Approximate Feedback Linearization Based on the Singular Perturbations Approach
https://doi.org/10.17587/mau.16.515-522
Abstract
One of the most common methods of synthesis of the nonlinear control systems is the method of a feedback linearization (FL). The idea of this method consists in conversion of the original nonlinear system into a linear one by means of a state feedback and coordinate transformation. Then, the methods of control theory for the linear systems are used for the system design. If the original nonlinear system cannot be linearized exactly by the state feedback, the method of the approximate feedback linearization (AFL) is used. The essence of AFL method lies in the feedback linearization only of a certain part of the original nonlinear system (not of the entire system). In this paper, the author proposes a method of an approximate feedback linearization control of the nonlinear singularly perturbed (SP) systems. The proposed method is based on a decomposition of the original SP system and construction of AFL control in the form of composite FL controls for the slow and fast subsystems. In general, a nonlinear SP system cannot be easily separated into slow and fast subsystems, because the conditions of Tikhonov theorem are not complied. In order to overcome this, the author proposes to perform the feedback linearization method at first for the system's part, which describes the fast state variables. Thus, a fast control is chosen, so that the conditions of Tikhonov theorem would be met. Then, using a standard singular perturbation technique, we obtain a slow subsystem. Further the problem of FL control for a slow subsystem is solved. The resulting AFL control is obtained in the form of a composite control. Application of the proposed approach is illustrated with two examples.
References
1. Мирошник И. В., Никифоров В. О., Фрадков А. Л. Нелинейное и адаптивное управление сложными динамическими системами. СПб.: Наука, 2000. 549 с.
2. Sastry S. Nonlinear systems: analysis, stability, and control. New York: Springer-Verlag, 2010.
3. Isidori A. Nonlinear control systems. New York: Springer-Verlag, 1995.
4. Henson M. A., Seborg D. E. Nonlinear process control. New Jersey: Prentice hall, 1997. 432 p.
5. Guardabassi G. O., Savaresi S. M. Approximate linearization via feedback: an overview // Automatica. 2001. Vol. 37. P. 1-15.
6. Krener A. J. Approximate linearization by state feedback and coordinate change // System & Control Letters. 1984. N 5. P. 181-185.
7. Kang W., Krener A. J. Extended quadratic controller normal form and dynamic state feedback linearization of nonlinear systems // SIAM Journal on Control and Optimization. 1992. N. 30. P. 1319-1337.
8. Kang W. Approximate linearization of nonlinear control systems / W. Kang // Systems & Control Letters. 1994. Vol. 23. P. 43-52.
9. Son J.-W., Lim J.-T. Stabilization of Approximately Feedback Linearizable Systems Using Singular Perturbation // IEEE Trans. on Automatic Control. 2008. Vol. 53, N 6. P. 1499-1503.
10. Kokotovic P. V., Khalil H. K., O'Reilly J. Singular perturbation methods in control: analysis and design. Orlando: Academic Press, 1986. 371 p.
11. Naidu D. S. Singular Perturbation Methodology in Control Systems. L.: P. Peregrinus on behalf of the Institution of Electrical Engineers, 1988. 304 p.
12. Дмитриев М. Г., Курина Г. А. Сингулярные возмущения в задачах управления // Автоматика и телемеханика. 2006. № 1. С. 3-51.
13. Khorasani K. On Linearization of Nonlinear Singularly Perturbed Systems // IEEE Trans. on Automatic Control. 1987. Vol. 32, N 3. P. 256-260.
14. Khorasani K. A Slow Manifold Approach to Linear Equivalents of Nonlinear Singularly Perturbed Systems // Automatica. 1989. Vol. 25, N 2. P. 301-306.
15. Кабанов А. А. Композиционный синтез нелинейных сингулярно возмущенных систем на основе метода линеаризации обратной связью // Тр. Х Междунар. конф. "Идентификация систем и задачи управления" SICPRO'15. Москва, 26-29 января 2015 г. М.: Изд. ИПУ РАН, 2015. С. 548-556.
16. Кабанов А. А. Управление нелинейными сингулярно возмущенными системами на основе метода линеаризации обратной связью // Матер. 4-го науч.-техн. семинара "Современные проблемы прикладной математики, информатики, автоматизации и управления", 23-27 сентября 2014 г., г. Севастополь. М.: Изд-во ИПИ РАН, 2014. С. 62-69.
17. Choi H.-L., Shin Y.-S., Lim J.-T. Control of nonlinear singularly perturbed systems using feedback linearization // IEE Proc.-Control Theory Appl. 2005. Vol. 152, N 1. P. 91-94.
18. Son J.-W., Lim J.-T. Feedback linearization of nonlinear singularly perturbed systems with non-separate slow-fast dynamics // IET Control Theory Appl. 2008. Vol. 2, N 8. P. 728-735.
19. Khalil H. K. Nonlinear systems. New Jersey: Prentice Hall, 2002. 750 p.
20. Дубовик С. А., Кабанов А. А. Мера устойчивости к сингулярным возмущениям и робастные свойства линейных систем // Проблемы управления и информатики. 2010. Вып. 3. С. 17-28.
21. Кабанов А. А. Синтез робастных систем стабилизации на основе теории сингулярных возмущений // Оптимизация произв. процессов: сб. науч. тр. Севастополь: изд-во СевНТУ, 2013. Вып. 14. С. 73-79.
Review
For citations:
Kabanov A.A. Approximate Feedback Linearization Based on the Singular Perturbations Approach. Mekhatronika, Avtomatizatsiya, Upravlenie. 2015;16(8):515-522. (In Russ.) https://doi.org/10.17587/mau.16.515-522
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