Adaptive Delayed Feedback Control Algorithm With an Iterated Feedback Gain in the Presence of Noise

We propose an adaptive modification of the delayed feedback control (DFC) method that automatically finds the optimal feedback gain of the DFC systems. The system under control is perturbed by a Gaussian white noise with a low amplitude, and the variance of the delayed difference is estimated by a simple integrator. The problem of minimization of the variance is substituted by a minimization of some quantity that represents a generalization of the Lyapunov exponent in the presence of noise. The generalization is derived by considering a simple Langevin equation. The numerical simulations for the controlled Rössler system show that the adaptively obtained optimal feedback gains are in good quantitative agreement with the corresponding exact values.