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

See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/224310574 D-OSKIL: A New Mechanism for Controlling Stick-Slip Oscillations in Oil Well Drillstirings ArticleinIEEE Transactions on Control Systems Technology · December 2008 DOI: 10.1109/TCST.2008.917873·Source: IEEE Xplore CITATIONS 60 READS 205 3 authors, including: Francisco R. Rubio Universidad de Sevilla 123 PUBLICATIONS3,077 CITATIONS SEE PROFILE All content following this page was uploaded by Francisco R. Rubio on 12 March 2015. The user has requested enhancement of the downloaded file. 1 DOSKIL: A New Mechanism for Controlling Stick-Slip Oscillations in Oil Well Drillstrings Carlos Canudas-de-Wit Laboratoire d’Automatique de Grenoble, INPG-CNRS, Grenoble, FRANCE. Email: carlos.canuda[email protected] Francisco R. Rubio Department of Automatic Control, University of Seville, Seville, SPAIN Email: [email protected] Miguel Angel Corchero Department of Automatic Control, University of Seville, Seville, SPAIN Email: macpe[email protected] Abstract Limit cycles occurring in oil well drillstrings result from the interaction between the drill bit and the rock during drilling operations. In this paper we propose to use the weight on the bit (WoB) force as an additional control variable to extinguish limit cycles when they occur. An approximate analysis based on the bias describing function and completed with some simulations, provides good evidence that the rotational dynamics of the oil well drillstring displays such a behavior. In particular, we propose an adaptation law for the WoB named D-OKILL mechanisms, which results from a variant of the oscillation killer (OSKIL) mechanism studied in detail in [6]. 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 [26], [34]. 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 June 6, 2007 DRAFT hal-00394990, version 1 - 13 Jun 2009 Author manuscript, published in "IEEE Transactions on Control Systems Technology 16, 6 (2008) 1177-1191" 2 a) b) Fig. 1. Oil drilling system in the field (a). Basic scheme of a vertical drilling system.(b) 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 [21]. 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 [14], [21], [24], [31]. 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 [3], [17]. 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 (WoB), modifying the drilling mud characteristics, introducing an additional friction at the bit [25], etc) is an effective strategy to suppress stick-slip motion [28]. However, this strategy depends too much on the personal skills of each drilling technician to be really effective. June 6, 2007 DRAFT hal-00394990, version 1 - 13 Jun 2009 3 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) [13] [28], 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 [15] which follows the same approach given in [13] and [28]. •Introducing a PID controller structure at the surface in order to control the rotary speed [1], [23], [24], [25]. •Using robust controllers, like the linear H∞control proposed in [29], to suppress stick-slip motion at the bit. •Using a controller based on an input-state feedback linearization of the nonlinear friction torque [2]. 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 [1] and linear approximations to stability of controlled drillstring has been studied in [24]. The value of the system weight measured at the bottom part, called Weight on Bit (WoB), has been proved to be an important parameter in the occurrence and possible avoidance of stick-slip oscillations (see [23] and [28]). Efficient drilling operation requires a certain amount of force (WoB) 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 WoB seems to be necessary to maintain a good drilling operation, thus, avoiding such oscillations (see [7] and [24]). 