- Email: [email protected]
Stabilization of torsional vibration in oilwell drillstring system
Accepted Manuscript
Stabilization of torsional vibration in oilwell drillstring system Elsevier, Samir Toumi, Lotfi Beji, Rhouma Mlayeh, Azgal Abichou PII: DOI: Reference:
S0947-3580(16)30061-9 10.1016/j.ejcon.2017.03.002 EJCON 200
To appear in:
European Journal of Control
Received date: Revised date: Accepted date:
12 July 2016 28 January 2017 14 March 2017
Please cite this article as: Elsevier, Samir Toumi, Lotfi Beji, Rhouma Mlayeh, Azgal Abichou, Stabilization of torsional vibration in oilwell drillstring system, European Journal of Control (2017), doi: 10.1016/j.ejcon.2017.03.002
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT
Stabilization of torsional vibration in oilwell drillstring system Radarweg 29, Amsterdam
CR IP T
Elsevier1 Samir Toumia,b,∗, Lotfi Bejia , Rhouma Mlayehb , Azgal Abichoub a are
AN US
with the IBISC-EA 4526 laboratory, University of Evry, 40 rue du Pelvoux, 91020, Evry, France. b are with the LIM laboratory, Polytechnic School of Tunisia, BP 743, 2078 La Marsa, Tunisia.
Abstract
Stick-slip oscillations also known as torsional vibrations appearing in oilwell drilling systems are a source of economic losses, borehole disruption, pipes disconnection, and prolonged drilling time. The torsional dynamics is modeled by
M
a damped wave equation. An important stability issue is to find a control law that reject perturbations due to torsional vibrations. Hence, the primary aim of this paper is to prove the well-posedness of the damped wave equation. In the
ED
next step, using the Riemann invariants, the damped wave equation is transformed into 2 × 2 first-order transport equations. The backstepping techniques
PT
combined with kernel equations and the Lyapunov theory are used to prove the local exponential stability of the transformed system, consequently the torsional dynamic. Simulation results are presented to illustrate the effectiveness of the
CE
control law.
Keywords: torsional vibrations, stability, drilling, partial
AC
differential equation
✩ Fully
documented templates are available in the elsarticle package on CTAN. author Email address: [email protected] (Samir Toumi ) 1 Since 1880.
∗ Corresponding
Preprint submitted to Journal of LATEX Templates
March 17, 2017
ACCEPTED MANUSCRIPT
1. Introduction For many years, the control and the model of drilling systems has been a very active research field. Vibrations dynamics on the drillstring system consti-
5
CR IP T
tute an important source of failures which affect penetration rates and increase drilling process costs, borehole disruption, pipes disconnection, and prolonged drilling time, are only some examples of results associated with drilling vibra-
tions. The process of drilling for oil is the creation of drilling over five thousand
feet deep in the soil until it reaches the gas and oil. During drilling operations, there exits three main types of vibrations caused by the drillstring [1]:
lateral vibrations (whirl motion due to the out-of-balance of the drillstring),
AN US
10
axial vibrations (bit bouncing phenomenon) and torsional vibrations (stick-slip phenomena). A detailed description of each mode of vibration is presented in [1].
Before the sixties, studies were focused on material strength of the drillstring components, however the trends have since changed to emphasize on its dynamic
M
15
behavior [1]. Since by the work of Bailey and Finnie of Shell Development Company in 1960 which are the first developed analytical and experimental study on
ED
torsional and axial drilling vibrations [2], several investigations on the modeling and control of drilling systems has been conducted [3, 4, 5, 6, 7, 8, 9, 10, 11]. In [6], Saldivar et al. are proposed energy function for the torsional distributed
PT
20
model allows to find a control law that ensures the energy dissipation during the drilling. Sagert et al. in [12] are introduced control law to avoid undesirable
CE
torsional oscillations of the drillstring in oilwell drilling systems using backstepping and flatness approaches. In [7], the authors are treated the control of
25
torsional-axial coupled drilling vibrations using flatness approach.
AC
The range of stability for the coupled axial-torsional motions in the presence of state-dependent delay as well as to investigate the effectiveness of a feedback controller to suppress torsional vibrations have been studied in [13]. Also, in [14], Liu et al. have been presented an original model considering time-delay
30
and stick-slip effects. In [15], the authors present a model-free active control
2
ACCEPTED MANUSCRIPT
method for the vibration isolation of a rigid item that is elastically mounted to a slender rod which is subject to axial vibrations. A modeling for control of torsional vibrations, a novel controller design strategies for drillstring sys-
35
CR IP T
tems, an experimental validation of the proposed controllers, and a modeling and analysis for the assessment of a downhole anti stick slip tool are presented in [11].
This paper tackle the torsional vibrations in oilwell drilling system. The dy-
namic of the torsional vibrations is modeled by a damped wave equation which encountered in drilling system dynamics that describe the torsional, axial, and
lateral vibrations. The control law in terms of angular velocities and axial-
AN US
40
lateral forces appears in the boundary conditions. Several research in drillstring dynamic analysis are studied in neglected the damping terms [16]. A long the infinite drillstring dynamics, the damping term is taking into account in our subsequent analysis. 45
The primary aim of this work is to prove the well posedness of our partial differ-
M
ential equation (PDE) which describes the torsional vibration. After this step, we are going to prove the stability of our system. In fact, to show the stability of
ED
PDEs in finite dimension there are two main techniques: from the exact solution if this solution is easy to be obtained or in general using the Lyapunov theory. 50
An exact solution can be obtained using the variable separation method or the
PT
Laplace transform technique. In the literature, the study of stability of hyperbolic PDEs is well described (see [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28]).
CE
The paper is organized as follows. The contribution in Section 2 is to show the well-posedness of torsional vibration system. In this section, we recall the PDE
55
with the boundary conditions that permit to describe the torsional vibration
AC
problem. Some useful formulation in terms of variable change and Riemann’s invariants are introduced. In Section 3, we find the control law using backstepping transformation, kernel equations, and a target system formulation. In this last section, the boundary control objectives are defined with the drillstring tor-
60
sional dynamic stabilizing problem. The simulation results are given in Section 4. Some concluding remarks and perspectives are also introduced. 3
ACCEPTED MANUSCRIPT
2. Controlled torsional system
||δ(t, .)||L2 ([0,1]) =
s
Z
1
χ(t, x)2 + ζ(t, x)2 dx
0
the norm of δ, and by Z
1
v1 (t, x)2 + v2 (t, x)2 dx
0
AN US
||E(t, .)||L2 ([0,1]) =
s
CR IP T
Firstly,we introduce some notations and norms used in the paper. For v1 (t, x) χ(t, x) and δ(t, x) = , we denote by E(t, x) = v2 (t, x) ζ(t, x)
the norm of E, and ∆ = {(x, ξ) ∈ R2 : 0 ≤ ξ ≤ x ≤ 1}. The norm ||A(., .)||∞ = max(x,ξ)∈∆ k|A(x, ξ)k|22 with k|A(., .)k|22 = supY ∈Rn ,kY k=1 kA(., .)Y k2 is the clas-
65
sical operator norm.
The key challenge in this section, is to study the well-possedness of the tor-
operator theory [29].
ED
2.1. Problem statement
M
sional drilling system in oil-gas industry. To achieve our objectives, we use the
During the drilling operation an excessive vibrations may arise and have a
70
major effect on drilling performance. The principle kinds of drilling vibration in
PT
oilwell drilling system is the stick-slip oscillations which may affect the rotation direction and the penetration of the drill bit as well as the damage of the drilling
CE
system. The rotary drilling system is known in oilwell drilling processes. Due to 75
interaction bit-rock, the acting torque in a circular shaft produces angular de-
AC
formation along the drill pipe, and a distributer parameter model can be derived.
