title: ÐÐµÑÐ¾Ð´ ÑÐ¸Ð½ÑÐµÐ·Ð° ÑÑÐ±Ð¾Ð¿ÑÐ¸Ð¼Ð°Ð»ÑÐ½ÑÑ ÑÐ¿ÑÐ°Ð²Ð»ÐµÐ½Ð¸Ð¹ Ð´Ð¸Ð½Ð°Ð¼Ð¸ÑÐµÑÐºÐ¾Ð¹ ÑÐ¸ÑÑÐµÐ¼Ñ Ñ Ð¼Ð°Ð»ÑÐ¼ Ð¿Ð°ÑÐ°Ð¼ÐµÑÑÐ¾Ð¼
title: Ð²ÑÐ¿ÑÑÐºÐ½Ð°Ñ ÐºÐ²Ð°Ð»Ð¸ÑÐ¸ÐºÐ°ÑÐ¸Ð¾Ð½Ð½Ð°Ñ ÑÐ°Ð±Ð¾ÑÐ° Ð±Ð°ÐºÐ°Ð»Ð°Ð²ÑÐ°
titleType: Subtitle
title: Method of synthesis of suboptimal controls of a dynamic system with a small parameter
Creators: name: ÐÐ¸, ÐÑÑÐµÑÐ»Ð°Ð²
Publisher: name: Ð¡Ð°Ð½ÐºÑ-ÐÐµÑÐµÑÐ±ÑÑÐ³ÑÐºÐ¸Ð¹ Ð¿Ð¾Ð»Ð¸ÑÐµÑÐ½Ð¸ÑÐµÑÐºÐ¸Ð¹ ÑÐ½Ð¸Ð²ÐµÑÑÐ¸ÑÐµÑ ÐÐµÑÑÐ° ÐÐµÐ»Ð¸ÐºÐ¾Ð³Ð¾
Year: 2023
Subjects: Ð´Ð¸Ð½Ð°Ð¼Ð¸ÑÐµÑÐºÐ°Ñ ÑÐ¸ÑÑÐµÐ¼Ð°
ÑÑÐ±Ð¾Ð¿ÑÐ¸Ð¼Ð°Ð»ÑÐ½Ð¾Ðµ ÑÐ¿ÑÐ°Ð²Ð»ÐµÐ½Ð¸Ðµ
Ð¼Ð°Ð»ÑÐ¹ Ð¿Ð°ÑÐ°Ð¼ÐµÑÑ
Ð°Ð»Ð³Ð¾ÑÐ¸ÑÐ¼
Ð¾Ð±ÑÐ°ÑÐ½Ð°Ñ ÑÐ²ÑÐ·Ñ
dynamic system
suboptimal control
small parameter
construction algorithm
feedback control

description: ÐÐ°Ð½Ð½Ð°Ñ ÑÐ°Ð±Ð¾ÑÐ° Ð¿Ð¾ÑÐ²ÑÑÐµÐ½Ð° Ð¸ÑÑÐ»ÐµÐ´Ð¾Ð²Ð°Ð½Ð¸Ñ Ð¼ÐµÑÐ¾Ð´Ð° ÑÐ¸Ð½ÑÐµÐ·Ð° ÑÑÐ±Ð¾Ð¿ÑÐ¸Ð¼Ð°Ð»ÑÐ½Ð¾Ð³Ð¾ ÑÐ¿ÑÐ°Ð²Ð»ÐµÐ½Ð¸Ñ Ð½ÐµÐ»Ð¸Ð½ÐµÐ¹Ð½Ð¾Ð¹ Ð´Ð¸Ð½Ð°Ð¼Ð¸ÑÐµÑÐºÐ¾Ð¹ ÑÐ¸ÑÑÐµÐ¼Ñ. ÐÐ°Ð´Ð°ÑÐ¸, ÐºÐ¾ÑÐ¾ÑÑÐµ ÑÐµÑÐµÐ½Ñ Ð²Â  ÑÐµÐ·ÑÐ»ÑÑÐ°ÑÐµ Ð¸ÑÑÐ»ÐµÐ´Ð¾Ð²Ð°Ð½Ð¸Ñ: ÐÐ±Ð·Ð¾Ñ Ð¸ Ð°Ð½Ð°Ð»Ð¸Ð· Ð¼ÐµÑÐ¾Ð´Ð¾Ð² Ð¸ÑÑÐ»ÐµÐ´Ð¾Ð²Ð°Ð½Ð¸Ñ Ð½ÐµÐ»Ð¸Ð½ÐµÐ¹Ð½ÑÑ ÑÐ¸ÑÑÐµÐ¼; Ð¡Ð¸Ð½ÑÐµÐ· ÑÑÐ±Ð¾Ð¿ÑÐ¸Ð¼Ð°Ð»ÑÐ½Ð¾Ð³Ð¾ ÑÐ¿ÑÐ°Ð²Ð»ÐµÐ½Ð¸Ñ Ð´Ð»Ñ Ð½ÐµÐ»Ð¸Ð½ÐµÐ¹Ð½Ð¾Ð¹ ÑÐ¸ÑÑÐµÐ¼Ñ Ð¼ÐµÑÐ¾Ð´Ð¾Ð¼ Ð¼Ð°Ð»Ð¾Ð³Ð¾ Ð¿Ð°ÑÐ°Ð¼ÐµÑÑÐ°; ÐÑÐ¸Ð¼ÐµÐ½ÐµÐ½Ð¸Ðµ ÑÐµÐ·ÑÐ»ÑÑÐ°ÑÐ° Ð´Ð»Ñ ÑÐ¸Ð½ÑÐµÐ·Ð° ÑÐ¿ÑÐ°Ð²Ð»ÐµÐ½Ð¸Ñ Ð´Ð»Ñ Ð½ÐµÐ»Ð¸Ð½ÐµÐ¹Ð½Ð¾Ð¹ Ð¼Ð¾Ð´ÐµÐ»Ð¸ ÑÐ»ÐµÐ´ÑÑÐµÐ¹ ÑÐ¸ÑÑÐµÐ¼Ñ Ñ Ð¿ÑÐ¸Ð²Ð¾Ð´Ð¾Ð¼. Ð ÑÐ°Ð±Ð¾ÑÐµ Ð¿ÑÐµÐ´Ð»Ð°Ð³Ð°ÐµÑÑÑ Ð°Ð»Ð³Ð¾ÑÐ¸ÑÐ¼, Ñ Ð¿Ð¾Ð¼Ð¾ÑÑÑ ÐºÐ¾ÑÐ¾ÑÐ¾Ð³Ð¾ Ð¼Ð¾Ð¶Ð½Ð¾ Ð¿Ð¾ÑÑÑÐ¾Ð¸ÑÑ Ð°ÑÐ¸Ð¼Ð¿ÑÐ¾ÑÐ¸ÑÐµÑÐºÐ¸ ÑÑÐ±Ð¾Ð¿ÑÐ¸Ð¼Ð°Ð»ÑÐ½Ð¾Ðµ ÑÐ¿ÑÐ°Ð²Ð»ÐµÐ½Ð¸Ðµ Ð² ÑÐ°ÑÑÐ¼Ð°ÑÑÐ¸Ð²Ð°ÐµÐ¼Ð¾Ð¹ Ð·Ð°Ð´Ð°ÑÐµ. Ð¡ÑÑÑ Ð°Ð»Ð³Ð¾ÑÐ¸ÑÐ¼Ð° ÑÐ¾ÑÑÐ¾Ð¸Ñ Ð² Ð¿Ð¾ÑÑÑÐ¾ÐµÐ½Ð¸Ð¸ Ð¿Ð¾Ð»Ð¸Ð½Ð¾Ð¼Ð¾Ð² Ð¢ÐµÐ¹Ð»Ð¾ÑÐ°, Ð¾Ð¿ÑÐµÐ´ÐµÐ»ÑÑÑÐ¸Ñ ÑÐ»ÐµÐ¼ÐµÐ½ÑÑ Ð¾Ð¿ÑÐ¸Ð¼Ð°Ð»ÑÐ½Ð¾Ð³Ð¾ ÑÐ¿ÑÐ°Ð²Ð»ÐµÐ½Ð¸Ñ. ÐÐ¿ÑÐµÐ´ÐµÐ»ÑÑÑÐ¸Ð¼Ð¸ ÑÐ»ÐµÐ¼ÐµÐ½ÑÐ°Ð¼Ð¸ Ð² ÑÐ°ÑÑÐ¼Ð°ÑÑÐ¸Ð²Ð°ÐµÐ¼Ð¾Ð¹ Ð·Ð°Ð´Ð°ÑÐµ ÑÐ²Ð»ÑÑÑÑÑ Ð½Ð°ÑÐ°Ð»ÑÐ½ÑÐµ ÑÑÐ»Ð¾Ð²Ð¸Ñ (Ð² Ð½Ð°ÑÐ°Ð»ÑÐ½ÑÐ¹ Ð¼Ð¾Ð¼ÐµÐ½Ñ Ð²ÑÐµÐ¼ÐµÐ½Ð¸ t_0) ÑÐ¾Ð¿ÑÑÐ¶ÐµÐ½Ð½ÑÑ Ð¿ÐµÑÐµÐ¼ÐµÐ½Ð½ÑÑ, ÐºÐ¾ÑÐ¾ÑÑÐµ Ð² ÑÐ¸Ð»Ñ Ð¿ÑÐ¸Ð½ÑÐ¸Ð¿Ð° Ð¼Ð°ÐºÑÐ¸Ð¼ÑÐ¼Ð° ÑÐ¾Ð¾ÑÐ²ÐµÑÑÑÐ²ÑÑÑ Ð¾Ð¿ÑÐ¸Ð¼Ð°Ð»ÑÐ½Ð¾Ð¼Ñ ÑÐ¿ÑÐ°Ð²Ð»ÐµÐ½Ð¸Ñ, Ð° ÑÐ°ÐºÐ¶Ðµ Ð´Ð»Ð¸ÑÐµÐ»ÑÐ½Ð¾ÑÑÑ Ð¿ÑÐ¾ÑÐµÑÑÐ°. Ð­ÑÐ¸ Ð¾Ð¿ÑÐµÐ´ÐµÐ»ÑÑÑÐ¸Ðµ ÑÐ»ÐµÐ¼ÐµÐ½ÑÑ, ÐºÐ°Ðº Ð¸ ÑÑÐ½ÐºÑÐ¸Ð¸ Ð¼Ð°Ð»Ð¾Ð³Ð¾ Ð¿Ð°ÑÐ°Ð¼ÐµÑÑÐ°, Ð¿ÑÐ¸Ð½Ð°Ð´Ð»ÐµÐ¶Ð°Ñ Ð¿ÑÐ¾ÑÑÑÐ°Ð½ÑÑÐ²Ñ C_[a,b]^p. Ð ÐµÐ·ÑÐ»ÑÑÐ°ÑÐ¾Ð¼ ÑÐ°Ð±Ð¾ÑÑ ÑÐ²Ð»ÑÐµÑÑÑ Ð°Ð»Ð³Ð¾ÑÐ¸ÑÐ¼ Ð¿Ð¾ÑÑÑÐ¾ÐµÐ½Ð¸Ñ ÑÑÐ±Ð¾Ð¿ÑÐ¸Ð¼Ð°Ð»ÑÐ½Ð¾Ð³Ð¾ ÑÐ¿ÑÐ°Ð²Ð»ÐµÐ½Ð¸Ñ, Ð° ÑÐ°ÐºÐ¶Ðµ ÑÐ¸ÑÐ»ÐµÐ½Ð½ÑÐ¹ ÑÐºÑÐ¿ÐµÑÐ¸Ð¼ÐµÐ½Ñ Ð´Ð»Ñ Ð¿ÑÐ¸Ð²ÐµÐ´ÐµÐ½Ð½ÑÑ Ð² ÑÐ°Ð±Ð¾ÑÐµ Ð¿ÑÐ¸Ð¼ÐµÑÐ¾Ð² Ð¾Ð¿Ð¸ÑÑÐ²Ð°ÐµÐ¼Ð¾Ð¹ Ð² Ð½ÐµÐ¹ Ð´Ð¸Ð½Ð°Ð¼Ð¸ÑÐµÑÐºÐ¾Ð¹ ÑÐ¸ÑÑÐµÐ¼Ñ Ð´ÐµÐ¼Ð¾Ð½ÑÑÑÐ¸ÑÑÑÑÐ¸Ð¹ ÑÐµÐ°ÐºÑÐ¸Ñ ÑÑÐ¾Ð¹ ÑÐ¸ÑÑÐµÐ¼Ñ Ð½Ð° ÑÐ°Ð·Ð»Ð¸ÑÐ½Ð¾Ð³Ð¾ Ð²Ð¸Ð´Ð° Ð²Ð¾Ð·Ð´ÐµÐ¹ÑÑÐ²Ð¸Ñ.
description: This work is devoted to the study of the method of synthesis of suboptimal control of a nonlinear dynamic system. The tasks that were solved during the research: Review and analysis of methods for analyzing nonlinear systems. Synthesis of suboptimal control for nonlinear system by small parameter method. Application of the result for control synthesis for a nonlinear model of a tracking system with an actuator. The paper proposes an algorithm that can be used to construct an asymptotically suboptimal control in the problem under consideration. The essence of the algorithm consists in the construction of Taylor polynomials of defining elements of the optimal control. The defining elements in the problem under consideration are the initial conditions (at the initial time moment t_0) of the conjugate variables, which by virtue of the maximum principle correspond to the optimal control, as well as the duration of the process. These defining elements, as well as small parameter functions, belong to the space C_[a,b]^p. The result of the work is an algorithm for constructing a suboptimal control, as well as a numerical experiment for the examples of the dynamic system described in the work, demonstrating the response of this system to various kinds of influences.