TY - GEN

T1 - Linear Interpolation of Program Control with Respect to a Multidimensional Parameter in the Convergence Problem

T2 - book chapter

AU - Ushakov, Vladimir

AU - Ershov, Aleksandr

AU - Ershova, Anna

AU - Alekseev, Aleksander V.

N1 - This research was supported by the Russian Science Foundation (grant no. 19-11-00105, https://rscf.ru/en/project/19-11-00105/).

PY - 2023/9/21

Y1 - 2023/9/21

N2 - We consider a control system containing a constant three-dimensional vector parameter, the approximate value of which is reported to the control person only at the moment of the movement start. The set of possible values of unknown parameter is known in advance. An convergence problem is posed for this control system. At the same time, it is assumed that in order to construct resolving control, it is impossible to carry out cumbersome calculations based on the pixel representation of reachable sets in real time. Therefore, to solve the convergence problem, we propose to calculate in advance several resolving controls, corresponds to possible parameter values in terms of some grid of nodes. If at the moment of the movement start it turns out that the value of the parameter does not coincide with any of the grid nodes, it is possible to calculate the program control using the linear interpolation formulas. However, this procedure can be effective only if a linear combination of controls corresponding to the same “guide” in the terminology of N.N. Krasovskii’s Extreme Aiming Method is used. In order to be able to effectively apply linear interpolation, for each grid cell, we propose to calculate 8 “nodal” resolving controls and use the method of dividing control into basic control and correcting control in addition. Due to the application of the latter method, the calculated solvability set turns out to be somewhat smaller than the actual one. But the increasing of accuracy of the system state transferring to the target set takes place.

AB - We consider a control system containing a constant three-dimensional vector parameter, the approximate value of which is reported to the control person only at the moment of the movement start. The set of possible values of unknown parameter is known in advance. An convergence problem is posed for this control system. At the same time, it is assumed that in order to construct resolving control, it is impossible to carry out cumbersome calculations based on the pixel representation of reachable sets in real time. Therefore, to solve the convergence problem, we propose to calculate in advance several resolving controls, corresponds to possible parameter values in terms of some grid of nodes. If at the moment of the movement start it turns out that the value of the parameter does not coincide with any of the grid nodes, it is possible to calculate the program control using the linear interpolation formulas. However, this procedure can be effective only if a linear combination of controls corresponding to the same “guide” in the terminology of N.N. Krasovskii’s Extreme Aiming Method is used. In order to be able to effectively apply linear interpolation, for each grid cell, we propose to calculate 8 “nodal” resolving controls and use the method of dividing control into basic control and correcting control in addition. Due to the application of the latter method, the calculated solvability set turns out to be somewhat smaller than the actual one. But the increasing of accuracy of the system state transferring to the target set takes place.

UR - http://www.scopus.com/inward/record.url?partnerID=8YFLogxK&scp=85174616752

U2 - 10.1007/978-3-031-43257-6_24

DO - 10.1007/978-3-031-43257-6_24

M3 - Conference contribution

SN - 978-3-031-43257-6

T3 - Communications in Computer and Information Science

SP - 324

EP - 337

BT - Mathematical Optimization Theory and Operations Research: Recent Trends

A2 - Khachay, Michael

A2 - Kochetov, Yury

PB - Springer Cham

CY - 978-3-031-43256-9

ER -