Experimental control of a chaotic pendulum with unknown dynamics using delay coordinates

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Abstract
Unstable periodic orbits (UPOs) of an experimental chaotic pendulum were stabilized using a semi- continuous control method (SCC), applying control actions several times per cycle. One advantage of this method, compared to a one-map-based control method such as the Ott-Grebogi-Yorke method [Phys. Rev. Lett. 64, 1196 (1990)], is the applicability to systems with relatively large unstable eigenvalues and/or high noise levels. Compared to a continuous type of feedback control as was proposed by Pyragas [Phys. Lett. A 170, 421 (1992)], the advantage is that the controller settings can be measured from experimental data. Because the control method uses delay coordinates, only one variable has to be measured. This paper describes an SCC method using delay coordinates, the extraction of UPOs from time series, how the effect of the control parameter can be measured, the effect on the control in case of an error in the estimate of the UPO, and how this error can be reduced to obtain more stable control.