Boundary observer design for hyperbolic PDE–ODE cascade systems

The paper presents an observer design for a class of hyperbolic PDE–ODE cascade systems with a boundary measurement. The cascade systems consist of coupled PDEs, featuring one rightward and one leftward convecting first-order transport PDEs, and a set of ODEs, which enter the PDEs through the left boundary of the systems. The design, which is based on the Volterra integral transformation, relies only on a single sensor at the right boundary of the system. The observer consists of a copy of the plant plus output injection terms both in the PDEs and the ODEs. The observer is constructed in a collocated setup, which means both sensing and actuation are located at the same boundary. The observer gains are computed analytically by solving Goursat-type PDEs in terms of Bessel function of the first kind. The observer design is tested against a field scale flow-loop test experiment in Stavanger by Statoil Oil Company. The results show that the observer converges to the actual values and that the design can be used as a process monitoring tool in oil well drilling.