Control of delayed measured systems and impulse length limitations in difference control

When stabilization of unstable periodic orbits or fixed points by the method given by Ott, Grebogi and Yorke (OGY) has to be based on a measurement delayed by τ orbit lengths, unmodified OGY method fails beyond a maximal Lyapunov number of λ_max = 1+1/τ. Therefore the question arises as to how the control of delayed measured chaotic systems can be improved. Apart from rhythmic control, OGY and difference control can be improved for this case by the memory methods discussed here, linear predictive logging control (LPLC) and memory difference control (MDC). These allow to overcome the measurement delay completely within the linear approximation around the orbit. Furthermore a new stability analysis of the two elementary Poincare-based chaos control schemes, OGY and difference control, is given by means of Floquet theory. This approach allows to calculate exactly the stability restrictions occurring for small measurement delays and for an impulse length shorter than the length of the orbit. As an unexpected result, while for OGY control the influence of the impulse length is marginal, difference control is shown to fail when the impulse length is taken longer than one half of the orbit length.