Skip to main content
SpringerLink
Log in

Guaranteed-accuracy approximation of reachable sets for a linear dynamic system subject to impulse actions

  • Published:
Computational Mathematics and Mathematical Physics Aims and scope Submit manuscript

Abstract

A method is proposed for approximating the reachable set of a dynamic system with a state space dimension no higher than six-eight considered on a finite time interval. The system is governed by linear differential equations with piecewise constant coefficients and impulse actions specified at prescribed times. The method is based on guaranteed-accuracy polyhedral approximations of reachable sets at researcher-specified times. Every approximation is constructed using the preceding one. A procedure is described for choosing parameters of the method that ensure the required accuracy with close-to-minimal time costs.

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. L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, and E. F. Mishchenko, The Mathematical Theory of Optimal Processes (Nauka, Moscow, 1983; Gordon & Breach, New York, 1986).

    MATH  Google Scholar 

  2. F. L. Chernous’ko, Estimation of the Phase State of Dynamic Systems (Nauka, Moscow, 1988) [in Russian].

    Google Scholar 

  3. A. B. Kurzhanski and I. Valyi, Ellipsoidal Calculus for Estimation and Control (Birkh@auser, Boston, 1996).

    Google Scholar 

  4. A. V. Lotov, Doctoral Dissertation in Mathematics and Physics (Vychisl. Tsentr, Akad. Nauk SSSR, Moscow, 1985).

    Google Scholar 

  5. A. V. Lotov, “Numerical Method for Constructing the Reachable Set of a Linear Control System with State Constraints,” Zh. Vychisl. Mat. Mat. Fiz. 15, 67–78 (1975).

    MathSciNet  Google Scholar 

  6. A. V. Lotov, V. A. Bushenkov, G. K. Kamenev, and O. L. Chernykh, Computer and Tradeoff Search: The Method of Reachable Goals (Nauka, Moscow, 1997).

    Google Scholar 

  7. A. V. Lotov, V. A. Bushenkov, and G. K. Kamenev, Interactive Decision Maps: Approximation and Visualization of Pareto Frontier (Kluwer Academic, Boston, 2004).

    MATH  Google Scholar 

  8. N. B. Brusnikina, “Calculation of the Support Function of the Reachable Set of a Control System with Guaranteed Error Estimate,” Vestn. Mosk. Univ., Ser. Vychisl. Mat. Kibern., No. 1, 42–48 (2006).

  9. N. B. Brusnikina, “Multistep Approximation of a Sequence of Reachable Sets for a Linear Autonomous Control System with Guaranteed Error Estimate,” Collected Papers by Young Researchers of the Faculty of Computational Mathematics and Cybernetics of Moscow State University (Mosk. Gos. Univ., Moscow, 2005), Vol. 2, 24–30.

    Google Scholar 

  10. V. I. Blagodatskikh, Introduction to Optimal Control (Vysshaya Shkola, Moscow, 2001) [in Russian].

    Google Scholar 

  11. R. Rockafellar, Convex Analysis (Princeton Univ. Press, Princeton, 1970; Mir, Moscow, 1973).

    MATH  Google Scholar 

  12. N. B. Brusnikina and G. K. Kamenev, “On the Complexity and Methods of Polyhedral Approximations of Convex Bodies with a Partially Smooth Boundary,” Zh. Vychisl. Mat. Mat. Fiz. 45, 1555–1565 (2005) [Comput. Math. Math. Phys. 45, 1500–1510 (2005)].

    MATH  MathSciNet  Google Scholar 

  13. N. S. Bakhvalov, N. P. Zhidkov, and G. M. Kobel’kov, Numerical Methods (Nauka, Moscow, 1987) [in Russian].

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Computational Mathematics and Cybernetics, Moscow State University, Leninskie gory, Moscow, 119992, Russia

    N. B. Brusnikina

  2. Dorodnicyn Computing Center, Russian Academy of Sciences, ul. Vavilova 40, Moscow, 119991, Russia

    A. V. Lotov

Authors
  1. N. B. Brusnikina
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. V. Lotov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. V. Lotov.

Additional information

Original Russian Text © N.B. Brusnikina, A.V. Lotov, 2007, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2007, Vol. 47, No. 11, pp. 1855–1864.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Brusnikina, N.B., Lotov, A.V. Guaranteed-accuracy approximation of reachable sets for a linear dynamic system subject to impulse actions. Comput. Math. and Math. Phys. 47, 1779–1787 (2007). https://doi.org/10.1134/S096554250711005X

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S096554250711005X

Keywords

Search

Navigation