Analysis and funnel control for nonlinear drill strings

TL;DR

This paper addresses output tracking for a nonlinear PDE-ODE drill string model using a novel funnel control approach that adapts to large wave traveling times, ensuring performance within a specified funnel.

Contribution

It introduces a new funnel control design for nonlinear boundary-coupled PDE-ODE systems with an analysis based on maximal monotone operator theory.

Findings

Existence of solutions established via maximal monotone operator framework.

Proposed funnel control ensures tracking within a pre-defined performance funnel.

Simulations demonstrate the effectiveness of the control approach.

Abstract

We study the output tracking problem for a vertically driven drill string system described by a nonlinear boundary-coupled PDE-ODE model. Solvability analysis of the drill string model is achieved by first casting the model in an abstract boundary value problem involving set-valued operators on an appropriate Hilbert space. The governing equation here consists of evolution and the damping part. Existence of solutions is established within the framework of maximal monotone operators where one first proves that the evolution operator is a linear skew-adjoint operator and the distributed damping term is a Nemytskii relation which is then proven to be maximal monotone. Maximal monotonicity of the combined operator is then a consequence of Rockafellar's theorem. Furthermore, we propose a novel funnel control design that ensures the angular velocity of the drill bit follows a dynamically adjusted reference trajectory, while the tracking error remains confined within a pre-specified performance funnel. The reference adjustment mechanism adapts in response to large wave traveling times that may cause performance degradation. The corresponding feasibility result is illustrated by some simulations.