Adaptive Control of Anti-stable Wave PDE Systems: Theory and Applications in Oil Drilling

—Adaptive control of PDEs is a nonlinear feedback design problem for an infinite-dimensional system. The author's early work for PDEs with unmatched uncertainties was limited to parabolic and first-order hyperbolic PDEs. However, problems in oil drilling, combustion dynamics, and other applications give rise to wave PDE models, which are not only unstable, but have all of their infinitely many eigenvalues at uncertain locations in the right half plane (anti-stable). For example, in oil drilling, the friction between the drill bit and the rock is characterized by a highly uncertain friction parameter (depending on the nature of the soil the drill is passing through), is of destabilizing (anti-damping) character, and appears on the opposite end of the PDE's domain relative to the boundary control input. The wave system being of second order in time requires a special adaptive design approach in order to avoid the appearance of the second time derivative of the parameter estimate in the error system. The design is performed using infinite-dimensional backstepping transformations and Lyapunov functionals constructed with such transformations.