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Algorithm of uniform filling of nonlinear dynamic system reachable set based on maximin problem solution

  • Alexandr Yu. Gornov
  • Evgeniya A. FinkelsteinEmail author
  • Tatiana S. Zarodnyuk
Original Paper
  • 36 Downloads

Abstract

In this paper we propose an algorithm of obtaining reachable points that uniformly fill the volume of the reachable set, and thus, result in a point cloud uniformly approximating a set even with a small number of points. To solve the task of finding each additional point is to solve the maximin optimal control problem. The design of the method allows considering reachable sets not only of two-dimensional systems but of multidimensional ones as well. The computational experiments conducted with the use of the proposed algorithm confirm the efficiency of the approach.

Keywords

Reachable set Optimal control Nonlinear dynamic system 

Notes

Acknowledgements

This work is partly supported by Grant No. 15-07-03827 of the Russian Foundation for Basic Research.

References

  1. 1.
    Tolstonogov, A.A.: Differential Inclusions in a Banach Space. Kluwer, Dordrecht (2000)CrossRefGoogle Scholar
  2. 2.
    Chernousko, F.L.: State Estimation for Dynamic Systems. CRC Press, Boca Raton (1994)Google Scholar
  3. 3.
    Panasyuk, A.I.: Differential equation for nonconvex attainment sets. Math. Notes Acad. Sci. USSR 37(5), 395–400 (1985)zbMATHGoogle Scholar
  4. 4.
    Lotov, A.V.: On the concept of generalized sets of accessibility and their construction for linear controlled systems. In: Smyshlyaev, A. (ed.) Proceedings of Task Force Meeting in Input-Output Modelling. CP-82-32, pp. 236–238. International Institute for Applied Systems Analysis, Laxenburg (1982)Google Scholar
  5. 5.
    Chernousko, F.L., Rokityanskii, D.Y.: Ellipsoidal bounds on reachable sets of dynamical systems with matrices subjected to uncertain perturbation. J. Optim. Theory Appl. 104(1), 1–19 (2000)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Kurzhanski, A.B., Valyi, I.: Ellipsoidal techniques for dynamic systems: control synthesis for uncertain systems. Dyn. Control 2(2), 87–111 (1992)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Botchkarev, O., Tripakis, S.: Verification of hybrid systems with linear differential inclusions using ellipsoidal approximations. In: Krogh, B., Lynch, N. (eds.) Hybrid Systems: Computation and Control, LNCS 1790, pp. 73–88. Springer, Berlin (2000)zbMATHGoogle Scholar
  8. 8.
    Kurzhanski, A.B., Varaiya, P.: On some nonstandard dynamic programming problems of control theory. In: Giannessi, F., Maugeri, A. (eds.) Variational Analysis and Applications. Kluwer, New York (2004)Google Scholar
  9. 9.
    Kurzhanski, A.B., Mitchell, I., Varaiya, P.: Control synthesis for state constrained systems and obstacle problems. In: Proceedings of the IFAC (NOLCOS) Symposium, pp. 657–662. Elsevier, Stuttgart (2004)Google Scholar
  10. 10.
    Gurman, V.I.: Extension Principle for Optimal Control Problems. Nauka, Moscow (1985). (in Russian) zbMATHGoogle Scholar
  11. 11.
    Dontchev, A.L., Hager, W.W.: Euler approximation of the feasible set. Numer. Funct. Anal. Optim. 15(3 & 4), 245–261 (1994)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Ushakov, V.N., Matviichuk, A.R., Ushakov, A.V.: Approximations of attainability sets and of integral funnels of differential inclusions. Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki 4, 23–29 (2011). (in Russian) CrossRefGoogle Scholar
  13. 13.
    Baier, R., Chahma, I.A., Lempio, F.: Stability and convergence of Euler’s method for state-constrained differential inclusions. SIAM J. Optim. 18(3), 1004–1026 (2007)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Baier, R., Gerdts, M., Xausa, I.: Approximation of reachable sets using optimal control algorithms. Numer. Algebra Control Optim. 3(3), 519–548 (2013)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Mitchell, I., Bayen, A., Tomlin, C.J.: Validating a Hamilton–Jacobi approximation to hybrid system reachable sets. In: DiBenedetto, M.D., Sangiovanni-Vincentelli, A. (eds.) Hybrid Systems: Computation and Control, LNCS 2034, pp. 418–431. Springer, Berlin (2001)Google Scholar
  16. 16.
    Tomlin, C.J., Lygeros, J., Sastry, S.: A game theoretic approach to controller design for hybrid systems. Proc. IEEE 88(7), 949–970 (2000)CrossRefGoogle Scholar
  17. 17.
    Mitchell, I.M., Bayen, A.M., Tomlin, C.J.: A time-dependent Hamilton–Jacobi formulation of reachable sets for continuous dynamic games. IEEE Trans. Autom. Control 50(7), 947–957 (2005)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Tiwari, A., Khanna, G.: Series of abstraction for hybrid automata. In: Tomlin, C.J., Greenstreet, M.R. (eds.) Hybrid Systems: Computation and Control, LNCS 2289, pp. 465–478. Springer, Berlin (2002)Google Scholar
  19. 19.
