Adaptive output feedback for oil drilling stick-slip instability modeled by wave PDE with anti-damped dynamic boundary

We develop an adaptive output-feedback controller for a wave PDE in one dimension with actuation 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 feature of the proposed controller is that it requires only the measurements of 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, with the first of the two delay elements being fed by control and the same element in turn feeding into a scalar ODE. This enables us to employ a prediction-based design for systems with 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.