Buy (600 RUB)
Generate QR code
Open Access
Subscription or Fee Access
Finite-Horizon Optimal Tracking Control for Nonlinear Systems Based on the SDC Method
https://doi.org/10.17587/mau.26.579-587
Abstract
The paper considers the problem of optimal tracking control for nonlinear systems on a finite time interval. In this case, the system is represented in the state space form with state-dependent coefficients (SDC) matrices. The problem of finding a solution to the tracking problem on a finite time interval in the nonlinear SDC formulation is associated with finding a solution to the state-dependent differential Riccati equation and the differential equation for the auxiliary feedforward vector, the initial conditions for which are usually specified at the right end. А typical approach to solving such problems uses the integration of these equations in the backward direction (from right to left), where the calculation of the SDC matrices of the system requires information on the state variables of the system and control, which is not available without additional measures. To overcome the indicated problem of the unknown state vector during backward integration, this paper proposes an approach based on deriving a solution through the corresponding differential equations for the Riccati matrix and the auxiliary vector, the initial conditions for which are uniquely specified at the left end of the time interval due to the use of a special Riccati transform, different from the typical one. This allows calculating the control through the integration of the corresponding differential equations in direct time, which removes the problem of the unknown state vector. The proposed approach is tested on the academic example of the Van der Pol oscillator, for which an additional study of the effectiveness of the proposed method in comparison with the most popular existing approaches is performed. The results of computer modeling confirmed the advantage of the proposed method, both in terms of the terminal accuracy of tracking the driving signal, and in terms of the mean square error of tracking.
About the Author
A. A. Kabanov
Sevastopol State University
Russian Federation
Russian Federation
Ph.D., Associate Professor,
Sevastopol, 299053.
References
1. Berkovitz L. D., Medhin N. G. Nonlinear optimal control theory, Boca Raton: CRC Press, Taylor & Francis Group, 2013.
2. Ashchepkov L. T., Dolgy D. V., Kim T., Agarwal R. P. Optimal control, Springer Cham, 2021.
3. Nekoo S. R. Tutorial and review on the state-dependent Riccati equation, Journal of Applied Nonlinear Dynamics, 2019, vol. 8, no. 2, pp. 109—166, DOI: 10.5890/JAND.2019.06.001.
4. Çimen T. Survey of State-dependent Riccati equation in nonlinear optimal feedback control synthesis, Journal of Guidance, Control, and Dynamics, 2012, vol. 35, no. 4, pp. 1025—1047, DOI: 10.2514/1.55821.
5. Çimen T. Systematic and effective design of nonlinear feedback controllers via the state-dependent Riccati equation (SDRE) method, Annual Reviews in Control, 2010, vol. 34, no. 1, pp. 32—51, DOI: 10.1016/J.ARCONTROL.2010.03.001.
6. Çimen T., Banks S. P. Global optimal feedback control for general nonlinear systems with nonquadratic performance criteria, Systems & Control Letters, 2004, vol. 53, no. 5, pp. 327—346, DOI: 10.1016/j.sysconle.2004.05.008.
7. Topputo F., Miani M., Bernelli-Zazzera F. Optimal selection of the coefficient matrix in state-dependent control methods, Journal of Guidance, Control, and Dynamics, 2015, vol. 38, no. 5, pp. 861—873, DOI: 10.2514/1.G000136.
8. Topputo F., Bernelli-Zazzera F. Approximate solutions to nonlinear optimal control problems in astrodynamics, ISRN Aerospace Engineer ing, 2013, vol. 2013, pp. 1—7.
9. Pearson J. D. Approximation methods in optimal control I. Sub-optimal control, International Journal of Electronics, 1962, vol. 13, no. 5, pp. 453—469.
10. Korayem M. H., Nekoo S. Finite-time state-dependent Riccati equation for time-varying nonaffine systems: Rigid and flexible joint manipulator control, ISA Transactions, 2015, vol. 54, pp. 125—144, DOI: 10.1016/j.isatra.2014.06.006.
11. Heydari A., Balakrishnan S. N. Closed-form solution to finite-horizon suboptimal control of nonlinear systems, Int. J. Robust. Nonlinear Control, 2015. vol. 25, no. 15, pp. 2687—2704.
12. Korayem M. Н., Nekoo S. R. Nonlinear optimal control via finite time horizon state-dependent Riccati equation, In 2014 second RSI/ISM international conference on robotics and mechatronics, 2014, pp. 878—883, DOI: 10.1109/ICRoM.2014.6991015.
13. Kabanov А. A. Modified SDRE method for finite-time nonlinear optimal control problem, Proceedings 2022 International Russian Automation Conference (RusAutoCon). Sochi, Russia, 5-11 Sept. 2022, pp. 639—643, DOI: 10.1109/RusAutoCon54946.2022.9896380.
14. Bryson А., Yu-Chi Ho. Applied optimal control, Washington, Hemisphere Pub. Corp., 1975.
15. Kwakernaak H., Sivan R. Linear optimal control systems. NY., Wiley-Interscience, 1972.
16. Naidu D. S. Optimal control systems. Electrical Engineering Handbook, Florida, Boca Raton, CRC Press, 2003.
17. Lewis F. L., Vrabie D. L., Syrmos V. L. Optimal control, Hoboken, NJ, Wiley, 2012.
18. Korayem M. Н., Nekoo S. R. State-dependent differential Riccati equation to track control of time-varying systems with state and control nonlinearities, ISA Transactions, 2015. vol. 57, pp. 117—135, DOI: 10.1016/j.isatra.2015.02.008.
19. Barbieri E., Alba-Flores R. On the innite-horizon LQ tracker, Syst. Control Lett., 2000, vol. 40, no. 2, pp. 77—82, DOI: 10.1016/S0167-6911(00)00004-9.
20. Alba-Flores R., Barbieri E. Real-time infinite horizon linear-quadratic tracking controller for vibration quenching in flexible beams, in Proc. 2006 IEEE Int. Conf. Syst., Man, Cybern., Taipei, Taiwan, Oct. 2006, pp. 38—43.
21. Khamis A., Naidu D. S. Nonlinear optimal tracking with incomplete state information using finite-horizon State Dependent Riccati Equation (SDRE), 2014 American Control Conference. Presented at the 2014 American Control Conference — ACC 2014, IEEE, Portland, OR, USA, pp. 2320—2325, DOI: 10.1109/ACC.2014.6858589.
22. Makarov D. A. Synthesis of control and state observer for weakly nonlinear systems based on the pseudo-linearization technique, Aut. Control Comp. Sci., 2019, vol. 53, pp. 823—829.
23. Mufti I. H., Chow C. K., Stock F. T. Solution of ill-conditioned linear two-point boundary value problems by the Riccati transformation, SIAM Rev., 1969, vol. 11, no. 4, pp. 616—619.
24. Sannuti P., Wason Н. Singular perturbation analysis of cheap control problems, The 22nd IEEE Conference on Decision and Control, 1983, pp. 231—236.
25. Gajic Z. Optimal control of singularly perturbed linear systems and applications (1st ed.), CRC Press, 2001.
26. Fossen T. I. Guidance and control of ocean vehicles, New York: John Wiley and Sons, 1999.
Review
For citations:
Kabanov A.A. Finite-Horizon Optimal Tracking Control for Nonlinear Systems Based on the SDC Method. Mekhatronika, Avtomatizatsiya, Upravlenie. 2025;26(11):579-587. (In Russ.) https://doi.org/10.17587/mau.26.579-587
Views:
32
.png)
Email this article 