Carlos Canudas-De-Wit
email: carlos.canudas-de-wit@inpg.fr

Francisco R Rubio
email: rubio@us.es

Miguel Angel Corchero

DOSKIL: A New Mechanism for Controlling Stick-Slip Oscillations in Oil Well Drillstrings

.

In opposition to the heuristic control structure proposed in [7], we show that the new Weight on Bit (WoB) control law results in a globally asymptotically stable closed loop-system. Simulations applying the D-OSKIL mechanism show that the stick-slip oscillations can be eliminated without requiring a re-design of the velocity rotary-table control.

I. INTRODUCTION

Oil well drillstrings (see Figure 1) are systems which present interesting features from the dynamical and control viewpoints as they pose many challenging technological problems [START_REF] Rabia | Oil Well Drilling Engineering: Principle and Practice[END_REF], [START_REF] William Ranney | Offshore Oil Technology Recent Development[END_REF]. The application of dynamic analysis and control techniques in a drilling system can lead to conclusions that allow us to propose new recommendations for drilling operations, drillstring design and control algorithm, which would produce economic benefits through a mix of lower development costs, higher production rates and improved recovery. Particularly, the presence of stickslip self-excited oscillations at the bottom part of the drillstrings as well as decreasing service life of drillstrings and downhole equipment, has drawn the attention of the control community in the last decade. The elimination of this kind of oscillations is a challenge for drillers and scientists since it can provide important cost savings in drilling operations, in terms on money and exploitation time [START_REF] Navarro-Lopez | Notas acerca del modelado, analisis y control de las vibraciones mecanicas en una sarta de perforacion[END_REF].

Different oscillations affecting the drillstring behavior are unavoidable. The occurrence of self-excited stick-slip vibrations (i.e., the top of the drillstring rotates with a constant rotary speed, whereas the bit (cutting device) rotary speed varies between zero and up to six times the rotary speed measured at the surface) as a common and damaging phenomena in drillstring systems has been highly described and analyzed in recent years, drawing the attention of the control community. For more information about drillstring oscillations and stick-slip phenomenon in oil well drillstrings, please see [START_REF] Harris | s Shock and Vibration Handbook[END_REF], [START_REF] Navarro-Lopez | Notas acerca del modelado, analisis y control de las vibraciones mecanicas en una sarta de perforacion[END_REF], [START_REF] Navarro-Lopez | Vibraciones mecánicas en una sarta de perforación: problemas de control[END_REF], [START_REF] Spanos | Oil well drilling: A vibrations perspective[END_REF].

Some causes of stick-slip oscillations are backlash between contacting parts, hysteresis, nonlinear damping and geometrical imperfections which are very difficult to model. However, the main cause of such vibrations in drillstrings is the friction appearing by contact with the rock formation [START_REF] Armstrong-Hélouvry | Stick-Slip Arising from Stribeck Friction[END_REF], [START_REF] Kyllingstad | A Study of Slip/stick Motion of the Bit[END_REF]. Consequently, a model describing the drillstring behavior should include a bit-rock friction torque model adequate enough to properly reproduce this effect.

Many ways of reducing these vibrations have been proposed, both from practical and theoretical viewpoints.

Historically, the experience of drillers has revealed that the manipulation of different drilling parameters (increasing the rotary speed, decreasing the weight-on-bit (W oB), modifying the drilling mud characteristics, introducing an additional friction at the bit [START_REF] Pavone | Application of High Sampling Rate Downhole Measurements for Analysis and Cure of Stick-slip in Drilling[END_REF], etc) is an effective strategy to suppress stick-slip motion [START_REF] Sananikone | A Field Method for Controlling Drillstring Torsional Vibrations[END_REF]. However, this strategy depends too much on the personal skills of each drilling technician to be really effective.

Usually, drilling systems are velocity-controlled to make them rotate at a constant velocity, but no specifications about vibration suppression or damping are considered. Another control option would be introducing new regulation methodologies (active, passive) in the loop, specifically aimed to compensate for drillstring vibrations. Among those,

the following examples can be pointed out:

• The so-called Soft Torque Rotary System (STRS) [START_REF] Halsey | Torque Feedback Used to Cure Slip-stick Motion[END_REF] [START_REF] Sananikone | A Field Method for Controlling Drillstring Torsional Vibrations[END_REF], that is a torque feedback at the top of the drillstring which makes the system behave in a "softer" way rather than as a fixed heavy flywheel, so that the torsional waves arriving at the surface are absorbed, breaking the harmful cycling motion.

• Introducing a vibration absorber at the top of the drillstring [START_REF] Jansen | Active Damping of Self-excited Torsional Vibrations in Oil Well Drillstrings[END_REF] which follows the same approach given in [START_REF] Halsey | Torque Feedback Used to Cure Slip-stick Motion[END_REF] and [START_REF] Sananikone | A Field Method for Controlling Drillstring Torsional Vibrations[END_REF].

• Introducing a PID controller structure at the surface in order to control the rotary speed [START_REF] Abbasssian | Application of Stability Approach to Torsional and Lateral Bit Dynamics[END_REF], [START_REF] Navarro-Lopez | Practical approach to modelling and controlling stick-slip oscillations in oilwell drillstrings[END_REF], [START_REF] Navarro-Lopez | Vibraciones mecánicas en una sarta de perforación: problemas de control[END_REF], [START_REF] Pavone | Application of High Sampling Rate Downhole Measurements for Analysis and Cure of Stick-slip in Drilling[END_REF].

• Using robust controllers, like the linear H ∞ control proposed in [START_REF] Serrarens | H∞ control for suppressing stick-slip in oil well drillstrings[END_REF], to suppress stick-slip motion at the bit.

• Using a controller based on an input-state feedback linearization of the nonlinear friction torque [START_REF] Abdulgalil | Nonlinear Control Design for Suppressing Stick-Slip in Oil Well Drillstrings[END_REF].

However, few works have provided a formal stability analysis of their proposed control strategies. For instance, analysis of the dynamical behavior of drillstring under vibrations has been explored in [START_REF] Abbasssian | Application of Stability Approach to Torsional and Lateral Bit Dynamics[END_REF] and linear approximations to stability of controlled drillstring has been studied in [START_REF] Navarro-Lopez | Vibraciones mecánicas en una sarta de perforación: problemas de control[END_REF].

The value of the system weight measured at the bottom part, called Weight on Bit (W oB ), has been proved to be an important parameter in the occurrence and possible avoidance of stick-slip oscillations (see [START_REF] Navarro-Lopez | Practical approach to modelling and controlling stick-slip oscillations in oilwell drillstrings[END_REF] and [START_REF] Sananikone | A Field Method for Controlling Drillstring Torsional Vibrations[END_REF]).

Efficient drilling operation requires a certain amount of force (W oB ) that may be incompatible with the low force range which may avoid stick-slip oscillations. This tradeoff between force magnitudes, provides a first indication that a regulation strategy of W oB seems to be necessary to maintain a good drilling operation, thus, avoiding such oscillations (see [START_REF] Canudas-De-Wit | A new mechanism for suppressing stick-slip in oil well drillstrings[END_REF] and [START_REF] Navarro-Lopez | Vibraciones mecánicas en una sarta de perforación: problemas de control[END_REF]).

This paper is focused on the problem of stick-slip oscillations produced at the bottom-hole assembly (BHA). The main idea is to use the weight on the bit (W oB) force as an additional control variable. In particular we adapt the oscillation killer (OSKIL) mechanism studied in [START_REF] Canudas-De-Wit | The oscillations killer: a mechanism to eliminate undesired limit cycles in nonlinear systems[END_REF], to the oil well drillstring systems (named here D-OSKIL 1 ) which has been shown to be particularly adapted for nonlinear systems displaying a local stable region with a stable limit set outside this local domain.

An approximate analysis based on the bias describing function provides good evidence that the rotational dynamics of the oil well drillstring display a similar behavior pattern. This analysis, although approximate, also gives a good intuition in the way that the W oB needs to be modified to suppress oscillations. An important property of the proposed D-OSKIL mechanism is that it allows recovering the nominal operation condition (the W oB recovers its nominal drilling value) while oscillations are suppressed.

In opposition to the heuristic control structure proposed in [START_REF] Canudas-De-Wit | A new mechanism for suppressing stick-slip in oil well drillstrings[END_REF], we show that the new proposed Weight on Bit (W oB ) control law results in a globally asymptotically stable closed loop-system. Therefore, the D-OSKIL mechanism eliminates the stick-slip oscillations without requiring a re-design of the velocity rotary-table control.

1 D-OSKIL stands for Drilling oscillation killer mechanism.

The paper is organized as follows. In Section II the basics about drillstring dynamics and vibrations are briefly introduced. In Section III, the drillstring model used in the paper is presented. The control loop used to regulate rotational velocity of the set and the a priori behavior of the closed loop system are shown in Sections IV and V, respectively. In Section VI, a Describing Function based analysis is made in order to obtain some information about stick-slip oscillations in the system, and next, in Section VII, the control mechanism named D-OSKIL obtained from the conclusions of the previous analysis is presented. In Section VIII, some simulations are shown. Section IX proposes an observed-based version of the same controller where only field existing measures are used. In the same section, we show also some simulations. And finally, Section X submits the conclusions and future research lines.

II. BASICS ON DRILLSTRING DYNAMICS AND VIBRATIONS

Standard rotatory drilling equipment, as shown in Figure 1 to depict what is commonly used by oil companies to extract gas and oil from the earth surface, uses a dill-bit (called bit) to crush the rock and make the hole in the ground. As the hole becomes deeper, some pipe sections (called drill pipes) are added, leaving the bit coupled at the bottom part of the set. These pipes, together with the drill bit, form the so-called drillstring. This drillstring is moved by means of a motor or system of motors in the surface. As it has been shown in the previous Section, operation of the drillstring looks just like that of a household electric drill, where a motor makes the bit rotate, and enough weight is applied to maintain the contact between the bit and the object to be drilled.