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 (WoB) force as an additional control variable. In particular we adapt the oscillation killer (OSKIL) mechanism studied in [6], to the oil well drillstring systems (named here D-OSKIL1) 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 WoB 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 [7], we show that the new proposed Weight on Bit (WoB) 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. 1D-OSKIL stands for Drilling oscillation killer mechanism. June 6, 2007 DRAFT hal-00394990, version 1 - 13 Jun 2009 4 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 [21] and [31] 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 [31]. June 6, 2007 DRAFT hal-00394990, version 1 - 13 Jun 2009 5 bit-bouncing whirl whirl stick-slip Fig. 2. Different types of vibrations in drillstring systems. The vibrations appearing in the drillstring can be divided into 3 different categories [14] [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 [19], 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 [22]. III. SYSTEM MODELLING Multiple kind of models have been used in literature to describe drillstring systems (see for example [19] and [31]). 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 [10]. 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: [17], [20], [27], [32] propose single-degree-of-freedom models, [1], [5], [22], [24] June 6, 2007 DRAFT hal-00394990, version 1 - 13 Jun 2009 6            v Jr ϕr dr kc ϕb db Jb ToB WoB Fig. 3. Drillstring two-coupled masses model. propose two-degree-of-freedom models including a linear controller, and [15], [29] 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 [22]. The model used here (depicted in Figure 3) is a two-degree-of-freedom model with two inertial masses Jr and Jb, locally damped by drand db. The inertias are coupled with each other by an elastic shaft of stiffness k and damping c. The variables ϕrand ϕbstand 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. The model equations are the following: Jr¨ϕr+c( ˙ϕr−˙ϕb) + k(ϕr−ϕb) + dr˙ϕr=v(1) Jb¨ϕb+c( ˙ϕb−˙ϕr) + k(ϕb−ϕr) + db˙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 [22]), and the normal force ucalled Weight on Bit (WoB), 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 June 6, 2007 DRAFT hal-00394990, version 1 - 13 Jun 2009 7 purposes. A. Model for simulations The state-space representation of the later model is the following: ˙x=Ax +Bv +Hµ(x, z)u(4) ˙z=f(x, z)(5) with A=     0 1 −1 −k Jr −(dr+c) Jr c Jr k Jb c Jb −(c+db) Jb      , B =     0 1 Jr 0      , H =     0 0 −1 Jb      where the state x= [x1x2x3]Tis defined as follows: x1=ϕr−ϕb x2= ˙ϕr(6) x3= ˙ϕb In this description, the state z∈Rrepresents 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 ( [12] and [18]). One possible model for Equation (5) is the LuGre friction model [8]: ˙z=x3−σ0|x3| g(x3)z, g(x3) = µC+ (µS−µC)e−(x3/vs)2(7) µ(x, z) = σ0z+σ1˙z, 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, µSS(x3) = g(x3)sgn(x3).In this model, σ0, σ1, vs, µC, µSare positive constants characterizing the friction physical properties. Also note that the torsional linear friction at the drill bit side is already incorporated in the Amatrix of the representation (4). B. Model for control Note that the previous model for the µ(x, z)includes an additional friction dynamics, zwhich 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, ˙x=Ax +Bv +Hµ(x3)u(8) June 6, 2007 DRAFT hal-00394990, version 1 - 13 Jun 2009 8 where here µ(x3)is a static map between the bit rotational velocity x3= ˙ϕband the normalized friction parameter µ, i.e. the steady-state form of the model (7), 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 ˙ϕrto a certain desired value ωd(a typical value for ωdis 5rad/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 [10], as shown by: v=k1+k2 s(ωd−˙ϕr)−k3( ˙ϕr−˙ϕb)(9) or equivalent v=k1(ωd−x2) + k2x4−k3(x2−x3) ˙x4= (ωd−x2) then the closed-loop equations take the form for the model for simulation, ˙x=Aclx+Bclωd+Hclµ(x, z)u(10) ˙z=f(x, z)(11) and the following one for the model for control analysis, ˙x=Aclx+Bclωd+Hclµ(x3)u(12) with the obvious observation that xis now of dimension four (due to the introduction of an integral term in the rotary table