THE LIMITS OF APPLICABILITY OF THE LINEARIZATION METHOD IN CALCULATING SMALL–TIME REACHABLE SETS

Mikhail I. Gusev     (Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, 16 S. Kovalevskaya Str., Ekaterinburg, 620990, Russian Federation)

Abstract


The reachable sets of nonlinear systems are usually quite complicated. They, as a rule, are non-convex and arranged to have rather complex behavior. In this paper, the asymptotic behavior of reachable sets of nonlinear control-affine systems on small time intervals is studied. We assume that the initial state of the system is fixed, and the control is bounded in the \(\mathbb{L}_2\)-norm. The subject of the study is the applicability of the linearization method for a sufficiently small length of the time interval. We provide sufficient conditions under which the reachable set of a nonlinear system is convex and asymptotically equal to the reachable set of a linearized system. The concept of asymptotic equality is defined in terms of the Banach-Mazur metric in the space of sets.  The conditions depend on the behavior of the controllability Gramian of the linearized system – the smallest eigenvalue of the Gramian should not tend to zero too quickly when the length of the time interval tends to zero.  The indicated asymptotic behavior occurs for a reasonably wide class of second-order nonlinear control systems but can be violated for systems of higher dimension.  The results of numerical simulation illustrate the theoretical conclusions of the paper.

Keywords


Nonlinear control systems; Small-time reachable sets; Asymptotics; Integral constraints; Linearization

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References


  1. Baier R., Gerdts M., Xausa I. Approximation of reachable sets using optimal control algorithms. Numer. Algebra Control Optim., 2013. Vol. 3, No. 3. P. 519–548. DOI: 10.3934/naco.2013.3.519
  2. Dmitruk A.V., Milyutin A.A., Osmolovskii N.P. Lyusternik’s theorem and the theory of extrema. Russian Math. Surveys, 1980. Vol. 35, No. 6. P. 11–51. DOI: 10.1070/RM1980v035n06ABEH001973
  3. Filippova T.F. Ellipsoidal estimates of reachable sets for control systems with nonlinear terms. IFAC-PapersOnLine, 2017. Vol. 50, No. 1. P. 15355–15360. DOI: 10.1016/j.ifacol.2017.08.2460
  4. Goncharova E., Ovseevich A. Small-time reachable sets of linear systems with integral control constraints: birth of the shape of a reachable set. J. Optim. Theory Appl., 2016. Vol. 168. P. 615–624. DOI: 10.1007/s10957-015-0754-4
  5. 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., 2007. Vol. 14. P. 57–73. DOI: 10.1007/s00030-006-4036-6
  6. Guseinov Kh.G., Nazlipinar A.S. Attainable sets of the control system with limited resources. Trudy Inst. Mat. i Mekh. UrO RAN, 2010. Vol. 16, No. 5. P. 261–268.
  7. Gusev M. On reachability analysis of nonlinear systems with joint integral constraints. In: Lecture Notes in Comput. Sci., vol. 10665: Large-Scale Scientific Computing. LSSC 2017. Lirkov I., Margenov S. (eds.) Cham: Springer, 2018. P. 219–227. DOI: 10.1007/978-3-319-73441-5_23
  8. Gusev M.I., Zykov I.V. On extremal properties of the boundary points of reachable sets for control systems with integral constraints. Proc. Steklov Inst. Math., 2018. Vol. 300. Suppl. 1. P. 114–125. DOI: 10.1134/S0081543818020116
  9. Gusev M.I. Estimates of the minimal eigenvalue of the controllability Gramian for a system containing a small parameter. In: Lecture Notes in Comput. Sci., vol. 11548: Int. Conf. Mathematical Optimization Theory and Operations Research. MOTOR 2019. Khachay M., Kochetov Y., Pardalos P. (eds.) Cham: Springer, 2019. P. 461–473. DOI: 10.1007/978-3-030-22629-9_32
  10. Gusev M.I., Osipov I.O. Asimptoticheskoye povedeniye mnozhestv dostizhimosti na malykh vremennykh promezhutkakh [Asymptotic behavior of reachable sets on small time intervals]. Trudy Inst. Mat. Mekh. UrO RAN, 2019. Vol. 25, No. 3. P. 86–99. DOI: 10.21538/0134-4889-2019-25-3-86-99 (in Russian)
  11. Kostousova E.K. On polyhedral estimates for reachable sets of discrete-time systems with bilinear uncertainty. Autom. Remote Control, 2011. Vol. 72. P. 1841–1851. DOI: 10.1134/S0005117911090062
  12. Krener A.J., Schättler H. The structure of small-time reachable sets in low dimensions. SIAM J. Control Optim., 1989. Vol. 27. No. 1. P. 120–147. DOI: 10.1137/0327008
  13. Kurzhanski A.B., Varaiya P. Dynamic optimization for reachability problems. J. Optim. Theory Appl., 2001. Vol. 108(2). P. 227–251. DOI: 10.1023/A:1026497115405
  14. Kurzhanski A.B., Varaiya P. Dynamics and Control of Trajectory Tubes. Theory and Computation. Systems Control Found. Appl., vol. 85. Basel: Birkh¨auser, 2014. 445 p. DOI: 10.1007/978-3-319-10277-1
  15. Patsko V.S., Pyatko S.G., Fedotov A.A. Three-dimensional reachability set for a nonlinear control system. J. Comput. Syst. Sci. Int., 2003. Vol. 42. No. 3. P. 320–328.
  16. Polyak B. T. Convexity of the reachable set of nonlinear systems under \(L_2\) bounded controls. Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 2004. Vol. 11. P. 255–267.
  17. Polyak B.T. Local programming. Comput. Math. Math. Phys., 2001. Vol. 41, No. 9. P. 1259–1266.
  18. Lee E.B. and Marcus L. Foundations of Optimal Control Theory. New York: J. Willey and Sons Inc., 1967. 576 p.
  19. Schättler H. Small-time reachable sets and time-optimal feedback control. In: Nonsmooth Analysis and Geometric Methods in Deterministic Optimal Control. Vol. 78: The IMA Volumes in Mathematics and its Applications. Mordukhovich B.S., Sussmann H.J. (eds.) New York: Springer, 1996. P. 203–225. DOI: 10.1007/978-1-4613-8489-2_9
  20. Tochilin P.A. On the construction of nonconvex approximations to reach sets of piecewise linear systems. Differ. Equ., 2015. Vol. 51, No. 11. P. 1499–1511. DOI: 10.1134/S0012266115110117
  21. Vdovin S.A., Taras’yev A.M., Ushakov V.N. Construction of the attainability set of a Brockett integrator. J. Appl. Math. Mech., 2004. Vol. 68, No. 5. P. 631–646. DOI: 10.1016/j.jappmathmech.2004.09.001
  22. Zykov I.V. On external estimates of reachable sets of control systems with integral constraints. Izv. IMI UdGU, 2019. Vol. 53. P. 61–72. DOI: 10.20537/2226-3594-2019-53-06 (in Russian)
  23. Walter W. Differential and Integral Inequalities. Berlin, Heidelberg: Springer-Verlag, 1970. 354 p. DOI: 10.1007/978-3-642-86405-6



DOI: http://dx.doi.org/10.15826/umj.2020.1.006

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