteration method”. Computers & Mathematics with Applications, 64(4), 604–610. https: //doi.org/10.1016/j.camwa.2011.12.066
[4] Betts, J. T. (1998). “Survey of numerical methods for trajectory optimization”. Journal of Guidance, Control, and Dynamics, 21(2), 193–207. https://doi.org/10.2514/2.
[5] Calafiore, G. C., Campi, M. C. (2006). “The scenario approach to robust control design”. IEEE Transactions on Automatic Control, 51(5), 742–753. https://doi.org/10.1109/ TAC.2006.8750411
[6] Campi, M. C., Garatti, S., Prandini, M. (2009). “The scenario approach for systems and control design”. Annual Reviews in Control, 33(2), 149–157. https://doi.org/10.1016/j. arcontrol.2009.07.001
[7] Campi, M. C., Garatti, S. (2018). Introduction to the Scenario Approach. SIAM, Philadelphia. https://doi.org/10.1137/1.9781611975444
[8] Chakraverty, S., Mahato, N. R., Karunakar, P., Rao, T. D. (2019). Advanced Numerical and Semi-Analytical Methods for Differential Equations. John Wiley & Sons, New Jersey. https://doi.org/10.1002/9781119423461
[9] Choi, J., Deo, A., Lagoa, C., Subramanyam, A. (2024). “Reduced sample complexity in scenario-based control system design via constraint scaling”. IEEE Control Systems Letters, 8, 2793–2798. https://doi.org/10.1109/LCSYS.2024.3515861
[10] Conway, B. A. (2012). “A survey of methods available for the numerical optimization of continuous dynamic systems”. Journal of Optimization Theory and Applications, 152(2), 271–306. https://doi.org/10.1007/s10957-011-9918-z
[11] Corriou, J.-P. (2018). Process Control: Theory and Applications. Springer, London. https://doi.org/10.1007/978-3-319-61143-3
[12] Wu, D., Chen, Y., Yu, C., Bai, Y., Teo, K. L. (2023). “Control parameterization approach to time-delay optimal control problems: A survey”. Journal of Industrial and Management Optimization, 19(5), 3750–3783. https://doi.org/10.3934/jimo.2022108
[13] Fikar, M., Latifi, M. A., Corriou, J. P., Creff, Y. (2000). “CVP-based optimal control of an industrial depropanizer column”. Computers & Chemical Engineering, 24(2), 909–915. https://doi.org/10.1016/S0098-1354(00)00355-0
[14] Goh, C. J., Teo, K. L. (1988). “Control parametrization: A unified approach to optimal control problems with general constraints”. Automatica, 24(1), 3–18. https://doi.org/ 10.1016/0005-1098(88)90003-9
[15] Goldberg, D. E. (1989). Genetic Algorithms in Search, Optimization, and Machine Learning. Addison-Wesley, Reading, MA.
[16] Hamdipoor, V., Nguyen, H. N., Mekhaldi, B., Parra, J., Badosa, J., Obaldia, F. C. (2025). “Experimental validation of scenario-based stochastic model predictive control of nanogrids”. Control Engineering Practice, 157, 106249. https://doi.org/10.1016/j. conengprac.2025.106249
[17] Hatamian, R., Hashemi, S. A. S. (2025). “A hybrid numerical approach for solving nonlinear optimal control problems”. Control and Optimization in Applied Mathematics, 10(1), 125–138. https://doi.org/10.30473/coam.2025.72875.1273
[18] He, J.-H. (1998). “Approximate analytical solution for seepage flow with fractional derivatives in porous media”. Computer Methods in Applied Mechanics and Engineering, 167(1), 57–68. https://doi.org/10.1016/S0045-7825(98)00108-X
[19] He, J.-H., Wu, X.-H. (2007). “Variational iteration method: New development and applications”. Computers & Mathematics with Applications, 54(7), 881–894. https://doi. org/10.1016/j.camwa.2006.12.083
[20] Idiri, G., Djennoune, S., Bettayeb, M. (2016). “Solving fixed final time fractional optimal control problems using the parametric optimization method”. Asian Journal of Control, 18(4), 1524–1536. https://doi.org/10.1002/asjc.1247
[21] Kustov, A. Y. (2023). “Parametrization of optimal anisotropic controllers”. Automation and Remote Control, 10, 59–71. https://doi.org/10.1134/S0005117923100077
[22] Lauer, F. (2024). “Margin-based scenario approach to robust optimization in high dimension”. IEEE Transactions on Automatic Control, 69(10), 7182–7189. https://doi.org/ 10.1109/TAC.2024.3393790
[23] Li, R., Teo, K. L., Wong, K. H., Duan, G. R. (2006). “Control parameterization enhancing transform for optimal control of switched system”. Mathematical and Computer Modelling, 43(11), 1393–1403. https://doi.org/10.1016/j.mcm.2005.08.012
[24] Lichtenecker, D., Rixen, D., Eichmeir, P., Nachbagauer, K. (2023). “On the use of adjoint gradients for time-optimal control problems regarding a discrete control parameterization”. Multibody System Dynamics, 59(3), 313–334. https://doi.org/10.1007/ s11044-023-09898-5