4
AN US
CR IP T
ACCEPTED MANUSCRIPT
M
Figure 1: Drilling system
The fundamentals of mechanical engineering, as the principal of angular
ED
momentum, the geometry of the drillstring, and the fact that the string only rotates around one fixed ς-axis, the governed differential equations of motion is
PT
derived for moments balance. As shown in (Fig. 1), ∂2ϑ (t, ς) + dτe + τ − (τ + dτ ) = 0 ∂t2 ∂ ∂ϑ where τ = GJ ∂ϑ ∂ς (t, ς), dτ = ∂ς GJ ∂ς (t, ς) dς, and a viscous friction term
CE
Idς
around the drillpipe elementary direction dς is added dτe = ddς ∂ϑ ∂t (t, ς). Conse-
AC
quently, the drillstring satisfies the following distributed model with respect to the torsional variable [9, 30, 3] , GJ
∂2ϑ ∂2ϑ ∂ϑ (t, ς) − I 2 (t, ς) − d (t, ς) = 0 2 ∂ς ∂t ∂t
(1)
with ς ∈ (0, L), t ∈ (0, +∞). Now, let us examine the boundary conditions and their corresponding surfaces 5
ACCEPTED MANUSCRIPT
(we can see [9, 6]). A top boundary condition is defined that integrates the applied velocity at the surface Ω(t), the fact that a local angular twist conducts to a velocity different, and the distributed parameter ϑ defined above, then at
GJ
∂ϑ ∂ϑ (t, 0) = ca ( (t, 0) − Ω(t)) ∂ς ∂t
CR IP T
the top (ς = 0) (2)
At the bottom or tip, i.e. (ς = L), it is important to take into account the friction caused by the bit-rock interaction which depends on the torsional angle
rate ϑ at ς = L, and integrates the bit moment of inertia, then, one may define the following at the tip
∂ϑ ∂2ϑ ∂ϑ (t, L) + Ib 2 (t, L) = −T ( (t, L)) ∂ς ∂t ∂t
AN US
GJ
(3)
where L is the drillstring length, ca the sliding torque coefficient, G the shear modulus, a lumped inertia Ib is chosen to represent the assembly at the bottom 80
hole, J the geometrical moment of inertia, d the drillstring damping, and Ω
M
the control input (angular velocity due to the rotary table). The inertia per unit length I is such that I = ρJ, where ρ is the area density. The following expression represent the model for the torque on the bit which is introduced in
∂ϑ ∂ϑ ∂ϑ ∂ϑ (t, L)) = cb (t, L) + Wob Rb µ( (t, L))sgn( (t, L)) ∂t ∂t ∂t ∂t
PT
T(
ED
[31], it allows to approximate the physical phenomenon at the bottom hole
∂ϑ ∂ϑ The both terms cb ∂ϑ ∂t (t, L) and Wob Rb µ( ∂t (t, L))sgn( ∂t (t, L)) represent respec-
tively, the viscous damping torque at the bit and the dry friction torque modeling
CE
the bit-rock contact. Wob > 0 is the weight on the bit and Rb > 0 is the bit
AC
radius. The bit dry friction µ is modeled as [6] µ(
γ ∂ϑ − b | ∂ϑ (t,L)| (t, L)) = µcb + (µsb − µcb )e υf ∂t ∂t
such that µsb ≥ µcb ∈ (0, 1) are the static and Coulomb friction coefficients,
0 < γb < 1 is a constant defining the velocity decrease rate, and the constant
velocity υf > 0 is introduced for the homogeneity of units. The following change of variable leads to an equivalent system, for which the
6
ACCEPTED MANUSCRIPT
function u(t, x) represent the angular position [8]: r I u(t, x) = ϑ(L t, L(1 − x)), x ∈ (0, 1), GJ
∂ 2 u(t, x) ∂t2 ∂u(t, 1) ∂t ∂ 2 u(t, 0) ∂t2 q 1 where λ = dL IGJ ,a=
= = =
CR IP T
we obtain
∂ 2 u(t, x) ∂u(t, x) −λ , x ∈ (0, 1) 2 ∂x ∂t r √ I I GJ ∂u(t, 1) L Ω(t) − GJ ca ∂x ∂u(t, 0) ∂u(t, 0) a + aF ( ) ∂x ∂t
LI Ib ,
(4)
(5)
(6)
AN US
85
r L 1 GJ ∂u(t, 0) T , GJ L I ∂t r r r 1 ∂u(t, 0) L 1 GJ ∂u(t, 0) 1 GJ ∂u(t, 0) = −cb − Wob Rb µ( )sgn( ) GJI ∂t GJ L I ∂t L I ∂t q ∂u(t,0) in which µ( L1 GJ I ∂t ) is as ∂u(t, 0) ∂t
r
GJ ∂u(t, 0) ) I ∂t
=
µcb + (µsb − µcb )e
γ
1 − υb | L f
√ GJ ∂u(t,0) I
∂t
|
ED
1 µ( L
= −
M
F
2.2. Well-posedness problem
Due to the presence of a nonlinear and complex relation resulting from the bit-rock interaction at the tip boundary, the well-posedness of the controlled
PT
90
torsional system becomes not trivial. Hence, in the following, one treats this
CE
issue using the semi-group theory (Further discussion in this theory in [29]).
AC
Let T > 0, the natural solution of the Cauchy problem is written in this form ∂ 2 u(t, x) = ∂t2 ∂u(t, 1) = ∂t ∂ 2 u(t, 0) = ∂t2 u(0, x) =
∂ 2 u(t, x) ∂u(t, x) −λ 2 ∂x ∂t r √ I I GJ ∂u(t, 1) L Ω(t) − GJ ca ∂x ∂u(t, 0) ∂u(t, 0) a + aF ( ) ∂x ∂t u0 (x), ut (0, x) = u1 (x)
7
(7) (8) (9) (10)
ACCEPTED MANUSCRIPT
where x ∈ (0, 1), t ∈ (0, T ), u0 ∈ K := {u ∈ H 1 (0, 1); u0 (0) = 0}, u1 ∈
95
L2 (0, 1), and Ω(t) ∈ R the control law.
It is obvious that K is a Hilbert space.
CR IP T
The vector space K is equipped with the following scalar product Z 1 1 ∂u (t, x) ∂u2 (t, x) hu1 (t, x), u2 (t, x)iK = dx ∂x ∂x 0
∂u(t,0) T Let us introduce Z = (u(t, x), ∂u(t,x) ∂t , u(t, 1), ∂t ) . Equations (7)-(10) can
Z(0) = Z0
A=
1
0
∂2 ∂x2
−λ
0
I GJ 0 ca hδ1 (x), .i
−ahδ00 (x), .i
0 0
0
0
0
0 0 , H(Z(t)) = 0 0 0 ∂u(t,0) aF ( ∂t ) 0 0
(11)
,
such as δ denotes the Dirac function for which
ED
0 q and f (t) = I L GJ Ω(t) 0
hδ10 (x), u(t, x)i = − ∂u(t,1) and hδ00 (x), u(t, x)i = − ∂u(t,0) ∂x ∂x .
PT
100
√
0
M
where
AN US
be compactly written as ˙ Z(t) = AZ(t) + H(Z(t)) + f (t)
At first, let us consider the problem (11) with H(Z) = 0 and f (t) = 0, accord-
CE
ingly we have the next Theorem. Theorem 1. The operator A generates a C0 semigroup S(t), t ≥ 0 of contrac-
AC
tions on X.
105
u(t, x)
∂u(t,x) ∂t Proof . Let X = { u(t, 1) ∂u(t,0) ∂t
R,
∂u(t,0) ∂t
∈ R}
, u(t, x) ∈ K,
8
∂u(t,x) ∂t
∈ L2 ([0, 1]), u(t, 1) ∈
ACCEPTED MANUSCRIPT
∂u (t,0) ∂t
CR IP T
This vector space X is equipped with the following inner-product u2 (t, x) u1 (t, x) 1 (t,x) ∂u2 (t,x) E D ∂u ∂t ∂u1 (t, x) ∂u2 (t, x) ∂t 1 2 , , iL2 [0,1] 2 X = hu (t, x), u (t, x)iK + h 1 ∂t ∂t u (t, 1) u (t, 1) 1 2 ∂u (t,0) ∂t
hu1 (t, 1), u2 (t, 1)iR + h
+
∂u1 (t, 0) ∂u2 (t, 0) , iR ∂t ∂t
We denote by k . k the norm in X associated to this scalar product.
AN US
Let A : D(A) ⊂ X →X be the linear operator defined by u(t, x) ∂u(t,x) ∂t ∈ X, u ∈ H 2 (0, 1), ∂u ∈ K, ∂u(t,1) = D(A) = { ∂t ∂x u(t, 1) ∂u(t,0) ∂t
∈ R} R, ∂u(t,0) ∂t
u(t, x)
∂u(t,0) ∂t
Moreover
=
∂u(t,x) ∂t ∂ 2 u(t,x) − λ ∂u(t,x) ∂x2 ∂t √ − IcaGJ ∂u(t,1) ∂x a ∂u(t,0) ∂x
ED
∂u(t,x) ∂t A u(t, 1)
M
We have
PT
hAZ, ZiX
= −λ
Z
y1
∂u(t,0) ∂x
u(t, x)
∂u(t,x) ∂t , ∀ u(t, 1) ∂u(t,0) ∂t
= 0, u(t, 1) ∈
∈ D(A)
1
∂u 2 ) dx ≤ 0, ∀ Z ∈ D(A) ∂t 0 (
w1
AC
CE
y2 w2 ∈ X, there exists w = ∈ D(A) It is easy to verify that ∀ y = y3 w3 y4 w4 such that w − Aw = y. Then, D(A) is dense in X and A is closed. Hence,
110
using the Lumer-Phillips theorem (see [32]) A is the infinitesimal generator of a strongly continuous group of isometries S(t), t ≥ 0, on X. Now, we are going to prove the existence and uniqueness of the system (11) with H(Z) and f (t) are different from zero. 9
ACCEPTED MANUSCRIPT
115
Theorem 2. Let f ∈ L1 ([0, T ], X) and Z0 ∈ D(A), then the problem 11 has a unique solution
given by Z(t) = S(t)Z(0) +
Z
\
C 0 ([0, T ], D(A))
CR IP T
Z ∈ C 1 ([0, T ], X)
t
S(t − s)F (Z(s)) + S(t − s)f (s)ds
0
To prove Theorem 2, we need the next lemmas:
120
AN US
Lemma 1. The nonlinear operator H(Z) is locally Lipschitz and dissipative. Proof . Recall that the nonlinear function F given by
µ(
1 L
r
M
GJ ∂u(t,0) I ∂t )
is as
GJ ∂u(t, 0) ) I ∂t
ED
q
due to the bit-rock contact
r L 1 GJ ∂u(t, 0) = − T , GJ L I ∂t r r r 1 ∂u(t, 0) L 1 GJ ∂u(t, 0) 1 GJ ∂u(t, 0) = −cb − Wob Rb µ( )sgn( ) GJI ∂t GJ L I ∂t L I ∂t
∂u(t, 0) F ∂t
where µ( L1
∂u(t,0) ∂t
=
γ
µcb + (µsb − µcb )e
1 − υb | L f
√ GJ ∂u(t,0) I
∂t
|
PT
After computing
AC
CE
hH(Z(t)), Z(t)iX
∂u(t, 0) ∂u(t, 0) ) ∂t r∂t 1 ∂u(t, 0) 2 ( ) = −cb GJI ∂t √ GJ ∂u(t,0) γ 1 L − υb | L | I ∂t f − Wob Rb µcb + (µsb − µcb )e GJ r 1 GJ ∂u(t, 0) ∂u(t, 0) × sgn( ) L I ∂t ∂t = F(
Consequently, hH(Z(t)), Z(t)iX ≤ 0 as µsb ≥ µcb . This implies that the operator
125
H(Z) is dissipative. It’s easy to verify that, H is locally Lipschitz. Accordingly, the operator H(Z) is locally Lipschitz and dissipative. 10
ACCEPTED MANUSCRIPT
Lemma 2. For any function f ∈ L1 ([0, T ], X) and any initial condition Z0 ∈ T D(A), the problem (11) has at most one solution in C 1 ([0, T ], X) C 0 ([0, T ], D(A)).
which satisfies the following system ˙ Z(t) = AZ(t) + H(Z(t)) Z(0) = 0
(12)
AN US
Since H(Z) and A are dissipative, we get
CR IP T
T Proof . Suppose Z1 and Z2 are two solutions of (11) in the class C 1 ([0, T ], X) C 0 ([0, T ], D(A)). T Then the difference Z = Z1 −Z2 is an element of C 1 ([0, T ], X) C 0 ([0, T ], D(A))
˙ hZ(t), Z(t)iX = hAZ(t), Z(t)iX + hH(Z(t)), Z(t)iX ≤ 0 130
Then
1 d k Z(t) kX ≤ 0 ⇒k Z(t) kX = 0 2 dt
then this one is unique.
M
Hence, problem (12) has a unique solution Z(t) = 0 for every Z0 ∈ D(A). Thus T proves that Z1 = Z2 and shows that (11) has a solution in C 1 ([0, T ], X) C 0 ([0, T ], D(A)),
135
ED
Proof (Theorem 2). By applying the two Lemmas given above and from results given in (Theorem 4.2 [33], [28, 29, 32]), it is easy to prove that our system
PT
(11) has a unique solution.
AC
CE
2.3. Drillstring modeling and control formulation q aL Let W (t) = cGJ Ω(t) − L1 GJ I ∂t u(t, 1) , we obtain ∂ 2 u(t, x) ∂t2 ∂u(t, 1) ∂x ∂ 2 u(t, 0) ∂t2
=
∂ 2 u(t, x) ∂u(t, x) −λ , x ∈ (0, 1) 2 ∂x ∂t
(13)
=
W (t)
(14)
=
a
∂u(t, 0) ∂u(t, 0) + aF ( ) ∂x ∂t
(15)
To linearize the boundary condition (15), we use the next form (as presented in [12]) u ¯(t, x) =
λwr 2 x − F (wr )x + wr t + u0 2 11
ACCEPTED MANUSCRIPT
∂u ¯(t,x) ∂t .
as a reference trajectory, such that wr = Then the system (13)-(15) becomes ∂ 2 u(t, x) ∂t2 ∂u(t, 1) ∂x ∂ 2 u(t, 0) ∂t2
where b =
∂F ∂w (wr )
and w(t) =
=
∂ 2 u(t, x) ∂u(t, x) −λ 2 ∂x ∂t
=
W (t)
=
a
(16)
CR IP T
140
(17)
∂u(t, 0) ∂u(t, 0) + ab ∂x ∂t
∂u(t,1) ∂t .
(18)
AN US
The local exponential stability of system (16)-(18) is easy to be obtained when we rewrite (16) in this form l r l ∂ ∂ = − λ ∂t ∂x r l 0 q ∂u(t,x) 1 with (r, l) = ( ∂u(t,x) , ) and λ = dL ∂x ∂t IGJ .
M
Introducing the Riemann invariants: y = r − l, z = r + l. Then the system (16)-(18) is transformed into the system (19)-(22)
PT
ED
∂y(t, x) ∂y(t, x) + = ∂t ∂x ∂z(t, x) ∂z(t, x) − = ∂t ∂x z(t, 1) = ∂z(t, 0) ∂y(t, 0) − = ∂t ∂t
λ (z(t, x) − y(t, x)) 2 λ − (z(t, x) − y(t, x)) 2 2W (t) − y(t, 1) a z(t, 0) + y(t, 0) + ab z(t, 0) − y(t, 0) .
(19) (20) (21) (22)
CE
Now, we are going to construct changes of variables for the 2 × 2 PDE states
AC
(19) -(22), which are facilitate our analysis. Hence, we introduce the notations λ
λ
φ(x) = e 2 x , β(x) = e− 2 x
and the new states v1 (t, x) = φ(x)y(t, x),
v2 (t, x) = β(x)z(t, x).
12
ACCEPTED MANUSCRIPT
Then the system (19)-(22) is transformed in the next form: =
0
=
v2 (t, 1)
=
∂v2 (t, 0) ∂v1 (t, 0) − ∂t ∂t 145
∂v1 (t, x) ∂v1 (t, x) λφ + − v2 (t, x) ∂t ∂x 2β ∂v2 (t, x) ∂v2 (t, x) λ β − − v1 (t, x) ∂t ∂x 2φ β(1) 2β(1)W (t) − v1 (t, 1) φ(1)
(23) (24)
CR IP T
0
(25)
= a(v2 (t, 0) + v1 (t, 0)) + ab(v2 (t, 0) − v1 (t, 0)) (26)
After transformation of the damped wave equation into a 2 × 2, first order, lin-
AN US
ear hyperbolic PDE, we propose a stabilizing boundary control scheme for the linearized system.
As suggested by the boundary conditions (25)-(26) for the drilling process, the objectives is to find a feedback control law W (t) that locally-exponentially sta150
bilizes the zero equilibrium of system (23)-(24).
