Output tracking for a flexible string system with unknown harmonic disturbances using adaptive internal model

A non‐collocated output tracking problem for a flexible string with tip payload at one end was studied in this paper, which is expected to describe by a wave equation coupled ordinary differential equation (ODE) system. The regulated output and the reference trajectory signal with unknown frequency and amplitude are non‐collocated with the boundary control input. An adaptive approach has been developed to determine the unknown frequency of a disturbance by constituting an auxiliary trajectory and an internal model structure. In order to regulate exponential output tracking, the proposed error‐based adaptive dynamic compensation controller is employed. Closed‐loop stability and well‐posedness is established by exploiting the principle of semigroup theory. Finally, simulation experiments are performed to show the proposed scheme is very effective.