The control of drilling vibrations: A coupled PDE-ODE modeling approach

Belem Saldivar; Sabine Mondié; Juan Carlos Avila Vilchis

International Journal of Applied Mathematics and Computer Science (2016)

  • Volume: 26, Issue: 2, page 335-349
  • ISSN: 1641-876X

Abstract

top
The main purpose of this contribution is the control of both torsional and axial vibrations occurring along a rotary oilwell drilling system. The model considered consists of a wave equation coupled to an ordinary differential equation (ODE) through a nonlinear function describing the rock-bit interaction. We propose a systematic method to design feedback controllers guaranteeing ultimate boundedness of the system trajectories and leading consequently to the suppression of harmful dynamics. The proposal of a Lyapunov-Krasovskii functional provides stability conditions stated in terms of the solution of a set of linear and bilinear matrix inequalities (LMIs, BMIs). Numerical simulations illustrate the efficiency of the obtained control laws.

How to cite

top

Belem Saldivar, Sabine Mondié, and Juan Carlos Avila Vilchis. "The control of drilling vibrations: A coupled PDE-ODE modeling approach." International Journal of Applied Mathematics and Computer Science 26.2 (2016): 335-349. <http://eudml.org/doc/280113>.

@article{BelemSaldivar2016,
abstract = {The main purpose of this contribution is the control of both torsional and axial vibrations occurring along a rotary oilwell drilling system. The model considered consists of a wave equation coupled to an ordinary differential equation (ODE) through a nonlinear function describing the rock-bit interaction. We propose a systematic method to design feedback controllers guaranteeing ultimate boundedness of the system trajectories and leading consequently to the suppression of harmful dynamics. The proposal of a Lyapunov-Krasovskii functional provides stability conditions stated in terms of the solution of a set of linear and bilinear matrix inequalities (LMIs, BMIs). Numerical simulations illustrate the efficiency of the obtained control laws.},
author = {Belem Saldivar, Sabine Mondié, Juan Carlos Avila Vilchis},
journal = {International Journal of Applied Mathematics and Computer Science},
keywords = {drilling vibrations; LMI approach; ultimate boundedness; coupled wave-ODE system},
language = {eng},
number = {2},
pages = {335-349},
title = {The control of drilling vibrations: A coupled PDE-ODE modeling approach},
url = {http://eudml.org/doc/280113},
volume = {26},
year = {2016},
}

TY - JOUR
AU - Belem Saldivar
AU - Sabine Mondié
AU - Juan Carlos Avila Vilchis
TI - The control of drilling vibrations: A coupled PDE-ODE modeling approach
JO - International Journal of Applied Mathematics and Computer Science
PY - 2016
VL - 26
IS - 2
SP - 335
EP - 349
AB - The main purpose of this contribution is the control of both torsional and axial vibrations occurring along a rotary oilwell drilling system. The model considered consists of a wave equation coupled to an ordinary differential equation (ODE) through a nonlinear function describing the rock-bit interaction. We propose a systematic method to design feedback controllers guaranteeing ultimate boundedness of the system trajectories and leading consequently to the suppression of harmful dynamics. The proposal of a Lyapunov-Krasovskii functional provides stability conditions stated in terms of the solution of a set of linear and bilinear matrix inequalities (LMIs, BMIs). Numerical simulations illustrate the efficiency of the obtained control laws.
LA - eng
KW - drilling vibrations; LMI approach; ultimate boundedness; coupled wave-ODE system
UR - http://eudml.org/doc/280113
ER -

