Feedback control of chaos in impact oscillator with multiple time-delays

Extended time -delayed feedback control (ETDF) has emerged as a promising control method for nonlinear systems, with applications in diverse fields such as energy harvesting, vibration mitigation, and chaos control. Whilst novel ETDF-based approaches incorporating multiple periods of a target orbit as time delays have been explored, a comprehensive analysis of the underlying phenomena and limitations of ETDF with such delays is lacking. In this study, we investigate the effects of various delays and the strength of the original instability of the target unstable periodic orbit on ETDF control using experimental data from an impact oscillator. Surprisingly, we observe that ETDF loses its ability to stabilize a target orbit when delays are even multiples of the target orbit's period. By combining Floquet theory, semi -analytical methods, experimental data, and numerical simulations, we unveil the fundamental mechanisms responsible for this loss of efficacy. Specifically, we demonstrate that stability cannot be achieved for delays that are even multiples of the target orbit period due to a shift in the imaginary parts of Floquet exponents as controller gains approach infinity. In contrast, by using odd multiples of the target orbit period as delays a more gradual degradation of stability is observed as their multiplicity increases. Furthermore, we propose a method to predict the maximum instability of an orbit that can be counterbalanced by ETDF. Our findings offer essential insights for the design of robust and efficient control strategies based on ETDF, while advancing our understanding of the underlying mechanisms governing such controllers.