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Adaptive Tuning of Feedback Gain in Time-Delayed Feedback Control

Alexander Fradkov, Eckehard Schoell, Philipp Hoevel, Valentin Flunkert, Peter Guzenko
We study the possibility to adaptively tune the feedback gain K in the well-known time-delayed feedback control. This adaptive modification of time-delayed feedback control is applied to the stabilization of an unstable fixed point and the stabilization of an unstable periodic orbit embedded in to a chaotic attractor. The adaptation algorithm is constructed using the speed-gradient method. For the stabilization of a fixed point, a generic two-variable normal form is used. The linear stability analysis
for the control applied to both one and two system variables is given. It is found by computer simulations that the adaptation algorithm can tune the feedback gain K from the initial value K=0 to some appropriate value in the domain of successful control. This final value may depend on the initial conditions for the system variables and adaptation-algorithm parameter. The adaptation algorithm is developed for both the standard and the extended time-delay autosynchronization control schemes. We find that in both cases the speed-gradient based adaptation algorithm
can ensure the successful control by finding an appropriate
feedback gain K.
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