Using delayed state feedback to stabilize periodic motions of an oscillator

The paper states how to stabilize the periodic motions of a linear, non-positively damped system of single degree of freedom through the use of delayed state feedback. The study indicates that the delayed state feedback works in certain frequency ranges. For a linear undamped system of single degree of freedom, the delayed displacement feedback is able to stabilize almost all periodic motions of the system provided that their fundamental frequencies are higher than the natural frequency of the system, but it only works in a series of narrower and narrower frequency bands lower than the natural frequency. The introduction of delayed velocity feedback can remarkably enlarge the working frequency ranges of delayed displacement feedback. However, even the delayed state feedback cannot stabilize a periodic motion if the corresponding period is an integral multiple of the natural period of system. The criteria of stability switches prove to be a powerful tool to analyze the stabilization problem of a linear system. However, the stabilization of a linear undamped system of single degree of freedom with delayed velocity feedback only is a degenerate case where the available criteria of stability switches fail to offer any useful information. A detailed study in the paper reveals the complexity of the degenerated case, where the stabilization conditions can be identified according to the second order derivative of the real part of an arbitrary characteristic root.