In order to make the study of a drilling system structure a bit more comprehensive, the following parts can be emphasized:

• Power System: A set of diesel and electric motors that provide the necessary energy to perform all the tasks.

• Supporting Structure: This is used to move the pipes in and out of the oil well, and so, to vary the weight applied during the process.

• The Rotatory System to make the system rotate. It is composed by:

-Swivel and Kelly: To connect the Supporting and Rotatory Systems.

-Rotatory Table : Also called turntable. It is a large disc-shaped inertia coupled to the drillstring that drives the rotating motion using power from electric motors.

-Drillstring: As shown before, it is a sequence of tubes that connect the rotatory table and the bit.

-Bit: The cutting device.

• Circulation System: It consists of a set of pipes and pumps which create a flow within the hole by drilling mud into it. This substance is aimed to lubricate and refrigerate the contact between the rock and the bit, and so to lift the rock cuttings from the drill bit to the surface.

A more exhaustive description of the rotatory system can be found, in [START_REF] Navarro-Lopez | Notas acerca del modelado, analisis y control de las vibraciones mecanicas en una sarta de perforacion[END_REF] and [START_REF] Spanos | Oil well drilling: A vibrations perspective[END_REF] among other papers.

One of the main problems is the appearance of oscillatory behaviors (limit cycles), that cause a decreasing of the drilling performance from the viewpoints of different parameters (rate of penetration at the surface, rotational speed of the bit, ...) and so provoking the mechanical failure of the drillstring or the breakage of any of the elements [START_REF] Spanos | Oil well drilling: A vibrations perspective[END_REF]. The vibrations appearing in the drillstring can be divided into 3 different categories [START_REF] Harris | s Shock and Vibration Handbook[END_REF] [10], (see Figure 2):

• Longitudinal vibrations are produced in a vertical direction from the drilling tower, causing rebounds of the bit at the bottom of the oil well, a phenomenon called bit-bouncing.

• Lateral vibrations are produced when the drillstring's mass center is displaced from the rotation axis, causing whirl-like movements and rebounds within the oil well walls, a phenomenon called whirling.

• Torsional vibrations are produced when the rotational velocities at the surface and the bottom of the drillstring are different, causing stick-slip movements.

Each oscillation phenomenon appears both at different times and different frequency ranges, and so, they can be studied separately. This work is focused on stick-slip vibrations.

The stick-slip oscillations are generally associated to typical dry friction profiles [START_REF] Leine | Stick-slip whirl interaction in drillstring dynamics[END_REF], i.e., when there is no movement, the friction torque (static friction) is larger than in non zero velocity cases (dynamic friction). The difference between those two magnitudes has been shown by many authors to be one of the most relevant variables that characterizes stick-slip oscillations [START_REF] Navarro-Lopez | Modelling and analysis of stick-slip behaviour in a drillstring under dry friction[END_REF].

III. SYSTEM MODELLING

Multiple kind of models have been used in literature to describe drillstring systems (see for example [START_REF] Leine | Stick-slip whirl interaction in drillstring dynamics[END_REF] and [START_REF] Spanos | Oil well drilling: A vibrations perspective[END_REF]). The type and the complexity of the model to be used are closely related to the aim pursued (modelling, simulation, model for control, etc). However, lumped parameters models have been shown to be valid enough to properly describe the stick-slip oscillation phenomena and easy enough to make the study not too complex [START_REF] Christoforou | Fully coupled vibrations of actively controlled drilstrings[END_REF].

The problem of modelling stick-slip phenomenon in a drillstring by means of a lumped-parameter model has been studied from several points of view. Most of them consider the drillstring as a torsional pendulum with different degrees of freedom, for instance: [START_REF] Kyllingstad | A Study of Slip/stick Motion of the Bit[END_REF], [START_REF] Lin | Stick-slip Vibration of Drill Strings[END_REF], [START_REF] Richard | Self-excited Stick-slip Oscillations of Drag Bits[END_REF], [START_REF] Van De Vrande | An Approximate Analysis of Dry-friction-induced Stick-slip Vibrations by a Smoothing Procedure[END_REF] propose single-degree-of-freedom models, [START_REF] Abbasssian | Application of Stability Approach to Torsional and Lateral Bit Dynamics[END_REF], [START_REF] Brett | The Genesis of Torsional Drillstring Vibrations[END_REF], [START_REF] Navarro-Lopez | Modelling and analysis of stick-slip behaviour in a drillstring under dry friction[END_REF], [START_REF] Navarro-Lopez | Vibraciones mecánicas en una sarta de perforación: problemas de control[END_REF] propose two-degree-of-freedom models including a linear controller, and [START_REF] Jansen | Active Damping of Self-excited Torsional Vibrations in Oil Well Drillstrings[END_REF], [START_REF] Serrarens | H∞ control for suppressing stick-slip in oil well drillstrings[END_REF] present two-degree-of-freedom models for the mechanical part of the system plus the model for the rotary table electric motor system. As it will be seen in subsequent sections, the bit-rock friction model is fundamental for properly reproducing stick-slip oscillations phenomenon. Many models have been proposed in literature, some of them summarized in [START_REF] Navarro-Lopez | Modelling and analysis of stick-slip behaviour in a drillstring under dry friction[END_REF]. The model used here (depicted in Figure 3) is a two-degree-of-freedom model with two inertial masses J r The model equations are the following:

J r φr + c( φr -φb ) + k(ϕ r -ϕ b ) + d r φr = v (1)
J b φb + c( φb -φr ) + k(ϕ b -ϕ r ) + d b φb = -T oB (2)
In constants above, the sub-script ′ r ′ , and ′ b ′ stands for rotary and bit, respectively.

A suitable model for T oB is essential, because the reproduction of stick-slip vibrations will strongly depend on the particular choice of the model for T oB. This torque represents the combined effects of reactive torque on the bit and nonlinear frictional forces along the drillstring. In our case, the T oB will be given by the product of µ( φb , z), which describes the normalized (dimensionless) torsional bit-rock friction (different bit-rock friction models are presented in [START_REF] Navarro-Lopez | Modelling and analysis of stick-slip behaviour in a drillstring under dry friction[END_REF]), and the normal force u called Weight on Bit (W oB), i.e.

T oB = µ( φb , z) • u (3)
Several forms for µ( φb , z) can be considered according the use of the model. Next, we describe the model for T oB used for simulations and for validating the control law, then a simplified model is introduced for control analysis purposes.

A. Model for simulations

The state-space representation of the later model is the following:

ẋ = Ax + Bv + Hµ(x, z)u (4) ż = f (x, z) (5)
with

A =      0 1 -1 -k Jr -(dr+c) Jr c Jr k J b c J b -(c+d b ) J b      , B =      0 1 Jr 0      , H =      0 0 -1 J b     
where the state x = [x 1 x 2 x 3 ] T is defined as follows:

x 1 = ϕ r -ϕ b x 2 = φr (6)
x 3 = φb

In this description, the state z ∈ R represents the internal friction state, and Equation ( 5) describes the friction dynamics. Various friction models have been shown to work properly to capture the typical friction phenomena (stiction, Stribeck effect, etc) which cause stick-slip oscillations ( [START_REF] Dupont | Single state elastoplastic friction models[END_REF] and [START_REF] Lampaert | Modification of the Leuven integrated friction model structure[END_REF]). One possible model for Equation [START_REF] Brett | The Genesis of Torsional Drillstring Vibrations[END_REF] is the LuGre friction model [START_REF] Canudas-De-Wit | A new model for control of systems with friction[END_REF]:

ż = x 3 -σ 0 |x 3 | g(x 3 ) z, g(x 3 ) = µ C + (µ S -µ C )e -(x3/vs) 2 (7) µ(x, z) = σ 0 z + σ 1 ż,
The function g(v) it mainly affect the steady-state characteristics of the friction model. In steady-state, the model predict the following friction value, µ SS (x 3 ) = g(x 3 )sgn(x 3 ). In this model, σ 0 , σ 1 , v s , µ C , µ S are positive constants characterizing the friction physical properties. Also note that the torsional linear friction at the drill bit side is already incorporated in the A matrix of the representation (4).

B. Model for control

Note that the previous model for the µ(x, z) includes an additional friction dynamics, z which is suited to describe motion at pre-sliding, and in particular to regularize the differential equation describing the system dynamics. An alternative is to use static description for µ(x) (maps without memory), which may be simple for control analysis.

The different between both models, may not be too significant, as long as computation issues are strongly simplified.

The model for control is then described by,

ẋ = Ax + Bv + Hµ(x 3 )u (8)
where here µ(x 3 ) is a static map between the bit rotational velocity x 3 = φb and the normalized friction parameter µ, i.e. the steady-state form of the model [START_REF] Canudas-De-Wit | A new mechanism for suppressing stick-slip in oil well drillstrings[END_REF], or any suitable approximation like the one shown in Figure 12.

IV. ROTATIONAL VELOCITY REGULATION LOOP

The first task to confront is designing a proper control law for motor torque v. This signal will be aimed mainly at regulating the rotational velocity of the rotatory table φr to a certain desired value ω d (a typical value for ω d is 5 rad/s). As we have shown in Section I, many architectures have been proposed for that purpose, from classic PID structures to a linear H ∞ robust control, although oil well drillstrings usually operate with reduced-order simple control laws.

