On Polyhedral Estimates of Reachable Sets of Discrete-Time Systems with Uncertain Matrices and Integral Bounds on Additive Terms

  • Elena K. KostousovaEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11974)


We consider discrete-time systems of bilinear type for the case when interval bounds on the coefficients of the system are imposed, additive input terms are restricted by integral non-quadratic constraints, and initial states belong to given sets, which are assumed to be parallelepipeds. An approach for estimating the reachable sets is presented. It is based on considering reachable sets in the “extended” space and constructing external and internal estimates of them in the form of polytopes of some special shape. The specific cross-sections of these polytopes provide the parallelepiped-valued or parallelotope-valued estimates of the reachable sets in the “initial” space. Evolution of the estimates in the “extended” space is determined by recurrence relations. All the estimates can be calculated by explicit formulas. The main attention is paid to internal estimates. Illustrative examples are presented.


Discrete-time systems Reachable sets Integral constraints Uncertain matrices Polyhedral estimates Parallelepipeds Parallelotopes 



The research was supported by the Russian Foundation for Basic Research (RFBR) under Project 18-01-00544a.


  1. 1.
    Baier, R., Donchev, T.: Discrete approximation of impulsive differential inclusions. Numer. Funct. Anal. Optim. 31(6), 653–678 (2010)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Baturin, V.A., Goncharova, E.V., Pereira, F.L., Sousa, J.B.: Measure-controlled dynamic systems: polyhedral approximation of their reachable set boundary. Autom. Remote Control 67(3), 350–360 (2006)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Chernousko, F.L., Rokityanskii, D.Y.: Ellipsoidal bounds on reachable sets of dynamical systems with matrices subjected to uncertain perturbations. J. Optim. Theory Appl. 104, 1–19 (2000)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Dykhta, V.A., Sumsonuk, O.N.: Optimal Impulse Control with Applications. Fizmatlit, Moscow (2000). (Russian)Google Scholar
  5. 5.
    Filippova, T.F.: Estimates of reachable sets of impulsive control problems with special nonlinearity. In: Todorov, M.D. (ed.) Application of Mathematics in Technical and Natural Sciences – AMiTaNS 2016. AIP Conference Proceedings, vol. 1773, pp. 100004–1–100004–10. Melville, New York (2016).
  6. 6.
    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). Scholar
  7. 7.
    Guseinov, K.G., Ozer, O., Akyar, E., Ushakov, V.N.: The approximation of reachable sets of control systems with integral constraint on controls. Nonlinear Differ. Equ. Appl. 14(1–2), 57–73 (2007)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Gusev, M.I.: On convexity of reachable sets of a nonlinear system under integral constraints. IFAC-PapersOnLine 51(32), 207–212 (2018)CrossRefGoogle Scholar
  9. 9.
    Jaulin, L., Kieffer, M., Didrit, O., Walter, É.: Applied Interval Analysis: With Examples in Parameter and State Estimation, Robust Control and Robotics. Springer, London (2001). Scholar
  10. 10.
    Kostousova, E.K.: Outer polyhedral estimates for attainability sets of systems with bilinear uncertainty. J. Appl. Math. Mech. 66(4), 547–558 (2002)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Kostousova, E.K.: External polyhedral estimates for reachable sets of linear discrete-time systems with integral bounds on controls. Int. J. Pure Appl. Math. 50(2), 187–194 (2009)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Kostousova, E.K.: State estimation for linear impulsive differential systems through polyhedral techniques. Discrete Continuous Dyn. Syst. (Issue Suppl.) 466–475 (2009)Google Scholar
  13. 13.
    Kostousova, E.K.: On polyhedral estimates for trajectory tubes of dynamical discrete-time systems with multiplicative uncertainty. Discrete Continuous Dyn. Syst. (Issue Suppl.) 864–873 (2011)Google Scholar
  14. 14.
    Kostousova, E.K.: State estimation for control systems with a multiplicative uncertainty through polyhedral techniques. In: Hömberg, D., Tröltzsch, F. (eds.) CSMO 2011. IFIPAICT, vol. 391, pp. 165–176. Springer, Heidelberg (2013). Scholar
  15. 15.
    Kostousova, E.K.: External polyhedral estimates of reachable sets of linear and bilinear discrete-time systems with integral bounds on additive terms. In: Proceedings of 2018 14th International Conference Stability and Oscillations of Nonlinear Control Systems (Pyatnitskiys Conference), STAB 2018, pp. 1–4. IEEE Xplore Digital Library (2018).
  16. 16.
    Kostousova, E.K.: State estimates of bilinear discrete-time systems with integral constraints through polyhedral techniques. IFAC-PapersOnLine 51(32), 245–250 (2018)CrossRefGoogle Scholar
  17. 17.
    Krasovskii, N.N.: Theory of Control of Motion: Linear Systems. Nauka, Moskow (1968). (Russian)Google Scholar
  18. 18.
    Kurzhanski, A.B., Dar’in, A.N.: Dynamic programming for impulse controls. Annu. Rev. Control 32, 213–227 (2008)CrossRefGoogle Scholar
  19. 19.
    Kurzhanski, A.B., Daryin, A.N.: Dynamic Programming for Impulse Feedback and Fast Controls. LNCIS, vol. 468. Springer, London (2020). Scholar
  20. 20.
    Kurzhanski, A.B., Vályi, I.: Ellipsoidal Calculus for Estimation and Control. Birkhäuser, Boston (1997)CrossRefGoogle Scholar
  21. 21.
    Kurzhanski, A.B., Varaiya, P.: Dynamics and Control of Trajectory Tubes: Theory and Computation. Birkhäuser, Basel (2014)CrossRefGoogle Scholar
  22. 22.
    Mazurenko, S.S.: Partial differential equation for evolution of star-shaped reachability domains of differential inclusions. Set Valued Var. Anal. 24, 333–354 (2016)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Pierce, J.G., Schumitzky, A.: Optimal impulsive control of compartment models, I: qualitative aspects. J. Optim. Theory Appl. 18(4), 537–554 (1976)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Polyak, B.T., Nazin, S.A., Durieu, C., Walter, E.: Ellipsoidal parameter or state estimation under model uncertainty. Automatica 40(7), 1171–1179 (2004)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Sinyakov, V.V.: Method for computing exterior and interior approximations to the reachability sets of bilinear differential systems. Differ. Equ. 51(8), 1097–1111 (2015)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Vdovina, O.I., Sesekin, A.N.: Numerical construction of attainability domains for systems with impulse control. In: Proc. Steklov Inst. Math. (Suppl. 1), S246–S255 (2005)Google Scholar
  27. 27.
    Veliov, V.M.: On the relationship between continuous- and discrete-time control systems. Cent. Eur. J. Oper. Res. 18(4), 511–523 (2010)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of SciencesEkaterinburgRussia

Personalised recommendations