References

top
  1. Anabtawiii, M. (2011). Practical stability of nonlinear stochastic hybrid parabolic systems of Itˆo-type: Vector Lyapunov functions approach, Nonlinear Analysis: Real World Applications 12(1): 1386-1400. 
  2. Bailey, J. and Finnie, I. (1960). An analytical study of drillstring vibration, Journal of Engineering for Industry, Transactions of the ASME 82(2): 122-128. 
  3. Ben-Tal, A. and Zibulevsky, M. (1997). Penalty/barrier multiplier methods for convex programming problems, SIAM Journal on Optimization 7(2): 347-366. Zbl0872.90068
  4. Boussaada, I., Mounier, H., Niculescu, S. and Cela, A. (2012). Analysis of drilling vibrations: A time delay system approach, 20th Mediterranean Conference on Control and Automation MED, Barcelona, Spain, pp. 610-614. 
  5. Canudas-de Wit, C., Rubio, F. and Corchero, M. (2008). D-OSKIL: A new mechanism for controlling stick-slip oscillations in oil well drillstrings, IEEE Transactions on Control Systems Technology 16(6): 1177-1191. 
  6. Challamel, N. (2000). Rock destruction effect on the stability of a drilling structure, Journal of Sound and Vibration 233(2): 235-254. 
  7. Detournay, E. and Defourny, P. (1992). A phenomenological model for the drilling action of drag bits, International Journal of Rock Mechanics, Mining Science and Geomechanical Abstracts 29(1): 13-23. 
  8. Fliess, M., Lévine, J., Martin, P. and Rouchon, P. (1995). Flatness and defect of non-linear systems: Introductory theory and examples, International Journal of Control 61(6): 1327-1361. Zbl0838.93022
  9. Fridman, E. and Dambrine, M. (2010). Control under quantization, saturation and delay: A LMI approach, Automatica 45(10): 2258-2264. Zbl1179.93089
  10. Fridman, E., Dambrine, M. and Yeganefar, N. (2008). Input to state stability of systems with time-delay: A matrix inequalities approach, Automatica 44(9): 2364-2369. Zbl1153.93502
  11. Fridman, E., Mondié, S. and Saldivar, B. (2010). Bounds on the response of a drilling pipe model, IMA Journal of Mathematical Control and Information 27(4): 513-526. Zbl1213.35150
  12. Grujić, L.T. (1973). On practical stability, International Journal of Control 17(4): 881-887. Zbl0254.93039
  13. Halsey, G., Kyllingstad, A. and Kylling, A. (1988). Torque feedback used to cure slip-stick motion, Proceedings of the 63rd Society of Petroleum Engineers Drilling Engineering Annual Technical Conference and Exhibition, Houston, TX, USA, pp. 277-282. 
  14. Jansen, J. (1993). Nonlinear Dynamics of Oilwell Drillstrings, Ph.D. thesis, Delft University of Technology, Delft. 
  15. Jansen, J. and van den Steen, L. (1995). Active damping of self-excited torsional vibrations in oil well drillstrings, Journal of Sound and Vibration 179(4): 647-668. 
  16. Javanmardi, K. and Gaspard, D. (1992). Application of soft torque rotary table in mobile bay, Technical Report IADC/SPE 23913, International Association of Drilling Contractors/Society of Petroleum Engineers, Dallas, TX. 
  17. Khalil, H. (2002). Nonlinear Systems, Third Edition, Prentice-Hall, Upper Saddle River, NJ. 
  18. Knuppel, T., Woittennek, F., Boussaada, I., Mounier, H. and Niculescu, S. (2014). Flatness-based control for a non-linear spatially distributed model of a drilling system, in A. Seuret et al. (Eds.), Low Complexity Controllers for Time Delay Systems: Advances in Delays and Dynamics, Volume 2, Springer, Cham, pp. 205-218. Zbl1328.93126
  19. Kǒcvara, M. and Stingl, M. (2003). PENNON-a code for nonlinear and convex semidefinite programming, Optimization Methods and Software 8(3): 317-333. Zbl1037.90003
  20. La Salle, J. and Lefschetz, S. (1961). Stability by Lyapunov's Direct Method: With Applications, Academic Press, New York, NY. Zbl0098.06102
  21. Lakshmikantham, V., Leela, S. and Martynyuk, A. (1990). Practical Stability of Nonlinear Systems, World Scientific Publishing Company, Singapore. Zbl0753.34037
  22. Levinson, N. (1944). Transformation theory of non-linear differential equations of the second order, Annals of Mathematics 45(4): 723-737. Zbl0061.18910
  23. Lu, H., Dumon, J. and de Wit, C.C. (2009). Experimental study of the D-OSKIL mechanism for controlling the stick-slip oscillations in a drilling laboratory testbed, 2009 IEEE Control Applications (CCA) & Intelligent Control (ISIC), St. Petersburg, Russia, pp. 1551-1556. 
