Estimates of Reachable Sets for Systems with Impulsive Control, Uncertainty and Nonlinearity
The problem of estimating trajectory tubes for nonlinear controlled dynamical systems 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 on state coordinates and the values of uncertain initial states and admissible controls are constrained by ellipsoidal restrictions. Matrix of linear terms in the state system velocities is also not exactly known, but it belongs to the known compact in the corresponding space, i.e. the dynamics of the system is complicated by the presence of bilinear components in the right-hand sides of the system of differential equations. We solve the problem of estimating the reachable sets of nonlinear control system of this kind, the results make it possible to construction of the corresponding numerical algorithms. We solve here the problem of estimating the reachable sets of nonlinear controlled system of this kind, the results make it possible to construct the corresponding numerical algorithms.
93B03, 93C41, 49N25
1. Gurman V. I. Printsip rasshireniya v zadachakh upravleniya [The Extension Principle in Control Problems]. Moscow, Nauka, 1985.
2. Gurman V. I., Sachkov Yu. L. Representation and Realization of the Generalized Solutions of the Unlimited-locus Controllable Systems. Autom. Remote Control, 2008, vol. 69, no 4, pp. 609–617.
3. Dykhta V. A., Samsonyuk O. N. Optimal’noe impul’snoe upravlenie s prilozheniyami [Optimal Impulsive Control with Applications]. Moscow, Fizmatlit, 2000.
4. Zavalishchin S. T., Sesekin A. N. Impul’snye processy: modeli i prilozhenija [Impulse Processes: Models and Applications]. Moscow, Nauka, 1991.
5. Krasovskii N. N. Teoriya upravleniya dvizheniem [Theory of motion control]. Moscow, Nauka, 1968.
6. Krotov V. F., Gurman V. I. Metody i zadachi optimal’nogo upravleniya [Methods and Problems of Optimal Control]. Moscow, Nauka, 1973.
7. Kurzhanski A. B. Upravlenie i nablyudenie v usloviyakh neopredelennosti [Control and Observation under Uncertainty]. Moscow, Nauka, 1977.
8. Filippova T. F. Construction of Set-valued Estimates of Reachable Sets for Some Nonlinear Dynamical Systems with Impulsive Control. Proceedings of the Steklov Institute of Mathematics (Supplementary issues), 2010, vol. 269, suppl. 1, pp. 95–102.
9. Filippova T. F., Matviichuk O. G. Algorithms to Estimate the Reachability Sets of the Pulse Controlled Systems with Ellipsoidal Phase Constraints. Autom. Remote Control, 2011, vol. 72, no 9, pp. 1911–1924.
10. Filippova T. F., Matviichuk O. G. Problems of Impulse Control under Uncertainty [Zadachi impul’snogo upravleniya v usloviyakh neopredelennosti]. Trudy XII vserossiyskogo soveshchaniya po problemam upravleniya [Proc. XII All-Russiaconference on control problems (VSPU-2014). IPU RAS, June 16–19 2014.]. Moscow, 2014, pp. 1024–1032.
11. Chernousko F. L. Otsenivanie fazovogo sostoyaniya dinamicheskikh sistem [Estimation of the Pphase State of Dynamical Systems]. Moscow, Nauka, 1988.
12. Chernousko F. L. Ellipsoidal’naya approksimatsiya mnozhestv dostizhimosti lineynoy sistemy s neopredelennoy matritsey [Ellipsoidal Approximation of the Attainability Sets of a Linear System with an Uncertain Matrix]. Prikladnayamatematika i mekhanika [Applied Mathematics and Mechanics], 1996, vol. 60, no 6, pp. 940–950.
13. Filippova T. F. Set-valued Dynamics in Problems of Mathematical Theory of Control Processes. International Journal of Modern Physics B (IJMPB), 2012, vol. 26, no. 25, pp. 1–8.
14. 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 Conference (ENOC 2014). Vienna, Austria, July 6-11,2014, H. Ecker, A. Steindl and S. Jakubek (Eds.), Institute of Mechanics and Mechatronics, TU–Vienna, Austria. CD-ROM volume (ISBN: 978-3-200-03433-4). Paper-ID 149. Vienna, 2014, pp. 1–2.
15. 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 (A.B. Kurzhanski,ed.) Progress in Systems and Control Theory. Boston, Birkhauser, 1993, vol. 17, pp. 122–188.
16. Kurzhanski A. B., Valyi I. Ellipsoidal Calculus for Estimation and Control. Boston, Birkh¨auser, 1997.
17. Kurzhanski A. B., Varaiya P. Dynamics and Control of Trajectory Tubes, Theory and Computation. Systems & Control:;Foundations & Applications,;vol. 85. Basel, Birkh¨auser, 2014.
18. Matviychuk O. G. Ellipsoidal Estimates of Reachable Sets of Impulsive Control Systems with Bilinear Uncertainty. Cybernetics and Physics Journal, 2016, vol. 5, no 3, pp. 96—104.
19. Rishel R. An Extended Pontryagin Principle for Control System whose Control Laws Contain Measures. SIAM J. Control, 1965, vol. 3, pp. 191–205.
20. Vinter R. B., Pereira F. L. A Maximum Principle for Optimal Processes with Discontinuous Trajectories. SIAM J. Contr. and Optimization, 1988, vol. 26, no 1, pp. 205–229.