Output Regulation for a Wave Equation With Unknown Exosystem

In this article, the output regulation for a 1-D wave equation where the disturbances generated from an unknown finite-dimensional exosystem appear in all possible channels is studied. The system is first transformed into a new system for which the disturbance appears in tracking error only. An adaptive observer approach is adopted in investigation to estimate all possible unknown frequencies that have entered into a transformed new system. By the estimates of the unknown frequencies, we are able to design a tracking-error-based feedback control to achieve output regulation and disturbance rejection for this partial differential equations (PDEs) in two different cases. In the first case, the derivative of the tracking error is allowed to be used in the control design, which leads to the exponential convergence of the tracking error. In the second case, the tracking error is solely used and the asymptotic convergence is achieved. A remarkable characteristic of the problem lies in the fact that the control operator is unbounded and is noncollocated with the regulated output, which represents a difficult situation for output regulation on PDEs. The proposed approach is potentially applicable to other PDEs.