  24. Ma, R., Dimirovski, G. and Zhao, J. (2013). Backstepping robust H control for a class of uncertain switched nonlinear systems under arbitrary switchings, Asian Journal of Control 15(1): 41-50. Zbl1327.93155
  25. Navarro-López, E. and Cortés, D. (2007a). Avoiding harmful oscillations in a drillstring through dynamical analysis, Journal of Sound and Vibration 307(1): 152-171. 
  26. Navarro-López, E. and Cortés, D. (2007b). Sliding-mode control of a multi-DOF oilwell drillstring with stick-slip oscillations, Proceedings of the 2007 American Control Conference, New York, NY, USA, pp. 3837-3842. 
  27. Navarro-López, E. and Licéaga-Castro, E. (2009). Non-desired transitions and sliding-mode control of a multi-DOF mechanical system with stick-slip oscillations, Chaos, Solitons and Fractals 41(4): 2035-2044. Zbl1198.34120
  28. Navarro-López, E. and Suárez, R. (2004). Practical approach to modelling and controlling stick-slip oscillations in oilwell drillstrings, Proceedings of the 2004 IEEE International Conference on Control Applications Taipei, Taiwan, pp. 1454-1460. 
  29. Pavone, D. and Desplans, J. (1994). Application of high sampling rate downhole measurements for analysis and cure of stick-slip in drilling, Technical Report SPE 28324, Society of Petroleum Engineers, Dallas, TX. 
  30. Rasvan, V. (2006). Three lectures on dissipativeness, IEEE International Conference on Automation, Quality and Testing, Robotics, Cluj-Napoca, Romania, pp. 167-177. 
  31. Saldivar, B., Knuppel, T., Woittennek, F., Boussaada, I., Mounier, H. and Niculescu, S. (2014). Flatness-based control of torsional-axial coupled drilling vibrations, 19th World Congress of the International Federation of Automatic Control, Cape Town, South Africa, pp. 7324-7329. 
  32. Saldivar, B. and Mondié, S. (2013). Drilling vibration reduction via attractive ellipsoid method, Journal of the Franklin Institute 350(3): 485-502. Zbl1268.93073
  33. Saldivar, B., Mondié, S., Loiseau, J. and Rasvan, V. (2013). Suppressing axial torsional coupled vibrations in oilwell drillstrings, Journal of Control Engineering and Applied Informatics 15(1): 3-10. 
  34. Serrarens, A., van de Molengraft, M., Kok, J. and van den Steeen, L. (1998). H control for suppressing stick-slip in oil well drillstrings, IEEE Control Systems 18(2): 19-30. 
  35. Skaugen, E. (1987). The effects of quasi-random drill bit vibrations upon drillstring dynamic behavior, Technical Report SPE 16660, Society of Petroleum Engineers, Dallas, TX. 
  36. Suh, Y., Kang, H. and Ro, Y. (2006). Stability condition of distributed delay systems based on an analytic solution to Lyapunov functional equations, Asian Journal of Control 8(1): 91-96. 
  37. Timoshenko, S. and Young, D. (1955). Vibrations Problems in Engineering, Third Edition, D. Van Nostrand Company, Princeton, NJ. 
  38. Tucker, R. and Wang, C. (1999). On the effective control of torsional vibrations in drilling systems, Journal of Sound and Vibration 224(1): 101-122. 
  39. Weaver, W., Timoshenko, S. and Young, D. (1990). Vibrations Problems in Engineering, Fifth Edition, John Wiley & Sons, New York, NY. 
  40. Wu, J., Li, S. and Chai, S. (2010). Exact controllability of wave equations with variable coefficients coupled in parallel, Asian Journal of Control 12(5): 650-655. 
  41. Yang, L. and Wang, J. (2014). Stability of a damped hyperbolic Timoshenko system coupled with a heat equation, Asian Journal of Control 16(2): 546-555. Zbl1290.35298
  42. Yoshizawa, T. (1960). Stability and boundedness of systems, Archive for Rational Mechanics and Analysis 6(1): 409-421. Zbl0096.28902
  43. Yoshizawa, T. (1966). Stability Theory by Lyapunov's Second Method, The Mathematical Society of Japan, Tokyo. Zbl0144.10802
  44. Zhang, X. and Zuazua, E. (2004). Polynomial decay and control of a 1-D hyperbolic-parabolic coupled system, Journal of Differential Equations 204(2): 380-438. Zbl1064.93008
  45. Zhou, Z. and Tang, S. (2012). Boundary stabilization of a coupled wave-ode system with internal anti-damping, International Journal of Control 85(11): 683-693. 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.