# Practical Stability Analysis of a Drilling Pipe under Friction with a PI-Controller

## TL;DR

This paper analyzes the exponential stability of a drilling pipe with friction controlled by a PI controller, using advanced Lyapunov functionals for coupled ODE/PDE models, and demonstrates limitations in reducing stick-slip oscillations.

## Contribution

It introduces a new Lyapunov functional approach for stability analysis of coupled ODE/PDE systems with PI control, addressing both linear and nonlinear torsional dynamics.

## Key findings

- Exponential stability established for linear torsional model using LMI framework.
- Nonlinear stick-slip oscillations are not mitigated by PI control.
- Numerical simulations confirm the theoretical results and limitations.

## Abstract

This paper deals with the exponential stability of a drilling pipe controlled by a PI controller. The model used leads to a coupled ODE / PDE and is consequently of infinite dimension. Using recent advances in time-delay systems, we derive a new Lyapunov functional based on an state extension made up of projections of the Riemann coordinates. Two cases will be considered. First, we will provide an exponential stability result expressed using the LMI framework. This result is dedicated to a linear version of the torsional dynamic. On a second hand, the nonlinear terms in the initial model, that generates the well-known stick-slip phenomenon is captured through a new stability theorem. Numerical simulations show the effectiveness of the method and that the stick-slip oscillations cannot be weaken using a PI controller.