M
3. BOUNDARY CONTROL DESIGN
This section shows the importance of backstepping transformations, well-
ED
posedness propriety of kernel equations, and the Lyapnov theory, providing a useful analysis for stability of oilwell drilling system. The backstepping tech155
niques is to eliminate the effects destabilizing terms that appear throughout the
PT
domain while the control is acting only from the boundary. The backstepping transformation as also known Volterra transformation has a lower triangular
CE
structure. The terms of instability in (23)-(25) are
λφ 2β v2
and
λβ 2 φ v1 ,
our main
objective is to eliminate these both terms to show the stablity of the target system.
AC
160
13
ACCEPTED MANUSCRIPT
3.1. Target system stability Our objectif is to design the control W (t) such that the transformed system (23)-(26) maps the next target system ∂χ(t, x) ∂x ∂ζ(t, x) = ∂x = 0
CR IP T
∂χ(t, x) ∂t ∂ζ(t, x) ∂t ζ(t, 1) ∂ζ(t, 0) ∂χ(t, 0) − ∂t ∂t
= −
(27) (28)
(29) = a ζ(t, 0) + χ(t, 0) − ab2 ζ(t, 0) − χ(t, 0) . (30)
ζ(0, x) = ζ0 (x), we get
ζ(t, x) =
AN US
Computing ζ by the method of characteristics such that the initial condition ζ0 (t + x) 0
0 ≤ t + x < 1, t + x ≥ 1.
The necessary relations between (χ, ζ) and (v1 , v2 ) will be identified in subsection 3.2.
M
165
Theorem 3. Consider system (27)-(30) with initial conditions (χ0 , ζ0 ) ∈ L2 ([0, 1]).
ED
Then the zero equilibrium of (27)-(30) is locally exponentially stable in the L2 sense. Further ζ go to zero in finite time.
PT
Proof . Let us introduce the next Lyapunov function Z 1 2 V (t) = M e−ηx χ2 (t, x) + N eηx ζ 2 (t, x)dx + ζ(t, 0) − χ(t, 0) ,
CE
0
where M , N and η are positive constants. N ≥ 2a ≥ M .
Differentiating V with respect to time, integrating by parts and using the bound-
AC
170
Choosing M and N such that
14
ACCEPTED MANUSCRIPT
CR IP T
ary conditions (29)-(30), we get Z 1 ∂χ ∂ζ ∂ζ ∂χ ˙ 2M χ e−ηx + 2N ζ eηx dx + 2( (t, 0) − V (t) = (t, 0))(ζ(t, 0) − χ(t, 0)) ∂t ∂t ∂t ∂t 0 Z 1 ηM e−ηx χ2 (t, x)dx + [N eηx ζ 2 (t, x)]10 = −[M e−ηx χ2 (t, x)]10 − 0 h i + 2 a(ζ(t, 0) + χ(t, 0)) − ab2 (ζ(t, 0) − χ(t, 0)) (ζ(t, 0) − χ(t, 0)) Z 1 N eηx ζ 2 (t, x)dx − η 0
= −M e−η χ2 (t, 1) − (2a − M )χ2 (t, 0) − (N − 2a)ζ 2 (t, 0) ≤
− min(η, 2ab2 )V (t).
AN US
− 2ab2 (ζ(t, 0) − χ(t, 0))2 − ηV (t)
This proves local exponential stability of (χ, ζ). Finally from (28) and (29),
M
basically, the solution of ζ satisfies ζ0 (t + x) ζ(t, x) = 0
0 ≤ t + x < 1, t + x ≥ 1,
ED
which implies that ζ tends to zero in finite-time. 3.2. Backstepping techniques and control law In order, to convert the original system (23)-(26) into the target system
175
PT
(27)-(30), we consider the next backstepping transformation (more details about
AC
CE
backstepping transformation and kernel equations in [25]): Z x Z x χ(t, x) = v1 (t, x) − Auu (x, ξ)v1 (t, ξ)dξ − Auv (x, ξ)v2 (t, ξ)dξ 0
0
1 + (b + b2 )e−abx (v2 (t, 0) − v1 (t, 0)). 2 Z x Z vu ζ(t, x) = v2 (t, x) − A (x, ξ)v1 (t, ξ)dξ − 0
+
λφ(x) 2β(x) ,
1 (b + b2 )eabx (v2 (t, 0) − v1 (t, 0)). 2
ε2 (x) =
λ β(x) 2 φ(x) ,
Avv (x, ξ)v2 (t, ξ)dξ
0
Let consider the notations: ε1 (x) =
(31) x
λ is defined above,
15
(32) v1 (t, x) , E(t, x) = v2 (t, x)
180
χ(t, x) Auu (x, ξ) Auv (x, ξ) , A(x, ξ) = , δ(t, x) = ζ(t, x) Avu (x, ξ) Avv (x, ξ) 1 2 −abx (b + b )e (v (t, 0) − v (t, 0)) 2 1 . and Φ(t, x) = 2 1 2 abx (b + b )e (v (t, 0) − v (t, 0)) 2 1 2
CR IP T
ACCEPTED MANUSCRIPT
In compact form, using (31) and (32) we have Z x δ(t, x) = E(t, x) − A(x, ξ)E(t, ξ)dξ + Φ(t, x). 0
(33)
Plugging (31)-(32) into (27)-(28), integrating by parts and using the boundary conditions, we obtain the next kernel equations
AN US
185
∂Auu (x, ξ) ∂Auu (x, ξ) − − ε2 (ξ)Auv (x, ξ) = 0 ∂x ∂ξ ∂Auv (x, ξ) ∂Auv (x, ξ) + − ε1 (ξ)Auu (x, ξ) = 0 − ∂x ∂ξ ∂Avu (x, ξ) ∂Avu (x, ξ) − − ε2 (ξ)Avv (x, ξ) = 0 ∂x ∂ξ ∂Avv (x, ξ) ∂Avv (x, ξ) + − ε1 (ξ)Avu (x, ξ) = 0 ∂x ∂ξ
(34) (35) (36) (37)
M
−
with the next boundary conditions
ED
−Auv (x, 0) = =
Avu (x, x)
=
PT
Auv (x, x)
1 (b + b2 )ae−abx 2
1 ε1 (x) 2 1 − ε2 (x) 2 Avu (x, 0) =
(38) (39) (40)
1 (b + b2 )aeabx . 2
(41)
CE
−Avv (x, 0) =
Auu (x, 0) =
The kernel of the backstepping transformation also known as Volterra transformation satisfies an interesting system of PDE which is easily solvable. Then,
AC
this equations are defined on the triangular domain ∆ = {(x, ξ) ∈ R2 : 0 ≤ ξ ≤
190
x ≤ 1}. By Theorem 5 (see the Appendix), we show the existence, uniqueness
and continuity of the solution for system (34)-(37) with boundary conditions (38)-(41). In order, to ensure that the closed-loop and the target systems have equivalent stability properties, the backstepping transformations (31) and (32) have to be 16
ACCEPTED MANUSCRIPT
195
invertible. Then, we study the invertibility of the transformation (31) and (32), we look for a transformation of the target system (27)- (28) into the closed-loop
x
Lχχ (x, ξ)χ(t, ξ)dξ +
v2 (t, x)
b + b2 −abx e (ζ(t, 0) − χ(t, 0)). 2 Z x
Lζχ (x, ξ)χ(t, ξ)dξ +
= ζ(t, x) +
0
b + b2 abx e (ζ(t, 0) − χ(t, 0)). 2
Denoting 200
Lχζ (x, ξ)ζ(t, ξ)dξ
(42)
Z
0
x
Lζζ (x, ξ)ζ(t, ξ)dξ
(43)
AN US
−
x
0
0
−
Z
CR IP T
system (23) - (26) as follows Z v1 (t, x) = χ(t, x) +
Lχχ (x, ξ) Lχζ (x, ξ) − 12 (b + b2 )e−abx (ζ(t, 0) − χ(t, 0)) and Λ(t, x) = . L(x, ξ) = Lζχ (x, ξ) Lζζ (x, ξ) − 21 (b + b2 )eabx (ζ(t, 0) − χ(t, 0)) We rewritten (42)-(43) in this form Z x E(t, x) = δ(t, x) + L(x, ξ)δ(t, ξ)dξ + Λ(t, x). (44)
M
0
Plugging (42)-(43) into (23)-(26), integrating by parts and using the boundary conditions we find the next kernel equations:
CE
PT
ED
∂Lχχ (x, ξ) ∂Lχχ (x, ξ) + − ε1 (x)Lζχ (x, ξ) = 0 ∂ξ ∂x ∂Lχζ (x, ξ) ∂Lχζ (x, ξ) − + − ε1 (x)Lζζ (x, ξ) = 0 ∂ξ ∂x ∂Lζχ (x, ξ) ∂Lζχ (x, ξ) − − ε2 (x)Lχχ (x, ξ) = 0 ∂ξ ∂x ∂Lζζ (x, ξ) ∂Lζζ (x, ξ) − − ε2 (x)Lχζ (x, ξ) = 0 − ∂ξ ∂x
(45) (46) (47) (48)
AC
with the next boundary conditions Lχχ (x, 0) = Lχζ (x, x) = Lζχ (x, x) = Lζζ (x, 0) =
Lχζ (x, 0) + a(b + b2 )e−abx ε1 (x) 2 ε2 (x) − 2 Lζχ (x, 0) − a(b + b2 )eabx , 17
(49) (50) (51) (52)
ACCEPTED MANUSCRIPT
