Construction of reachability sets for nonlinear control systems by grid algorithm with an apriori reduction procedure

I. V. Izmestyev ; V. N. Ushakov

Čelâbinskij fiziko-matematičeskij žurnal, Tome 9 (2024) no. 3, pp. 364-374.

Résumé

The problem of constructing reachable sets for nonlinear control systems is considered. To solve this problem, a grid algorithm is proposed, in which the procedures for calculating the next reachable set and reducing are combined. This approach allows for more efficient use of computer resources when performing calculations. A program that implements this algorithm is written in the C++ programming language using the OpenMP parallel computing technology. Model calculations have been perfomed.

Détail

Keywords: control, numerical method, reachability set.

Bibliographie

[1] Krasovskii N.N., Subbotin A.I., Game-theoretical control problems, Springer, New York, 1988, 517 pp. | MR | MR | Zbl

[2] Nikol'skii M.S., “Approximation of the attainability set for a controlled process”, Mathematical Notes, 41:1 (1987), 44–48 | DOI | MR | Zbl

[3] Chernous'ko F.L., Estimation of the phase state of dynamic systems. Ellipsoid method, Nauka, Moscow, 1988 (In Russ.)

[4] Kurzhanski A. B., Valyi I., Ellipsoidal calculus for estimation and control, Birkhäeuser, Basel, 1997 | MR | Zbl

[5] Anan'evskii I.M., “Control of a nonlinear vibratory system of the fourth order with unknown parameters”, Automation and Remote Control, 62:1 (2001), 343–355 | DOI | MR | Zbl

[6] Filippova T.F., “External estimates for reachable sets of a control system with uncertainty and combined nonlinearity”, Proceedings of the Steklov Institute of Mathematics, 301:1 (2018), 32–43 | DOI | MR

[7] Gusev M.I., Zykov I.V., “On the geometry of reachable sets for control systems with isoperimetric constraints”, Proceedings of the Steklov Institute of Mathematics, 304:1 (2019), S76–S87 | DOI | MR | Zbl

[8] Zimovets A.A., Matviichuk A.R., “A parallel algorithm for constructing approximate attainable sets of nonlinear control systems”, Bulletin of the Udmurt University. Mathematics. Mechanics. Computer science, 25:4 (2015), 459–472 (In Russ.) | Zbl

[9] Ushakov V.N., Ukhobotov V.I., Ushakov A.V., Parshikov G.V., “On solving approach problems for control systems”, Proceedings of the Steklov Institute of Mathematics, 291 (2015), 263–278 | DOI | DOI | MR | MR | Zbl

[10] Ushakov V. N., Ukhobotov V. I., Ushakov A. V., Parshikov G. V., “On solution of control problems for nonlinear systems on finite time interval”, IFAC-PapersOnLine, 49:18 (2016), 380–385 | DOI

[11] Matviychuk A.R., Ukhobotov V.I., Ushakov A.V., Ushakov V.N., “The approach problem of a nonlinear controlled system in a finite time interval”, Journal of Applied Mathematics and Mechanics, 81:2 (2017), 114–128 | DOI | MR | Zbl

[12] Ushakov V.N., Ukhobotov V.I., Ushakov A.V., Izmest'ev I.V., “Control systems of variable structure. Attainability sets and integral funnels”, Journal of Mathematical Sciences, 260 (2022), 820–832 | DOI | MR | Zbl

[13] Ushakov V.N., Ushakov A.V., Izmest'ev I.V., Anfalov E.D., Program for solving the problem of performance and calculating the integral funnel of a nonlinear controlled system using parallel computing, Certificate of state registration of a computer program, registration date: 24.01.2023, registration number (certificate): 2023611643 (In Russ.)

[14] Beznos A.V., Grishin A.A., Lenskii A.V., Okhozimskii D.E., Formalskii A.M., “The control of pendulum using flywheel”, Workshop on Theoretical and Applied Mechanics, Moscow State University, Moscow, 2009, 170–195 (In Russ.) | MR

[15] Ukhobotov V. I., Izmest'ev I. V., “Control problem with disturbance and unknown moment of change of dynamics”, Lecture Notes in Control and Information Sciences. Stability, Control and Differential Games, eds. A. Tarasyev, V. Maksimov, T. Filippova, Springer, Cham, 2020, 335–344 | MR | Zbl

[16] Sinyakov V.V., Computational methods for reachability and synthesis problems of controls under nonlinear conditions, PhD thesis, Lomonosov Moscow State University, Moscow, 2015 (In Russ.)