Delayed Measured Systems in Poincare-based Chaos Control

Abstract The generic problem discussed here is the effect of measurement delay on Poincare-based chaos control and possibilities for improved control. When stabilization of unstable periodic orbits or fixed points by the method given by Ott, Grebogi, and Yorke (OGY) must be based on a measurement delayed by tau orbit lengths, the performance of unmodified OGY method is expected to decline. For experimental considerations, it is desired to know the range of stability with minimal knowledge of the system. Unmodified OGY control fails beyond a maximal Lyapunov number of λmax = 1+ (1/τ). The area of stability is investigated both for OGY control of known fixed points and for difference control of unknown or inaccurately known fixed points. Hence, both unmodified Ott-GrebogiYorke control and difference control can be successfully applied only for a certain range of Lyapunov numbers depending on the delay time. Herefrom arises the question how the control of delayed measured chaotic systems can be improved, i.e. what extensions must be considered if one wants to stabilize fixed points with Lyapunov numbers above λmax. This limitation can be overcome by at least two classes of methods, namely, by rhythmic control and by the memory methods of linear predictive logging control and memory difference control. Counterintuitively, for the case of difference control it is also possible to give an explicit deadbeat control scheme that allows, within linear approximation, to perform control without principal limitations in delay time, dimension, and Ljapunov numbers.