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This paper considers a class of infinite dimensional systems with structured perturbation. Such a perturbation is assumed to be expressed in terms of the output operator and an unknown matrix. The proposed adaptive observers include a coupling term which penalizes the disagreement of the estimates. The enforcement of consensus is applied to both state and parameter estimates, thereby constituting the main contribution of this work. Due to the specific operator Lyapunov equation that the nominal plant operator satisfies, the convergence of the estimation errors along with the asymptotic convergence of the state and parameter deviations from the mean are established. Extensive simulation studies examine also the case of adapting the consensus gains, which describe the case where the consensus gain is adjusted according to the disagreement of the estimates.