To prove existence, uniqueness and continuity of the system (45)-(48), with 205
boundary conditions (49)-(52) we use the Theorem 5 (see the Appendix). By plugging the transformation (32) into (29), the control law is obtained. So,
ζ(t, 1)
= v2 (t, 1) − +
Z
0
1
Avu (1, ξ)v1 (t, ξ)dξ −
1 (b + b2 )eab (v2 (t, 0) − v1 (t, 0)) = 0. 2
Hence
+
0
1
Avv (1, ξ)v2 (t, ξ)dξ
Z 1 Z 1 β(1) v1 (t, 1) − Avu (1, ξ)v1 (t, ξ)dξ − Avv (1, ξ)v2 (t, ξ)dξ φ(1) 0 0 1 (b + b2 )eab (v2 (t, 0) − v1 (t, 0)) = 0. 2
AN US
2β(1)W (t) −
Z
CR IP T
plugging (32) into (29), and using (25) we get
M
From this equality, the control law can be easily computed, which is given by: Z Z 1 1h 1 1 1 vu W (t) = v1 (t, 1) + A (1, ξ)v1 (t, ξ)dξ + Avv (1, ξ)v2 (t, ξ)dξ 2 φ(1) β(1) 0 0 i 1 (b + b2 )eab (v2 (t, 0) − v1 (t, 0)) . (53) − 2
ED
210
Theorem 4. Consider system (23)-(26), with initial conditions (v10 , v20 ) and the control law (53) where the kernels Avu and Avv are obtained from (34)-(41)
PT
with Λ(t, x), Φ(t, x) ∈ L2 ([0, 1]). Then the equilibrium (v1 , v2 ) = (0, 0) is locally
exponentially stable in the L2 sense. Further, the equilibrium v2 vanishes to zero in finite time.
CE
215
AC
Proof . Recall that E(t, x) = δ(t, x) +
Z
x
L(x, ξ)δ(t, ξ)dξ + Λ(t, x),
0
and δ(t, x)
=
E(t, x) −
Z
x
A(x, ξ)E(t, ξ)dξ + Φ(t, x).
0
18
ACCEPTED MANUSCRIPT
By Theorem 5 (see the Appendix) the kernel A and L are continuous. Consequently, Z . L(., ξ)δ(t, ξ)dξ ||δ(t, .)||L2 ([0,1]) + kΛ(t, .)kL2 ([0,1]) +
≤
0
≤
||δ(t, .)||L2 ([0,1]) + kΛ(t, .)kL2 ([0,1]) + |||L|||∞ ||δ(t, .)||L2 ([0,1])
≤ C1 (t)||δ(t, .)||L2 ([0,1])
where C1 (t) = Besides
n kΛ(t,.)k
L2 ([0,1])
||δ(t,.)||L2 ([0,1])
||δ(t, .)||L2 ([0,1])
o + 1 + |||L|||∞ .
Z . ||E(t, .)||L2 ([0,1]) + kΦ(t, .)kL2 ([0,1]) + A(., ξ)E(t, ξ)dξ
≤
AN US
220
L2 ([0,1])
CR IP T
||E(t, .)||L2 ([0,1])
0
≤
||E(t, .)||L2 ([0,1]) + kΦ(t, .)kL2 ([0,1]) + |||A|||∞ ||E(t, .)||L2 ([0,1])
≤ C2 (t)||E(t, .)||L2 ([0,1])
where C2 (t) = If t = 0 then
n kΦ(t,.)k
L2 ([0,1])
||E(t,.)||L2 ([0,1])
o + 1 + |||A|||∞ .
M
||δ(0, .)||L2 ([0,1]) ≤ C2 (0)||E(0, .)||L2 ([0,1]) , n kΦ(0,.)k o 2 such that C2 (0) = ||E(0,.)||L 2([0,1]) + 1 + |||A|||∞ . L ([0,1])
ED
By Theorem 3, there exists c > 0 and κ > 0 such that
||δ(t, .)||L2 ([0,1]) ≤ c||δ(0, .)||L2 ([0,1]) e−κt .
PT
Hence, there exists a constant C > 0 and κ > 0 such that
CE
||E(t, .)||L2 ([0,1]) ≤ C||E(0, .)||L2 ([0,1]) e−κt .
It is straightforward E(t, x) = (v1 (t, x), v2 (t, x)) tends local exponentially to zero. Finally, we know that as ζ go to zero in finite time, then also v2 go to zero in
AC
finite time. This ends the proof.
225
L2 ([0,1])
4. SIMULATION Controlling drillstring vibrations requires a good understanding of the bottom hole assembly (BHA) dynamics and its interactions with the drillstring. 19
ACCEPTED MANUSCRIPT
The drillstring vibrations can be self-excited and induced by nonlinear dynamic interactions between drilled formation and drill bit, and between drillstring 230
and borehole which are often characterized by erratic patterns and behavior.
CR IP T
Uncontrolled vibrations can decrease the rate of penetration, and accordingly increase the well cost. Furthermore, excessive oscillations can interfere with
measurement while drilling tools or even cause their damage. They can also cause a significant waste of drilling energy. Vibrations often induce well bore 235
instabilities that can worsen the condition of the well and reduce the directional control. In the last years, the dynamics of drilling has been studied numerically
AN US
and experimentally but in most researches local problems, for example, torsional vibrations or stick-slip have been focussed [34].
In this section, simulation results of the torsional vibrations model are presented. 240
The strategy developed in section 2 and 3 is applied to find the synthesis of a stabilizing controller ensuring the suppression of the torsional vibrations and consequently the reduction of the bit bouncing phenomenon. As the drillstring
M
system (1)-(3) is equivalent to system of study (16)-(19), it is important to test the effectiveness of the boundary control law for the local exponential stability
ED
of system (16)-(19). The next physical parameters are used in simulation: Variable
Value
Description
L
2000 m
Drillstring length
PT
245
I
0.095 kg.m
Inertia per unit length
Ib
311 kg.m2
Inertia at the drillstring bottom 4
1.19.10 m
ca
2000 N m.s.rad−1
CE
J
d
AC
5
0.09 kg.m.s
−1
Geometrical moment of inertia Sliding torque coefficient Drillstring damping
Table 1: Different physical parameters
As shown by figures 2 and 3, the states v1 (t, x) and v2 (t, x) reach zero local.
20
AN US
CR IP T
ACCEPTED MANUSCRIPT
AC
CE
PT
ED
M
Figure 2: Stabilization of the state v1 (t, x).
Figure 3: Stabilization of the state v2 (t, x).
21
ACCEPTED MANUSCRIPT
The simulation results of (Fig. 2)-(Fig. 3) show that the stick-slip vibrations are reduced by means of the application of the control law (53) depending on the angular velocity at the bottom extremity.
CR IP T
250
Conclusion
In this paper, the oilwell drilling system described by a damped wave equa-
tion was transformed into 2 × 2 first-order linear hyperbolic PDEs. The gen-
erally neglected damping term is considered here, and the boundary conditions 255
are defined for the two first-order hyperbolic PDEs. Basically this makes our
AN US
stabilizing problem different. The fundamental tools such that the backstepping technique, the well-posedness propriety of the kernel equations, and the Lyapunov theory were shown their importance in the transformation procedure and
the control law design. The oilwell drilling system is shown locally exponentially 260
stable from the stick-slip suppression. Future investigations should focus on the
M
performances of the controllers. It is important to evaluate their robustness to changes in operating conditions, such as variations of the weight-on-bit Wob . This, along with the design of an observer for the backstepping controller, will
APPENDIX
PT
265
ED
be topic of future works.