    Alur, R., Dang, T., Ivancic, F.: Reachability analysis of hybrid systems via predicate abstraction. In: Tomlin, C.J., Greenstreet, M.R. (eds.) Hybrid Systems: Computation and Control, LNCS 2289, pp. 35–48. Springer, Berlin (2002)zbMATHGoogle Scholar
  20. 20.
    Hwang, I., Balakrishnan, H., Ghosh, R., Tomlin, C.J.: Reachability analysis of deltanotch lateral inhibition using predicate abstraction. In: Sahni, S., Prasanna, V.K., Shukla, U. (eds.) High Performance Computing-HiPC2002, LNCS 2552, pp. 715–724. Springer, Berlin (2002)Google Scholar
  21. 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)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Phillipova, 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)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Brockett, R.W.: On the Reachable Set for Bilinear Systems. Lecture Notes in Economics and Mathematical Systems, vol. 111, pp. 54–63 (1975)Google Scholar
  24. 24.
    Chentsov, A.G.: Relaxation of reachable sets and extension constructions. Cybern. Syst. Anal. 28(4), 554–561 (1992)CrossRefGoogle Scholar
  25. 25.
    Frankowska, H.: Contingent cones to reachable sets of control systems. SIAM J. Control Optim. 27(1), 170–198 (1989)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Goncharova, E., Ovseevich, A.: Limit behavior of reachable sets of linear time-invariant systems with integral bounds on control. J. Optim. Theory Appl. 157(2), 400–415 (2013)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Gornov, A.Yu.: On a class of algorithms for constructing internal estimates of reachable set. In: Proceedings of the International Workshop, Pereslavl-Zalessky (1998)Google Scholar
  28. 28.
    Gornov, AYu.: Computational Technologies for Solving Optimal Control Problems. Nauka, Novosib (2009). (in Russian) Google Scholar
  29. 29.
    Nikol’skii, M.S.: A method of approximating an attainable set for a differential inclusion. USSR Comput. Math. Math. Phys. 28(4), 192–194 (1988)CrossRefGoogle Scholar
  30. 30.
    Cellina, A., Orneals, A.: Representation of the attainable set for Lipschitzian differential inclusions. Rocky Mt. J. Math. 22(1), 117–124 (1992)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Donchev, A.L., Lempio, F.: Difference methods for differential inclusions: a survey. SIAM Rev. 34(2), 263–294 (1992)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Kastner-Maresch, A.: Implicit Runge–Kutta methods for differential inclusions. Numer. Funct. Anal. Optim. 11(9/10), 937–958 (1991)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Komarov, V.A., Pevchikh, K.E.: A method of approximating attainability sets for differential-inclusions with a specified accuracy. USSR Comput. Math. Math. Phys. 31(1), 109–112 (1991)zbMATHGoogle Scholar
  34. 34.
    Raczynski, S.: Differential inclusions in system simulation. Trans. Soc. Comput. Simul. 13(1), 47–54 (1996)Google Scholar
  35. 35.
    Wolenski, P.: The exponential formula for the reachable set of Lipschitz differential inclusion. SIAM J. Control Optim. 28(5), 1148–1161 (1990)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Zhigljavsky, A., Zilinskas, A.: Stochastic Global Optimization. Springer, Berlin (2008)zbMATHGoogle Scholar
  37. 37.
    Charalamous, C., Bandler, J.W.: Nonlinear minimax optimization as a sequence of least pth optimization with finite values of p. Int. J. Syst. Sci. 7, 377–391 (1976)CrossRefGoogle Scholar
  38. 38.
    Bertsekas, D.P.: Approximation procedures based on the method of multipliers. J. Optim. Theory Appl. 23, 487–510 (1977)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Kort, B.W., Bertsakas, D.P.: A new penalty function algorithm for constrained minimization. In: Proceedings of the 1972 IEEE Conferences on Decision and Control, New Orlean, Louisiana (1972)Google Scholar
  40. 40.
    Li, X.S.: An aggregate function method for nonlinear programming. Sci. China (A) 34, 1467–1473 (1991)MathSciNetzbMATHGoogle Scholar
  41. 41.
    Patsko, V.S., Pyatko, S.G., Fedotov, A.A.: Three-dimensional reachability set for a nonlinear control system. J. Comput. Syst. Sci. Int. 42(3), 320–328 (2003)MathSciNetzbMATHGoogle Scholar
  42. 42.
    Smirnov, A.: Attainability Analysis of the DICE Model. Interim Report IR-05-000. International Institute for Applied Systems Analysis, Laxenburg (2005)Google Scholar
  43. 43.
    Gornov, A.Yu., Zarodnyuk, T.S., Madzhara, T.I., Daneyeva, A.V., Veyalko, I.A.: A collection of test multiextremal optimal control problems. In: Optimization, Simulation and Control, Springer Optimization and Its Applications, vol. 76, pp. 257–274 (2013)Google Scholar
  44. 44.
    Gornov, AYu., Zarodnyuk, T.S., Finkelstein, E.A., Anikin, A.S.: The method of uniform monotonous approximation of the reachable set border for a controllable system. J. Glob. Optim. 66(1), 53–64 (2016)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Laboratory of Optimal ControlMatrosov Institute for System Dynamics and Control Theory SB RASIrkutskRussia

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