A. Rotary table velocity control loop

In this work, the structure of the velocity controller is inspired by the one presented in [START_REF] Christoforou | Fully coupled vibrations of actively controlled drilstrings[END_REF], as shown by:

v = k 1 + k 2 s (ω d -φr ) -k 3 ( φr -φb ) (9)
or equivalent

v = k 1 (ω d -x 2 ) + k 2 x 4 -k 3 (x 2 -x 3 ) ẋ4 = (ω d -x 2 )
then the closed-loop equations take the form for the model for simulation,

ẋ = A cl x + B cl ω d + H cl µ(x, z)u (10) ż = f (x, z) (11)
and the following one for the model for control analysis,

ẋ = A cl x + B cl ω d + H cl µ(x 3 )u (12)
with the obvious observation that x is now of dimension four (due to the introduction of an integral term in the rotary table control), i.e. x = [x 1 , x 2 , x 3 , x 4 ], and with the A cl , B cl and H cl given as:

A cl =         0 1 -1 0 -k Jr -(dr+c+k1+k3) Jr (c+k3) Jr k2 Jr k J b c J b -(c+d b ) J b 0 0 -1 0 0         , B cl =         0 k1 Jr 0 1         , H cl =         0 0 -1 J b 0        
The steady-state value of x, considering µ * = µ(x * 3 ), is:

x * 1 = u 0 µ * + d b ω d k (13) x * 2 = ω d (14)
x *

3

= ω d (15) Therefore, rotatory table and bit velocities stabilize at the desired rotational velocity, revealing a proper feature in our feedback control law.

x * 4 = (d b + d r )ω d + µ * u 0 k 2 (16)
There are many possible ways of adjusting the gains k 1 , k 2 and k 3 . One possible option is to use a classic two-time-scales separation method to accomplish this (see, for example [START_REF] Canudas-De-Wit | Theory of Robot Control[END_REF]). Further details are given in Appendix A.

B. Closed-loop system model validation

Once the control torque has been designed and tuned, the global performance of the model in terms of reproducing the behavior of real drilling equipment, particularly the ability of reproducing stick-slip oscillations, must be validated.

In Figure 4, a comparison between stick-slip oscillations measured in the field (Figure obtained with permission from [START_REF] Serrarens | H∞ control for suppressing stick-slip in oil well drillstrings[END_REF]) and the ones produced by simulating the closed-loop system (10)-( 11) can be seen. Note that, although the model does not include the lateral and vertical motion dynamics, it is able to reproduce trajectories that are qualitatively close to the ones reported from field data in terms on frequency, amplitude, and shape. Irregularities in the oscillation amplitudes of the field measures are probably due to the vertical bouncing oscillations not included in our model.

V. PRELIMINARY ANALYSIS OF STICK-SLIP OSCILLATIONS BEHAVIOR

With regards to the oscillatory behavior of drillstring system, it is assumed that, depending on certain working parameters like the instantaneous value of W oB, the rotatory speed of the set, the velocity control gains values, and so forth, the drillstring system can enter into oscillation.

There will be cases in which the controller can suppress such oscillation, and so, the system recovers its nominal working regime, and other cases in which the system enters into sustained oscillation. This behavior is observed in real drillstring equipment, as seen in Figure 4.

A particular feature of the model presented here (for simulations) is that it succeeds in reproducing such behavior, as it can be seen in Figure 5, where the W oB parameter is perturbed away from its nominal operation value; in the left (upper) corner of Figure 5, the W oB changes from 40000N to 45000N and the controller is still able to correctly regulate the rotatory speed, whereas in the left (lower) corner of Figure 5 the W oB is perturbed enough (from 40000N to 50000N ) so that the velocity controller cannot compensate the perturbation and the system enters into stick-slip oscillation.

A. Intuitive behaviour

Although the system at hand is more complex, the following explanation of the system behaviour gives some intuition on how the system may operated. The system trajectories can intuited to behave like the ones in Figure 6.

From this figure, we expect that the behavior of a nonlinear system will display a local stable region with a stable limit set outside this local attraction domain (some more formal statements will be given latter). This is the pattern display with our system when u is assumed to be constant. Indeed, the system behavior is even more complex when u can take on different values. For instance, a saddle-node bifurcation of periodic orbits may be exhibited if the value of u is decreased enough. In that case, two limit cycles emerge at the bifurcation point, one of them being stable while the other is unstable. As the system has an equilibrium point, its local stability is not affected by the emergence of such limit cycles. Furthermore, for values of u smaller than the one corresponding to the bifurcation point, limit cycles disappear and the equilibrium becomes globally stable.

Equilibrium Domain

Stable Orbit

Fig. 6. Figure presents the general idea of a local equilibrium encircled by a stable limit cycle. When the perturbation is not too large system trajectories return to equilibrium. When the perturbation is large enough, the trajectory converge to a limit cycle.

VI. STICK-SLIP OSCILLATIONS PREDICTION BASED ON DF-ANALYSIS

The oscillation pattern shown by Figure 4 reflects two main characteristics: first oscillations will present a bias term, and second they are dominated by a main harmonic with a relatively well-defined period. Therefore, the SBDF (Sinusoid plus bias describing function) method can then be used as a first approximation to predict possible limit cycle and to study its stability. To further simplify the computation of the SBDF corresponding to the nonlinear dynamic friction map [START_REF] Canudas-De-Wit | A new mechanism for suppressing stick-slip in oil well drillstrings[END_REF], we would rather use a simpler approximation of the resulting steady-state characteristic of [START_REF] Canudas-De-Wit | A new mechanism for suppressing stick-slip in oil well drillstrings[END_REF], see Figure 7. This approximation is valid because the friction dynamics are much faster than the ones of the drillstring mechanism. Applying harmonic balance, the necessary condition to produce maintained oscillations in the system is:

G(jω 0 ) = -1 N 1 (A 0 , ω 0 , y 0 ) (17)
p 0 p 1 A = 0 A = a 1 A = a 2 A → ∞ a 2 > a 1 a) b)
y 0 [1 + G(0)N 0 (A 0 , y 0 )] = W (0)ω d (18)
A 0 and ω 0 are the particular values satisfying both equalities. They represent the amplitude and frequency of the predicted oscillations, if any. y 0 is the output bias, and N 1 and N 0 are given in the Appendix B.

Note that G(jω) depends on the control parameters, hence the assigned bandwidth ω n .

Since the observed oscillations are clearly asymmetrical, the SBDF is used to compute N 1 (A, ω, y 0 ), see [START_REF] Atherton | Nonlinear Control Engineering. Describing Function: Analysis and Design[END_REF].

N 1 (A, y 0 ) will have only a real part because the friction torque characteristic has an odd symmetry.

Note that the frequency locus of G(jω 0 ) can only cross the real axis at one point, then ω 0 can be uniquely computed from Im {G(jω 0 )} = 0, ∀ ω 0 = 0, |ω 0 | < ∞, and hence it does not depend on the other two unknown variables A 0 and y 0 . The introduction of a integral action in the velocity control results in G(0) = 0, and

W (0) = 1.
Therefore, the output bias can be straightforwardly computed from equation [START_REF] Lampaert | Modification of the Leuven integrated friction model structure[END_REF]. Yielding

y 0 = W (0) [1 + G(0)N 0 (A 0 , y 0 )] ω d = ω d
With y 0 = ω d known and independent of frequency and amplitude, the prediction of the limit cycle and its associated amplitude A 0 , can be obtained only from equation ( 17), i.e.

Re {G(jω

0 )} = - 1 N (A 0 ) with N (A 0 ) = N 1 (A, ω d ).
Figure 8 shows how -1 N (A) changes as a function of A. The main feature of this amplitude locus, pertinent to oscillation predictions, is the location of points p 0 and p 1 . These points derive from the maximum and minimum values of N (A). In particular, it is interesting to notice that these points have the following property:

|p 0 | ∼ 1 W oB and |p 1 | ∼ 1 W oB
Then, low values of W oB make these two points larger in magnitude (its sign remains unchanged), reducing the possibility to intersect the Nyquist of G(jω). Inversely, when the weight on bit force increases, the probability of oscillation is higher.

It is also interesting to note, that for almost every realistic choice of ω n , two sets of limit cycles will be predicted; one stable set and another unstable. The stable set arises at amplitudes A = A s , whereas the unstable one occurs for A = A u . In all cases A s > A u . This means that for all parameter combinations (W oB -ω n ) where an intersection of G(jω) and -1/N (A) takes place, there exists a local (attractive) stable domain delimited by A u . This property is further exploited in the proposed D-OSKIL mechanism.

It is also of interest to compute the set of possible combinations between system bandwidth and weight on bit force for which stable limit cycles are predicted. Figure 9 shows the area in the (W oB -ω n )-plane giving rise to possible stable oscillations. This analysis allow us to determine the range of proper values of controller closed-loop bandwidth to avoid oscillations as a function of the operating W oB. As the figure shows, and on the basis of this analysis, oscillations may be eliminated either by changing ω n , and/or by reducing the W oB magnitude.

One possible control strategy could be then to modify the rotary table bandwidth to reduce the possibility of entering into oscillation. From Figure 9, we can see that suitable choices for ω n are either small or large values.

Small values are not suited because they yield to poor performance, and large values are limited by noise. Typical values for ω n are in the range [START_REF] Lin | Stick-slip Vibration of Drill Strings[END_REF][START_REF] Shorten | Spectral conditions for positive realness of single-input-single-output systems[END_REF] rad/s.

Another alternative will be to keep the value of W oB below the oscillation zone. However, this will have two drawbacks: first, high-performance drilling operations require magnitudes for W oB larger than the ones indicated by the oscillation limits, and second, this strategy will require a precise knowledge of this map, which in addition will vary according to drilling depth. Keeping W oB low, will thus be a too conservative strategy, which may cause exploitation time to increase.

High-performance drilling operation takes place in a region in the plane (W oB -ω n ), where potential oscillations may occur. Therefore, in this paper we account for this particularity assuming that the nominal system operation parameters are taken within this zone. The design of the D-OSKIL mechanisms presented next, will be performed in light of the following two observations:

• oscillation can be eliminated by decreasing the Weight on Bit force, and

• if W oB is reduced from its nominal value, then after the oscillation has been eliminated, the value of W oB needs to be brought back to its nominal value again to restart an efficient drilling operation.

