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

TL;DR

This paper investigates how high-frequency periodic modulation of delay in feedback control can stabilize unstable steady states, expanding the controllability domain through distributed delay effects and multiple delay schemes.

Contribution

It introduces a novel analysis of delayed feedback control with high-frequency delay modulation, demonstrating increased stabilization domain and effects of phase coupling and latency.

Findings

High-frequency delay modulation creates an effective distributed delay.

Variable delay broadens the parameter space for stabilization.

Multiple delay feedback schemes improve control robustness.

Abstract

We analyze the stabilization of unstable steady states by delayed feedback control with a periodic time-varying delay in the regime of a high-frequency modulation of the delay. The average effect of the delayed feedback term in the control force is equivalent to a distributed delay in the interval of the modulation, and the obtained distribution depends on the type of the modulation. In our analysis we use a simple generic normal form of an unstable focus, and investigate the effects of phase-dependent coupling and the influence of the control loop latency on the controllability. In addition, we have explored the influence of the modulation of the delays in multiple delay feedback schemes consisting of two independent delay lines of Pyragas type. A main advantage of the variable delay is the considerably larger domain of stabilization in parameter space.