Finite dimensional observer design for a class of ODE-PDE systems involving delay

We study the state estimation problem for a ODE-PDE delayed cascaded coupling. The delayed output of a nonlinear system in triangular form is the boundary condition of a 1-D linear parabolic system at one side, and the available measurement is taken at the opposite side. The observer of the parabolic system is based on the modal decomposition approach and it is finite dimensional. The nonlinear observer compensates the delay in the measurement. We prove that under a suitable delay bound the estimation error is bounded even when the dynamic of the parabolic system is unbounded.