The studying of the existence, uniqueness and continuity of the solution
CE
to systems (34)-(37), (45)-(48), with boundary conditions respectively (38)(41), (49)-(52), are similar. Because both systems have similar structures. We consider the ”generalized Goursat” problem in which the direct kernel (34)(37) with boundary conditions (38)-(41) and the inverse kernel equations (38)-
AC
270
(41) with boundary conditions, (49)-(52), are a particular case. Defining for
22
ACCEPTED MANUSCRIPT
i = 1, .., 4 the next system equations: ∂G1 (x, ξ) ∂G1 (x, ξ) + c1 (ξ) ∂x ∂ξ 2
c1 (x) c2 (x)
4 X
ε1i (x, ξ)Gi (x, ξ) + g1 (x, ξ), (54)
i=1
2
∂G (x, ξ) ∂G (x, ξ) − c2 (ξ) ∂x ∂ξ
=
∂G3 (x, ξ) ∂G3 (x, ξ) − c1 (ξ) ∂x ∂ξ
=
4 X i=1
4 X i=1
4
4
c2 (x)
=
∂G (x, ξ) ∂G (x, ξ) + c2 (ξ) ∂x ∂ξ
=
4 X i=1
ε3i (x, ξ)Gi (x, ξ) + g3 (x, ξ) (56)
ε4i (x, ξ)Gi (x, ξ) + g4 (x, ξ), (57)
AN US
with the next boundary conditions
ε2i (x, ξ)Gi (x, ξ) + g2 (x, ξ), (55)
CR IP T
c1 (x)
G1 (x, 0)
= q1 (x)G2 (x, 0) + h1 (x),
(58)
G2 (x, x)
= h2 (x),
(59)
G3 (x, x)
= h3 (x),
(60)
G4 (x, 0)
=
q2 (x)G3 (x, 0) + h4 (x).
(61)
275
M
This system is defined on the triangular domain ∆ = {(x, ξ) : 0 ≤ ξ ≤ x ≤ 1}. We use the method of characteristics to prove the existence and uniqueness of
ED
the equations. Hence, to prove the existence and uniqueness we prefer to use the next theorem:
Theorem 5. [35] Consider system (54)-(57), with boundary conditions (58)C N ([0, 1]). Then there exists a unique C N (∆) solution Gi for i = 1, 2, 3, 4.
CE
280
PT
(61) such that hi , qi ∈ C N ([0, 1]), gi , εij ∈ C N (∆), i, j = 1, 2, 3, 4 and c1 , c2 ∈
References
AC
References
[1] J. D. Jansen, Nonlinear dynamics of oilwell drillstrings, Delft University of
285
Technology, 1993. doi:10.1007/s10596-016-9576-1.44.
[2] J. J. Bailey, I. Finnie, An analytical study of drill-string vibration, Journal of Engineering for Industry 82 (2) (1960) 122–127. doi:10.1115/1. 3663017. 23
ACCEPTED MANUSCRIPT
[3] S. Toumi, L. Beji, R. Mlayeh, A. Abichou, Stability analysis of coupled torsional vibration and pressure in oilwell drillstring system, International Journal of Control (2017)doi:10.1080/00207179.2016.1278269.
290
CR IP T
[4] S. Dwars, Recent advances in soft torque rotary systems, Society of Petroleum Engineers (2015)doi:http://dx.doi.org/10.2118/ 173037-MS.
[5] X. Liu, N. Vlajic, X. Long, G. Meng, B. Balachandran, State-dependent
delay influenced drill-string oscillations and stability analysis, Journal of
295
AN US
Vibration and Acoustics 136 (5) (2014) 051008. doi:10.1115/1.4027958. [6] B. Saldivar, S. Mondi´e, J. J. Loiseau, V. Rasvan, Stick-slip oscillations in
oillwell drilstrings: distributed parameter and neutral type retarded model approaches, IFAC Proceedings Volumes 44 (1) (2011) 284–289. doi:10. 3182/20110828-6-IT-1002.00084.
300
M
[7] B. Saldivar, T. Kn¨ uppel, F. Woittennek, I. Boussaada, H. Mounier, S.I. Niculescu, Flatness-based control of torsional-axial coupled drilling vidoi:
ED
brations, IFAC Proceedings Volumes 47 (3) (2014) 7324–7329. 10.3182/20140824-6-ZA-1003.02205. 305
[8] S. Toumi, L. Beji, R. Mlayeh, A. Abichou, Stabilization of stick-slip os-
PT
cillations integrating fluid injection in oilwell drillstring system, European Control Conference (2016) 352–357doi:10.1109/ECC.2016.7810310.
CE
[9] B. Saldivar, I. Boussaada, H. Mounier, S. Niculescu, Analysis and Control of Oilwell Drilling Vibrations, A Time-Delay Systems Approach, Advances in Industrial Control. Springer, New York, 2015. doi:10.1007/
AC
310
978-3-319-15747-4.
[10] S. Toumi, R. Mlayeh, L. Beji, A. Abichou, Stability analysis of oilwell drilling torsional vibrations, Control and Automation (MED), 24th Mediterranean Conference (2016) 677–682doi:10.1109/MED.2016.
315
7535945. 24
ACCEPTED MANUSCRIPT
[11] T. Vromen, Control of stick-slip vibrations in drilling systems Eindhoven, Technische Universiteit Eindhoven, 2015. [12] C. Sagert, F. D. Meglio, M. Krstic, P. Rouchon, Backstepping and
CR IP T
flatness approaches for stabilization of the stick-slip phenomenon for
drilling, IFAC Proceedings Volumes 46 (2) (2013) 779–784. doi:10.3182/
320
20130204-3-FR-2033.00126.
[13] X. Liu, N. Vlajic, X. Long, G. Meng, B. Balachandran, Coupled axialtorsional dynamics in rotary drilling with state dependent delay: Stability
AN US
and control, Nonlinear Dynamics 78 (3) (2014) 1891–1906. doi:10.1007/ s11071-014-1567-y.
325
[14] X. Liu, N. Vlajic, X. Long, G. Meng, Nonlinear oscillations of a flexible rotor with a drill bit: Stick-slip and delay effects, Nonlinear Dynamics 72 (1-2) (2013) 61–77. doi:10.1007/s11071-012-0690-x.
M
[15] E. Kreuzer, L. Krumm, M.-A. Pick, E. Solowjow, M. Steidl, Active vibration isolation via decomposition of traveling waves, 11th International Con-
330
ED
ference on Vibration Problems, Lisbon, Portugal (9–12 September 2013). [16] P. Rouchon, Flatness and stick-slip stabilization, Technical Report 492. France: Ecole des Mines de Paris (1998).
PT
[17] S. Toumi, L. Beji, R. Mlayeh, A. Abichou, Boundary observer design for hyperbolic pde in rotary drilling systems, IEEE 55th Conference on
335
CE
Decision and Control (CDC), Las Vegas, USA (2016) 2128–2133doi: 10.1109/CDC.2016.7798578.
AC
[18] A. Smyshlyaev, E. Cerpa, M. Krstic, Boundary stabilization of a 1-d wave
340
equation with in-domain antidamping, SIAM Journal on Control and Optimization 48 (6) (2010) 4014–4031. doi:10.1137/080742646.
[19] A. Diagne, G. Bastin, J. M. Coron, Lyapunov exponential stability of linear hyperbolic systems of balance laws, IFAC Proceedings Volumes 44 (1) (2011) 13320–13325. doi:10.3182/20110828-6-IT-1002.01506. 25
ACCEPTED MANUSCRIPT
[20] F. D. Meglio, R. Vazquez, M. Krstic, N. Petit, Backstepping stabilization of an underactuated 3× 3 linear hyperbolic system of fluid flow equations,
345
American Control Conference (ACC) (2012) 3365–3370doi:10.1.1.400.
CR IP T
2886. [21] J. M. Coron, R. Vazquez, M. Krstic, G. Bastin, Local exponential h2
stabilization of a 2×2 quasilinear hyperbolic system using backstepping, SIAM Journal on Control and Optimization 51 (3) (2013) 2005–2035.
