ew Features of Positive Time-Delayed Feedbacks in Vibration Control

Compared with negative feedbacks that are used intensively, positive feedbacks have been used relatively few in control theory and control engineering, because positive feedbacks enlarge state variations so that they deteriorate the system stability. This is not the case if a delay involves in the feedback path. On the basis of stability switches and the rightmost characteristic roots that determine the system stability, this paper investigates the control effect of positive feedbacks in stabilizing unstable motion as well as in improving stability of the typical linear vibration system. One of our observations is that a time-delayed positive displacement feedback works more effectively than the corresponding negative feedback as shown in the following three aspects: (1). the delay values that are used for stabilization via positive feedback are smaller than the admissible delays via negative feedbacks; (2). the admissible delay range for stabilization via positive feedback is broader than that via negative feedbacks; (3). in a given range of delay, the minimal value of the real parts of the rightmost characteristic roots of the closed-loop under positive feedback control is much smaller than that for the case of negative feedback, so that with such a delay, the state of the closed-loop under positive feedback approaches to the equilibrium much more quickly than the case of negative feedback, if the states start from the same initial conditions. For the same stabilization problem, however, negative feedback works more effectively than positive feedback if a delayed velocity feedback or a delayed acceleration feedback is performed. In addition, with the same feedback gain, a positive delayed displacement feedback is preferable than a positive delayed velocity feedback or a positive delayed acceleration feedback. Keywords stabilization, vibration control, positive delayed feedback, stability switches, characteristic roots