Feedback control with finite accuracy: more knowledge and better control for free

It is known [Phys. Rev. Lett. 63 (11) (1990) 1196–1199] that a system can be controlled near an unstable period-k point by applying a small perturbation signal to some parameter in the system. The required perturbation is calculated as a function of the current system state. We consider that, in applications, such stabilizing algorithms can be implemented with only finite accuracy. The error associated with the finite accuracy will grow exponentially, requiring repeated application of the parameter perturbation to keep the system near the fixed point. We show that under a repeated perturbation control algorithm, the resultant dynamical system is a piecewise expanding map that is well approximated by a Renyı́ transformation. Additionally, we show that by an analysis of the kneading sequences, the system state can be known with greater accuracy than is measured. Furthermore, we modify the standard parametric control algorithm to provide better time averaged control of the system without increasing the complexity of implementation. We demonstrate application of these principles to the 2D case by considering a saddle fixed point of the Ikeda map and the OGY [Phys. Rev. Lett. 63 (11) (1990) 1196–1199] algorithm.