M.G. Dmitriev, D.A. Makarov Iterative Control Synthesis Algorithm in a Singular Perturbed Nonlinear Problem Based on the SDRE Technology
M.G. Dmitriev, D.A. Makarov Iterative Control Synthesis Algorithm in a Singular Perturbed Nonlinear Problem Based on the SDRE Technology

The paper presents an iterative method for solving the stabilization problem for one class of nonlinear dynamical systems with “fast” and “slow” motions, where the systems are formally linear, but their coefficients depend on the state. The stabilizing controller is constructed on the basis of the movements separation principle and modification of the SDRE approach, which consists in using the iterative method of solving the Riccati matrix equation for a singularly perturbed optimal control problem.


singular perturbations, stabilization algorithm, nonlinear systems, separation of motions, iterative methods, system, Riccati matrix equation.

PP. 76-84.

DOI 10.14357/20718632200108

1. Dmitriev M. G., Kurina G. A. Singular perturbations in control problems // Automation and Remote Control. 2006. Vol. 67(1). Pp. 1-43.
2. Kurina G. A., Dmitriev M. G., Naidu D. S. Discrete singularly perturbed control problems (A survey) // Dynamics of Continuous, Discrete and Impulsive Systems Series B: Applications and Algorithms. 2017. Vol. 24(5). Pp. 335-370.
3. Naidu D. S., Calise A. J. Singular perturbations and time scales in guidance and control of aerospace systems: A survey // Journal of Guidance, Control, and Dynamics. 2001. Vol. 24(6). Pp. 1057-1078.
4. Gornov A. Yu., Dmitriev M. G., Tyatyushkin A. I. Experience in solving optimal control problems with a boundary layer [Opyt resheniya zadach optimal'nogo upravleniya s pogranichnym sloem], Deposited in Dep. in VINITI [Dep. v VINITI] 27.11.85, 28441-1385, Computing Center of the Siberian Branch of the Academy of Sciences of the USSR in Krasnoyarsk [VC SO AN SSSR v g. Krasnoyarske], in Russian.
5. Çimen T. Survey of state-dependent Riccati equation in nonlinear optimal feedback control synthesis. Journal of Guidance, Control, and Dynamics. 2012. Vol. 35(4). Pp. 1025-1047.
6. Nekoo S. R. Tutorial and Review on the State-dependent Riccati Equation // Journal of Applied Nonlinear Dynamics. 2019. Vol. 8(2). Pp. 109-166.
7. Çimen T. State-dependent Riccati equation (SDRE) control: A survey // IFAC Proceedings Volumes. 2008. Vol. 41(2). Pp. 3761-3775.
8. Ghadami S. M., Amjadifard R., Khaloozadeh H. Designing SDRE-based controller for a class of nonlinear singularly perturbed systems // International Journal of Robotics and Automation. 2013. Vol. 4. Pp. 1-18.
9. Ghadami S. M., Amjadifard R., Khaloozadeh H. Optimizing a class of nonlinear singularly perturbed systems using SDRE technique // Journal of Dynamic Systems, Measurement, and Control. 2014. Vol. 136 (1). Pp. 011003-1 - 011003-13.
10. Zhang Y. Naidu D.S, Caia C., Zoua Y. Composite control of a class of nonlinear singularly perturbed discrete-time systems via D-SDRE // International Journal of Systems Science. 2016. Vol. 47(11). Pp. 2632-2641.
11. Dmitriev M.G., Makarov D.A. The stabilizing composite control in a weakly nonlinear singularly perturbed control system // 21st International Conference on System Theory, Control and Computing (ICSTCC 2017), Sinaia, Romania, October 19 - 21, 2017. Pp. 594-599. 10.1109/ICSTCC.2017.8107099.
12. Dmitriev M.G., Makarov D.A. Smooth nonlinear controller in a weakly nonlinear control system with state-dependent coefficients [Gladkij nelinejnyj regulyator v slabo nelinejnoj sisteme upravleniya s koefficientami, zavisyashchimi ot sostoyaniya]. // Proceedings of the Institute for System Analysis of RAN [Trudy Instituta sistemnogo analiza RAN]. 2014. Vol. 64(4). Pp. 53-58. In Russian.
13. Danik YU.E., Dmitriev M.G., Makarov D.A. An algorithm for constructing regulators for nonlinear systems with the formal small parameter [Odin algoritm postroeniya regulyatorov dlya nelinejnyh sistem s formal'nym malym parametrom] // Information Technology and Computing Systems [Informacionnye tekhnologii i vychislitel'nye sistemy]. 2015. №4. Pp. 35-44.
14. Gajic Z., Shen X. Parallel algorithms for optimal control of large scale linear systems. Springer Science & Business Media, 2012. ISBN-13: 978-1-4471-3221-9. DOI: 10.1007/978-1-4471-3219-6.
15. Dmitriev M.G., Klishevich A.M. Iterative solution of optimal control problems with fast and slow motions // Systems & control letters. 1984. Vol. 4(4). Pp. 223-226.
16. Kokotovic P.V., Yackel R.A. Singular perturbation on linear regulators: basic theorems // IEEE Trans. Automat. Control. 1972. Vol. 17(1). Pp. 29-37.
17. Glizer V.YA., Dmitriev M.G. Singular perturbations in the linear optimal control problem with a quadratic functional [Singulyarnye vozmushcheniya v linejnoj zadache optimal'nogo upravleniya s kvadratichnym funkcionalom] // Reports of the USSR Academy of Sciences [DAN SSSR]. 1975. Vol. 225(5). Pp. 997-1000, in Russian.

2023 / 01
2022 / 04
2022 / 03
2022 / 02

© ФИЦ ИУ РАН 2008-2018. Создание сайта "РосИнтернет технологии".