control), i.e. x= [x1, x2, x3, x4], and with the Acl,Bcl and Hcl given as: Acl =         0 1 −1 0 −k Jr −(dr+c+k1+k3) Jr (c+k3) Jr k2 Jr k Jb c Jb −(c+db) Jb0 0−1 0 0         , Bcl =         0 k1 Jr 0 1         , Hcl =         0 0 −1 Jb 0         The steady-state value of x, considering µ∗=µ(x∗ 3), is: x∗ 1=u0µ∗+dbωd k(13) x∗ 2=ωd(14) x∗ 3=ωd(15) x∗ 4=(db+dr)ωd+µ∗u0 k2 (16) June 6, 2007 DRAFT hal-00394990, version 1 - 13 Jun 2009 15           kek WoB WoBmin u0 ˜u attractive limit cycle repellent limit cycle a b b b b Fig. 10. Possible drillstring system trajectories. b) Behavior beyond nominal operations: To suppress such oscillations, the u=WoB must be reduced by means of the control signal ˜u, 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 WoB must be recovered in a proper slow manner, to continue with the drilling task, i.e. u→u0. Significantly enough, the variation of ˜u 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 ˜u), will be here of the form: ˙ ˜u=P0 −u0{−σ˜u+ Φ(·)} where σ > 0, can be understood as a time-constant of the controller, and P0 −u0is a projector operator ensuring that solutions of the above equation makes ˜ustay in the range (−u0,0], and Φ(·)is a nonlinear function which must be designed to ensure system stability, that is: ˙ϕr→ωdand ˜u→0 In order to make the presentation simpler, we will trop the explicit use of the projection operation Pin the following section. However, the reader should keep in mind that ˜uis a bounded signal in the prescribed range. With this in mind, the complete closed-loop equations are: ˙x=Aclx+Bclωd+Hclµ(x3)(u0+ ˜u)(20) ˙ ˜u=−σ˜u+ Φ(·)(21) where matrices Acl, Bcl, Hcl have been defined previously. June 6, 2007 DRAFT hal-00394990, version 1 - 13 Jun 2009 16     - -    Ψ ∆ G(s) Γ y ˜u2 ˜u1 Fig. 11. Error system block diagram. B. Error equations Error equations can be obtained by applying the change of coordinates e=x−x∗, and considering the steady-state values µ(e3)ss =µ∗and ˜uss = 0. This yields2: ˙e=Acle+Hcl [µ(y)˜u+ ˜µ(y)u0](22) ˙ ˜u=−σ˜u+ Φ(y)(23) y=Ce =e3(24) where the term ˜µ(y)is defined as follows: ˜µ(y) = µ(y)−µ∗(25) and we assume that the update rule for ˜uis 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: G(s) : Ψ 7→ y(26) Γ : y7→ ˜u1=µ(y)˜u(27) ∆ : y7→ ˜u2= ˜µ(y)u0(28) where Ψ = −(˜u1+ ˜u2). C. Error equation properties 1) PR condition on G(s):The map G(s)is: G(s) = −C(sI −Acl)Hcl (29) 2With an abuse of notation, we will use µ(y)to denote the expression of µ(x3), in the shift coordinate y+ωd, i.e. µ(x3) = µ(y+ωd) = µ(y). Note that at y= 0, we have µ(y= 0) = µ(ωd) = µ∗. June 6, 2007 DRAFT hal-00394990, version 1 - 13 Jun 2009 17 with C= [0 0 1 0], or equivalently: G(s) = s(a2s2+a1s+a0) b4s4+b3s3+b2s2+b1s+b0 (30) The error equation and updating rule of ˜uhave 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 (Acl, Hcl, C)as demonstrated in [30] 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 −Acl)Hcl.G(s)is PR if and only if: 1) CAclHcl >0,2)Acl is stable, 3) the matrix Acl(I−(1/CAclHcl)AclHclC)Acl has no eigenvalues on the open negative real axis (−∞,0). The first condition is easy to compute in terms of model parameters. This gives CAclHcl = (c+db)/J2 b>0, and always holds from the physics of the system. The second condition is also verified since Acl 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 [16], the following property holds: ∃P=PT>0, Q =LTL≥0such that: AT clP+P Acl =−Q=−LTL≤0(31) PHcl =−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) = eTP e, and •G(s)has a finite L2-gain: γ2(G) = supω|G(jω)|<∞ 2) Boundedness of signal Ψ(t):From the definitions of ˜u1and ˜u2in Equations (27) and (28), together with the assumption that the adaptation mechanism yields values in the range ˜u∈(−u0,0], it follows that both signals, ˜u1 and ˜u2, are bounded, that is: ||˜u1||∞= sup t≥0|˜u1| ≤ u0<∞,(33) ||˜u2||∞= sup t≥0|˜u2| ≤ 2·u0<∞.(34) Hence ||Ψ(t)||∞≤3·u0. 