350
doi:10.1137/120875739.
AN US
[22] C.-Z. Xu, G. Sallet, Exponential stability and transfer functions of processes
governed by symmetric hyperbolic systems, ESAIM: Control, Optimisation and Calculus of Variations 7 (2002) 421–442. doi:10.1051/cocv:2002062. 355
[23] J. Rauch, M. E. Taylor, Exponential decay of solutions to hyperbolic equations in bounded domains, Indiana university Mathematics journal 21 (1)
M
(1974) 79–86. doi:238228429.
[24] F. D. Meglio, R. Vazquez, M. Krstic, Stabilization of a system of cou-
ED
pled first-order hyperbolic linear pdes with a single boundary input, IEEE Transactions on Automatic Control 58 (12) (2013) 3097–3111.
360
doi:
10.1109/TAC.2013.2274723.
PT
[25] M. Krstic, A. Smyshlyaev, Backstepping boundary control for first-order hyperbolic pdes and application to systems with actuator and sensor de-
CE
lays, Systems & Control Letters 57 (9) (2008) 750–758. doi:10.1016/j. sysconle.2008.02.005.
365
AC
[26] J. M. Coron, B. d’Andrea Novel, G. Bastin, A strict lyapunov function
370
for boundary control of hyperbolic systems of conservation laws, IEEE transactions on automatic control 52 (1) (2007) 2–11. doi:10.1109/TAC. 2006.887903.
[27] R. Vazquez, M. Krstic, J. M. Coron, Backstepping boundary stabiliza-
26
ACCEPTED MANUSCRIPT
tion and state estimation of a 2× 2 linear hyperbolic system (2011) 4937– 4942doi:10.1109/CDC.2011.6160338. [28] T. D. Nguyen, O. Egeland, Stabilization of towed cables, Decision and
CR IP T
Control, 2004. CDC. 43rd IEEE Conference on 5 (2004) 5059–5064. doi: 10.1109/CDC.2004.1429609.
375
[29] A. Pazy, Semigroups of linear operators and applications to partial differential equations, Vol. 44, Springer New York, 1983. 978-1-4612-5561-1.
doi:10.1007/
AN US
[30] N. Challamel, Rock destruction effect on the stability of a drilling structure,
Journal of Sound and Vibration 233 (2) (2000) 235–254. doi:10.1006/
380
jsvi.1999.2811.
[31] E. Navarro-L´ opez, R. Su´ arez, Practical approach to modelling and controlling stick-slip oscillations in oilwell drillstrings, Control Applications. Pro-
M
ceedings of the 2004 IEEE International Conference 2 (2004) 1454–1460. doi:10.1109/CCA.2004.1387580.
385
ED
[32] J. M. Coron, Control and nonlinearity, no. 136, American Mathematical Soc, 2007. doi:1098765432112111009.
PT
[33] M. Slemrod, Feedback stabilization of a linear control system in hilbert space with ana priori bounded control, Mathematics of Control, Signals and Systems 2 (3) (1989) 265–285. doi:10.1007/BF02551387.
CE
390
[34] N. Mihajlovic, A. van Veggel, N. van de Wouw, H. Nijmeijer, Analysis of
AC
friction induced limit cycling in an experimental drill-string system, Journal
395
of dynamic systems, measurement, and control 126 (4) (2004) 709. doi: 10.1115/1.1850535.
[35] R. Vazquez, J. M. Coron, M. Krstic, G. Bastin, Local exponential h2 stabilization of a 2× 2 quasilinear hyperbolic system using backstepping (2011) 1329–1334doi:10.1109/CDC.2011.6161075.
27
Stabilization of torsional vibration in oilwell drillstring system Elsevier, Samir Toumi, Lotfi Beji, Rhouma Mlayeh, Azgal Abichou PII: DOI: Reference:
S0947-3580(16)30061-9 10.1016/j.ejcon.2017.03.002 EJCON 200
To appear in:
European Journal of Control
Received date: Revised date: Accepted date:
12 July 2016 28 January 2017 14 March 2017
Please cite this article as: Elsevier, Samir Toumi, Lotfi Beji, Rhouma Mlayeh, Azgal Abichou, Stabilization of torsional vibration in oilwell drillstring system, European Journal of Control (2017), doi: 10.1016/j.ejcon.2017.03.002
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT
Stabilization of torsional vibration in oilwell drillstring system Radarweg 29, Amsterdam
CR IP T
Elsevier1 Samir Toumia,b,∗, Lotfi Bejia , Rhouma Mlayehb , Azgal Abichoub a are
AN US
with the IBISC-EA 4526 laboratory, University of Evry, 40 rue du Pelvoux, 91020, Evry, France. b are with the LIM laboratory, Polytechnic School of Tunisia, BP 743, 2078 La Marsa, Tunisia.
Abstract
Stick-slip oscillations also known as torsional vibrations appearing in oilwell drilling systems are a source of economic losses, borehole disruption, pipes disconnection, and prolonged drilling time. The torsional dynamics is modeled by
M
a damped wave equation. An important stability issue is to find a control law that reject perturbations due to torsional vibrations. Hence, the primary aim of this paper is to prove the well-posedness of the damped wave equation. In the
ED
next step, using the Riemann invariants, the damped wave equation is transformed into 2 × 2 first-order transport equations. The backstepping techniques
PT
combined with kernel equations and the Lyapunov theory are used to prove the local exponential stability of the transformed system, consequently the torsional dynamic. Simulation results are presented to illustrate the effectiveness of the
CE
control law.
Keywords: torsional vibrations, stability, drilling, partial
AC
differential equation
✩ Fully
documented templates are available in the elsarticle package on CTAN. author Email address: [email protected] (Samir Toumi ) 1 Since 1880.
∗ Corresponding
Preprint submitted to Journal of LATEX Templates
March 17, 2017
ACCEPTED MANUSCRIPT
1. Introduction For many years, the control and the model of drilling systems has been a very active research field. Vibrations dynamics on the drillstring system consti-
5
CR IP T
tute an important source of failures which affect penetration rates and increase drilling process costs, borehole disruption, pipes disconnection, and prolonged drilling time, are only some examples of results associated with drilling vibra-
tions. The process of drilling for oil is the creation of drilling over five thousand
feet deep in the soil until it reaches the gas and oil. During drilling operations, there exits three main types of vibrations caused by the drillstring [1]:
lateral vibrations (whirl motion due to the out-of-balance of the drillstring),
AN US
10
axial vibrations (bit bouncing phenomenon) and torsional vibrations (stick-slip phenomena). A detailed description of each mode of vibration is presented in [1].
Before the sixties, studies were focused on material strength of the drillstring components, however the trends have since changed to emphasize on its dynamic
M
15
behavior [1]. Since by the work of Bailey and Finnie of Shell Development Company in 1960 which are the first developed analytical and experimental study on
ED
torsional and axial drilling vibrations [2], several investigations on the modeling and control of drilling systems has been conducted [3, 4, 5, 6, 7, 8, 9, 10, 11]. In [6], Saldivar et al. are proposed energy function for the torsional distributed
PT
20
model allows to find a control law that ensures the energy dissipation during the drilling. Sagert et al. in [12] are introduced control law to avoid undesirable
CE
torsional oscillations of the drillstring in oilwell drilling systems using backstepping and flatness approaches. In [7], the authors are treated the control of
25
torsional-axial coupled drilling vibrations using flatness approach.
AC
The range of stability for the coupled axial-torsional motions in the presence of state-dependent delay as well as to investigate the effectiveness of a feedback controller to suppress torsional vibrations have been studied in [13]. Also, in [14], Liu et al. have been presented an original model considering time-delay
30
and stick-slip effects. In [15], the authors present a model-free active control
2
ACCEPTED MANUSCRIPT
method for the vibration isolation of a rigid item that is elastically mounted to a slender rod which is subject to axial vibrations. A modeling for control of torsional vibrations, a novel controller design strategies for drillstring sys-
35
CR IP T
tems, an experimental validation of the proposed controllers, and a modeling and analysis for the assessment of a downhole anti stick slip tool are presented in [11].
