External estimates for reachable sets of a control system with uncertainty and combined nonlinearity

T. F. Filippova

Trudy Instituta matematiki i mehaniki, Tome 23 (2017) no. 1, pp. 262-274

Résumé

The problem of estimating the trajectory tubes of a nonlinear control dynamic system with uncertainty in the initial data is studied. It is assumed that the dynamic system has a special structure in which the nonlinear terms are defined by quadratic forms in the state coordinates and the values of uncertain initial states and admissible controls are subject to ellipsoidal constraints. The matrix of the linear terms in the velocities of the system is not known exactly; it belongs to a given compact set in the corresponding space. Thus, the dynamics of the system is complicated by the presence of bilinear components in the right-hand sides of the differential equations of the system. We consider a complex case and generalize the author's earlier results. More exactly, we assume the simultaneous presence in the dynamics of the system of bilinear functions and quadratic forms (without the assumption of their positive definiteness), and we also take into account the uncertainty in the initial data and the impact of the control actions, which may also be treated here as undefined additive disturbances. The presence of all these factors greatly complicates the study of the problem and requires an adequate analysis, which constitutes the main purpose of this study. The paper presents algorithms for estimating the reachable sets of a nonlinear control system of this type. The results are illustrated by examples.

Détail

Keywords: control system, reachable set, state estimation, uncertainty.

Bibliographie

[1] Krasovskii N.N., Teoriya upravleniya dvizheniem, Nauka, M., 1968, 476 pp.

[2] Krasovskii N.N., Subbotin A.I., Pozitsionnye differentsialnye igry, Nauka, M., 1974, 458 pp.

[3] Kurzhanskii A.B., Upravlenie i nablyudenie v usloviyakh neopredelennosti, Nauka, M., 1977, 392 pp. | MR

[4] Ushakov V.N., “K zadache postroeniya stabilnykh mostov v differentsialnoi igre sblizheniya-ukloneniya”, Izv. AN SSSR. Tekhn. kibernetika, 1980, no. 4, 29–36 | Zbl

[5] Subbotina N.N., “Universalnye optimalnye strategii v pozitsionnykh differentsialnykh igrakh”, Differents. uravneniya, 19:11 (1983), 1890–1896 | MR | Zbl

[6] Subbotina N.N., Kolpakova E.A., “Connections between optimal control problems and generalized solutions of PDEs of the first order”, IFAC Proc. Volumes (IFAC-PapersOnline), 19:3 (2014), 11381–11384 | DOI

[7] Kurzhanski A.B., Filippova T.F., “On the theory of trajectory tubes - a mathematical formalism for uncertain dynamics, viability and control”, Advances in Nonlinear Dynamics and Control: a Report from Russia, Progress in Systems and Control Theory, 17, ed. A.B. Kurzhanski, Birkhauser, Boston, 1993, 122–188 | DOI | MR

[8] Ushakov V.N., Matviichuk A.R., Ushakov A.V., “Approksimatsiya mnozhestv dostizhimosti i integralnykh voronok differentsialnykh vklyuchenii”, Vestn. Udmurt. un-ta. Matematika, Mekhanika, Kompyut. nauki, 2011, no. 4, 23–39 | Zbl

[9] Kurzhanski A.B., Valyi I., Ellipsoidal calculus for estimation and control, Birkhauser, Boston, 1997, 321 pp. | MR | Zbl

[10] Chernousko F.L., Otsenivanie fazovogo sostoyaniya dinamicheskikh sistem, Nauka, M., 1988, 320 pp.

