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On Ellipsoidal Estimates for Reachable Sets of the Control System

  • Oxana G. MatviychukEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11548)

Abstract

The problem of the ellipsoidal estimation of the reachable set of the control system under uncertainties is considered. The matrix included in the differential equations of the system dynamics is uncertain and only bounds on admissible values of this matrix coefficients are known. It is assumed that the initial states of the system are unknown but belong to a given star-shaped symmetric nondegenerate polytope. This polytope may be a non-convex set. Under such conditions, the dynamical system is a nonlinear and reachable set loses convexity property. A Minkowski function is used in the investigation to describe the trajectory tubes and their set-valued estimates. The step by step algorithm for constructing external and internal ellipsoidal estimates of reachable sets for such bilinear control systems is proposed. Numerical experiments were performed. The results of these numerical experiments are included.

Keywords

Control system Ellipsoidal calculus Estimation Reachable set 

References

  1. 1.
    Boscain, U., Chambrion, T., Sigalotti, M.: On some open questions in bilinear quantum control. In: European Control Conference (ECC), Zurich, Switzerland, 17–19 July 2013, pp. 2080–2085. IEEE Xplore Digital Library (2013).  https://doi.org/10.23919/ECC.2013.6669238
  2. 2.
    Boussaïd, N., Caponigro, M., Chambrion, T.: Total variation of the control and energy of bilinear quantum systems. In: 52nd IEEE Conference on Decision and Control, Florence, Italy, 10–13 December 2013, pp. 3714–3719. IEEE Xplore Digital Library (2014).  https://doi.org/10.1109/CDC.2013.6760455
  3. 3.
    Chernousko, F.L.: State Estimation for Dynamic Systems. CRC Press, Boca Raton (1994)Google Scholar
  4. 4.
    Filippova, T.F.: Set-valued dynamics in problems of mathematical theory of control processes. Int. J. Mod. Phys. Ser. B (IJMPB) 26(25), 1–8 (2012)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Filippova, T.F., Lisin, D.V.: On the estimation of trajectory tubes of differential inclusions. Proc. Steklov Inst. Math. Probl. Control Dyn. Syst. (Suppl. 2), S28–S37 (2000)Google Scholar
  6. 6.
    Filippova, T.F., Matviychuk, O.G.: Estimates of reachable sets of control systems with bilinear-quadratic nonlinearities. Ural Math. J. 1(1), 45–54 (2015).  https://doi.org/10.15826/umj.2015.1.004CrossRefzbMATHGoogle Scholar
  7. 7.
    Kostousova, E.K.: On tight polyhedral estimates for reachable sets of linear differential systems. AIP Conf. Proc. 1493, 579–586 (2012).  https://doi.org/10.1063/1.4765545CrossRefGoogle Scholar
  8. 8.
    Kurzhanski, A.B.: Control and Observation Under Conditions of Uncertainty. Nauka, Moscow (1977). [in Russian]Google Scholar
  9. 9.
    Kurzhanski, A.B., Valyi, I.: Ellipsoidal Calculus for Estimation and Control. Birkhäuser, Boston (1997)CrossRefGoogle Scholar
  10. 10.
    Kurzhanski, A.B., Filippova, T.F.: On the theory of trajectory tubes – a mathematical formalism for uncertain dynamics, viability and control. In: Kurzhanski, A.B. (ed.) Advances in Nonlinear Dynamics and Control: a Report from Russia. Progress in Systems and Control Theory, vol. 17, pp. 22–188. Birkhäuser, Boston (1993)CrossRefGoogle Scholar
  11. 11.
    Matviychuk, O.G.: Ellipsoidal estimates of reachable sets of impulsive control systems with bilinear uncertainty. Cybern. Phys. 5(3), 96–104 (2016)Google Scholar
  12. 12.
    Matviychuk, O.G.: Internal ellipsoidal estimates for bilinear systems under uncertainty. AIP Conf. Proc. 1789, 1–7 (2016).  https://doi.org/10.1063/1.4968500. Article No. P. 060008CrossRefGoogle Scholar
  13. 13.
    Matviychuk, O.G.: Estimates of the trajectory tubes of impulsive control systems. In: 14th International Conference “Stability and Oscillations of Nonlinear Control Systems” (Pyatnitskiy’s Conference) (STAB), pp. 1–4. IEEE Xplore Digital Library (2018).  https://doi.org/10.1109/STAB.2018.8408379
  14. 14.
    Matviychuk, O.G.: Estimation techniques for bilinear control systems. IFAC-PapersOnLine 51(32), 877–882 (2018).  https://doi.org/10.1016/j.ifacol.2018.11.434CrossRefGoogle Scholar
  15. 15.
    Matviychuk, A.R., Matviychuk, O.G.: A method of approximate reachable set construction on the plane for a bilinear control system with uncertainty. AIP Conf. Proc. 2025, 1–6 (2018).  https://doi.org/10.1063/1.5064933. Article No. 100004CrossRefGoogle Scholar
  16. 16.
    Mazurenko, S.: Partial differential equation for evolution of star-shaped reachability domains of differential inclusions. Set-Valued Var. Anal. 24(2), 333–354 (2006).  https://doi.org/10.1007/s11228-015-0345-4MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Polyak, B.T., Nazin, S.A., Durieu, C., Walter, E.: Ellipsoidal parameter or state estimation under model uncertainty. Automatica 40(7), 1171–1179 (2004).  https://doi.org/10.1016/j.automatica.2004.02.014MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Schweppe, F.: Uncertain Dynamic Systems. Prentice-Hall, Englewood Cliffs (1973)Google Scholar
  19. 19.
    Tidmore, F.E.: Extremal structure of star-shaped sets. Pac. J. Math. 29(2), 461–465 (1969)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Vzdornova, O.G.: Ellipsoidal techniques in state estimation problem for linear impulsive control systems. In: Proceedings of the 3rd International Conference on Physics and Control (PhysCon 2007), Potsdam, Germany, 03–07 September 2007, pp. 1–6. Universitätsverlag Potsdam, Potsdam (2007). http://lib.physcon.ru/doc?id=5fc7ce762ef6
  21. 21.
    Walter, E., Pronzato, L.: Identification of Parametric Models from Experimental Data. Springer, Heidelberg (1997)zbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Krasovskii Institute of Mathematics and Mechanics, Ural Branch of Russian Academy of SciencesEkaterinburgRussia
  2. 2.Ural Federal UniversityEkaterinburgRussia

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