Adaptive Observer Design for an n + 1 Hyperbolic PDE with Uncertainty and Sensing on Opposite Ends

An adaptive observer design for a system of n + 1 coupled 1-D linear hyperbolic partial differential equations with an uncertain boundary condition is presented, extending previous results by removing the need for sensing collocated with the uncertainty. This modification is important and motivated by applications in oil & gas drilling where information about the down-hole situation is crucial in order to prevent or deal with unwanted incidents. Uncertainties are usually present down-hole while measurements are available top-side at the rig, only. Boundedness of the state and parameter estimates is proved in the general case, while convergence to true values requires bounded system states and, for parameter convergence, persistent excitation. The central tool for analysis is the infinitedimensional backstepping method applied in two steps, the first of which is time-invariant, while the second is time-varying induced by the time-varying parameter estimates.