ESTONIAN ACADEMY
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eesti teaduste
akadeemia kirjastus
PUBLISHED
SINCE 1952
 
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proceedings
of the estonian academy of sciences
ISSN 1736-7530 (Electronic)
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Feedback linearization of possibly non-smooth systems; pp. 109–123

Full article in PDF format | https://doi.org/10.3176/proc.2017.2.01

Authors
Arvo Kaldmäe, Ülle Kotta, Alexey Shumsky, Alexey Zhirabok

Abstract

 

The algebraic approach known as functions’ algebra is used to develop the necessary and sufficient conditions for the existence of state transformation and static state feedback that linearize the system equations. The advantage of this method is that it allows considering also non-smooth systems. The main object in functions’ algebra is the set of vector functions, divided into equivalence classes, which form a lattice. Both discrete- and continuous-time cases are considered. The solutions to the feedback linearization problem are expressed in terms of a finite sequence of vector functions, which contain all the independent functions having certain relative degrees. The theoretical results are illustrated by numerous examples.

 


References

 

1. Aranda-Bricaire, E., Kotta, Ü., and Moog, C. H. Linearization of discrete-time systems. SIAM J. Control Optim., 1996, 34(6), 1999–2023.
https://doi.org/10.1137/S0363012994267315

2. Califano, C., Monaco, S., and Normand-Cyrot, D. On the problem of feedback linearization. Syst. Control Lett., 1999, 36(1), 61–67.
https://doi.org/10.1016/S0167-6911(98)00070-X

3. Chen, W. and Ballance, D. On a switching control scheme for nonlinear systems with ill-defined relative degree. Syst. Control Lett., 2002, 47(2), 159–166.
https://doi.org/10.1016/S0167-6911(02)00186-X

4. Conte, G., Moog, C. H., and Perdon, A. M. Algebraic Methods for Nonlinear Control Systems. Theory and Applications. Springer, London, 2007.
https://doi.org/10.1007/978-1-84628-595-0

5. Filaretov, V. F., Lebedev, A. V., and Yukhimets, D. A. Control Devices and Systems of Underwater Robots. Nauka, Moscow, 2005 (in Russian).

6. Glad, T. and Ljung, L. Control Theory. Multivariable and Nonlinear Methods. Taylor & Francis, London, 2000.

7. Grizzle, J. Feedback linearization of discrete-time systems. In Analysis and Optimization of Systems: Proceedings of the Seventh International Conference on Analysis and Optimization of Systems (Bensoussan, A. and Lions, J. L., eds). Springer, Berlin, 1986, 273–281.
https://doi.org/10.1007/BFb0007564

8. Hartmanis, J. and Stearns, R. The Algebraic Structure Theory of Sequential Machines. Prentice-Hall, New York, 1966.

9. Isidori, A. Nonlinear Control Systems. Springer, London, 1995.
https://doi.org/10.1007/978-1-84628-615-5

10. Jakubczyk, B. and Respondek, W. On the linearization of control systems. Bull. Acad. Pol. Sci. Ser. Sci. Math., 1980, 28, 517–522.

11. Kaldmäe, A., Kotta, Ü., Shumsky, A., and Zhirabok, A. Measurement feedback disturbance decoupling in discrete-time nonlinear systems. Automatica, 2013, 49(9), 2887–2891.
https://doi.org/10.1016/j.automatica.2013.06.013

12. Kaparin, V., Kotta, Ü., Shumsky, A., Tõnso, M., and Zhirabok, A. Implementation of the tools of functions algebra: first steps. In Proceedings of the 7th Vienna Conference on Mathematical Modelling, Vienna, Austria 14–17 February, 2012 (Troch, I. and Breitenecker, F., eds). IFAC-Elsevier, 2012, 1231–1236.
https://doi.org/10.3182/20120215-3-at-3016.00218

13. Kotta, Ü. and Mullari, T. Discussion on: “Unified approach to the problem of full decoupling via output feedback”. Eur. J. Control, 2010, 16(4), 326–328.
https://doi.org/10.1016/S0947-3580(10)70660-9

14. Kotta, Ü., Tõnso, M., Belikov, J., Kaldmäe, A., Kaparin, V., Shumsky, A., and Zhirabok, A. A symbolic software package for nonlinear control systems. In Proceedings of the 2013 International Conference on Process Control (PC): Strbske Pleso, Slovakia, June 18–21, 2013 (Fikar, M. and Kvasnica, M., eds). IEEE, 2013, 101–106.
https://doi.org/10.1109/pc.2013.6581391

15. Kotta, Ü., Tõnso,M., Shumsky, A., and Zhirabok, A. Feedback linearization and lattice theory. Syst. Control Lett., 2013, 62(3), 248–255.
https://doi.org/10.1016/j.sysconle.2012.11.014

16. Krener, A. J. On the equivalence of control systems and the linearization of nonlinear systems. SIAM J. Control, 1973, 11(4), 670–676.
https://doi.org/10.1137/0311051

17. Lee, H. G., Arapostathis, A., and Marcus, S. I. Linearization of discrete time nonlinear systems. In Proceedings of the 1987 American Control Conference, Minneapolis, USA. IEEE, 1987, 857–862.

18. Marino, R. On the largest feedback linearizable subsystem. Syst. Control Lett., 1986, 6(5), 345–351.
https://doi.org/10.1016/0167-6911(86)90130-1

19. Shumsky, A. and Zhirabok, A. Nonlinear diagnostic filter design: algebraic and geometric points of view. Int. J. Appl. Math. Comput. Sci., 2006, 16(1), 101–113.

20. Shumsky, A. and Zhirabok, A. Unified approach to the problem of full decoupling via output feedback. Eur. J. Control, 2010, 16(4), 313–325.
https://doi.org/10.3166/ejc.16.313-325

21. Simões, C., Nijmeijer, H., and Tsinias, J. Nonsmooth stabilizability and feedback linearization of discrete-time systems. Int. J. Robust Nonlin., 1996, 6, 171–188.
https://doi.org/10.1002/(SICI)1099-1239(199604)6:3<171::AID-RNC140>3.0.CO;2-0
https://doi.org/10.1002/(SICI)1099-1239(199604)6:3<171::AID-RNC140>3.3.CO;2-S

22. Sira-Ramirez, H. and Agrawal, S. K. Differentially Flat Systems. Marcel Dekker, New York, 2004.

23. Tomlin, C. J. and Sastry, S. S. Switching through singularities. Syst. Control Lett., 1998, 35(3), 145–154.
https://doi.org/10.1016/S0167-6911(98)00046-2

24. Zhirabok, A. and Shumsky, A. The Algebraic Methods for Analysis of Nonlinear Dynamic Systems. Dalnauka, Vladivostok, 2008 (in Russian).

 


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