VII. THE D-OSKIL MECHANISM DESIGN

This section presents the design of the D-OSKIL mechanism, which stands for Drilling OScillation KILler.

First we introduce some intuitive ideas completing the understanding on the system behaviors gained in previous sections. Then we present the mathematical form for this algorithm and the different steps involved in their stability properties.

A. Basic Ideas

Let us first assume that the normal force u (i.e. the weight on Bit W oB ), is given as the sum of the nominal weight of the drillstring (u 0 ) and the D-OSKIL control signal (ũ):

u = u 0 + ũ ( 19
)
Figure 10 is a schematic description of what we expected to be the behavior of the system. Although it is not formal, we may think of this Figure as a bifurcation diagram, where in the horizontal axis we have the "bifurcation parameter" u, and in the vertical axis a measure of the oscillations (i.e. the Euclidian norm of the error vector, to be defined later). As previously discussed, u 0 is here the nominal value for u of an efficient drilling operation.

The upper solid line describes a possible family of stable oscillations, whereas the lower dotted line is a family of unstable ones. The lines with arrows describe possible trajectories of the closed-loop system parameterized as a function of u. These trajectories are extrapolated from the a priori behavior of the drillstring system.

There are two interesting cases to be analyzed: a) Nominal behavior: Assume that u = u 0 , and the system trajectories are at equilibrium. Then if a perturbation arrives:

• Either the perturbation is not large enough to divert the system trajectories from their local attraction domain (a curve in figure). Therefore the velocity controller succeeds in regulating the trajectories back to their equilibrium. Or,

• The perturbation is large enough to divert the system from equilibrium domain. In this case (b curves in figure), the trajectories are attracted by the corresponding stable limit cycle and the system enters into sustained oscillation.

b) Behavior beyond nominal operations:

To suppress such oscillations, the u = W oB must be reduced by means of the control signal ũ, until its trajectory reaches the bifurcation point, and the local controller is able to return the system trajectories to the equilibrium. Thereafter, the nominal value of the W oB must be recovered in a proper slow manner, to continue with the drilling task, i.e. u → u 0 . Significantly enough, the variation of ũ should be restricted to a valid domain, and in particular restricted to a positive values other than zero. Without this restriction, it is clear that drilling may not be efficient, or it will be impractical.

The general structure of the variation law for u (or equivalent for ũ), will be here of the form:

u = P 0 -u0 {-σũ + Φ(•)}
where σ > 0, can be understood as a time-constant of the controller, and P 0 -u0 is a projector operator ensuring that solutions of the above equation makes ũ stay in the range (-u 0 , 0], and Φ(•) is a nonlinear function which must be designed to ensure system stability, that is:

φr → ω d and ũ → 0
In order to make the presentation simpler, we will trop the explicit use of the projection operation P in the following section. However, the reader should keep in mind that ũ is a bounded signal in the prescribed range.

With this in mind, the complete closed-loop equations are:

ẋ = A cl x + B cl ω d + H cl µ(x 3 )(u 0 + ũ) (20) u = -σũ + Φ(•) (21)
where matrices A cl , B cl , H cl have been defined previously.

with C = [0 0 1 0], or equivalently:

G(s) = s(a 2 s 2 + a 1 s + a 0 ) b 4 s 4 + b 3 s 3 + b 2 s 2 + b 1 s + b 0 (30)
The error equation and updating rule of ũ have been designed in such a way that the resulting map G(s) has relative degree one. This condition is necessary to obtain PR and SPR functions. The verification that G(s) is PR can be done directly on the triplets of matrices (A cl , H cl , C) as demonstrated in [START_REF] Shorten | Spectral conditions for positive realness of single-input-single-output systems[END_REF] for SPR function. Adapting this result to PR functions and using our notation at hand, results in the following relaxed conditions: consider the transfer function

G(s) = -C(sI -A cl )H cl . G(s) is PR if and only if: 1) CA cl H cl > 0, 2)A cl is stable, 3) the matrix A cl (I -(1/CA cl H cl )A cl H cl C)A cl has no eigenvalues on the open negative real axis (-∞, 0).
The first condition is easy to compute in terms of model parameters. This gives CA cl H cl = (c + d b )/J 2 b > 0, and always holds from the physics of the system. The second condition is also verified since A cl is designed to be stable. The last condition is more involved, but it can be easily checked numerically. For typical values of drillstring system model parameters and v-control gains considered in this paper, it is possible to show that condition 3) holds.

Note also that by continuity of the eigenvalues with respect to the matrices parameters, there will exists a certain degree of robustness of this condition with respect the model uncertainly.

Consequently we have that G(s) is a Positive Real (PR) function, and hence from the Kalman-Yacubovich-Popov Lemma [START_REF] Khalil | Nonlinear Systems[END_REF], the following property holds: ∃P = P T > 0, Q = L T L ≥ 0 such that:

A T cl P + P A cl = -Q = -L T L ≤ 0 (31)
P H cl = -C (32)
Therefore, as a consequence we have the following two properties for the linear map G(s):

• G(s) is a passive relative to V (e) = e T P e, and

• G(s) has a finite L 2 -gain: γ 2 (G) = sup ω |G(jω)| < ∞
2) Boundedness of signal Ψ(t): From the definitions of ũ1 and ũ2 in Equations ( 27) and ( 28), together with the assumption that the adaptation mechanism yields values in the range ũ ∈ (-u 0 , 0], it follows that both signals, ũ1 and ũ2 , are bounded, that is:

||ũ 1 || ∞ = sup t≥0 |ũ 1 | ≤ u 0 < ∞, (33)
||ũ 2 || ∞ = sup t≥0 |ũ 2 | ≤ 2 • u 0 < ∞. (34)
Hence

||Ψ(t)|| ∞ ≤ 3 • u 0 .
3) Boundedness of the output y(t): Since G(s) : Ψ → y is a lineal stable map, the output signal y is also bounded, i.e.

Ψ ∈ L ∞ ⇒ y ∈ L ∞ (35)
4) Sector condition on ∆: With regard to Figure 11, the output of the map ∆ can be seen as a disturbance acting on the closed loop system resulting from the operators G(s) in feedback connection with nonlinear operator Γ. This operator belongs to the cone sector [a, b] as displayed in the same Figure (see [START_REF] Vidyasagar | Nonlinear Systems Analysis[END_REF] for further discussion on sector definitions). Formally this is stated as follows.

The nonlinear operator ∆(y) belongs to the sector [a, b] if the following holds true:

• ∆(0) = 0 • a ≤ ∆(y)
y ≤ b , ∀y ≥ 0, or equivalently,

• ay 2 ≤ y∆(y) ≤ by 2 , ∀y ∈ ℜ

In our case, the values for a and b are:

a = - µ S -µ C ω d u 0 (36) b = µ S + µ C ω d u 0 (37)
and consequently, the map ∆ has also finite L 2 -gain, which is bounded by:

γ 2 (∆) ≤ max[|a|, |b|] (38)

5) Block transformation:

As it can be seen in Figure 12-(b), the map ∆ is almost passive, since almost the whole diagram is within the first and third quadrants. This characteristic is generic, as the difference between break-away and Coulomb friction levels is generally small (a is small when compared to b).

Figure 13 shows a possible block transformation, where the following new operators, ∆ * , and Γ * are defined:

∆ * : y → (ũ 2 + εy) Γ * : y → (ũ 1 -εy) + + + - - - ∆ ε G(s) Γ ε y ∆ * Γ * Ψ Fig. 13. Modified Block Diagram.
With this transformation, it can be easily proved that the map ∆ * is passive if the value of ε is taken such that ε = |a|, as can be seen in Figure 14.

ε = |a| ⇒ t 0 y(ũ 2 + εy)dt ≥ 0 (39)
With this new feedback configuration, the problem of designing a stable update law for ũ(t) is equivalent to finding a function Φ(y), and parameter conditions, such that the transformed operator Γ * defines a passive map.

This design strategy results from well known properties of feedback interconnected passive systems.

The next subsection uses such a result to demonstrate the stability properties of one possible candidate update rule.

D. D-OSKIL updating law

Under the premise that the complete form of the update law should also include a suited projection operator ensuring that the variation of ũ is limited to the admissible parameters range, the following updating rule will be analyzed.

E. Stability analysis

Lemma 1: Let ρ > 0 be an arbitrarily positive constant, and λ, σ be such that the following design inequality holds,

λ σ ≥ µ S µ C -1 u 0 ω d + σ µ C ρ (41)
where µS µC ≥ 1. Then map Γ * : y → (µ(y)ũ -εy) is strictly input passive, i.e.

I = t 0 (µ(y)ũ -εy)y ≥ ρ t 0 y 2 -β 0 (42)
with β 0 = ymax σ 2u 0 > 0.

Proof: Let I define the integral of the input-output product of the operator Γ * , i.e. We are now in a position to establish the main stability result.

Theorem 7.1: Consider the closed-loop system of Figure 13 with G(s), Γ * , ∆ * holding the following properties:

(i) G(s) is a PR operator satisfying ( 31)- [START_REF] Van De Vrande | An Approximate Analysis of Dry-friction-induced Stick-slip Vibrations by a Smoothing Procedure[END_REF] (ii) ∆ * is a passive map satisfying (39) (iii) Γ * is a strictly input passive map satisfying (42), i.e. design parameters are such that the condition (41) holds.

Then, (e * , ũ * ) = (0, 0) is a globally asymptotically stable equilibrium of the considered closed-loop system.

Proof: Let us take the following scalar function:

V (e, ũ) = 1 2 e T P e + t 0 y(ũ 2 + εy) + I(ũ, y)

+ β 0 -ρ t 0 y 2 (43)
From ( 39) and (42), we have that V (e, ũ) is semi-positive definite.

