Sensitivity of the transmission zeros of flexible space structures

Abstract

The uncertainties inherent in the dynamics of flexible spacecraft make sensitivity and robustness questions very important when designing vibration isolation systems for these vehicles. One technique recently proposed for this problem is pole/zero cancellation, in which state feedback is used to make as many closed-loop modes as possible unobservable at the points on the structure where vibrations are to be minimized. Study of the robustness properties of this approach first requires an analysis of the sensitivity of the transmission zeros of second-order structural models. These questions are addressed in this paper, where it is proved (first in terms of partial derivatives and then condition numbers) that the sensitivities of the transmission zeros of any structure with collocated sensors and actuators are closely related to those of the poles of the system. Furthermore, the closed-loop poles produced by applying pole/zero cancellation to such a structure are shown to have sensitivities approaching those of the zeros, so these too are approximately given from the sensitivities of the open-loop poles. Finally, these points are illustrated by simple examples.