Flexible structures containing a large number of modes can benefit from adaptive control techniques which are well suited to applications that have unknown modeling parameters and poorly known operating conditions. In this paper, we focus on a direct adaptive control approach that has been extended to adaptively reject persistent disturbances. This theory will be extended to accommodate troublesome modal subsystems of a plant that might inhibit the adaptive controller. In some cases the plant in does not satisfy the adaptive controller requirements of Almost Strict Positive Realness. Instead, there maybe be a modal subsystem that inhibits this property. In this paper we will modify the adaptive controller with a Residual Mode Filter (RMF) to compensate for the troublesome modal subsystem, or the Q modes. This paper addresses leakage, or propagation, of the disturbances into the Q modes. We will apply the above theoretical results to a flexible structure example to illustrate the behavior with and without the residual mode filter.