Sensitivity of the zeros of flexible structures to sensor and actuator location

Flexible space structures (FSS) must often be controlled by means of non-collocated sensors and actuators, so as to avoid the performance limitations inevitable in collocated control. But this nearly always leads to right half-plane transmission zeros: Furthermore, non-collocated zeros have been observed to be, in some cases, extremely sensitive to the choice of FSS model order, as well as to the exact positions of sensors and actuators. However, a quantitative explanation of the ill-conditioning of non-collocated zeros has not yet been obtained. This question is addressed in this paper, where it is shown that ill-conditioning occurs when the zero direction constrained mode shape and its dual are nearly orthogonal. Two new condition numbers are then derived that quantify the sensitivity of the zeros to sensor and actuator position variations, and conclusions drawn concerning the zeros of near-collocated structures. Finally, these results are used to study the difficult non-collocated sensor/actuator placement problem