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Estimation of Star-Shaped Reachable Sets of Nonlinear Control Systems

  • Tatiana F. Filippova
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10665)

Abstract

The problem of estimating reachable sets of nonlinear dynamical control systems with uncertainty in initial states is studied when it is assumed that only the bounding set for initial system positions is known and any additional statistical information is not available. We study the case when the system nonlinearity is generated from one side by bilinear terms in the matrix elements included in the state velocities and from the other side by quadratic functions in the right-hand part of system differential equations. Using results of the theory of trajectory tubes of control systems and techniques of differential inclusions theory and also results of ellipsoidal calculus we find set-valued estimates of reachable sets of such nonlinear uncertain control system.

Keywords

Control systems Nonlinearity Uncertainty Ellipsoidal estimates 

Notes

Acknowledgments

The research was supported by Russian Science Foundation (RSF Project No.16-11-10146).

References

  1. 1.
    Apreutesei, N.C.: An optimal control problem for a prey-predator system with a general functional response. Appl. Math. Lett. 22(7), 1062–1065 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    August, E., Lu, J., Koeppl, H.: Trajectory enclosures for nonlinear systems with uncertain initial conditions and parameters. In: Proceedings of the 2012 American Control Conference, Fairmont Queen Elizabeth, Montréal, Canada, pp. 1488–1493. June 2012Google Scholar
  3. 3.
    Baier, R., Büskens, C., Chahma, I.A., Gerdts, M.: Approximation of reachable sets by direct solution methods of optimal control problems. Optim. Methods Softw. 22, 433–452 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Boscain, U., Chambrion, T., Sigalotti, M.: On some open questions in bilinear quantum control. In: European Control Conference (ECC), Zurich, Switzerland, pp. 2080–2085. July 2013Google Scholar
  5. 5.
    Ceccarelli, N., Di Marco, M., Garulli, A., Giannitrapani, A.: A set theoretic approach to path planning for mobile robots. In: Proceedings of the 43rd IEEE Conference on Decision and Control, Atlantis, Bahamas, pp. 147–152. December 2004Google Scholar
  6. 6.
    Chernousko, F.L.: State Estimation for Dynamic Systems. Nauka, Moscow (1988)Google Scholar
  7. 7.
    Chernousko, F.L., Ovseevich, A.I.: Properties of the optimal ellipsoids approximating the reachable sets of uncertain systems. J. Optim. Theory Appl. 120(2), 223–246 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Demyanov, V.F., Rubinov, A.M.: Quasidifferential Calculus. Optimization Software, New York (1986)CrossRefzbMATHGoogle Scholar
  9. 9.
    Filippova, T.F.: Set-valued solutions to impulsive differential inclusions. Math. Comput. Modell. Dyn. Syst. 11(2), 149–158 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Filippova, T.F.: Construction of set-valued estimates of reachable sets for some nonlinear dynamical systems with impulsive control. Proc. Steklov Inst. Math. 269(Suppl. 2), 95–102 (2010)CrossRefzbMATHGoogle Scholar
  11. 11.
    Filippova, T.F.: Differential equations of ellipsoidal state estimates in nonlinear control problems under uncertainty. In: Discrete and Continuous Dynamical Systems, Supplement 2011, Dynamical Systems, Differential Equations and Applications, vol. 1. pp. 410–419, Springfield, American Institute of Mathematical Sciences (2011)Google Scholar
  12. 12.
    Filippova, T.F.: State estimation for uncertain systems with arbitrary quadratic nonlinearity. In: Proceedings of the PHYSCON 2015, Istanbul, Turkey, August 1922, pp. 1–6 (2015)Google Scholar
