Variable-delay feedback control of unstable steady states in retarded time-delayed systems

Abstract

We study the stability of unstable steady states in scalar retarded time-delayed systems subjected to a variable-delay feedback control. The important aspect of such a control problem is that timedelayed systems are already infinite-dimensional before the delayed feedback control is turned on. When the frequency of the modulation is large compared to the system's dynamics, the analytic approach consists of relating the stability properties of the resulting variable-delay system with those of an analogous distributed delay system. Otherwise, the stability domains are obtained by a numerical integration of the linearized variable-delay system. The analysis shows that the control domains are significantly larger than those in the usual time-delayed feedback control, and that the complexity of the domain structure depends on the form and the frequency of the delay modulation.

Amplitude and phase dynamics in oscillators with distributed-delay coupling

“…At the same time, in many realistic systems, the time delays themselves are not constant [53,54] and may either vary depending on the values of system variables (state-dependent delays) or just not be explicitly known. In order to account for such situations mathematically, one can use the formalism of distributed time delays, where the time delay is represented through an integral kernel describing a particular delay distribution [55][56][57].…”

Delayed feedback control of unstable steady states with high-frequency modulation of the delay

“…In the limiting case of a high-frequency modulation [43,49], when the parameter ν becomes sufficiently large compared to the uncontrolled system's dynamics described by an intrinsic frequency ω, the linearized system (2) with a modulated delay has the same asymptotic stability properties as the averaged linear distributed-delay systeṁ…”

“…Another improvement of the method was achieved by introducing a time-varying delay into the delayed argument of the control force, which could be realized experimentally by changing some characteristics of the delay line in a deterministic (periodic) or a stochastic fashion [42][43][44]. For example, in coupled laser systems the delay time could be modulated by periodically changing the distance of the lasers or external resonator, e.g., by piezoelectric modulation.…”

Local Dynamics of a Laser with Rapidly Oscillating Parameters

“…С целью расширения области эффективного управле-ния в многомерных системах предлагались различные модификации ОС [3,4], учет многократного запаздывания [5], дополнительная периодическая модуляция уров-ня ОС [6], периодическая и стохастическая модуляция времени задержки [7][8][9]. В последнем случае, по аналогии с классическими результатами в задаче о динамике маятника с вибрирующим подвесом [10,11], можно ожидать стабилизации неустой-чивого равновесия при высокочастотной модуляции параметров ОС.…”