Degree of controllability for close modes of flexible space structures

It is pointed out that the controllability of an isolated mode of a flexible structure is straightforward to check, as is the controllability of exactly repeated modes. However, a case that arises often in practical applications, e.g. in connection with flexible space structures, is that of modes which are close but not exactly equal. Quantifying the controllability of such systems is a non-trivial task. The author studies this question and derives simple expressions for the degrees of controllability of near-repeated modes of flexible structures. These expressions allow the role of damping and frequency separation in controllability (and, by duality, observability) to be quantified, given insight into such problems as balancing and model reduction for flexible spacecraft