Skip to main content
SpringerLink
Log in

Algorithm for piecewise-linear approximation of the reachable set boundary

  • Nonlinear Systems
  • Published:
Automation and Remote Control Aims and scope Submit manuscript

Abstract

The studies and approaches to estimating the reachable sets of the control systems were reviewed in brief. An algorithm for piecewise-linear approximation of the boundary of the reachable set was proposed on the basis of solving a special problem of optimal control in terms of the criterion of volume maximum of the corresponding estimate of the reachable set. The results of computer experiments were presented.

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

Similar content being viewed by others

References

  1. Tolstonogov, A.A., Differentsial’nye vklyucheniya v banakhovom prostranstve (Differential Inclusions in the Banach Space), Novosibirsk: Nauka, 1986.

    Google Scholar 

  2. Chernous’ko, F.L., Otsenivanie fazovogo sostoyaniya dinamicheskikh sistem (Estimation of the Phase State of Dynamic Systems), Moscow: Nauka, 1988.

    MATH  Google Scholar 

  3. Panasyuk, A.I., Differential Equations of Nonconvex Reachable Sets, Mat. Zametki, 1985, vol. 37, no. 5, pp. 717–726.

    MATH  MathSciNet  Google Scholar 

  4. Khrustalev, M.M., Exact Description of Reachable Sets and Global Optimality Conditions for Dynamic Systems, Autom. Remote Control, 1988, vol. 49, no. 5, part 1, pp. 597–604; no. 7, part 1, pp. 874–881.

    MATH  MathSciNet  Google Scholar 

  5. Lotov, A.V., On the Notion of Generalized Reachable Sets and Their Construction for the Linear Control Systems, Dokl. Akad. Nauk SSSR, 1980, vol. 250, no. 5, pp. 1081–1083.

    MathSciNet  Google Scholar 

  6. Gurman, V.I., Printsip rasshireniya v zadachakh optimal’nogo upravleniya (Principle of Extension in Optimal Control Problems), Moscow: Nauka, 1985.

    Google Scholar 

  7. Kurzhanski, A.B. and Valyi, I., Ellipsoidal Techniques for Dynamic Systems. Control Synthesis for Uncertain Systems, Dynam. Control, 1992, vol. 2, no. 2, pp. 87–11.

    Article  MATH  MathSciNet  Google Scholar 

  8. Ushakov, V.N., Matviichuk, A.R., and Ushakov, A.V., Approximation of the Reachable Sets and Integral Funnels, Vestn. Udmurt. Univ., Mat., 2011, no. 4, pp. 23–39.

    Google Scholar 

  9. Nikol’skii, M.S., On One Method of Approximation of the Reachable Set for Control Process, Zh. Vychisl. Mat. Mat. Fiz., 1988, vol. 28, no. 8, pp. 1252–1254.

    Google Scholar 

  10. Chentsov, A.G., Permissible Sets and Their Relaxations. I. Functional-Differential Equations, in Mezhvuz. sb. nauchn. tr. (Interuniv. Coll. Sci. Works), Perm: Perm. Politekh. Inst., 1990, pp. 185–196.

    Google Scholar 

  11. Komarov, V.A. and Pevchikh, K.E., On One Method of Approximation of the Reachable Sets of Differential Inclusions with Given Precision, Zh. Vychisl. Mat. Mat. Fiz., 1991, vol. 31, no. 1, pp. 152–157.

    MathSciNet  Google Scholar 

  12. Konstantinov, G.N., Normirovanie vozdeistvii na dinamicheskie sistemy (Normalization of Actions on the Dynamic Systems), Irkutsk: Irkutsk. Gos. Univ., 1983.

    Google Scholar 

  13. Dontchev, A.L. and Hager, W.W., Euler Approximation of the Feasible Set, Numer. Funct. Anal. Optim., 1994, vol. 15(3–4), pp. 245–261.

    Article  MATH  MathSciNet  Google Scholar 

  14. Dykhta, V.A. and Samsonyuk, O.N., Estimates of the Reachable Sets and Optimality Conditions for the Nonlinear Control Systems with Discontinuous Trajectories, Vestn. Tambov. Univ., Ser. Estestv. Tekhn., 2009, vol. 14, no. 4, pp. 707–709.