Computing the time-derivative of V (e, ũ), and using the properties (i) -(iii) of the theorem, results in:

V (e, ũ) = - 1 2 e T L T Le -ρy 2 ≤ 0 , ∀e, ũ (44)
Therefore, from last Equation, we have that y → 0 with a rate depending on the value of ρ. The rest of the proof follows from the application of the LaSalle's invariance principle. From Equation ( 23), we can see that if y → 0, together with the fact that σ is a positive constant and Φ(0) = 0, gives that ũ → 0. Finally, this implies that the two last terms in Equation ( 22) also tends to zero, i.e. So it can be concluded that e → 0, and hence that e * = 0, and ũ * = 0 are a globally asymptotically stable equilibria.

VIII. SIMULATION EXAMPLE

In order to demonstrate the behavior of the proposed adaptive law, simulations of the drillstring system controlled under the D-OSKIL mechanism designed in Section VII-D are shown in Figures 15 and16. The values 3 for system model parameters used in the simulations are presented in Table I. In these Figures, the typical profiles in terms on rotatory velocity, both in surface and downhole, and system W oB are shown.

As it can be observed, with the nominal weight u 0 = 40000N, the system is under a sustained oscillation regime. The D-OSKIL mechanism is activated at t = 50s, and in both cases, the controller is able to extinguish such oscillations, although the W oB profiles obtained are quite different. In Figure 15 (with control parameters λ = 2.000 and σ = 0.3) the D-OSKIl mechanism is able to extinguish the oscillations with a soft evolution in the W oB control signal.

On the other hand, in Figure 16 (with control parameters λ = 2.500 and σ = 1.0), the transition from oscillation regime to stabilization period is faster than the one obtained in Figure 15. The stabilization time for the value of W oB is also faster in Figure 16, but in this case some oscillations occur during the transition. This issue is due to the λ value, when large values of the λ are chosen a sharp W oB value transition is obtained in the switching instant of time (in our simulations t = 50s).

From a practical point of view, and in order to avoid oscillations in the control signal, the parameter values proposed in Figure 15 seem to be more appropriate.

IX. OBSERVER-BASED DESIGN

In this section we present some extensions of the previous control which has been studied and designed under the hypothesis of the measure of the bit rotational velocity φb . In this section we first provide an alternative way to get this measure trough a state observer.

A. Observer Design

The observer is designed on the basis of open-loop equation [START_REF] Canudas-De-Wit | A new model for control of systems with friction[END_REF]. That is on

ẋ = Ax + Bv + (Hu) • µ (45) y o = C o x = φr (46)
Following [START_REF] Zhang | Adaptive observer for MIMO linear time varying systems[END_REF] we propose to use the following observer based on the observation of the rotary angular velocity

y o = φr : ẋ = Ax + Bv + (Hu) • μ + K o + βΓΓ T C T o (y o -C o x) (47) μ = βΓ T C T o (y o -C o x) (48) Γ = [A -K o C o ] Γ + (Hu) (49)
where: Γ is a (3 × 1) time-varying vector, β > 0 is a positive scalar, x, and μ are the state and friction coefficient estimates respectively, y o is the measured output, K o is a (3 × 1) observer vector gain, and

C o = [0, 1, 0]. Note that C o = C.
The observer not only provides an estimate for the bit rotational velocity x3 , but also provide an estimate of the friction coefficient μ which can be useful for other monitoring purposes. The observer, as indicated in [START_REF] Zhang | Adaptive observer for MIMO linear time varying systems[END_REF], results in a globally exponentially stable observer providing the following hypothesis hold:

• µ is constant

• Exists a matrix K o such that (A -K o C o
) is strictly stable matrix, i.e. The constant pair (A, C o ) is detectable.

• u(t) is persistently exciting, i.e. ∃δ, T > 0 such that the following inequality is satisfied:

t+T t Γ(τ ) T C T o C o Γ(τ )dτ > δ > 0
Let comment the practical implication of the previous hypothesis.

The first hypothesis assumes that the friction coefficient is constant, or eventually slow-time variant4 μ ≈ 0. Note that this approximation is often assumed in the context of observer design with unknown inputs, but also in the context of adaptive control. Here this hypothesis means that the rate of variation of the rock friction coefficient does not exhibit substantial changes during the drill-operation. Even if the drilled surfaces may have different friction characteristics, the rate of penetration (drilling-speed) remains small.

The second hypothesis correspond to the necessity observation property need to build the observed. By inspecting this condition, we can see that the system observability is invariant with respect the matric A and C o .

Finally, the last property is necessary to the observer to converge. Note that as the "adaptation" is done under a single parameter µ, the required condition is weak and will be simple to fulfill. To see this note that the pair

(A -K o C o , H
) is controllable and the pair (A -K o C o , C o ) is observable, then the persistently exciting condition is easily verified if the WoB force is not equal to zero, i.e. u(t) > 0. A detailed justification of this can be found in [START_REF] Dolcini | Observer-based Optimal Control of Dry Clutch Engagement[END_REF].

B. Simulation with the observed-based controller

The original controller has the form

u = -σũ + λ(x 3 -ω d ) • sgn(µ(x 3 -ω d )) (50)
On the other hand, by substituting the controller structure in Equation (53), the following expression is obtained:

ǫ 2 z = - z J eq -ǫ 2 ( c J eq + d b J b ) ż + ( d b J b - d r J r ) φr + + v s J r + ǫ J r v f + T oB J b (61)
Due to two-time scales separation, it can be assumed that the slow variable has reached its steady state value respecting to the variations of the fast variable z, and so, rewriting Equation (61) as:

ǫ 2 z = - z J eq -ǫ 2 ( c J eq + d b J b ) ż + ǫ J r v f + T oB J b + ρ( φ * r , ϕ * r ) (62)
where the superscript * means steady-state value, and:

ρ( φ * r , ϕ * r ) = ( d b J b - d r J r ) φ * r + v s ( φ * r , ϕ * r ) J r = z * J eq - T oB J b (63)
By rewriting Equation (62) in fast error coordinates ζ = z -z * , we have:

ζ + ( c J eq + d b J b ) ζ + 1 ǫ 2 J eq ζ = 1 J r ǫ v f (64)
Taking

v f = -k 3 ζ: ζ + ( c J eq + d b J b + k 3 J r ǫ ) ζ + 1 ǫ 2 J eq ζ = 0 (65)
and so, imposing certain damping value for torsion dynamics (δ tor ), the value of k 3 can be computed:

k 3 = J r √ k • 2δ tor √ kJ eq - c J eq - d b J b (66)
B. COMPUTATION OF THE FUNCTIONS N 0 , AND N 1 N 0 (A 0 , y 0 ) = 1 2πy 0

x(t)d(θ)

N 1 (A 0 , y 0 ) = 1 πA 0 x(t)cos(θ)d(θ) (68)

N 0 (A 0 , y 0 ) = 2F s y o g( y 0 A 0 ) - A 0 (F s -F c ) 2y 0 v s • •[f 0 ( v s + y 0 A 0 ) -f 0 ( v s -y 0 A 0 )] (69)
N 1 (A 0 , y 0 ) = 4F s πA 0 D 1 (

y 0 A 0 ) - (F s -F c ) 2v s • •[f 1 ( v s + y 0 A 0 ) + f 1 ( v s -y 0 A -0 )] (70)
with:

g(x) =          arcsin(x) π | x |≤ 1 -1 2
x < -1

1 2
x > 1

         (71) f 0 (x) =    2 π (xarcsin(x) + √ 1 -x 2 ) | x |≤ 1 | x | | x |> 1    (72) f 1 (x) =          2 π (arcsin(x) + x √ 1 -x 2 ) | x |≤ 1 -1 x < -1 1 x > 1          (73) D 1 (x) =    √ 1 -x 2 | x |≤ 1 0 | x |> 1    (74)

Fig. 1 .

 1 Fig. 1. Oil drilling system in the field (a). Basic scheme of a vertical drilling system.(b)

Fig. 2 .

 2 Fig. 2. Different types of vibrations in drillstring systems.

Fig. 3 .

 3 Fig. 3. Drillstring two-coupled masses model.

  and J b , locally damped by d r and d b . The inertias are coupled with each other by an elastic shaft of stiffness k and damping c. The variables ϕ r and ϕ b stand for the rotary and the bit angle. The rotary torque control signal v used to regulate the rotary angular velocity φr . The T oB (Torque on Bit) represents the total friction torque over the drill bit.

Fig. 4 .

 4 Fig. 4. Stick-slip measured in the field (a) (taken from [29]) and simulation profiles obtained with the closed-loop model (b).

Fig. 5 .

 5 Fig. 5. Different behaviors obtained by perturbing W oB. Figures show the time-profiles of the table and bit speed.

-Fig. 7 .

 7 Fig. 7. System block diagram and static friction torque.

Fig. 8 .

 8 Fig. 8. Describing function N (A) (a), and evolution of -1 N(A) (b). |p 0 | is two or three magnitude orders lower than |p 1 |, depending on value of W oB.

Fig. 9 .

 9 Fig. 9. Oscillations frequency domain. Border obtained by simulations and border predicted from the SBDF analysis.

Fig. 10 .

 10 Fig. 10. Possible drillstring system trajectories.

Fig. 12 .

 12 Fig. 12. Normalized friction function (a). ∆(y) lies at the interval [a, b] (b).

Figure 12 -

 12 Figure 12-(a) shows the memoryless friction map used for this study. Note that as the steady-state rotatory bit speed (ω d ) is in general much larger than the Stribeck velocity v s , we can then assume that µ * = µ C . Therefore, taking into account that y = e 3 = φb -ω d , the output of block ∆ will have the profile shown in Figure 12-(b).

Fig. 14 .

