Degrees of controllability and observability for close modes of flexible space structures

Flexible structures usually possess vibration modes with close, but not exactly repeated, natural frequencies. Quantifying the controllability and observability of such modes is a nontrivial task; this contrasts with the case of modes that are either exactly repeated or widely separated. The paper derives expressions for the degrees of controllability and observability of near-repeated modes by examining the closed-form Grammians of flexible structures. The results obtained quantify the role of damping and frequency separation in controllability and observability