Abstract
On a finite time horizon, we consider a control system described by a vector differential equation with right-hand side that changes its structure at some times spaced by a distance that cannot be less than a certain given value. In between two adjacent structure change instants, the right-hand side is a function that is Lipschitz in state variables, continuous in time, and linear in the control and perturbation, which take values in some convex closed sets. It is assumed that at the structure change instants the solution of the system may experience a jump by a certain vector of which only the direction is known. A uniform mesh is specified on the system operation interval. The values of the state vector are measured (with an error) at the mesh points. We solve the problem of constructing an algorithm for the formation of a system control that ensures bringing the system trajectory to the minimum possible neighborhood of the goal set at the end time. A solution algorithm is indicated that is based on the constructions of positional control theory and is resistant to information interferences and computational errors.
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REFERENCES
Krasovskii, N.N. and Subbotin, A.I., Pozitsionnye differentsial’nye igry (Positional Differential Games), Moscow: Fizmatlit, 1974.
Krasovskii, N.N., Upravlenie dinamicheskoi sistemoi (Control of a Dynamical System), Moscow: Nauka, 1985.
Subbotin, A.I. and Chentsov, A.G., Optimizatsiya garantii v zadachakh upravleniya (Optimization of Guarantee in Control Problems), Moscow: Nauka, 1981.
Izbrannye trudy Osipova Yu.S. (Selected Works of Yu.S. Osipov), Moscow: Izd. Mosk. Univ., 2009.
Grigorenko, N.L., Matematicheskie metody upravleniya neskol’kimi dinamicheskimi protsessami (Mathematical Methods of Control of Several Dynamic Processes), Moscow: Izd. Mosk. Univ., 1990.
Ushakov, V.N., To the construction of stable bridges in a differential pursuit–evasion game, Izv. Akad. Nauk SSSR. Tekh. Kibern., 1980, no. 4, pp. 29–36.
Kryazhimskii, A.V., To the theory of positional differential pursuit–evasion games, Dokl. Akad. Nauk SSSR, 1978, vol. 239, no. 4, pp. 123–128.
Osipov, Yu.S. and Kryazhimskii, A.V., Inverse Problems for Ordinary Differential Equations: Dynamical Solutions, London–Basel: Gordon & Breach, 1995.
Osipov, Yu.S., Control packages: an approach to solution of positional control problems with incomplete information, Russ. Math. Surv., 2006, vol. 61, no. 4, pp. 611–661.
Kryazhimskii, A.V. and Maksimov, V.I., On combination of the processes of reconstruction and guaranteeing control, Autom. Remote Control, 2013, vol. 74, no. 8, pp. 1235–1248.
Maksimov, V.I., Differential guidance game with incomplete information on the state coordinates and unknown initial state, Differ. Equations, 2015, vol. 51, no. 12, pp. 1656–1665.
Emel’yanov, S.V., Sistemy avtomaticheskogo upravleniya s peremennoi strukturoi (Systems of Automatic Control with Variable Structure), Moscow: Nauka, 1967.
Zavalishchin, S.E. and Sesekin, A.N., Impul’snye protsessy: modeli i prilozheniya (Impulse Processes: Models and Application), Yekaterinburg: Ural. Otd. Ross. Akad. Nauk, 1991.
Miller, B.M. and Rubinovich, E.Ya., Optimizatsiya dinamicheskikh sistem s impul’snymi upravleniyami (Optimization of Dynamical Systems with Impulse Controls), Moscow: Nauka, 2004.
Semerov, V.V., Repin, V.M., and Zhurina, N.E., Algoritmizatsiya protsessov upravleniya letatel’nymi apparatami v klasse logiko-dinamicheskikh sistem (Algorithmization of Aircraft Control Processes in the Class of Logical-Dynamical Systems), Moscow: Mosk. Aviats. Inst., 1987.
Bortakovskii, A.S., Optimizatsiya pereklyuchayushchikhsya sistem (Optimization of Switching Systems), Moscow: Izd. Mosk. Aviats. Inst., 2016.
Engell, S., Frehse, G., and Schnieder, E., Modeling, Analysis and Design of Hybrid System, Berlin–Heidelberg: Springer, 2002.
Savkin, A.V. and Evans, R.J., Hybrid Dynamical Systems: Controller and Sensor Switching Problems, Boston: Birkhäuser, 2002.
Vasil’ev, S.N. and Malikov, A.I., On some results on the stability of mode-switching and hybrid systems, in Aktual’nye problemy mekhaniki sploshnoi sredy. K 20-letiyu IMM KazNTs RAN (Topical Problems of Continuum Mechanics. To the 20th Anniversary of the Institute of Mathematics and Mechanics of the Kazan Scientific Center of the Russian Academy of Sciences), Kazan, 2011, pp. 23–81.
Axelsson, H., Boccadoro, V., Egerstedt, M., Valigi, P., and Wardi, Y., Optimal mode-switching for hybrid systems with varying initial states, J. Nonlinear Anal. Hybrid Syst. Appl., 2008, vol. 2, no. 3, pp. 765–772.
Maksimov, V.I., The tracking of the trajectory of a dynamical system, J. Appl. Math. Mech., 2011, vol. 75, no. 6, pp. 667–674.
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Osipov, Y.S., Maksimov, V.I. Feedback in a Control Problem for a System with Discontinuous Right-Hand Side. Diff Equat 57, 533–552 (2021). https://doi.org/10.1134/S0012266121040091
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DOI: https://doi.org/10.1134/S0012266121040091