    Google Scholar 

  15. Raczynski, S., Differential Inclusions in System Simulation, Trans. SCS, 1996, vol. 13(1), pp. 47–54.

    Google Scholar 

  16. Wolenski, P., The Exponential Formula for the Reachable Set of Lipschitz Differential Inclusion, SIAM J. Control Optim., 1990, vol. 28, no. 5, pp. 1148–1161.

    Article  MATH  MathSciNet  Google Scholar 

  17. Donchev, A.L. and Lempio, F., Difference Methods for Differential Inclusions: A Survey, SIAM Rev., 1992, vol. 34(2), pp. 263–294.

    Article  MathSciNet  Google Scholar 

  18. Kastner-Maresch, A., Implicit Runge-Kutta Methods for Differential Inclusions, Numer. Funct. Anal. Optim., 1991, vol. 11, pp. 937–958.

    Article  MATH  MathSciNet  Google Scholar 

  19. Cellina, A. and Orneals, A., Representation of the Attainable Set for Lipschitzian Differential Inclusions, Rocky Mountain J. Math., 1992, vol. 22, no. 1, pp. 117–124.

    Article  MATH  MathSciNet  Google Scholar 

  20. Frankowska, H., Contingent Cones to Reachable Sets of Control Systems, SIAM J. Control Optim., 1989, vol. 27, no. 1, pp. 170–198.

    Article  MATH  MathSciNet  Google Scholar 

  21. Brockett, R.W., On the Reachable Set for Bilinear Systems, Lecture Notes Econ. Math. Syst., 1975, vol. 111, pp. 54–63.

    Article  MathSciNet  Google Scholar 

  22. Kostousova, E.K., On Boundedness and Unboundedness of the External Polyhedral Estimates of the Reachable Sets of Linear Differential Systems, Tr. Inst. Mat. Mekh. UrO RAN, 2009, vol. 15, no. 4, pp. 134–145.

    Google Scholar 

  23. Filippova, T.F., Differential Equations of Ellipsoidal Estimates of the Reachable Sets of Nonlinear Dynamic Control System, Tr. Inst. Mat. Mekh. UrO RAN, 2010, vol. 16, no. 1, pp. 223–232.

    Google Scholar 

  24. Gornov, A.Yu., On a Class of Algorithms for Constructing Internal Estimates of Reachable Set, in Proc. Int. Workshop, Pereslavl-Zalessky, Sept. 7–11, 1998.

  25. Gornov, A.Yu., Vychislitel’nye tekhnologii resheniya zadach optimal’nogo upravleniya (Computing Technologies for Solution of the Optimal Control Problems), Novosibirsk: Nauka, 2009.

    Google Scholar 

  26. Zhigljavsky, A. and Zilinskas, A., Stochastic Global Optimization, New York: Springer, 2008.

    MATH  Google Scholar 

  27. Gordon, V.O. and Sementsov-Ognevskii, M.A., Kurs nachertatel’noi geometrii (Course of Descriptive Geometry), Moscow: Vysshaya Shkola, 2000.

    Google Scholar 

  28. Hilbert, D. and Cohn-Vossenn, S., Anschauliche Geometrie, Berlin: Verlang von J. Springer, 1932. Translated into English under the title Geometry and the Imagination, New York: Chelsea, 1952 (reprints of 1983, 1990), and Russian under the title Naglyadnaya geometriya (Descriptive Geometry), Moscow, Leningrad: ONTI, 1936 (reprints of 1983, Gostekhizdat, and 2010, Editorial URSS).

    Book  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of System Dynamics and Control Theory, Siberian Branch, Russian Academy of Sciences, Irkutsk, Russia

    A. Yu. Gornov & E. A. Finkel’shtein

Authors
  1. A. Yu. Gornov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. E. A. Finkel’shtein
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. Yu. Gornov.

Additional information

Original Russian Text © A.Yu. Gornov, E.A. Finkel’shtein, 2015, published in Avtomatika i Telemekhanika, 2015, No. 3, pp. 22–31.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gornov, A.Y., Finkel’shtein, E.A. Algorithm for piecewise-linear approximation of the reachable set boundary. Autom Remote Control 76, 385–393 (2015). https://doi.org/10.1134/S0005117915030030

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0005117915030030

Keywords

Search

Navigation