Pseudospectral Method for Finding Optimal Control of Trajectory Bundles Based on Multi-Agent Optimization Algorithms



A class of problems of optimal control of nonlinear continuous deterministic systems under conditions of uncertainty is considered. To solve the problem, a numerical algorithm for finding the optimal control is formed, in which the parameterization of the control law is used, which depends on time and a set of coordinates of the state vector available for measurement. This approach is based on the approximation of the control law by a series using a system of basis functions with unknown coefficients. The search for unknown coefficients in the expansion of the control law is implemented using multi-agent optimization methods: a hybrid multi-agent interpolation search algorithm and a multi-agent algorithm based on the use of linear controllers for controlling the movement of agents. A software has been developed and two model examples and an applied problem of stabilizing a satellite with the help of engines installed on it have been solved.

General Information

Keywords: optimal control, multi-agent optimization algorithms, trajectory bundle, Chebyshev polynomials, pseudospectral method

Journal rubric: Optimization Methods

Article type: scientific article


Received: 17.03.2023


For citation: Karane M.S. Pseudospectral Method for Finding Optimal Control of Trajectory Bundles Based on Multi-Agent Optimization Algorithms. Modelirovanie i analiz dannikh = Modelling and Data Analysis, 2023. Vol. 13, no. 2, pp. 99–122. DOI: 10.17759/mda.2023130206. (In Russ., аbstr. in Engl.)


