Adaptive search for the optimal feedback gain of time-delayed feedback controlled systems in the presence of noise

We propose two adaptive algorithms for the time-delayed feedback control method to tune the feedback gain to an optimal value in the presence of noise. By the optimal value we mean the value of the feedback gain that minimizes the mean square of the control signal. The first algorithm is model independent; it uses trial values of the feedback gain and defines the optimal value by the least-squares polynomial fitting. The second algorithm is based on the gradient descent method and requires the knowledge of the system equations. Here any initial value of the feedback gain is continuously adjusted towards the optimal value without any trials. The efficacy of the algorithms is demonstrated with different specific models, namely, a simple linear map, the Rössler system and the normal form of the subcritical Hopf bifurcation.

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