[11] Kurzhanski A.B., Varaiya P., Dynamics and control of trajectory tubes, theory and computation, Birkhauser, Basel, 2014, 445 pp. | DOI | MR | Zbl

[12] Filippova T.F., “Trajectory tubes for impulsive control problems”, 6th European Control Conference (ECC2001), 2001, 7076349, 2766–2769

[13] Filippova T., “Differential equations of ellipsoidal state estimates in nonlinear control problems under uncertainty”, AIMS J. Discrete Contin. Dyn. Syst., 8th AIMS Conf. on Dyn. Syst., Diff. Eq. and Appl., Suppl. Vol. I, 2011, 410–419 | MR | Zbl

[14] Filippova T.F., “Set-valued dynamics in problems of mathematical theory of control processes”, International Journal of Modern Physics B, 26:25 (2012), 1–8 | DOI | MR

[15] Filippova T.F., Berezina E.V., “On state estimation approaches for uncertain dynamical systems with quadratic nonlinearity: Theory and computer simulations”, Proc. of the International Conf. on Large-Scale Scientific Computing, Springer, Berlin, 2008, 326–333 | DOI | MR | Zbl

[16] Filippova T.F., Matviichuk O.G., “Algoritmy otsenivaniya mnozhestv dostizhimosti impulsnykh upravlyaemykh sistem s ellipsoidalnymi fazovymi ogranicheniyami”, Avtomatika i telemekhanika, 2011, no. 9, 127–141 | Zbl

[17] Filippova T.F., Matviichuk O.G., “Zadachi impulsnogo upravleniya v usloviyakh neopredelennosti”, Tr. XII Vseros. soveschaniya po problemam upravleniya (VSPU-2014), Institut problem upravleniya RAN, Moskva, 2014, 1024–1032

[18] Filippova T.F., Matviychuk O.G., “Algorithms of estimating reachable sets of nonlinear control systems with uncertainty”, Proc. of the 7th Chaotic Modeling and Simulation Internat. Conf., ed. Ch. H. Skiadas, Internat. Society for the Advancement of Science and Technology, 2014, 115–124

[19] Chernousko F.L., “Ellipsoidalnaya approksimatsiya mnozhestv dostizhimosti lineinoi sistemy s neopredelennoi matritsei”, Prikl. matematika i mekhanika, 60:6 (1996), 940–950 | MR | Zbl

[20] Filippova T.F., “Differential equations of ellipsoidal state estimates for bilinear-quadratic control systems under uncertainty”, 9th Chaotic Modeling and Simulation Intern. Conf. (CHAOS 2016), Book Abstr., London, 2016, 32

[21] Filippova T.F., Matviychuk O.G., Kostousova E.K., “Estimation techniques for uncertain dynamical systems with bilinear and quadratic nonlinearities”, Dynamical Systems: Control and Stability, Proc. of the 13th Internat. Conf. on Dynamical Systems: Theory and Applications (DSTA-2015), eds. J.M.J. Awrejcewicz, M. Kaźmierczak and P. Olejnik, 2015, 185–196

[22] Filippova T.F., Matviychuk O.G., “Estimates of reachable sets of control systems with bilinear-quadratic nonlinearities”, Ural Math. J., 1:1 (2015), 45–54 | DOI

[23] Filippova T.F., “Asymptotic behavior of the ellipsoidal estimates of reachable sets of nonlinear control systems with uncertainty”, Proc. of the 8th European Nonlinear Dynamics Conf. (ENOC2014), CD-ROM vol., eds. H. Ecker, A. Steindl, S. Jakube, 2014, Paper-ID 149, 1–2

[24] Gusev M.I., “Vneshnie otsenki mnozhestv dostizhimosti nelineinykh upravlyaemykh sistem”, Avtomatika i telemekhanika, 2012, no. 3, 39–51 | Zbl

[25] Kostousova E.K., “On tight polyhedral estimates for reachable sets of linear differential systems”, AIP Conf. Proc., 1493 (2012), 579–586 | DOI

[26] Kostousova E.K., “State estimation for control systems with a multiplicative uncertainty through polyhedral techniques”, IFIP Advances in Information and Communication Technology (IFIP AICT), 391, 2013, 165–176 | DOI | MR | Zbl