[25] Lin, Q., Loxton, R., Teo, K. (2014). “The control parameterization method for nonlinear optimal control: A survey”. Journal of Industrial and Management Optimization, 10(1), 275–309. https://doi.org/10.3934/jimo.2014.10.275
[26] Puschke, J., Djelassi, H., Kleinekorte, J., Hannemann-Tamás, R., Mitsos, A. (2018). “Robust dynamic optimization of batch processes under parametric uncertainty: Utilizing approaches from semi-infinite programs”. Computers & Chemical Engineering, 116(9), 253–267. https://doi.org/10.1016/j.compchemeng.2018.05.025
[27] Rafiei, Z., Kafash, K., Karbassi, S. M. (2017). “A computational method for solving optimal control problems and their applications”. Control and Optimization in Applied Mathematics, 2(1), 1–13. https://mathco.journals.pnu.ac.ir/article_4819.html
[28] Ramos, J. I. (2008). “On the variational iteration method and other iterative techniques for nonlinear differential equations”. Applied Mathematics and Computation, 199(1), 39– 69. https://doi.org/10.1016/j.amc.2007.09.024
[29] Rao, S. S. (2009). Engineering Optimization: Theory and Practice. John Wiley & Sons, New Jersey. https://doi.org/10.1002/9780470549124
[30] Rehbock, V., Teo, K. L., Jennings, L. S., Lee, H. W. J. (1999). “A survey of the control parametrization and control parametrization enhancing methods for constrained optimal control problems”. Progress in Optimization: Contributions from Australasia, 30, 247– 275. https://doi.org/10.1007/978-1-4613-3285-5_13
[31] Rocchetta, R., Mey, A., Oliehoek, F. A. (2024). “A survey on scenario theory, complexity, and compression-based learning and generalization”. IEEE Transactions on Neural Networks and Learning Systems, 35(12), 16985–16999. https://doi.org/10.1109/ TNNLS.2023.3308828
[32] Sargent, R. W. H. (2000). “Optimal control”. Journal of Computational and Applied Mathematics, 124(1), 361–371. https://doi.org/10.1016/S0377-0427(00)00418-0
[33] Si, W., Gao, S., Zhang, M., Wen, T., Bai, Y., Wang, H. (2025). “Approximate optimal control for uncertain nonlinear systems: A reinforcement relearning framework”. Nonlinear Dynamics, 113(18), 24821–24848. https://doi.org/10.1007/ s11071-025-11393-9
[34] Stengel, R. F. (1994). Optimal Control and Estimation. Dover Publications, New York.
[35] Tatari, M., Dehghan, M. (2007). “On the convergence of He’s variational iteration method”. Journal of Computational and Applied Mathematics, 207(1), 121–128. https: //doi.org/10.1016/j.cam.2006.07.017
[36] Teo, K. L., Goh, C. J., Wong, K. H. (1991). A Unified Computational Approach to Optimal Control Problems. Longman Scientific & Technical, New York.
[37] Teo, K. L., Jennings, L. S., Lee, H. W. J., Rehbock, V. (1999). “The control parameterization enhancing transform for constrained optimal control problems”. The Journal of the Australian Mathematical Society. Series B. Applied Mathematics, 40, 314–335. https://doi.org/10.1017/S0334270000010936
[38] Terkmani, R., Maidi, A., Guermah, S., Aidene, M. (2020). “Receding horizon control based on the variational iteration method”. International Journal of Systems, Control and Communications, 11(3), 242–260. https://doi.org/10.1504/IJSCC.2020.109072
[39] Trélat, E. (2012). “Optimal Control and Applications to Aerospace: Some Results and Challenges”. Journal of Optimization Theory and Applications, 154(3), 713–758. https: //doi.org/10.1007/s10957-012-0050-5
[40] Wadi, A., Vamvoudakis, K. G. (2026). “Trajectory-informed machine learning for quantum optimal control of uncertain systems”. Automatica, 185, 112756. https://doi.org/ 10.1016/j.automatica.2025.112756
[41] Wang, X., Weyer, E. (2024). “Scenario based optimisation over uncertain system identification models”. Automatica, 165, 111686. https://doi.org/10.1016/j.automatica. 2024.111686
[42] Zagorowska, M., Falugi, P., O’Dwyer, E., Kerrigan, E. C. (2024). “Automatic scenario generation for efficient solution of robust optimal control problems”. International Journal of Robust and Nonlinear Control, 34(2), 1370–1396. https://doi.org/10.1002/rnc. 7038
[43] Zeidler, E. (1995). Applied Functional Analysis: Applications to Mathematical Physics. Springer, New York. https://doi.org/10.1007/978-1-4612-0815-0
[44] Zhang, L., Liu, C., Qian, C., Hua, C. (2026). “Prescribed-time optimal control of a class of uncertain nonlinear systems with unknown control direction”. Neurocomputing, 672, 132683. https://doi.org/10.1016/j.neucom.2026.132683
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