 14 Fig. 14. Profile of map ∆ * .

2 2 u 0 y 2 Remark 1 :

 22021 Substituting ũ from Equation[START_REF] Navarro-Lopez | Practical approach to modelling and controlling stick-slip oscillations in oilwell drillstrings[END_REF] in the above expression gives, From sections VII-C.2, and VII-C.3, signals y and µ(y) have been shown to be bounded. Let note these bounds as: |y| < y max , |µ(y)| < 1. Therefore, Taking now Φ(y) = λ y sgn(µ(y)), gives |a|)y 2 -β 0 where the last inequality is obtained by using the lower bound on |µ(y)|, i.e., |µ(y)| ≥ µ C , the definition of ε = |a| (with a as given in Equation (36)) and the constant β 0 = ymax σ 2u 0 . Finally, introducing the condition (41) in the above expression, gives the following lower bound on I: |a|)y 2 -β 0 ≥ ρ t -β 0 which proves the lemma. The condition (41) exhibits several interesting practical features. It relates the design parameters (σ, λ) as a function of physical drillstring system characteristics such as the nominal W oB (u 0 ), the rock friction features (µ S -µ C ), and the desired rotational velocity (ω d ). The parameter ρ as shown latter, provides a measure of the convergence rate of the output y to zero.

Fig. 15 .

 15 Fig. 15. Simulation of D-OSKIL scheme with σ = 0.3 and λ = 2.000.

  y)ũ + μ(y)u 0 ] = [µ(0)0 + μ(0)u 0 ] = 0 since µ(0) = µ * = µ C , and μ(0) = µ(0) -µ * = µ * -µ * = 0. Therefore, this results in ė = A cl e + H cl lim t→∞ [•] = A cl e

Fig. 16 .

 16 Fig. 16. Simulation of D-OSKIL scheme with σ = 1.0 and λ = 2.500.

Fig. 18 .

 18 Fig.[START_REF] Lampaert | Modification of the Leuven integrated friction model structure[END_REF]. N 0 (A 0 , y 0 ) and N 1 (A 0 , y 0 ) functions in the case that y 0 = 5 rad/s.

[START_REF] Armstrong-Hélouvry | Stick-Slip Arising from Stribeck Friction[END_REF] DRAFT

The numerical values of the drilling system parameters according to a 2000m long drillstring have been taken from[START_REF] Serrarens | H∞ control for suppressing stick-slip in oil well drillstrings[END_REF].[START_REF] Armstrong-Hélouvry | Stick-Slip Arising from Stribeck Friction[END_REF] DRAFT

In this case, it can also be shown that stability (not asymptotic) follows is preserved

B. Error equations

Error equations can be obtained by applying the change of coordinates e = x-x * , and considering the steady-state values µ(e 3 ) ss = µ * and ũss = 0. This yields 2 :

where the term μ(y) is defined as follows:

and we assume that the update rule for ũ is designed on the basis of the output y. The error system can be described by the block diagram in Figure 11, with the following definitions:

∆ : y → ũ2 = μ(y)u 0 [START_REF] Sananikone | A Field Method for Controlling Drillstring Torsional Vibrations[END_REF] where Ψ = -(ũ 1 + ũ2 ).

C. Error equation properties 1) PR condition on G(s):

The map G(s) is:

2 With an abuse of notation, we will use µ(y) to denote the expression of µ(x 3 ), in the shift coordinate y+ω d , i.e. µ(x 3 ) = µ(y+ω d ) = µ(y).

Note that at y = 0, we have µ(y = 0) = µ(ω d ) = µ * . replacing x 3 by its estimate x3 , and noting that sgn(µ(x 3 -ω d )) = sgn(x 3 -ω d ), we get the final, quite simple, control law structure to be implemented together with the observer,

In Figure 17, we repeat the experiment shown previously in Figure 15 (with the parameters given in Table I) with the observed-based control law instead. The additional parameter for the observer used in this simulation where:

X. CONCLUSIONS

We have proposed to use the weight on the bit (WoB) force as an additional control variable to extinguish limit cycles when they occur. An adaptation law, named D-OSKIL, of the oscillation killer (OSKIL) mechanism, to oil well drillstring systems has been proposed. An approximate analysis based on the bias describing function did provide a good insight of the slip-stick behavior and in the control design.

We have then presented a stability analysis of a variant of the D-OSKIL control mechanism, introduced in [START_REF] Canudas-De-Wit | A new mechanism for suppressing stick-slip in oil well drillstrings[END_REF], to remove stick-slip oscillations in drillstring systems. The presented stability analysis based on passivity has shown that this algorithm is globally asymptotically stable. Simulations applying such an algorithm showed that stick-slip oscillations can be effectively eliminated.

The implementation of the proposed control algorithm requires the sign of a friction torque on the bit, µ(y), which can be computed if the sign of rotational velocity measured at the bit is known. It has also been shown in the paper, that the measurement of this velocity can be undertaken by the proper design of an nonlinear observer, without jeopardizing the performance of the closed-loop system.

ACKNOWLEDGMENT

The authors would like to acknowledge MCYT-FEDER for funding this work under grants DPI2004-06419, HF2003-0237 and SAB2003-0085. Thanks for funding are also due to the program PICASSO No. 07261YJ(EGIDE), of the French Ministry of Foreign Affaires.

APPENDIX A. TUNING METHOD FOR CONTROLLER GAINS

From the system model Equations ( 1) and ( 2), and defining a new variable z and constant ǫ in the way:

the ϕ r and z dynamics are:

Since the torsional stiffness k is usually very large, the assumption ǫ = 0 can be made, and so:

where the control signal is supposed to be split in slow (v s ) and fast (v f ) modes:

Therefore, solving Equation (55):

And so, Equation (54) results in: And so, by imposing some damping (δ r ) and natural frequency (ω n ) in the closed loop dynamics T oB → φr , the values of k 1 and k 2 can be computed from:


Sachiyo Oishi

Daniel Wendling

Jean Sibilia

Chantal

Chantal Job-Deslandre

Loic Guillevin

Jacques Benichou

Marc Flipo

Carole Duquenne

Francis Guillemin

Alain Saraux
email: alain.saraux@chu-brest.fr

Treatment of active rheumatoid arthritis: comparison of patients younger vs older than 75 years (CORPUS cohort)

Keywords: Rheumatoid arthritis, glucocorticoids, prednisone, biologics, elderly

Objectives: Little information is available on the characteristics of elderly patients starting TNFα antagonist treatment for rheumatoid arthritis (RA). The objective of this work was to compare prescription patterns in RA patients younger vs. older than 75 years. Methods: Biologic-naive patients with active RA (DAS28 > 3.2) despite first-line therapy were included between 2007 and 2009 in the prospective, multicentre, longitudinal, observational, population-based CORPUS-RA cohort. TNFα antagonist users were defined as having received at least one TNFα antagonist during the first study year. The groups < 75 years and ≥ 75 years were compared regarding comorbidities, inflammation (CRP and ESR), disease activity (DAS28), disability (HAQ-DI), number of physician visits, and treatment. To verify the impact of the cut off, we also compared patients aged 70 years or more to patients younger than 70 years. Results: Of 543 RA patients, 382 had complete one-year follow-up data, including 114 TNFα antagonist users, 3 (6%) among the 49 patients aged 75 years or over and 111 (32%) of the 333 patients younger than 75 years (p < 0.01). Disease activity in the two age groups was similar at inclusion and after one year. Comorbidities and a history of auto-immunity were more common in the older group. Compared to their younger counterparts, the older patients received glucocorticoids more often (p = 0.003) and synthetic disease-modifying anti-rheumatic drugs less often (p = 0.01). Conclusion: TNFα antagonists are used less often and glucocorticoids more often in elderly patients with active RA compared to their younger counterparts. The fact that this study was performed in 2007-9 is a limitation in terms of relevance to today's patients and further studies should be conducted in new cohorts of active RA.

Introduction

Rheumatoid arthritis (RA) is among the most common chronic inflammatory joint diseases. It manifests as polyarthritis with joint destruction that causes functional disability, thereby adversely affecting quality of life. The current ageing of the general population is mirrored by ageing of the population with RA. Thus, cases of RA diagnosed after 60 years of age are on the rise. Most older patients with RA have an active lifestyle that requires good disease control.

