Output regulation for a heat equation with unknown exosystem

In this paper, we consider output regulation for a 1-d heat equation where the disturbances generated from an unknown finite-dimensional exosystem enter all possible channels. We adopt adaptive observer internal model approach which has been well developed for lumped parameter systems over two decades to estimate all possible unknown frequencies that have entered into a transformed 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 PDE. A significance of the problem lies in the fact that both the control and observation operators are unbounded. The proposed approach is potentially applicable to other PDEs.