Model-based feedback control of vibration in flexible structures can be complicated by the possibility that interaction with an external body occurs. If not accounted for, instability or poor performance may result. In this paper, a method is proposed for achieving robust vibration control of flexible structures under contact. The method uses robust linear state feedback, coupled with a state estimation scheme utilizing contact force measurement. Uncertain contact characteristics are modelled by a sector-bounded non-linear function, such that state feedback gains can be synthesized using a matrix inequality formulation of the Popov stability criterion. A separation theorem is used to establish a robust H cost bound for the closed loop system. Experimental results from a multi-mode flexible structure testbed confirm that vibration attenuation and stability can be maintained over a broad range of contact characteristics, in terms of compliance and clearance.