Skip to main content
SpringerLink
Log in

The method of uniform monotonous approximation of the reachable set border for a controllable system

  • Published:
Journal of Global Optimization Aims and scope Submit manuscript

Abstract

A numerical method of a two-dimensional non-linear controllable system reachable set boundary approximation is considered. In order to approximate the boundary right piecewise linear closed contours are used: a set of broken lines on a plane. As an application of the proposed technique a method of finding linear functional global extremum is described, including its use for systems with arbitrary dimensionality.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

References

  1. Baier, R., Gerdts, M., Xausa, I.: Approximation of reachable sets using optimal control algorithms. Numer. Algebra Control Optim. 3(3), 519–548 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  2. Barton, P.I., Lee, C.K., Yunt, M.: Optimization of hybrid systems. Comput. Chem. Eng. 30(10–12), 1576–1589 (2006)

    Article  Google Scholar 

  3. Bellman, R.: Dynamic Programming. Princeton University Press, Princeton, New Jersey (1957)

    MATH  Google Scholar 

  4. Bressan, A., Colombo, G.: Generalized Baire category and differential inclusions in Banach spaces. J. Differ. Equ. 76, 135–158 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  5. Brockett, R.W.: On the reachable set for bilinear systems. Lect. Notes Econ. Math. Syst. 111, 54–63 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  6. Cellina, A., Ornelas, A.: Representation of the attainable set for Lipschitzian differential inclusions. Rocky Mt. J. Math. 22(1), 117–124 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chachuat, B., Singer, A.B., Barton, P.I.: Global methods for dynamic optimization and mixed-integer dynamic optimization. Ind. Eng. Chem. Res. 45(25), 8373–8392 (2006)

    Article  Google Scholar 

  8. Chernousko, F.L.: State Estimation for Dynamic Systems. CRC Press, Boca Raton (1994)

    MATH  Google Scholar 

  9. Donchev, A.L., Lempio, F.: Difference methods for differential inclusions: a survey. SIAM Rev. 34(2), 263–294 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  10. Dykhta, V.A., Sorokin, S.P.: Hamilton–Jacobi inequalities and the optimality conditions in the problems of control with common end constraints. Autom. Remote Control. 72(9), 1808–1821 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  11. Floudas, C.A., Gounaris, C.E.: A review of recent advances in global optimization. J. Glob. Optim. 45, 3–38 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  12. Frankowska, H.: Contingent cones to reachable sets of control systems. SIAM J. Control Optim. 27(1), 170–198 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  13. Gornov, A.Yu.: Implementation of the random multistart method for optimal control problem. In: Proceedings of Lyapunov readings, vol. 38 (2003) (in Russian)

  14. Gornov, A.Yu.: Computational Technologies for Solving Optimal Control Problems. Nauka, Novosibirsk (2009). (in Russian)

  15. Gornov, A.Yu., Zarodnyuk, T.S., Madzhara, T.I., Daneyeva, A.V., Veyalko, I.A.: A collection of test multiextremal optimal control problems. Springer Optim. Appl. 76, 257–274 (2013)

  16. Guseinov, Kh.G., Moiseyev, A.A., Ushakov, V.N.: On the approximation of reachable domains of control systems. J. Appl. Math. Mech. 62, 169–175 (1998)

  17. Hackl G.: Reachable sets, control sets and their computation. Dissertation, Universitat Augsburg, Augsburger Mathematische Schriften Bend (1996)

  18. Hajek, O.: Control Theory in the Plane. Lecture notes in control and information sciences. Springer, Berlin (2008)

    Google Scholar 

  19. Kastner-Maresch, A.: Implicit Runge–Kutta methods for differential inclusions. Numer. Funct. Anal. Optim. 11, 937–958 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  20. Khrustalev, M.M.: Exact description of reachable sets and global optimality conditions for dynamic systems. Autom. Remote Control. 5, 597–604 (1988). 7: 874–881

    MathSciNet  MATH  Google Scholar 

  21. Kostousova, E.K.: On the boundedness of outer polyhedral estimates for reachable sets of linear systems. Comput. Math. Math. Phys. 48(6), 918–932 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  22. Krasovskii, N.N.: Theory of Motion Control. Nauka, Moscow (1968) (in Russian)

  23. Krener, A.J., Schattler, H.: The structure of small-time reachable sets in low dimensions. SIAM J. Control Optim. 27(1), 120–147 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  24. Krotov, V.F.: Global Methods in Optimal Control Theory. Marcel Dekker Inc., New York (1996)

