title: ÐÑÐ¾ÐµÐºÑÐ¸Ð¾Ð½Ð½ÑÐ¹ Ð¼ÐµÑÐ¾Ð´ Ð¾Ð¿ÑÐ¸Ð¼Ð¸Ð·Ð°ÑÐ¸Ð¸ Ð´Ð¸Ð½Ð°Ð¼Ð¸ÑÐµÑÐºÐ¸Ñ ÑÐ¸ÑÑÐµÐ¼: ÑÐ¾ÑÑÐ¾ÑÐ½Ð¸Ðµ Ð¸ Ð¿ÐµÑÑÐ¿ÐµÐºÑÐ¸Ð²Ñ ÑÐ°Ð·Ð²Ð¸ÑÐ¸Ñ
title: Projection method for optimizing dynamic systems: state and development prospects
Creators: name: ÐÐ°ÑÐ¸Ð»ÑÐµÐ², Ð®ÑÐ¸Ð¹
name: ÐÑÑÐµÐ¼Ð¾Ð², ÐÑÑÐµÐ¼
name: ÐÐ¾Ð·Ð»Ð¾Ð², ÐÐ»Ð°Ð´Ð¸Ð¼Ð¸Ñ
name: "Ð¡Ð¸ÑÑÐµÐ¼Ð½ÑÐ¹ Ð°Ð½Ð°Ð»Ð¸Ð· Ð² Ð¿ÑÐ¾ÐµÐºÑÐ¸ÑÐ¾Ð²Ð°Ð½Ð¸Ð¸ Ð¸ ÑÐ¿ÑÐ°Ð²Ð»ÐµÐ½Ð¸Ð¸", Ð¼ÐµÐ¶Ð´ÑÐ½Ð°ÑÐ¾Ð´Ð½Ð°Ñ Ð½Ð°ÑÑÐ½Ð°Ñ Ð¸ ÑÑÐµÐ±Ð½Ð¾-Ð¿ÑÐ°ÐºÑÐ¸ÑÐµÑÐºÐ°Ñ ÐºÐ¾Ð½ÑÐµÑÐµÐ½ÑÐ¸Ñ (25; 2021; Ð¡Ð°Ð½ÐºÑ-ÐÐµÑÐµÑÐ±ÑÑÐ³)
Publisher: name: ÐÐÐÐÐ¢ÐÐ¥-ÐÐ ÐÐ¡Ð¡
Year: 2021
Subjects: ÐÐ¸Ð½Ð°Ð¼Ð¸ÑÐµÑÐºÐ¸Ðµ ÑÐ¸ÑÑÐµÐ¼Ñ
ÐÐ²ÑÐ¾Ð¼Ð°ÑÐ¸ÑÐµÑÐºÐ¾Ðµ ÑÐ¿ÑÐ°Ð²Ð»ÐµÐ½Ð¸Ðµ
Ð¿ÑÐ¾ÐµÐºÑÐ¸Ð¾Ð½Ð½ÑÐ¹ Ð¼ÐµÑÐ¾Ð´ Ð¾Ð¿ÑÐ¸Ð¼Ð¸Ð·Ð°ÑÐ¸Ð¸
Ð¾Ð¿ÐµÑÐ°ÑÐ¾ÑÑ Ð¼Ð¸Ð½Ð¸Ð¼Ð¸Ð·Ð°ÑÐ¸Ð¸ ÑÑÐ½ÐºÑÐ¸Ð¾Ð½Ð°Ð»Ð¾Ð²
ÑÑÑÐ¾Ð¹ÑÐ¸Ð²Ð¾ÑÑÑ ÑÐ¸ÑÑÐµÐ¼ ÑÐ¿ÑÐ°Ð²Ð»ÐµÐ½Ð¸Ñ

description: Ð Ð°ÑÑÐ¼Ð¾ÑÑÐµÐ½Ñ Ð¼Ð°ÑÐµÐ¼Ð°ÑÐ¸ÑÐµÑÐºÐ¸Ðµ Ð¼Ð¾Ð´ÐµÐ»Ð¸ ÑÑÐ°ÑÐ¸Ð¾Ð½Ð°ÑÐ½ÑÑ Ð¸ Ð¿ÐµÑÐµÑÐ¾Ð´Ð½ÑÑ ÑÐ¾ÑÑÐ¾ÑÐ½Ð¸Ð¹ Ð´Ð¸Ð½Ð°Ð¼Ð¸ÑÐµÑÐºÐ¸Ñ ÑÐ¸ÑÑÐµÐ¼, Ð¾Ð¿ÑÐµÐ´ÐµÐ»ÐµÐ½Ð½ÑÐµ Ð»Ð¸Ð½ÐµÐ¹Ð½ÑÐ¼Ð¸ Ð¸ Ð½ÐµÐ³Ð»Ð°Ð´ÐºÐ¸Ð¼Ð¸ Ð´Ð¸ÑÑÐµÑÐµÐ½ÑÐ¸Ð°Ð»ÑÐ½ÑÐ¼Ð¸, ÑÐ°Ð·Ð½Ð¾ÑÑÐ½ÑÐ¼Ð¸ Ð¸Ð»Ð¸ Ð°Ð»Ð³ÐµÐ±ÑÐ°Ð¸ÑÐµÑÐºÐ¸Ð¼Ð¸ ÑÑÐ°Ð²Ð½ÐµÐ½Ð¸ÑÐ¼Ð¸ Ð¾Ð±ÑÐµÐºÑÐ¾Ð². ÐÐ³ÑÐ°Ð½Ð¸ÑÐµÐ½Ð½ÑÐµ ÑÐ¿ÑÐ°Ð²Ð»ÐµÐ½Ð¸Ñ Ð´Ð»Ñ Ð¾Ð±ÑÐµÐºÑÐ¾Ð² ÑÐ¾ÑÐ¼Ð¸ÑÑÑÑÑÑ Ð¿ÑÐ¾ÐµÐºÑÐ¸Ð¾Ð½ÑÐ¼Ð¸ Ð¾Ð¿ÐµÑÐ°ÑÐ¾ÑÐ°Ð¼Ð¸ Ð¼Ð°ÑÐµÐ¼Ð°ÑÐ¸ÑÐµÑÐºÐ¾Ð³Ð¾ Ð¿ÑÐ¾Ð³ÑÐ°Ð¼Ð¼Ð¸ÑÐ¾Ð²Ð°Ð½Ð¸Ñ, Ð¿ÑÐµÐ¾Ð±ÑÐ°Ð·ÑÑÑÐ¸Ð¼Ð¸ Â«ÑÐµÐ»ÐµÐ²ÑÐµ Ð²ÐµÐºÑÐ¾ÑÑÂ» ÑÑÐ½ÐºÑÐ¸Ð¾Ð½Ð°Ð»Ð¾Ð² Ð¸ Ð¾Ð³ÑÐ°Ð½Ð¸ÑÐµÐ½Ð¸Ñ Ð² Ð²ÐµÐºÑÐ¾ÑÑ ÑÐ¿ÑÐ°Ð²Ð»ÐµÐ½Ð¸Ð¹ Ñ Ð¾Ð±ÑÐ°ÑÐ½Ð¾Ð¹ ÑÐ²ÑÐ·ÑÑ. ÐÑÐ¾ÐµÐºÑÐ¸Ð¾Ð½Ð½ÑÐµ Ð¾Ð¿ÐµÑÐ°ÑÐ¾ÑÑ Ð¼Ð¸Ð½Ð¸Ð¼Ð¸Ð·Ð¸ÑÑÑÑ Ð»Ð¸Ð½ÐµÐ¹Ð½ÑÐµ Ð¸Ð»Ð¸ ÐºÐ²Ð°Ð´ÑÐ°ÑÐ¸ÑÐ½ÑÐµ ÑÑÐ½ÐºÑÐ¸Ð¾Ð½Ð°Ð»Ñ Ð½Ð° ÐºÐ¾Ð¼Ð¿Ð°ÐºÑÐ½ÑÑ Ð¼Ð½Ð¾Ð¶ÐµÑÑÐ²Ð°Ñ Ð¿ÑÐ¸ Ð¾Ð³ÑÐ°Ð½Ð¸ÑÐµÐ½Ð¸ÑÑ ÑÐ¸Ð¿Ð° ÑÐ°Ð²ÐµÐ½ÑÑÐ² Ð¸ Ð½ÐµÑÐ°Ð²ÐµÐ½ÑÑÐ² Ð½Ð° Ð¿ÐµÑÐµÐ¼ÐµÐ½Ð½ÑÐµ, Ð¾Ð¿ÑÐµÐ´ÐµÐ»ÑÑÑÐ¸Ðµ ÐºÐ¾Ð¾ÑÐ´Ð¸Ð½Ð°ÑÑ Ð¸ ÑÐ¿ÑÐ°Ð²Ð»ÐµÐ½Ð¸Ñ. ÐÐ° Ð¾ÑÐ½Ð¾Ð²Ðµ Ð¿ÑÐ¾ÐµÐºÑÐ¸Ð¾Ð½Ð½ÑÑ Ð¾Ð¿ÐµÑÐ°ÑÐ¾ÑÐ¾Ð² ÑÐ¸Ð½ÑÐµÐ·Ð¸ÑÐ¾Ð²Ð°Ð½Ñ Ð´Ð¸ÑÐºÑÐµÑÐ½ÑÐµ ÑÐ¸ÑÑÐµÐ¼Ñ Ñ Ð¾Ð³ÑÐ°Ð½Ð¸ÑÐµÐ½Ð½ÑÐ¼Ð¸ Ð»Ð¾ÐºÐ°Ð»ÑÐ½Ð¾ Ð¸Ð»Ð¸ Ð¸Ð½ÑÐµÑÐ²Ð°Ð»ÑÐ½Ð¾ Ð´Ð¾Ð¿ÑÑÑÐ¸Ð¼ÑÐ¼Ð¸ Ð¸ Ð¾Ð¿ÑÐ¸Ð¼Ð°Ð»ÑÐ½ÑÐ¼Ð¸ ÑÐ¿ÑÐ°Ð²Ð»ÐµÐ½Ð¸ÑÐ¼Ð¸. Ð¡ÑÐ¾ÑÐ¼ÑÐ»Ð¸ÑÐ¾Ð²Ð°Ð½Ñ ÑÑÐ»Ð¾Ð²Ð¸Ñ ÑÑÑÐ¾Ð¹ÑÐ¸Ð²Ð¾ÑÑÐ¸ Ð½Ð° Ð¾ÑÐ½Ð¾Ð²Ðµ ÑÑÐ»Ð¾Ð²Ð¸Ð¹ ÑÐ¶Ð°ÑÐ¸Ñ Ð¾Ð¿ÐµÑÐ°ÑÐ¾ÑÐ¾Ð² ÑÐ¸ÑÑÐµÐ¼ Ð¸ Ð¼ÐµÑÐ¾Ð´Ð° ÑÑÐ½ÐºÑÐ¸Ð¹ ÐÑÐ¿ÑÐ½Ð¾Ð²Ð° Ð´Ð»Ñ Ð»Ð¸Ð½ÐµÐ¹Ð½ÑÑ, Ð½ÐµÐ»Ð¸Ð½ÐµÐ¹Ð½ÑÑ Ð»Ð¾ÐºÐ°Ð»ÑÐ½Ð¾ Ð¸ Ð¸Ð½ÑÐµÑÐ²Ð°Ð»ÑÐ½Ð¾ Ð¾Ð¿ÑÐ¸Ð¼Ð°Ð»ÑÐ½ÑÑ Ð´Ð¸ÑÐºÑÐµÑÐ½ÑÑ ÑÐ¸ÑÑÐµÐ¼. Ð ÐµÐ·ÑÐ»ÑÑÐ°ÑÑ Ð¸Ð»Ð»ÑÑÑÑÐ¸ÑÐ¾Ð²Ð°Ð½Ñ Ð¿ÑÐ¸Ð¼ÐµÑÐ°Ð¼Ð¸ ÑÐ¿ÑÐ°Ð²Ð»ÐµÐ½Ð¸Ñ ÑÐµÐ³ÑÐ»ÑÑÐ½ÑÐ¼Ð¸ Ð¸ ÑÐ°Ð¾ÑÐ¸ÑÐµÑÐºÐ¸Ð¼Ð¸ ÑÐµÐ¶Ð¸Ð¼Ð°Ð¼Ð¸ ÑÐ½ÐµÑÐ³Ð¾ÑÐ¸ÑÑÐµÐ¼.
description: Mathematical models of stationary and transient states of dynamical systems, determined by linear and no smooth differential, difference, or algebraic equations of objects, are considered. Restricted controls for objects are formed by projection operators of mathematical programming that transform the âtarget vectorsâ of functional and constraints into vectors of feedback controls. Projection operators minimize linear or quadratic functionals on compact sets under equality and inequality constraints on variables defining coordinates and controls. Based on projection operators, discrete systems with locally or interval-limited admissible and optimal controls are synthesized. The stability conditions are formulated based on the conditions of contraction of system operators and the method of Lyapunov functions for linear, nonlinear locally and interval optimal discrete systems. The results are illustrated by examples of control of regular and chaotic regimes of power systems.