To obtain a remission or low disease activity without further joint destruction, synthetic disease-modifying antirheumatic drugs (sDMARDs) should be prescribed as soon as possible after the diagnosis of RA. If these drugs fail, biologics such as TNFα antagonists may provide disease control. TNFα antagonists are licensed for use in older patients. However, the small number of older patients included in randomised controlled trials of TNFα antagonists [START_REF] Maini | Infliximab (chimeric anti-tumour necrosis factor a monoclonal antibody) versus placebo in rheumatoid arthritis patients receiving concomitant methotrexate: a randomised phase III trial[END_REF][START_REF] Elliott | Randomised double-blind comparison of chimeric monoclonal antibody to tumour necrosis factor[END_REF][START_REF] Weinblatt | Adalimumab, a fully human antitumor necrosis factor alpha monoclonal antibody, for the treatment of rheumatoid arthritis in patients taking concomitant methotrexate: the ARMADA trial[END_REF][START_REF] Klareskog | Therapeutic effect of etanercept and methotrexate compared with each treatment alone in patients with rheumatoid arthritis: double blind randomized controlled trial[END_REF] has left gaps in our knowledge of the benefits and risks in this age group. In the few meta-analyses of randomised controlled trials comparing biologics (TNFα antagonists, abatacept, tocilizumab, or anakinra) with or without methotrexate to a placebo in patients with RA, mean age was less than 55 years. [START_REF] Pincus | Relative efficiencies of the 7 rheumatoid arthritis Core Data Set measures to distinguish active from control treatments in 9 comparisons from clinical trials of 5 agents[END_REF][START_REF] Michaud | The comparative safety of tumor necrosis factor inhibitors in rheumatoid arthritis: a meta-analysis update of 44 trials[END_REF][START_REF] Jansen | Comparative efficacy of biologics as monotherapy and in combination with methotrexate on patient reported outcomes (PROs) in rheumatoid arthritis patients with inadequate response to conventional DMARDs-a systematic review and network meta-analysis[END_REF] Compared to their younger counterparts, patients older than 75 years had a heavier burden of comorbidities (e.g., kidney dysfunction and heart failure) and more often took multiple chronic medications. These differences suggest a greater risk of adverse drug effects that may lead physicians to limit the aggressiveness of the treatments they prescribe to older patients. [START_REF] Tutuncu | Do patients with olderonset rheumatoid arthritis receive less aggressive treatment?[END_REF][START_REF] Ogasawara | Observational cross-sectional study revealing less aggressive treatment in Japanese elderly than nonelderly patients with rheumatoid arthritis[END_REF] Although driven by concern over patient safety, this attitude may deprive older patients of optimal disease control and quality of life. Several trials comparing patients younger than 65 years and 65 years or over showed similar drug safety profiles and treatment discontinuation rates. [START_REF] Alivernini | Leflunomide treatment in elderly patients with rheumatoid or psoriatic arthritis; retrospective analysis of safety and adherence to treatment[END_REF][START_REF] Ranganath | Disease-modifying antirheumatic drug use in the elderly rheumatoid arthritis patient[END_REF][START_REF] Radovits | Tumour necrosis factor-alpha antagonists in the management of rheumatoid arthritis in the elderly: a review of their efficacy and safety[END_REF][START_REF] Soubrier | Rheumatoid arthritis in the elderly in the era of tight control rheumatoid arthritis in the elderly in the era of tight control[END_REF][START_REF] Bathon | Safety and efficacy of etanercept treatment in elderly subjects with rheumatoid arthritis[END_REF] Efficacy also seems unrelated to age, [START_REF] Bathon | Safety and efficacy of etanercept treatment in elderly subjects with rheumatoid arthritis[END_REF][START_REF] Köller | Response of elderly patients with rheumatoid arthritis to methotrexate or TNF inhibitors compared with younger patients[END_REF] although questions have been raised about the efficacy of second-line drugs. [START_REF] Fleischmann | Response to etanercept (Enbrel) in elderly patients with rheumatoid arthritis: a retrospective analysis of clincal trial results[END_REF][START_REF] Lahaye | Overview of biologic treatments in the elderly[END_REF] In France, although fully reimbursed by the statutory health insurance system, TNFα antagonists may be under-prescribed. In the subgroup of CORPUS cohort patients who had active RA, [START_REF] Saraux | Which patients with rheumatoid arthritis, spondyloarthritis, or juvenile idiopathic arthritis receive TNF-α antagonists in France? The CORPUS cohort study[END_REF] TNFα antagonist initiation was more strongly associated with younger age, longer disease duration, glucocorticoid use, and poorer quality of life than with higher disease activity.

This study is a post hoc analysis of the data of the French Corpus cohort conducted at the request of the French health authorities to assess TNFα antagonist prescription patterns in patients with active RA. We have chosen this cohort because it is an opportunity to have a longitudinal prospective populationbased cohort study of active RA. The primary objective of this work was to compare two age groups, < 75 years and ≥ 75 years, regarding second-line drugs used to treat RA insufficiently controlled by first-line therapy. Secondary objectives were to compare the two age groups among patients started on TNFα antagonist therapy, regarding glucocorticoid dosage and changes in C-reactive protein (CRP) levels, disease activity, and disability over the first study year.

Results

Study population

Patients with RA were recruited by 80 rheumatologists working throughout continental France. Of 550 patients with RA, 382 (69.5%) had complete follow-up data and were naive to biologics, including 333 (87.2%) younger than 75 years and 49 (13%) 75 years or older (Figure 1).

Data at study inclusion (Table 1)

Few characteristics differed significantly between the two age groups. In particular, all patients in both age groups were taking sDMARD therapy, usually methotrexate (74.6%) or leflunomide (18.5%). However, comorbidities and a history of auto-immunity were more common in the older group, which had a higher proportion of patients on glucocorticoid therapy, although the dosage was not different between groups.

Data at follow-up (Table 2)

At follow-up, the older group had fewer patients receiving TNFα antagonist therapy and sDMARD therapy but more patients receiving glucocorticoids. However, the proportion of older patients on glucocorticoid therapy was lower at follow-up than at inclusion, and the glucocorticoid dosage at follow-up was similar in the two age groups. Despite these treatment differences, the two age groups were not significantly different regarding inflammation severity, disease activity, or disability. The main sDMARDs used were methotrexate (81%) and leflunomide (14%).

Older patients receiving TNFα antagonist therapy (Table 3)

Of the 49 patients 75 years or older, only 3 were prescribed TNFα antagonist therapy during the study (Table 3). None Continuation of the first-line drug was more common in the younger patients (207/250 (82.8%) vs. 55/79 (69.6%); p = 0.02), whereas glucocorticoid therapy was less common (128/249 vs. 60/ 79; p < 0.01).

Discussion

This study identified significant differences in the treatment of RA between patients < 75 and ≥ 75 years, during the first year of follow-up. TNFα antagonist therapy was used more often in the younger patients, despite similar disease activity in the two age groups. Glucocorticoid therapy was used more often and sDMARD therapy less often in the older age group.

Short-term glucocorticoid therapy is often prescribed to patients with RA, to control flares and while waiting for DMARD therapy to take effect. Long-term glucocorticoid therapy, in contrast, is not recommended, as its risk/benefit ratio remains unclear. In everyday practice, however, 50% to 60% of patients with RA take glucocorticoid therapy. [START_REF] Sokka | QUEST-RA: quantitative clinical assessment of patients with rheumatoid arthritis seen in standard rheumatology care in 15 countries[END_REF][START_REF] Sokka | men, and rheumatoid arthritis: analyses of disease activity, disease characteristics, and treatments in the QUEST-RA study[END_REF] A systematic literature review showed that long-term low-dose glucocorticoid therapy combined with DMARD therapy was generally effective but that glucocorticoid-related adverse events, although dose-related, occurred even with low doses [START_REF] Kavanaugh | Benefits and risks of low-dose glucocorticoid treatment in the patient with rheumatoid arthritis[END_REF] and were of greatest concern in older patients with multiple comorbidities. Of six studies that assessed major cardiovascular events, four found an increase associated with low-dose glucocorticoid therapy. [START_REF] Kavanaugh | Benefits and risks of low-dose glucocorticoid treatment in the patient with rheumatoid arthritis[END_REF] In a study of mortality in RA, taking more than 5 mg/d of glucocorticoids was associated with a significantly higher risk of death. [START_REF] Ruyssen-Witrand | Cardiovascular risk induced by low-dose corticosteroids in rheumatoid arthritis: a systematic literature review[END_REF] Glucocorticoid therapy is also associated with a higher frequency of serious infections. [START_REF] Kavanaugh | Benefits and risks of low-dose glucocorticoid treatment in the patient with rheumatoid arthritis[END_REF][START_REF] Ruyssen-Witrand | Cardiovascular risk induced by low-dose corticosteroids in rheumatoid arthritis: a systematic literature review[END_REF][START_REF] Listing | Mortality in rheumatoid arthritis: the impact of disease activity, treatment with glucocorticoids, TNFα inhibitors and rituximab[END_REF] The differences in first-line therapy between our two age groups are consistent with earlier reports. In a registry study reported in 2006, onset of RA after 60 years of age was associated with significantly less use of biologics (25% vs. 33.1% for onset < 60 years) and DMARDs (30.9% vs. 40.5%) despite similar disease duration, activity, and severity. [START_REF] Tutuncu | Do patients with olderonset rheumatoid arthritis receive less aggressive treatment?[END_REF] A cross-sectional study done in 2008 found that treatment was less aggressive in older patients with RA compared to their younger counterparts; in particular, initial methotrexate therapy was significantly less commonly prescribed (47% vs. 57% of patients) and methotrexate dosages were lower (5.46 ± 1.66 mg/week vs. 5.96 ± 1.77 mg/week). [START_REF] Ogasawara | Observational cross-sectional study revealing less aggressive treatment in Japanese elderly than nonelderly patients with rheumatoid arthritis[END_REF] In general, the risk of adverse drug events increases with advancing age, due to the gradual accumulation of comorbidities such as heart failure and kidney dysfunction, as well as to the ageassociated decline in immune function. Also, drugs are often less effective in older patients. In a review of four randomised and five open-label studies of etanercept, treatment efficacy was independent from age. [START_REF] Köller | Response of elderly patients with rheumatoid arthritis to methotrexate or TNF inhibitors compared with younger patients[END_REF] Similarly, an analysis of data from two randomised trials showed no age-related differences in the efficacy of methotrexate alone or of TNFα antagonist therapy with or without methotrexate. [START_REF] Bathon | Safety and efficacy of etanercept treatment in elderly subjects with rheumatoid arthritis[END_REF] In four randomised trials and two long-term extensions, etanercept used to treat RA was slightly less effective in patients older than 65 years than in younger patients, but the difference was small and the frequency of adverse events was similar in the two groups. [START_REF] Fleischmann | Response to etanercept (Enbrel) in elderly patients with rheumatoid arthritis: a retrospective analysis of clincal trial results[END_REF] These data suggest that older patients with RA should be treated aggressively if needed to achieve good disease control and quality of life. A prospective study evaluated a treat-totarget strategy in 151 patients with elderly-onset RA (mean age, 74.9 years). [START_REF] Sugihara | Structural and functional outcomes of a therapeutic strategy targeting low disease activity in patients with elderly-onset rheumatoid arthritis: a prospective cohort study (CRANE)[END_REF] Adherence to the strategy was 83.4% after 6 months and 75.5% after 1 year. Importantly, after 1 year, nearly 50% of patients were in structural remission and 63% were in functional remission. Similarly, Cho et al shown the retention rate of TNF alpha was comparable in the elderly and younger patients although the major cause of discontinuation was AEs in the elderly patients, while it was drug ineffectiveness in younger patients. [START_REF] Cho | Drug retention and safety of TNF inhibitors in elderly patients with rheumatoid arthritis[END_REF] A 2005 review article indicated that sDMARDs and TNFα antagonists were similarly tolerated by patients younger than 65 years and those aged 65 years or over. [START_REF] Ranganath | Disease-modifying antirheumatic drug use in the elderly rheumatoid arthritis patient[END_REF] Nevertheless, older patients were at higher risk for non-serious infections and tuberculosis reactivation. The risk of cardiac events is higher in patients with NYHA class II, II, or IV heart failure treated with TNFα antagonists, and the risk of lymphoma increased with age although there is no proven association with biologics. In patients with RA aged 65 years or over, the higher risk of infections associated with etanercept therapy seemed related to age-related comorbidities rather than to the drug itself. [START_REF] Bathon | Safety and efficacy of etanercept treatment in elderly subjects with rheumatoid arthritis[END_REF] The first strength of this study is its nationwide, longitudinal, prospective design. The second strength is that it included patients based on disease activity and not on prescription criteria. This point allowed us to assess the rates and reasons of prescribing or not TNFα antagonist therapy.