  13. 13.
    Filippova, T.F.: Estimates of reachable sets of impulsive control problems with special nonlinearity. In: Proceedings of the AIP Conference, vol. 1773, p. 100004, 1–8 (2016)Google Scholar
  14. 14.
    Filippova, T.F., Berezina, E.V.: On state estimation approaches for uncertain dynamical systems with quadratic nonlinearity: theory and computer simulations. In: Lirkov, I., Margenov, S., Waśniewski, J. (eds.) LSSC 2007. LNCS, vol. 4818, pp. 326–333. Springer, Heidelberg (2008).  https://doi.org/10.1007/978-3-540-78827-0_36 CrossRefGoogle Scholar
  15. 15.
    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, pp. S28–S37 (2000)Google Scholar
  16. 16.
    Filippova, T.F., Matviychuk, O.G.: Reachable sets of impulsive control system with cone constraint on the control and their estimates. In: Lirkov, I., Margenov, S., Waśniewski, J. (eds.) LSSC 2011. LNCS, vol. 7116, pp. 123–130. Springer, Heidelberg (2012).  https://doi.org/10.1007/978-3-642-29843-1_13 CrossRefGoogle Scholar
  17. 17.
    Filippova, T.F., Matviychuk, O.G.: Estimates of reachable sets of control systems with bilinearquadratic nonlinearities. Ural Math. J. 1(1), 45–54 (2015)CrossRefGoogle Scholar
  18. 18.
    Gusev, M.I.: Application of penalty function method to computation of reachable sets for control systems with state constraints. In: Proceedings of the AIP Conference, vol. 1773 p. 050003, 1–8 (2016)Google Scholar
  19. 19.
    Kurzhanski, A.B.: Control and Observation Under Conditions of Uncertainty. Nauka, Moscow (1977)Google Scholar
  20. 20.
    Kurzhanski, A.B., Veliov, V.M. (eds.): Set-valued Analysis and Differential Inclusions: Progress in Systems and Control Theory, vol. 16. Birkhäuser, Boston (1990)Google Scholar
  21. 21.
    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. 122–188. Birkhäuser, Boston (1993).  https://doi.org/10.1007/978-1-4612-0349-0_4 CrossRefGoogle Scholar
  22. 22.
    Kurzhanski, A.B., Valyi, I.: Ellipsoidal Calculus for Estimation and Control. Birkhäuser, Boston (1997)CrossRefzbMATHGoogle Scholar
  23. 23.
    Kurzhanski, A.B., Varaiya, P.: Dynamics and Control of Trajectory Tubes: Theory and Computation. Systems & Control, Foundations & Applications, vol. 85. Basel, Birkhäuser (2014)zbMATHGoogle Scholar
  24. 24.
    Krastanov, M.I., Veliov, V.M.: High-order approximations to nonholonomic affine control systems. In: Lirkov, I., Margenov, S., Waśniewski, J. (eds.) LSSC 2009. LNCS, vol. 5910, pp. 294–301. Springer, Heidelberg (2010).  https://doi.org/10.1007/978-3-642-12535-5_34 CrossRefGoogle Scholar
  25. 25.
    Matviychuk, O.G.: Ellipsoidal estimates of reachable sets of impulsive control systems with bilinear uncertainty. Cybern. Phys. 5(3), 96–104 (2016)Google Scholar
  26. 26.
    Matviychuk, O.G.: Internal ellipsoidal estimates for bilinear systems under uncertainty. In: Proceedings of the AIP Conference, vol. 1789, p. 060008, 1–8 (2016)Google Scholar
  27. 27.
    Mazurenko, S.S.: A differential equation for the gauge function of the star-shaped attainability set of a differential inclusion. Doklady Math. 86(1), 476–479 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Polyak, B.T., Nazin, S.A., Durieu, C., Walter, E.: Ellipsoidal parameter or state estimation under model uncertainty. Automatica 40, 1171–1179 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Schweppe, F.C.: Uncertain Dynamical Systems. Prentice-Hall, Englewood Cliffs (1973)Google Scholar
  30. 30.
    Veliov, V.: Second-order discrete approximation to linear differential inclusions. SIAM J. Numer. Anal. 29(2), 439–451 (1992)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  1. 1.Krasovskii Institute of Mathematics and Mechanics of RASEkaterinburgRussian Federation

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