Observer Design for a Coupled ODE-PDE System from a Wellbore Reservoir Drilling Model

Abstract : The problem of state estimation for a coupled ODE-PDE system is addressed here by means of the back-stepping method for PDEs. The ODE is a finite-dimensional, linear, and time-invariant system and the PDE is a linear radial diffusion equation with Neumann and Robin boundary conditions. The coupling appears at one of the boundaries of the PDE and is bidirectional. More precisely, the ODE state appears in one of the boundary conditions of the PDE and the value of the PDE state at the boundary is an input to the ODE. Measurements of the ODE output are available, while the state of the PDE is out of sight. The estimate is defined as the state of an observer; constructed as a copy of the coupled system ODE-PDE with output error feedback. This study is motivated by the influx estimation problem from a wellbore-reservoir model used in managed pressured drilling applications.