3) Boundedness of the output y(t):Since G(s) : Ψ 7→ yis a lineal stable map, the output signal yis 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. June 6, 2007 DRAFT hal-00394990, version 1 - 13 Jun 2009 18      µ( ˙ϕb) ˙ϕb µS µC vs     ∆ y −ay by a) b) Fig. 12. Normalized friction function (a). ∆(y)lies at the interval [a, b](b). Figure 12-(a) shows the memoryless friction map used for this study. Note that as the steady-state rotatory bit speed (ωd) is in general much larger than the Stribeck velocity vs, we can then assume that µ∗=µC. Therefore, taking into account that y=e3= ˙ϕb−ωd, the output of block ∆will have the profile shown in Figure 12-(b). This operator belongs to the cone sector [a, b]as displayed in the same Figure (see [33] 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, •ay2≤y∆(y)≤by2,∀y∈ ℜ In our case, the values for aand bare: a=−µS−µC ωd u0(36) b=µS+µC ωd u0(37) and consequently, the map ∆has also finite L2-gain, which is bounded by: γ2(∆) ≤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 (ais small when compared to b). June 6, 2007 DRAFT hal-00394990, version 1 - 13 Jun 2009 19 Figure 13 shows a possible block transformation, where the following new operators, ∆∗, and Γ∗are defined: ∆∗:y7→ (˜u2+εy) Γ∗:y7→ (˜u1−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 εis taken such that ε=|a|, as can be seen in Figure 14. ε=|a| ⇒ Zt 0 y(˜u2+εy)dt ≥0(39) With this new feedback configuration, the problem of designing a stable update law for ˜u(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 ˜uis limited to the admissible parameters range, the following updating rule will be analyzed. June 6, 2007 DRAFT hal-00394990, version 1 - 13 Jun 2009 20    y ∆∗ ε Fig. 14. Profile of map ∆∗. Let us consider the nonlinear function Φ(y): Φ(y) = λ y sgn(µ(y)) λ≥0(40) Note that this choice is conditioned by the ability of computing the sign of µ(y). Considering the form of the friction model for this study, the sign of µ(y) = µ( ˙ϕb−ωd), can be computed if ˙ϕbcan be measured, or at least, observed as shown in simulations later on. We proceed according to this hypothesis in what follows. E. Stability analysis Lemma 1: Let ρ > 0be an arbitrarily positive constant, and λ,σbe such that the following design inequality holds, λ σ≥µS µC−1u0 ωd +σ µC ρ(41) where µS µC≥1. Then map Γ∗:y7→ (µ(y)˜u−εy)is strictly input passive, i.e. I=Zt 0 (µ(y)˜u−εy)y≥ρZt 0 y2−β0(42) with β0=ymax σ2u0>0. Proof: Let Idefine the integral of the input-output product of the operator Γ∗, i.e. I=Zt 0 (µ(y)˜u−εy)y=Zt 0 µ(y)˜uy −Zt 0 εy2 Substituting ˜ufrom Equation (23) in the above expression gives, I=1 σZt 0 µ(y)Φ(y)y−1 σZt 0 µ(y)y˙ ˜u−Zt 0 εy2 From sections VII-C.2, and VII-C.3, signals yand µ(y)have been shown to be bounded. Let note these bounds as: |y|< ymax,|µ(y)|<1. Therefore, June 6, 2007 DRAFT hal-00394990, version 1 - 13 Jun 2009 21 I≥1 σZt 0 µ(y)yΦ(y)−Zt 0 εy2−ymax σZt 0 ˙ ˜u Taking now Φ(y) = λ y sgn(µ(y)), gives I≥Zt 0 (λ σ|µ(y)|−ε)y2−ymax σ|˜u(t)−˜u(0)| ≥Zt 0 (λ σ|µ(y)|−ε)y2−ymax σ2u0 ≥Zt 0 (λ σµC−|a|)y2−β0 where the last inequality is obtained by using the lower bound on |µ(y)|, i.e., |µ(y)| ≥ µC, the definition of ε=|a|(with aas given in Equation (36)) and the constant β0=ymax σ2u0. Finally, introducing the condition (41) in the above expression, gives the following lower bound on I: I≥Zt 0 (λ σµC−|a|)y2−β0≥ρZt 0 y2−β0 which proves the lemma. Remark 1: 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 WoB (u0), the rock friction features (µS−µC), and the desired rotational velocity (ωd). The parameter ρas shown latter, provides a measure of the convergence rate of the output yto zero. 