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This article deals with the stability analysis of a drilling pipe controlled by a PI controller. The model is a coupled ordinary differential equation/partial differential equation (PDE) and is consequently of infinite dimension. Using recent advances in time-delay systems, we derive a new Lyapunov functional based on a state extension made up of projections of the Riemann coordinates. First, we will provide an exponential stability result expressed using the linear matrix inequality (LMI) framework. This result is dedicated to a linear version of the torsional dynamic. On the other hand, the influence of the nonlinear friction force, which may generate the well-known stick-slip phenomenon, is analyzed through a new stability theorem. Numerical simulations show the effectiveness of the method and that the stick-slip oscillations cannot be weakened using a PI controller.

Practical Stability Analysis of a Drilling Pipe Under Friction With a PI-Controller