  1. Kurzhanskii A.B. Upravlenie i nablyudenie v usloviyakh neopredelennosti [Control and observation in conditions of uncertainty]. Moscow: Publ. Nauka, 1977. 392 p.
  2. Ovsyannikov D.A., Mizintseva M.A., Balabanov M.Yu., Durkin A.P., Edamenko N.S., Kotina E.D., Ovsyannikov A.D. Optimizatsiya dinamiki puchkov traektorii c ispol'zovaniem gladkikh i negladkikh funktsionalov [Optimization of dynamics of bundles of trajectories using smooth and non-smooth functionals]. Ch.1. Vestn. SPbGU. Ser.10. Prikladnaya matematika. Informatika. Protsessy upravleniya = Part 1. Bulletin of St. Petersburg State University. Series 10. Applied Mathematics. Computer science. Control processes, 2020. Vol.16. no.1. P. 73–84. (In Russ.)
  3. Henrion D., Korda  M. Convex Computation of the Region of Attraction of Polynomial Control Systems // European Control Conf, (ECC). Zurich, 2013. P. 676–681.
  4. Deng H., Zhang W., Shen C. Stability Analysis of Optimal Trajectory for Nonlinear Optimal Control Problems // Journal of Mathematics. 2020. P.1–5. doi:10.1155/2020/1392705.
  5. Bortakovskii A.S. Optimal'noe i suboptimal'noe upravleniya puchkami traektorii determinirovannykh sistem avtomatnogo tipa [Optimal and suboptimal control of bundles of trajectories of deterministic systems of automatic type]. Izv. RAN TiSU = Proceedings of the RAS TiSU. 2016. No. 1. P. 5–26.
  6. Panteleev A.V. Metaevristicheskie algoritmy optimizatsii zakonov upravleniya dinamicheskimi sistemami [Metaheuristic algorithms for optimizing control laws for dynamic systems]. Moscow: Publ. Faktorial, 2020. 564 p.
  7. Afanas'ev V. N., Frolova N. A. Differentsial'naya igra v zadache upravleniya nelineinym ob"ektom s ogranicheniyami na upravlyayushchie vozdeistviya [Differential game in the control problem for a non-linear object with restrictions on control actions]. Trudy Instituta sistemnogo analiza RAN = Proceedings of the Institute of System Analysis of the RAS. 2020. V. 70. no 3. P. 56-64. doi:10.14357/20790279200307
  8. Liu M., Zhao J., Hoogendoorn S., Wang M. An optimal control approach of integrating traffic signals and cooperative vehicle trajectories at intersections // Transportmetrica B: Transport Dynamics. 2021. No. 10. P.1–17. doi:10.1080/21680566.2021.1991505.
  9. Rybakov K.A. Applying Spectral Form of Mathematical Description for Representation of Iterated Stochastic Integrals // Differencialnie Uravnenia i Protsesy Upravlenia. 2019. No. 4. P. 1–31.
  10. Kosnikov Yu.N. Osobennosti primeneniya radial'nykh bazisnykh funktsii v geometricheskom modelirovanii trekhmernykh ob"ektov vizualizatsii [Features of application of radial basis functions in geometric modeling of three-dimensional visualization objects]. Modeli, sistemy, seti v ekonomike, tekhnike, prirode i obshchestve = Models, systems, networks in economics, technology, nature and society 2020. №4.
  11. Garg D., Patterson M., Hager W., Rao A., Benson D. An overview of three pseudospectral methods for the numerical solution of optimal control problems // Advances in the Astronautical Sciences. 2017. V.135, P.1–17.
  12. Tang X., Hu Z.L. New results on pseudospectral methods for optimal control // Automatica. Vol.65. 2016, P.160-163. doi:10.1016/j.automatica.2015.11.035
  13. Mehrpouya M. A., Peng H. A robust pseudospectral method for numerical solution of nonlinear optimal control problems // International Journal of Computer Mathematics. 2021, no.98(6), P.1146–1165, doi: 10.1080/00207160.2020.1807521
  14. Hager W. W., Liu J., Mohapatra S., Rao A. V., Wang X. -S. A pseudospectral method for optimal control based on collocation at the Gauss points // IEEE Conference on Decision and Control (CDC). 2018, P. 2490–2495, doi: 10.1109/CDC.2018.8618929.
  15. Li Y., Chen W., Yang L. Linear Pseudospectral Method with Chebyshev Collocation for Optimal Control Problems with Unspecified Terminal Time //Aerospace 2022, no. 9. doi:10.3390/aerospace9080458
  16. Gendreau M., Potvin J-Y. Handbook of Metaheuristic. N.Y.: Springer, 2019. 604 p.
  17. Panteleev A., Karane M. Application of Multi-agent Optimization Methods Based on the Use of Linear Regulators and Interpolation Search for a Single Class of Optimal Deterministic Control Systems // Applied Mathematics and Computational Mechanics for Smart Applications. Singapore: Springer, 2021. P. 217–244
  18. Panteleev A., Karane M. Multi-agent Optimization Algorithms for a Single Class of Optimal Deterministic Control Systems // Advances in Theory and Practice of Computational Mechanics. Singapore: Springer, 2020. pp. 271–291
  19. Karane M., Panteleev A. Benchmark Analysis of Novel Multi-agent Optimization Algorithm Using Linear Regulators for Agents Motion Control // IOP Conf. Series: Materials Science and Engineering. Alushta, 2020. doi:10.1088/1757-899X/927/1/012023
  20. Wang D., Tan D., Liu L. Particle swarm optimization algorithm: an overview // Soft Computing, 2018, 22(2), P.387–408. doi:10.1007/s00500-016-2474-6.
  21. Wang G., Gandomi A., Alavi A., Gong D. A comprehensive review of krill herd algorithm: variants, hybrids and applications // Artificial Intelligence Review. 2019, no. 51, P.119–148. doi:10.1007/s10462-017-9559-1.
  22. Davendra D., Zelinka I. Self-Organizing Migrating Algorithm. Methodology and Implementation. Studies in Computational Intelligence. Springer 2016. 626 p.
  23. Finkel'shtein E.A. Vychislitel'nye tekhnologii approksimatsii mnozhestva dostizhimosti upravlyaemoi sistemy: Avtoref. diss. kand. tekhn. nauk. [Computational technologies for approximating the reachability set of a controlled system. PhD (Engineering) Thesis] Irkutsk, 2018. 19 p.
  24. Krylov I.A. Chislennoe reshenie zadachi ob optimal'noi stabilizatsii sputnik [Numerical solution of the problem of optimal satellite stabilization]. ZhVM i MF = Journal of Comp. Math. and Math. Physics. 1968. V.8. no.1. P. 203–208. (In Russ.)
  25. Karane, M.M., Panteleev, A.V. Multiagent Algorithms for Optimizing Bundles of Trajectories of Deterministic Systems with Incomplete Instant Feedback // J. Comput. Syst. Sci. Int. 61. 2022. P.751–775. doi:10.1134/S1064230722050082

Information About the Authors

Mary Magdalene S. Karane, Master's Degree Student, Moscow Aviation Institute (National Research University), Moscow, Russia, e-mail:



Total: 41
Previous month: 10
Current month: 2


Total: 26
Previous month: 3
Current month: 0