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

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.

[1]  Stephen P. Timoshenko,et al.  Vibration problems in engineering , 1928 .

[2]  N. Levinson,et al.  Transformation Theory of Non-Linear Differential Equations of the Second Order , 1944 .

[3]  R. Courant,et al.  Methods of Mathematical Physics , 1962 .

[4]  K. Weiss.  Vibration Problems in Engineering , 1965, Nature.

[5]  A. F. Filippov.  Classical Solutions of Differential Equations with Multi-Valued Right-Hand Side , 1967 .

[6]  Dean Karnopp,et al.  Computer simulation of stick-slip friction in mechanical dynamic systems , 1985 .

[7]  Brian Armstrong-Hélouvry,et al.  Stick-slip arising from Stribeck friction , 1990, Proceedings., IEEE International Conference on Robotics and Automation.

[8]  Arthur J. Helmicki,et al.  Ill-posed distributed parameter systems: a control viewpoint , 1991 .

[9]  Carlos Canudas de Wit,et al.  A survey of models, analysis tools and compensation methods for the control of machines with friction , 1994, Autom..

[10]  Omer Morgiil,et al.  On the Stabilization and Stability Robustness Against Small Delays of Some Damped Wave Equations , 1995 .

[11]  Kim-Chuan Toh,et al.  SDPT3 -- A Matlab Software Package for Semidefinite Programming , 1996 .

[12]  M.J.G. van de Molengraft,et al.  H/sub /spl infin// control for suppressing stick-slip in oil well drillstrings , 1998 .

[13]  Charles Wang,et al.  On the Effective Control of Torsional Vibrations in Drilling Systems , 1999 .

[14]  N. Challamel.  ROCK DESTRUCTION EFFECT ON THE STABILITY OF A DRILLING STRUCTURE , 2000 .

[15]  Jack K. Hale,et al.  Effects of Small Delays on Stability and Control , 2001 .

[16]  Ahmet S. Yigit,et al.  Fully coupled vibrations of actively controlled drillstrings , 2003 .

[17]  Johan Efberg,et al.  YALMIP : A toolbox for modeling and optimization in MATLAB , 2004 .

[18]  J. Lofberg,et al.  YALMIP : a toolbox for modeling and optimization in MATLAB , 2004, 2004 IEEE International Conference on Robotics and Automation (IEEE Cat. No.04CH37508).

[19]  R. Suarez,et al.  Practical approach to modelling and controlling stick-slip oscillations in oilwell drillstrings , 2004, Proceedings of the 2004 IEEE International Conference on Control Applications, 2004..

[20]  E.M. Navarro-Lopez,et al.  Sliding-mode control of a multi-DOF oilwell drillstring with stick-slip oscillations , 2007, 2007 American Control Conference.

[21]  J. Coron.  Control and Nonlinearity , 2007 .

[22]  Emmanuel M Detournay,et al.  A simplified model to explore the root cause of stick–slip vibrations in drilling systems with drag bits , 2007 .

[23]  Carlos Canudas-de-Wit,et al.  D-OSKIL: A New Mechanism for Controlling Stick-Slip Oscillations in Oil Well Drillstrings , 2008, IEEE Transactions on Control Systems Technology.

[24]  M. Krstić.  Delay Compensation for Nonlinear, Adaptive, and PDE Systems , 2009 .

[25]  G. Weiss,et al.  Observation and Control for Operator Semigroups , 2009 .

[26]  Eva M. Navarro-López,et al.  An alternative characterization of bit-sticking phenomena in a multi-degree-of-freedom controlled drillstring , 2009 .

[27]  Emilia Fridman,et al.  Bounds on the response of a drilling pipe model , 2010, IMA J. Math. Control. Inf..

[28]  Sophie Tarbouriech,et al.  Stability and Stabilization of Linear Systems with Saturating Actuators , 2011 .

[29]  Pierre Rouchon,et al.  Backstepping and flatness approaches for stabilization of the stick-slip phenomenon for drilling , 2013 .

[30]  P. Olver.  Nonlinear Systems , 2013 .

[31]  Dimitri Peaucelle,et al.  S-Variable Approach to LMI-Based Robust Control , 2014 .

[32]  Miroslav Krstic,et al.  Output-feedback adaptive control of a wave PDE with boundary anti-damping , 2014, Autom..

[33]  Guang Meng,et al.  Coupled axial-torsional dynamics in rotary drilling with state-dependent delay: stability and control , 2014 .

[34]  Frédéric Gouaisbaut,et al.  Hierarchy of LMI conditions for the stability analysis of time-delay systems , 2015, Syst. Control. Lett..

[35]  Sabine Mondié,et al.  The control of drilling vibrations: A coupled PDE-ODE modeling approach , 2016, Int. J. Appl. Math. Comput. Sci..

[36]  Sophie Tarbouriech,et al.  Wave Equation With Cone-Bounded Control Laws , 2016, IEEE Transactions on Automatic Control.

[37]  Delphine Bresch-Pietri,et al.  Backstepping observer based-control for an anti-damped boundary wave PDE in presence of in-domain viscous damping , 2016, 2016 IEEE 55th Conference on Decision and Control (CDC).

[38]  Sabine Mondié,et al.  A control oriented guided tour in oilwell drilling vibration modeling , 2016, Annu. Rev. Control..

[39]  G. Bastin,et al.  Stability and Boundary Stabilization of 1-D Hyperbolic Systems , 2016 .

[40]  Ole Morten Aamo,et al.  Linear stability analysis of self-excited vibrations in drilling using an infinite dimensional model , 2016 .

[41]  Halil Ibrahim Basturk,et al.  Observer-Based Boundary Control Design for the Suppression of Stick–Slip Oscillations in Drilling Systems With Only Surface Measurements , 2017 .

[42]  Alexandre Seuret,et al.  Tractable sufficient stability conditions for a system coupling linear transport and differential equations , 2017, Syst. Control. Lett..

[43]  Roman J. Shor,et al.  Torsional vibrations with bit off bottom: Modeling, characterization and field data validation , 2017 .

[44]  Frédéric Gouaisbaut,et al.  Exponential Lyapunov Stability Analysis of a Drilling Mechanism , 2018, 2018 IEEE Conference on Decision and Control (CDC).

[45]  Luca Zaccarian,et al.  Global Asymptotic Stability of a PID Control System With Coulomb Friction , 2016, IEEE Transactions on Automatic Control.

[46]  Alexandre Terrand-Jeanne,et al.  Regulation of the downside angular velocity of a drilling string with a P-I controller , 2018, 2018 European Control Conference (ECC).

[47]  F. Gouaisbaut,et al.  Input/output stability of a damped string equation coupled with ordinary differential system , 2018, International Journal of Robust and Nonlinear Control.

[48]  Matthieu Barreau,et al.  Lyapunov Stability Analysis of a String Equation Coupled With an Ordinary Differential System , 2017, IEEE Trans. Autom. Control..

[49]  Matthieu Barreau,et al.  Stability analysis of coupled ordinary differential systems with a string equation : application to a drilling mechanism. (Analyse de stabilité de systèmes différentiels ordinaires couplés avec une équation des ondes : application aux mécanismes de forage) , 2019 .

[50]  Alexandre Terrand-Jeanne,et al.  Regulation of Inhomogeneous Drilling Model With a P-I Controller , 2020, IEEE Transactions on Automatic Control.