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, Tome 26 (2016), p. 335-349 / Harvested from The Polish Digital Mathematics Library

Résumé

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.

Détail

Publié le : 2016-01-01
EUDML-ID : urn:eudml:doc:280113

Bibliographie

[000] 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.

[001] 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.

[002] Ben-Tal, A. and Zibulevsky, M. (1997). Penalty/barrier multiplier methods for convex programming problems, SIAM Journal on Optimization 7(2): 347-366. | Zbl 0872.90068

[003] 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.

[004] 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.

[005] Challamel, N. (2000). Rock destruction effect on the stability of a drilling structure, Journal of Sound and Vibration 233(2): 235-254.

[006] 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.

[007] 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. | Zbl 0838.93022

[008] Fridman, E. and Dambrine, M. (2010). Control under quantization, saturation and delay: A LMI approach, Automatica 45(10): 2258-2264. | Zbl 1179.93089

[009] 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. | Zbl 1153.93502

[010] 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. | Zbl 1213.35150

[011] Grujić, L.T. (1973). On practical stability, International Journal of Control 17(4): 881-887. | Zbl 0254.93039

[012] 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.

[013] Jansen, J. (1993). Nonlinear Dynamics of Oilwell Drillstrings, Ph.D. thesis, Delft University of Technology, Delft.

[014] 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.

[015] 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.

[016] Khalil, H. (2002). Nonlinear Systems, Third Edition, Prentice-Hall, Upper Saddle River, NJ.

[017] 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. | Zbl 1328.93126

[018] Kǒcvara, M. and Stingl, M. (2003). PENNON-a code for nonlinear and convex semidefinite programming, Optimization Methods and Software 8(3): 317-333. | Zbl 1037.90003

[019] La Salle, J. and Lefschetz, S. (1961). Stability by Lyapunov's Direct Method: With Applications, Academic Press, New York, NY. | Zbl 0098.06102

[020] Lakshmikantham, V., Leela, S. and Martynyuk, A. (1990). Practical Stability of Nonlinear Systems, World Scientific Publishing Company, Singapore. | Zbl 0753.34037

[021] Levinson, N. (1944). Transformation theory of non-linear differential equations of the second order, Annals of Mathematics 45(4): 723-737. | Zbl 0061.18910

[022] 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.

[023] 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. | Zbl 1327.93155

[024] 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.

[025] 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.

[026] 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. | Zbl 1198.34120

[027] 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.

[028] 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.

[029] Rasvan, V. (2006). Three lectures on dissipativeness, IEEE International Conference on Automation, Quality and Testing, Robotics, Cluj-Napoca, Romania, pp. 167-177.

[030] 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.

[031] Saldivar, B. and Mondié, S. (2013). Drilling vibration reduction via attractive ellipsoid method, Journal of the Franklin Institute 350(3): 485-502. | Zbl 1268.93073

[032] 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.

[033] 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.

[034] 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.

[035] 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.

[036] Timoshenko, S. and Young, D. (1955). Vibrations Problems in Engineering, Third Edition, D. Van Nostrand Company, Princeton, NJ.

[037] 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.

[038] Weaver, W., Timoshenko, S. and Young, D. (1990). Vibrations Problems in Engineering, Fifth Edition, John Wiley & Sons, New York, NY.

[039] 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.

[040] 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. | Zbl 1290.35298

[041] Yoshizawa, T. (1960). Stability and boundedness of systems, Archive for Rational Mechanics and Analysis 6(1): 409-421. | Zbl 0096.28902

[042] Yoshizawa, T. (1966). Stability Theory by Lyapunov's Second Method, The Mathematical Society of Japan, Tokyo. | Zbl 0144.10802

[043] 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. | Zbl 1064.93008

[044] 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.