ON SOLVING AN ENHANCED EVASION PROBLEM FOR LINEAR DISCRETE–TIME SYSTEMS

Elena K. Kostousova     (Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, 16 S. Kovalevskaya Str., Ekaterinburg, 620108, Russian Federation)

Abstract


We consider the problem of an enhanced evasion for linear discrete-time systems, where there are two conflicting bounded controls and the aim of one of them is to be guaranteed to avoid the trajectory hitting a given target set at a given final time and also at intermediate instants. First we outline a common solution scheme based on the construction of so called solvability tubes or repulsive tubes. Then a much more quick and simple for realization method based on the construction of the tubes with parallelepiped-valued cross-sections is presented under assumptions that the target set is a parallelepiped and parallelotope-valued constraints on controls are imposed. An example illustrating this method is considered.


Keywords


Linear control systems, Discrete-time systems, Uncertainty, Evasion problem, Parallelepipeds

Full Text:

PDF

References


  1. Ananyev B.I. An application of motion correction methods to the alignment problem in navigation. Ural Math. J., 2016. Vol. 2, No. 2. P. 16–26. DOI: 10.15826/umj.2016.2.002
  2. Bertsekas D.P., Rhodes I.B. On the minimax reachability of target sets and target tubes. Automatica, 1971. Vol. 7, No. 2. P. 233–247. DOI: 10.1016/0005-1098(71)90066-5
  3. Botkin N., Martynov K., Turova V., Diepolder J. Generation of dangerous disturbances for flight systems. Dyn. Games and Appl., 2019. Vol. 9. P. 628–651. DOI: 10.1007/s13235-018-0259-5
  4. Chernousko F.L. Estimation for Dynamic Systems. Boca Raton: CRC Press, 1994. 304 p.
  5. Filippova T.F. Control and estimation for a class of impulsive dynamical systems. Ural Math. J., 2019. Vol. 5, No. 2. P. 21–30. DOI: 10.15826/umj.2019.2.003
  6. Filippova T.F., Matviychuk O.G. Optimality principles for solving nonlinear control problems under uncertainty. In: CHAOS 2021: Proc. 14th Chaotic Modeling and Simulation Int. Conf., Skiadas C.H., Dimotikalis Y. (eds). Springer Pros. Complexity, 2022. P. 143–154. DOI: 10.1007/978-3-030-96964-6_12
  7. Gusev M.I., Zykov I.V. External estimates and comparison principle for trajectory tubes of nonlinear control systems. AIP Conf. Proc., 2019. Vol. 2164. Art. no. 060008. 10 p. DOI: 10.1063/1.5130810
  8. Kamneva L.V., Patsko V.S. Construction of the solvability set in differential games with simple motion and nonconvex terminal set. Proc. Steklov Inst. Math., 2018. Vol. 301, Suppl. 1. P. S57–S71. DOI: 10.1134/S008154381805005X
  9. Kostousova E.K. State estimation for dynamic systems via parallelotopes: Optimization and parallel computations. Optim. Methods Softw., 1998. Vol. 9, No. 4. P. 269–306. DOI: 10.1080/10556789808805696
  10. Kostousova E.K. On polyhedral estimates in problems of the synthesis of control strategies in linear multistep systems. In: Algoritmy i programmnye sredstva parallel’nykh vychislenii [Algorithms and Software for Parallel Computing]. Ekaterinburg: IMM UrO RAN, 2006. Vol. 9. P. 84–105. (in Russian)
  11.  Kostousova E.K. On the polyhedral method of solving problems of control strategy synthesis. Proc. Steklov Inst. Math., 2016. Vol. 292, Suppl. 1. P. S140–S155. DOI: 10.1134/S0081543816020127
  12. Kostousova E.K. On polyhedral control synthesis for dynamical discrete-time systems under uncertainties and state constraints. Discrete Contin. Dyn. Syst., 2018. Vol. 38, No. 12. P. 6149–6162. DOI: 10.3934/dcds.2018153
  13. Kostousova E.K. On a polyhedral method of solving an evasion problem for linear dynamical systems. AIP Conf. Proc., 2019. Vol. 2164. Art. no. 060009. 10 p. DOI: 10.1063/1.5130811
  14. Kostousova E.K. On a polyhedral method for solving an evasion problem for linear discrete-time systems. In: Proc. of 2020 15th Int. Conf. on Stability and Oscillations of Nonlinear Control Systems (Pyatnitskiy’s Conference) (STAB). IEEE Xplore, 2020. P. 1–4. DOI: 10.1109/STAB49150.2020.9140471
  15. Kostousova E.K. External polyhedral estimates of reachable sets of discrete-time systems with integral bounds on additive terms. Math. Control Relat. Fields, 2021. Vol. 11, No. 3. P. 625-641. DOI: 10.3934/mcrf.2021015
  16. Krasovskii N.N., Subbotin A.I. Game-Theoretical Control Problems. New York: Springer, 1988. 517 p.
  17. Kurzhanski A.B., Daryin A.N. Dynamic Programming for Impulse Feedback and Fast Controls: The Linear Systems Case. Lect. Notes Control Inf. Sci., vol. 468. London: Springer, 2020. 275 p. DOI: 10.1007/978-1-4471-7437-0
  18. Kurzhanski A.B., Vályi I. Ellipsoidal Calculus for Estimation and Control. Boston: Birkhäuser, 1997. 321 p.
  19. Kurzhanski A.B., Varaiya P. Dynamics and Control of Trajectory Tubes. Theory and Computation. Systems Control Found. Appl., vol. 85. Basel: Birkhäuser, 2014. 445 p. DOI: 10.1007/978-3-319-10277-1
  20. Kurzhanskiy A.A., Varaiya P. Theory and computational techniques for analysis of discrete-time control systems with disturbances. Optim. Methods Softw., 2011. Vol. 26, No. 4–5. P. 719–746. DOI: 10.1080/10556788.2010.548867
  21. Martynov K., Botkin N., Turova V., Diepolder J. Real-time control of aircraft take-off in windshear. Part I: Aircraft model and control schemes. In: Proc. of 2017 25th Mediterranean Conf. on Control and Automation (MED 2017). IEEE Xplore, 2017. P. 277–284. DOI: 10.1109/MED.2017.7984131
  22. Martynov K., Botkin N., Turova V., Diepolder J. Quick construction of dangerous disturbances in conflict control problems. In: Advances in Dynamic Games. Ramsey D.M., Renault J. (eds). Ann. Internat. Soc. Dynam. Games, vol. 17. Cham: Birkhäuser, 2020. P. 3–24. DOI: 10.1007/978-3-030-56534-3_1
  23. Schneider R.G. Convex Bodies: The Brunn-Minkowski Theory. Cambridge: Cambridge Univ. Press, 1993. 490 p.
  24. Ushakov V.N., Uspenskii A.A., Matviychuk A.R., Malev A.G. Stability defect of sets in game problems of approaching. IFAC-PapersOnline, 2012. Vol. 45, No. 25. P. 89–93. DOI: 10.3182/20120913-4-IT-4027.00063




DOI: http://dx.doi.org/10.15826/umj.2022.1.006

Article Metrics

Metrics Loading ...

Refbacks

  • There are currently no refbacks.