The adaptive observer has an important role on the modern control theory since it reconstructs a state of unknown plant from input and output measuring. However, it is restricted to the finite dimensional linear plant and is not applicable to the continuous-time linear plant with time delays. The restriction is not so practical that many plants have their own time delays and processing in the controller and measuring instrument may add the time delay to the plant which has no time delay.
This paper presents a continuous-time adaptive observer for the linear plant with input and output time delays. It consists of an adptive state estimator and an adaptive state predictor, so that the global convergence of the reconstructed state to the true value can be proved with the sufficiently richness condition. The estimator, almost equal to the ordinary adaptive observer, gives an estimate of the past state, from which the predictor reconstructs the required state as the observed value. A numerical simulation is performed for the purpose of verifying the convergence proof.