Adaptive Output Regulation for a Vibrating String With Output Delay: Exponential Tracking With a Single Measurement

ABSTRACT This paper addresses the output regulation problem for an unstable wave equation subject to both output delay and all‐channel disturbances of unknown frequencies and amplitudes. The problem is first reformulated into a stabilization framework via regulator equations, for which a backstepping‐based feedforward regulator is developed. A key feature of the control design is the partitioning of the time domain, which facilitates the independent synthesis of observers and predictors. The observer stage incorporates a state observer with a dynamic compensator and an adaptive observer based on the internal model principle to handle the unknown disturbances. The resulting error‐based feedback control law requires only a single measurement and ensures exponential convergence of the tracking error to zero. To the best of our knowledge, this is the first solution to the output regulation problem for distributed parameter systems that simultaneously accommodates output delay and unknown disturbances. The efficacy of the proposed scheme is demonstrated through numerical simulations.