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)-(32) (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∗,˜u∗) = (0,0) is a globally asymptotically stable equilibrium of the considered closed-loop system. Proof: Let us take the following scalar function: V(e, ˜u) = 1 2eTPe +Zt 0 y(˜u2+εy) + I(˜u, y) + β0−ρZt 0 y2(43) From (39) and (42), we have that V(e, ˜u)is semi-positive definite. Computing the time-derivative of V(e, ˜u), and using the properties (i)−(iii)of the theorem, results in: ˙ V(e, ˜u) = −1 2eTLTLe −ρy2≤0,∀e, ˜u(44) Therefore, from last Equation, we have that y→0with 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, June 6, 2007 DRAFT hal-00394990, version 1 - 13 Jun 2009 22 0 50 100 150 0 10 20 30 40 Time[s] Rotatory speed [rad/s] Operation under DOSKIL Surface Downhole 0 50 100 150 0 1 2 3 4x 104 Time[s] Weigth on Bit [N] Fig. 15. Simulation of D-OSKIL scheme with σ= 0.3and λ= 2.000. together with the fact that σis a positive constant and Φ(0) = 0, gives that ˜u→0. Finally, this implies that the two last terms in Equation (22) also tends to zero, i.e. lim t→∞ [µ(y)˜u+ ˜µ(y)u0] = [µ(0)0 + ˜µ(0)u0] = 0 since µ(0) = µ∗=µC, and ˜µ(0) = µ(0) −µ∗=µ∗−µ∗= 0. Therefore, this results in ˙e=Acle+Hcl lim t→∞ [·] = Acle So it can be concluded that e→0, and hence that e∗= 0, and ˜u∗= 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 and 16. The values3for 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 WoB are shown. As it can be observed, with the nominal weight u0= 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 WoB profiles obtained are quite different. 3The numerical values of the drilling system parameters according to a 2000mlong drillstring have been taken from [29]. June 6, 2007 DRAFT hal-00394990, version 1 - 13 Jun 2009 23 0 50 100 150 0 10 20 30 40 Time[s] Rotatory speed [rad/s] Operation under DOSKIL Surface Downhole 0 50 100 150 2.5 3 3.5 4 x 104 Time[s] Weigth on Bit [N] Fig. 16. Simulation of D-OSKIL scheme with σ= 1.0and λ= 2.500. 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 WoB 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 WoB is also faster in Figure 16, but in this case some oscillations occur during the transition. This issue is due to the λvalue, when large values of the λare chosen a sharp WoB 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 (8). That is on ˙x=Ax +Bv + (Hu)·µ(45) yo=Cox= ˙ϕr(46) June 6, 2007 DRAFT hal-00394990, version 1 - 13 Jun 2009 24 Following [35] we propose to use the following observer based on the observation of the rotary angular velocity yo= ˙ϕr: ˙ ˆx=Aˆx+Bv + (Hu)·ˆµ+Ko+βΓΓTCT o(yo−Coˆx)(47) ˙ ˆµ=βΓTCT o(yo−Coˆx)(48) ˙ Γ = [A−KoCo] Γ + (Hu)(49) where: Γis a (3 ×1) time-varying vector, β > 0is a positive scalar, ˆx, and ˆµare the state and friction coefficient estimates respectively, yois the measured output, Kois a (3 ×1) observer vector gain, and Co= [0,1,0]. Note that Co6=C. The observer not only provides an estimate for the bit rotational velocity ˆx3, but also provide an estimate of the friction coefficient ˆµwhich can be useful for other monitoring purposes. The observer, as indicated in [35], results in a globally exponentially stable observer providing the following hypothesis hold: •µis constant •Exists a matrix Kosuch that (A−KoCo)is strictly stable matrix, i.e. The constant pair (A, Co)is detectable. •u(t)is persistently exciting, i.e. ∃δ, T > 0such that the following inequality is satisfied: Zt+T t Γ(τ)TCT oCoΓ(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 Aand Co. 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−KoCo, H)is controllable and the pair (A−KoCo, Co)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 [11]. B. Simulation with the observed-based controller The original controller has the form ˙ ˜u=−σ˜u+λ(x3−ωd)·sgn(µ(x3−ωd)) (50) 4In this case, it can also be shown that stability (not asymptotic) follows is preserved June 6, 2007 DRAFT hal-00394990, version 1 - 13 Jun 2009 31 [31] Spanos, P.D., Chevallier, A.M., Politis N.P. and Payne M.L. Oil well drilling: A vibrations perspective. The Shock and Vibration Digest, 35(2):81–99, 2003. [32] van de Vrande, B.L., van Campen, D.H., de Kraker, A. An Approximate Analysis of Dry-friction-induced Stick-slip Vibrations by a Smoothing Procedure. Nonlinear Dynamics, 19:157–169, 1999. [33] Vidyasagar, M. Nonlinear Systems Analysis. Prentice-Hall, second edition, 1993. [34] William Ranney, M. Offshore Oil Technology Recent Development. Noyes Data Corporation, 1979. [35] Zhang, Q. Adaptive observer for MIMO linear time varying systems. INRIA report, No.4111, 2001. 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