    MATH  Google Scholar 

  25. Kurzhanski, A.B., Valyi, I.: Ellipsoidal techniques for dynamic systems: control synthesis for uncertain systems. Dyn. Control. 2(2), 87–111 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  26. Lempio, F.: Modified Euler methods for differential inclusions. In: Proceedings of Workshop on Set-Valued Anal. 17–21, Sept, Bulgaria, Pamporovo, pp. 131–148 (1990)

  27. Lin, Y.D., Stadtherr, M.A.: Deterministic global optimization of nonlinear dynamic systems. AIChE J. 53(4), 866–875 (2007)

    Article  Google Scholar 

  28. Lopez Cruz I.L.: Efficient evolutionary algorithms for optimal control. PhD-Thesis, Wageningen University, Netherlands (2002)

  29. Lotov, A.V.: Generalized reacable sets method in multiple criteria problems. Methodology and Software for Interactive Decision Support, vol. 337, pp. 250–256. Springer, New York (1986)

    Google Scholar 

  30. Luus, R.: Direct search Luus–Jaakola optimization procedure. In: Floudas, C.A., Pardalos P.M. (eds.) Encyclopedia of Optimization, 2nd edn, pp. 735–739. Springer, New York (2009)

  31. Mitchell, I.M., Bayen, A.M., Tomlin, C.J.: A time-dependent Hamilton–Jacobi formulation of reachable sets for continuous dynamic games. IEEE Trans Automat. Control 50(7), 947–957 (2005)

    Article  MathSciNet  Google Scholar 

  32. Nikolskii, M.S.: Approximation of the attainability set for a controlled process. Math. Notes Acad. Sci. USSR 41(1), 44–48 (1987)

    Article  Google Scholar 

  33. Panasyuk, A.I.: Differential equation for nonconvex attainment sets. Math. notes Acad. Sci. USSR 37(5), 395–400 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  34. Philippova, T.F.: Differential equations for ellipsoidal estimates for reachable sets of a nonlinear dynamical control system. Proc. Steklov Inst. Math. 271(1), 75–84 (2010)

    Article  Google Scholar 

  35. Pontryagin, L.S., Boltyanskii, V.G., Gamkrelidze, R.V., Mishchenko, E.F.: The Mathematical Theory of Optimal Processes. Wiley, New York-London (1962)

    Google Scholar 

  36. Raczynski, S.: Differential inclusions in system simulation. Trans. SCS 13(1), 47–54 (1996)

    Google Scholar 

  37. Tolstonogov, A.A.: Differential Inclusions in a Banach Space. Mathematics and its Applications, vol. 524. Kluwer Academic Publishers, Dordrecht (2000)

    Book  MATH  Google Scholar 

  38. Ushakov, V.N., Matviychuk, A.R., Ushakov, A.V.: Approximations of attainability sets and of integral funnels of differential inclusions. Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki 4, 23–39 (2011). (in Russian)

    Article  MATH  Google Scholar 

  39. Valyi, I.: Ellipsoidal approximations in problems of control. Model. Adapt. Control. 105, 361–384 (1988)

  40. Veliov, V.M.: Second order discrete approximation to strongly convex differential inclusions. Syst. Control Lett. 13(3), 263–269 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  41. Winter, R.: A characterization of the reachable set for non-linear control systems. SIAM J. Control Optim. 18(6), 599–610 (1980)

    Article  MathSciNet  Google Scholar 

  42. Wolenski, P.: The exponential formula for the reachable set of lipschitz differential inclusion. SIAM J. Control Optim. 28(5), 1148–1161 (1990)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

This work is partly supported by Grants N 14-01-31296 and N 15-37-20265 of the Russian Foundation for Basic Research

Author information

Authors and Affiliations

  1. Laboratory of Optimal Control, Institute for Systems Dynamics and Control Theory SB RAS, 134 Lermontov Str., 664033, Irkutsk, Russia

    Alexandr Yu. Gornov, Tatiana S. Zarodnyuk, Evgeniya A. Finkelstein & Anton S. Anikin

Authors
  1. Alexandr Yu. Gornov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Tatiana S. Zarodnyuk
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Evgeniya A. Finkelstein
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Anton S. Anikin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Evgeniya A. Finkelstein.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gornov, A.Y., Zarodnyuk, T.S., Finkelstein, E.A. et al. The method of uniform monotonous approximation of the reachable set border for a controllable system. J Glob Optim 66, 53–64 (2016). https://doi.org/10.1007/s10898-015-0346-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10898-015-0346-8

Keywords

Search

Navigation