On solving terminal approach and evasion problems for linear discrete-time systems under state constraints

E. K. Kostousova

Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, Tome 34 (2024) no. 2, pp. 204-221

Résumé

The paper is devoted to elaboration of polyhedral techniques for solving two control problems for linear discrete-time systems with uncertainties under state constraints, namely, the terminal approach problem and the terminal evasion one. Such problems arise in systems with two controls, where the aim of the first is to steer the trajectory onto a given terminal set at a given instant without violating the state constraints, the aim of the other is opposite. It is assumed that the terminal set is a parallelepiped, the controls are bounded by parallelotope-valued constraints, and the state constraints are given in the form of so-called zones. We present techniques for solving both problems basing on polyhedral (parallelotope-valued or parallelepiped-valued) tubes. The techniques for solving the approach problem were proposed by the author earlier, but here additional properties of them are investigated. In particular, for the case without state constraints, guaranteed estimates are found for the trajectory that ensure that it is inside the tube. Convenient sufficient conditions are given to guarantee the obtaining of nondegenerate cross-sections during the calculations. For the evasion problem, a common solution scheme is considered, and then polyhedral techniques are proposed. The whole parametric families of external and internal polyhedral estimates for the solvability tubes for both problems are presented and compared. An illustrative example is given.

Détail

Keywords: systems with uncertainties, control synthesis, approach problem, evasion problem, polyhedral methods, parallelepipeds
Mots-clés : parallelotopes

Bibliographie

[1] Krasovskii N.N., Subbotin A.I., Game-theoretical control problems, Springer, New York, 1988 | MR | MR | Zbl

[2] Kurzhanski A.B., Vályi I., Ellipsoidal calculus for estimation and control, Birkhäuser, Boston, 1997 | MR | Zbl

[3] Kurzhanski A.B., Varaiya P., Dynamics and control of trajectory tubes. Theory and computation, Birkhäuser, Cham, 2014 | DOI | MR | Zbl

[4] Ushakov V.N., “Unification in the game problem of convergence and the property of stability”, Chelyabinskiy Fiziko-Matematicheskiy Zhurnal, 7:1 (2022), 54–79 (in Russian) | DOI | MR | Zbl

[5] Kamneva L., “A scheme for calculating solvability sets “up to moment” in linear differential games”, Journal of Dynamical and Control Systems, 29:3 (2023), 989–1018 | DOI | MR | Zbl

[6] Botkin N., Martynov K., Turova V., Diepolder J., “Generation of dangerous disturbances for flight systems”, Dynamic Games and Applications, 9:3 (2019), 628–651 | DOI | MR | Zbl

[7] Chernousko F.L., State estimation for dynamic systems, CRC Press, Boca Raton, 1994

[8] Kurzhanskiy A.A., Varaiya P., “Theory and computational techniques for analysis of discrete-time control systems with disturbances”, Optimization Methods and Software, 26:4–5 (2011), 719–746 | DOI | MR | Zbl

[9] Filippova T.F., “HJB-inequalities in estimating reachable sets of a control system under uncertainty”, Ural Mathematical Journal, 8:1 (2022), 34–42 | DOI | MR | Zbl

[10] Gusev M.I., Osipov I.O., “On a local synthesis problem for nonlinear systems with integral constraints”, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp’yuternye Nauki, 32:2 (2022), 171–186 (in Russian) | DOI | MR | Zbl

[11] Vicino A., Zappa G., “Sequential approximation of feasible parameter sets for identification with set membership uncertainty”, IEEE Transactions on Automatic Control, 41:6 (1996), 774–785 | DOI | MR | Zbl

[12] Kostousova E.K., “State estimation for dynamic systems via parallelotopes optimization and parallel computations”, Optimization Methods and Software, 9:4 (1998), 269–306 | DOI | MR | Zbl

[13] Kostousova E.K., “Polyhedral estimates for attainability sets of linear multistage systems with integral constraints on the control”, Computational Technologies, 8:4 (2003), 55–74 (in Russian) | MR | Zbl | Zbl

[14] Martynov K., Botkin N.D., Turova V.L., Diepolder J., “Quick construction of dangerous disturbances in conflict control problems”, Advances in Dynamic Games, Annals of the International Society of Dynamic Games, 17, Birkhäuser, Cham, 2020, 3–25 | DOI | MR | Zbl

[15] Kostousova E.K., “On the polyhedral method of solving problems of control strategy synthesis”, Proceedings of the Steklov Institute of Mathematics, 292:suppl. 1 (2016), S140–S155 | DOI | MR | Zbl

[16] Kostousova E.K., “On feedback target control for uncertain discrete-time systems through polyhedral techniques”, Computational Technologies, 22:2 (2017), 19–36 | MR | Zbl | Zbl

[17] Kostousova E.K., “On the polyhedral method of control synthesis in the problem of target evasion in discrete-time systems”, Trudy Instituta Matematiki i Mekhaniki UrO RAN, 27:3 (2021), 101–114 (in Russian) | DOI | MR

[18] Martynov K., Botkin N., Turova V., Diepolder J., “Real-time control of aircraft take-off in windshear. Part I: Aircraft model and control schemes”, 2017 25th Mediterranean Conference on Control and Automation (MED), 2017, 277–284 | DOI

[19] Zaitsev V., Kim I., “Exponential stabilization of linear time-varying differential equations with uncertain coefficients by linear stationary feedback”, Mathematics, 8:5 (2020), 853 | DOI | MR

[20] Petrov N.N., “The problem of simple group pursuit with phase constraints in time scales”, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp’yuternye Nauki, 30:2 (2020), 249–258 (in Russian) | DOI | MR | Zbl

[21] Kostousova E.K., “On a control synthesis in an enhanced evasion problem for linear discrete-time systems”, Trudy Instituta Matematiki i Mekhaniki UrO RAN, 29:1 (2023), 111–126 (in Russian) | DOI | MR

[22] Schneider R., Convex bodies. The Brunn–Minkowski theory, Cambridge University Press, Cambridge, 1993 | DOI | MR | Zbl

[23] Lankaster P., Theory of matrices, Academic Press, New York–London, 1969 | MR

[24] Kostousova E.K., “On solving an enhanced evasion problem for linear discrete-time systems”, Ural Mathematical Journal, 8:1 (2022), 55–63 | DOI | MR | Zbl