Adaptive output-feedback for wave PDE with anti-damping - application to surface-based control of oil drilling stick-slip instability

We develop an adaptive output-feedback controller for a wave PDE in one dimension with actuation and measurement on one boundary and with an unknown anti-damping dynamics on the opposite boundary. This model is representative of drill string torsional instabilities arising in deep oil drilling, for which the model of bottom interaction with the rock is poorly known. The key achievement of the proposed controller is that it requires only the measurements of top-boundary values and not of the entire distributed state of the system. Our approach is based on employing Riemann variables to convert the wave PDE into a cascade of two delay elements and to reconstruct a delayed version of the unmeasured boundary. This enables us to employ a prediction-based design for systems with output and input delays, suitably converted to the adaptive output-feedback setting. The result's relevance and ability to suppress undesirable torsional vibrations of the drill string in oil well drilling systems is illustrated with simulation example.