This study had two main limitations. One is the small number of patients with RA older than 75 years and treated by TNFα antagonists. The other is the moderate relevance of this study performed in 2007-9, to today's patients as new biologics agents are now available. Nevertheless, adalimumab and etanercept remain the most frequently used anti TNF in France.

In conclusion, older patients (≥ 75 years or ≥ 70 years) with active RA receive sDMARDs and TNFα antagonists less often, and glucocorticoids more often, than do their younger counterparts. A personalised evaluation of the risk/benefit ratio of various drugs is mandatory, with a special regard to comorbidities in older patients with RA. Further studies should be conducted in new cohorts of active RA.

Materials and methods

Study design and population

CORPUS is a French, prospective, observational, multicentre, longitudinal, population-based cohort of biologics-naive patients with inflammatory joint disease (RA, spondyloarthritis, or juvenile idiopathic arthritis) recruited in private practices and hospitals between 2007 and 2009 by 102 rheumatologists, internists, and paediatricians. [START_REF] Saraux | Which patients with rheumatoid arthritis, spondyloarthritis, or juvenile idiopathic arthritis receive TNF-α antagonists in France? The CORPUS cohort study[END_REF] This cohort was established at the request of French health authorities to assess TNFα antagonist use in patients with these disorders. The patients were monitored prospectively for at least one year.

Before inclusion, all patients gave their written consent to participate in the CORPUS study. The study was approved by the Ethics Committee of Nancy, France.

This study was conducted in the subset of CORPUS patients [START_REF] Saraux | Which patients with rheumatoid arthritis, spondyloarthritis, or juvenile idiopathic arthritis receive TNF-α antagonists in France? The CORPUS cohort study[END_REF] older than 18 years who met American College of Rheumatology (ACR) criteria for RA, [START_REF] Arnett | The American Rheumatism Association 1987 revised criteria for the classification of rheumatoid arthritis[END_REF] had active disease (DAS28 > 3.2) at inclusion despite first-line therapy, had never taken biologics, and underwent at least one follow-up evaluation 3 to 24 months after inclusion. All treatments were at the discretion of the managing rheumatologists. TNFα antagonist users were defined as patients who received at least one TNFα antagonist injection between inclusion and 3 months before the follow-up visit. Between 2007 and 2009, the TNFα antagonists available in France were etanercept, adalimumab, and infliximab.

Data collection

The study data were collected on standardized forms by the rheumatologists who recruited the patients to the cohort. Each patient was evaluated twice, at inclusion and 12 months later. Each evaluation consisted of a standardised interview, general physical examination, laboratory tests, and self-administered questionnaires.

The following data were recorded at inclusion: age; sex; disease duration; anti-citrullinated peptide antibody (ACPA) and rheumatoid factor (RF) status; ACR criteria for RA; history of treatment with sDMARDs, biologics, and glucocorticoids; extra-articular signs of RA; and medical history and comorbidities. Data recorded at both visits were the tender and swollen joint counts, plasma CRP level, erythrocyte sedimentation rate (ESR), patient visual analogue scale (VAS) score for global disease activity, DAS28 (calculated with ESR if available and CRP otherwise), [START_REF] Prevoo | Modified disease activity scores that include twenty-eight-joint counts. Development and validation in a prospective longitudinal study of patients with rheumatoid arthritis[END_REF] Health Assessment Questionnaire Disability Index (HAQ-DI), [START_REF] Fries | Measurement of patient outcome in arthritis[END_REF] radiographic erosions, and treatments (sDMARDs, biologics, and glucocorticoids) with their start and stop dates and dosages.

Figure 1 .

 1 Figure 1. Patient flow chart.

Table 1 .

 1 Patient characteristics at inclusion into the CORPUS cohort.

		< 75 years of	≥ 75 years of
		age	age
		(mean age,	(mean age,
		56.7 years)	80.8 years)	p
		n = 333	n = 49	value
	Women, n (%)	265/333 (79)	41/49 (84)	0.57
	CRP, mg/L, mean (SD)	9.67 (12.9)	17.7 (34.1)	0.13
	CRP > 10 mg/L, n (%)	126/290 (43.4) 17/39 (43.6)	0.99
	ACPA-positive, n (%)	212/333 (63.7) 28/49 (57.1)	0.43
	RF-positive, n (%)	230/330 (69.7) 32/49 (65.3)	0.62
	ESR> 20 mm, n (%)	161/325 (49.5) 25/47 (53.2)	0.75
	DAS28, mean (SD)	3.71 (1.40)	3.89 (1.51)	0.38
	HAQ-DI, mean (SD)	0.75 (0.68)	0.94 (0.75)	0.19
	Physician visits, n (%)	158/185 (85.4) 24/30 (80.0)	0.42
	Admissions, n (%)	33/186 (17.7)	5/30 (16.6)	1.00
	Heart failure, n (%)	3/330 (0.9)	3/49 (6.1)	0.03
	Kidney failure, n (%)	1/330 (0.3)	2/49 (4.1)	0.04
	Respiratory failure, n (%)	10/330 (3.0)	5/49 (10.2)	0.03
	History of cancer, n (%)	4/329 (1.2)	2/49 (4.1)	0.18
	History of infections, n (%) High risk of infection ¥ , n (%)	24/329 (7.3) 1/330 (0.3)	4/49 (8.2) 1/49 (2.0)	0.77 0.24
	History of stroke, n (%)	4/330 (1.2)	0/49 (0.0)	1.00
	History of autoimmunity, n (%) Hypertension • , n (%)	5/330 (1.5) 36/330 (10.9)	5/49 (10.2) 9/49 (18.4)	< 0.01 0.15
	History of thrombosis, n (%)	5/330 (1.5)	3/49 (6.1)	0.07
	Glucocorticoid treatment, n (%)	238/330 (72.1) 40/46 (86.7)	0.03
	Daily prednisone-equivalents at	18.1 (48.6)	15.6 (20.4)	0.86
	inclusion, mg, mean (SD)
	sDMARD therapy, n (%)	330/330	49/49 (100.0) 1.00
		(100.0)
	¥ defined by chronic ulcer skin, suspected prosthetic joint infection, long-term
	indwelling urinary catheter or other implanted material
	• defined as systolic blood pressure > 140 mmHg and diastolic blood pressure
	> 90 mmHg

Table 3 .

 3 Characteristics of the three patients 75 years or older who received TNFα antagonist therapy.

	Patient 1	Patient 2	Patient 3

Table 2 .

 2 Characteristics of patient at follow-up.

		< 75 years of	≥ 75 years of	p
		age	age	value
	CRP> 10 mg/L, n (%)	65/270 (24.1)	13/36 (36.1)	0.15
	ESR> 20 mm, n (%)	111/305 (36.4) 14/46 (30.4)	0.51
	DAS28, mean (SD)	3.69 (1.42)	4.04 (1.36)	0.08
	HAQ-DI, mean (SD)	0.40 (0.42)	0.60 (0.50)	0.08
	sDMARDs, n (%)	235/287 (81.8) 27/42 (64.3)	0.01
	Glucocorticoid treatment, n (%)	155/286 (54.2) 33/42 (78.6) < 0.01
	Daily prednisone-equivalents at	7.8 (9.67)	6.04 (5.10)	0.26
	inclusion, mg, mean (SD)
	TNFα antagonist treatment, n (%)	111/333 (33.3)	3/49 (6.1)	< 0.01
	First TNFα antagonist	Etanercept: 55	Adalimumab:2
		Adalimumab:41	Infliximab: 1
		Infliximab: 14
		Unknown: 1

Funding

The CORPUS study was prescribed by the French government and funded by Abbott France, Schering Plough, and Wyeth Pharmaceuticals, and conducted with the support of Inserm Transfert.

Statistical analyses

The data collected at inclusion and at follow-up were described as mean± SD or median (range) for quantitative data and n (%) for qualitative data (missing data were excluded when computing percentages).

The groups < 75 years and ≥ 75 years were compared regarding comorbidities, inflammation (CRP and ESR), disease activity (DAS28), disability (HAQ-DI), number of physician visits and admissions, and first-line treatment. TNFα antagonist use was compared in the two age groups. Univariate analyses were performed to identify variables associated with TNFα antagonist use in each age group, using the chi-square test or Fisher's exact test, as appropriate, for qualitative variables and the Mann-Whitney test for quantitative variables.

To verify the impact of the cut off to separate old and young patients, we also compared patients aged 70 years or more to patients younger than 70 years.

All tests were two-sided and p values ≤ 0.05 were considered significant. Statistical analyses were performed using SPSS 23.0 (SPSS Inc., Chicago, IL, USA).

Abbreviations

Disclosure of potential conflicts of interest

No potential conflict of interest was reported by the authors.