Regulation of Inhomogeneous Drilling Model With a P-I Controller

In this paper, we demonstrate that a proportional integral controller allows the regulation of the angular velocity of a drill-string despite unknown frictional torque and measuring only the angular velocity at the surface. Our model is an one-dimensional damped inhomogeneous wave equation subject to an unknown dynamic at one side while the control and the measurement are in the other side. After writing this system of balance laws into the Riemann coordinates, we design a Lyapunov functional to prove the exponential stability of the closed loop and show how it implies the regulation of the angular velocity.