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Trudy Inst. Mat. i Mekh. UrO RAN, 2010, Volume 16, Number 1, Pages 223–232 (Mi timm539)  

This article is cited in 17 scientific papers (total in 17 papers)

Differential equations of ellipsoidal estimates for reachable sets of a nonlinear dynamical control system

T. F. Filippova

Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences

Abstract: The problem of estimating trajectory tubes of a nonlinear control system with uncertainty in initial data is considered. It is assumed that the dynamical system has a special structure, in which nonlinear terms are quadratic in phase coordinates and the values of the uncertain initial states and admissible controls are subject to ellipsoidal constraints. Differential equations are found that describe the dynamics of the ellipsoidal estimates of reachable sets of the nonlinear dynamical system under consideration. To estimate reachable sets of the nonlinear differential inclusion corresponding to the control system, we use results from the theory of ellipsoidal estimation and the theory of evolution equations for multivalued states of dynamical systems under uncertainty.

Keywords: reachable set, trajectory tubes, set-valued estimates, differential inclusions, ellipsoidal estimation, control systems, dynamical systems.

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English version:
Proceedings of the Steklov Institute of Mathematics (Supplementary issues), 2010, 271, suppl. 1, S75–S84

Bibliographic databases:

Document Type: Article
UDC: 517.977
Received: 28.12.2009

Citation: T. F. Filippova, “Differential equations of ellipsoidal estimates for reachable sets of a nonlinear dynamical control system”, Trudy Inst. Mat. i Mekh. UrO RAN, 16, no. 1, 2010, 223–232; Proc. Steklov Inst. Math. (Suppl.), 271, suppl. 1 (2010), S75–S84

Citation in format AMSBIB
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\paper Differential equations of ellipsoidal estimates for reachable sets of a~nonlinear dynamical control system
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\pages 223--232
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    Citing articles on Google Scholar: Russian citations, English citations
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    2. V. N. Ushakov, A. R. Matviichuk, P. D. Lebedev, “Defekt stabilnosti v igrovoi zadache o sblizhenii v moment”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 2010, no. 3, 87–103  mathnet  elib
    3. Ushakov V.N., “On the stability property in a game-theoretic approach problem with fixed terminal time”, Differ. Equ., 47:7 (2011), 1046–1058  crossref  mathscinet  zmath  isi  elib  elib  scopus
    4. A. A. Davydov, V. M. Zakalyukin, “Controllability of non-linear systems: generic singularities and their stability”, Russian Math. Surveys, 67:2 (2012), 255–280  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    5. P. D. Lebedev, A. V. Ushakov, “Approximating sets on a plane with optimal sets of circles”, Autom. Remote Control, 73:3 (2012), 485–493  mathnet  crossref  isi
    6. V. N. Ushakov, P. D. Lebedev, A. R. Matviychuk, A. G. Malev, “Differential games with fixed terminal time and estimation of the instability degree of sets in these games”, Proc. Steklov Inst. Math., 277 (2012), 266–277  mathnet  crossref  mathscinet  isi
    7. Matviychuk O.G., “Estimation Problem for Impulsive Control Systems Under Ellipsoidal State Bounds and with Cone Constraint on the Control”, Applications of Mathematics in Engineering and Economics (AMEE'12), AIP Conference Proceedings, 1497, eds. Pasheva V., Venkov G., Amer Inst Physics, 2012, 3–12  crossref  adsnasa  isi  scopus
    8. Ushakov V.N., Matviichuk A.R., Ushakov A.V., Kazakov A.L., “O postroenii reshenii zadachi o sblizhenii fiksirovannyi moment vremeni”, Izvestiya irkutskogo gosudarstvennogo universiteta. seriya: matematika, 5:4 (2012), 95–115  mathnet  mathscinet  zmath  elib
    9. August E., Koeppl H., “Computing Enclosures for Uncertain Biochemical Systems”, IET Syst. Biol., 6:6 (2012), 232–240  crossref  isi  elib  scopus
    10. August E., Lu J., Koeppl H., “Trajectory Enclosures for Nonlinear Systems with Uncertain Initial Conditions and Parameters”, 2012 American Control Conference (Acc), Proceedings of the American Control Conference, IEEE Computer Soc, 2012, 1488–1493  crossref  isi
    11. V. N. Ushakov, A. R. Matviichuk, A. V. Ushakov, G. V. Parshikov, “Invariantnost mnozhestv pri konstruirovanii reshenii zadachi o sblizhenii v fiksirovannyi moment vremeni”, Tr. IMM UrO RAN, 19, no. 1, 2013, 264–283  mathnet  mathscinet  elib
    12. V. N. Ushakov, A. R. Matviychuk, G. V. Parshikov, “A method for constructing a resolving control in an approach problem based on attraction to the solvability set”, Proc. Steklov Inst. Math. (Suppl.), 284, suppl. 1 (2014), 135–144  mathnet  crossref  mathscinet  isi  elib
    13. T. F. Filippova, “Estimates of reachable sets of control systems with nonlinearity and parametric perturbations”, Proc. Steklov Inst. Math. (Suppl.), 292, suppl. 1 (2016), 67–75  mathnet  crossref  mathscinet  isi  elib
    14. A. Yu. Gornov, E. A. Finkel'shtein, “Algorithm for piecewise-linear approximation of the reachable set boundary”, Autom. Remote Control, 76:3 (2015), 385–393  mathnet  crossref  isi  elib  elib
    15. Filippova T.F., “Ellipsoidal Estimates of Reachable Sets For Control Systems With Nonlinear Terms”, IFAC PAPERSONLINE, 50:1 (2017), 15355–15360  crossref  isi  scopus
    16. Shao L., Zhao F., Cong Yu., “Approximation of Convex Bodies By Multiple Objective Optimization and An Application in Reachable Sets”, Optimization, 67:6 (2018), 783–796  crossref  mathscinet  zmath  isi  scopus
    17. Filippova T.F., “The Hjb Approach and State Estimation For Control Systems With Uncertainty”, IFAC PAPERSONLINE, 51:13 (2018), 7–12  crossref  isi  scopus
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