This paper tackle the torsional vibrations in oilwell drilling system. The dy-
namic of the torsional vibrations is modeled by a damped wave equation which encountered in drilling system dynamics that describe the torsional, axial, and
lateral vibrations. The control law in terms of angular velocities and axial-
AN US
40
lateral forces appears in the boundary conditions. Several research in drillstring dynamic analysis are studied in neglected the damping terms [16]. A long the infinite drillstring dynamics, the damping term is taking into account in our subsequent analysis. 45
The primary aim of this work is to prove the well posedness of our partial differ-
M
ential equation (PDE) which describes the torsional vibration. After this step, we are going to prove the stability of our system. In fact, to show the stability of
ED
PDEs in finite dimension there are two main techniques: from the exact solution if this solution is easy to be obtained or in general using the Lyapunov theory. 50
An exact solution can be obtained using the variable separation method or the
PT
Laplace transform technique. In the literature, the study of stability of hyperbolic PDEs is well described (see [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28]).
CE
The paper is organized as follows. The contribution in Section 2 is to show the well-posedness of torsional vibration system. In this section, we recall the PDE
55
with the boundary conditions that permit to describe the torsional vibration
AC
problem. Some useful formulation in terms of variable change and Riemann’s invariants are introduced. In Section 3, we find the control law using backstepping transformation, kernel equations, and a target system formulation. In this last section, the boundary control objectives are defined with the drillstring tor-
60
sional dynamic stabilizing problem. The simulation results are given in Section 4. Some concluding remarks and perspectives are also introduced. 3
ACCEPTED MANUSCRIPT
2. Controlled torsional system
||δ(t, .)||L2 ([0,1]) =
s
Z
1
χ(t, x)2 + ζ(t, x)2 dx
0
the norm of δ, and by Z
1
v1 (t, x)2 + v2 (t, x)2 dx
0
AN US
||E(t, .)||L2 ([0,1]) =
s
CR IP T
Firstly,we introduce some notations and norms used in the paper. For v1 (t, x) χ(t, x) and δ(t, x) = , we denote by E(t, x) = v2 (t, x) ζ(t, x)
the norm of E, and ∆ = {(x, ξ) ∈ R2 : 0 ≤ ξ ≤ x ≤ 1}. The norm ||A(., .)||∞ = max(x,ξ)∈∆ k|A(x, ξ)k|22 with k|A(., .)k|22 = supY ∈Rn ,kY k=1 kA(., .)Y k2 is the clas-
65
sical operator norm.
The key challenge in this section, is to study the well-possedness of the tor-
operator theory [29].
ED
2.1. Problem statement
M
sional drilling system in oil-gas industry. To achieve our objectives, we use the
During the drilling operation an excessive vibrations may arise and have a
70
major effect on drilling performance. The principle kinds of drilling vibration in
PT
oilwell drilling system is the stick-slip oscillations which may affect the rotation direction and the penetration of the drill bit as well as the damage of the drilling
CE
system. The rotary drilling system is known in oilwell drilling processes. Due to 75
interaction bit-rock, the acting torque in a circular shaft produces angular de-
AC
formation along the drill pipe, and a distributer parameter model can be derived.
4
AN US
CR IP T
ACCEPTED MANUSCRIPT
M
Figure 1: Drilling system
The fundamentals of mechanical engineering, as the principal of angular
ED
momentum, the geometry of the drillstring, and the fact that the string only rotates around one fixed ς-axis, the governed differential equations of motion is
PT
derived for moments balance. As shown in (Fig. 1), ∂2ϑ (t, ς) + dτe + τ − (τ + dτ ) = 0 ∂t2 ∂ ∂ϑ where τ = GJ ∂ϑ ∂ς (t, ς), dτ = ∂ς GJ ∂ς (t, ς) dς, and a viscous friction term
CE
Idς
around the drillpipe elementary direction dς is added dτe = ddς ∂ϑ ∂t (t, ς). Conse-
AC
quently, the drillstring satisfies the following distributed model with respect to the torsional variable [9, 30, 3] , GJ
∂2ϑ ∂2ϑ ∂ϑ (t, ς) − I 2 (t, ς) − d (t, ς) = 0 2 ∂ς ∂t ∂t
(1)
with ς ∈ (0, L), t ∈ (0, +∞). Now, let us examine the boundary conditions and their corresponding surfaces 5
ACCEPTED MANUSCRIPT
(we can see [9, 6]). A top boundary condition is defined that integrates the applied velocity at the surface Ω(t), the fact that a local angular twist conducts to a velocity different, and the distributed parameter ϑ defined above, then at
GJ
∂ϑ ∂ϑ (t, 0) = ca ( (t, 0) − Ω(t)) ∂ς ∂t
CR IP T
the top (ς = 0) (2)
At the bottom or tip, i.e. (ς = L), it is important to take into account the friction caused by the bit-rock interaction which depends on the torsional angle
rate ϑ at ς = L, and integrates the bit moment of inertia, then, one may define the following at the tip
∂ϑ ∂2ϑ ∂ϑ (t, L) + Ib 2 (t, L) = −T ( (t, L)) ∂ς ∂t ∂t
AN US
GJ
(3)
where L is the drillstring length, ca the sliding torque coefficient, G the shear modulus, a lumped inertia Ib is chosen to represent the assembly at the bottom 80
hole, J the geometrical moment of inertia, d the drillstring damping, and Ω
M
the control input (angular velocity due to the rotary table). The inertia per unit length I is such that I = ρJ, where ρ is the area density. The following expression represent the model for the torque on the bit which is introduced in
∂ϑ ∂ϑ ∂ϑ ∂ϑ (t, L)) = cb (t, L) + Wob Rb µ( (t, L))sgn( (t, L)) ∂t ∂t ∂t ∂t
PT
T(
ED
[31], it allows to approximate the physical phenomenon at the bottom hole
∂ϑ ∂ϑ The both terms cb ∂ϑ ∂t (t, L) and Wob Rb µ( ∂t (t, L))sgn( ∂t (t, L)) represent respec-
tively, the viscous damping torque at the bit and the dry friction torque modeling
CE
the bit-rock contact. Wob > 0 is the weight on the bit and Rb > 0 is the bit
AC
radius. The bit dry friction µ is modeled as [6] µ(
γ ∂ϑ − b | ∂ϑ (t,L)| (t, L)) = µcb + (µsb − µcb )e υf ∂t ∂t
such that µsb ≥ µcb ∈ (0, 1) are the static and Coulomb friction coefficients,
0 < γb < 1 is a constant defining the velocity decrease rate, and the constant
velocity υf > 0 is introduced for the homogeneity of units. The following change of variable leads to an equivalent system, for which the
6
ACCEPTED MANUSCRIPT
function u(t, x) represent the angular position [8]: r I u(t, x) = ϑ(L t, L(1 − x)), x ∈ (0, 1), GJ
∂ 2 u(t, x) ∂t2 ∂u(t, 1) ∂t ∂ 2 u(t, 0) ∂t2 q 1 where λ = dL IGJ ,a=
= = =
CR IP T
we obtain
∂ 2 u(t, x) ∂u(t, x) −λ , x ∈ (0, 1) 2 ∂x ∂t r √ I I GJ ∂u(t, 1) L Ω(t) − GJ ca ∂x ∂u(t, 0) ∂u(t, 0) a + aF ( ) ∂x ∂t
LI Ib ,
(4)
(5)
(6)
AN US
85
r L 1 GJ ∂u(t, 0) T , GJ L I ∂t r r r 1 ∂u(t, 0) L 1 GJ ∂u(t, 0) 1 GJ ∂u(t, 0) = −cb − Wob Rb µ( )sgn( ) GJI ∂t GJ L I ∂t L I ∂t q ∂u(t,0) in which µ( L1 GJ I ∂t ) is as ∂u(t, 0) ∂t
r
GJ ∂u(t, 0) ) I ∂t
=
µcb + (µsb − µcb )e
γ
1 − υb | L f
√ GJ ∂u(t,0) I
∂t
|
ED
1 µ( L
= −
M
F
2.2. Well-posedness problem
Due to the presence of a nonlinear and complex relation resulting from the bit-rock interaction at the tip boundary, the well-posedness of the controlled
PT
90
torsional system becomes not trivial. Hence, in the following, one treats this
CE
issue using the semi-group theory (Further discussion in this theory in [29]).
AC
Let T > 0, the natural solution of the Cauchy problem is written in this form ∂ 2 u(t, x) = ∂t2 ∂u(t, 1) = ∂t ∂ 2 u(t, 0) = ∂t2 u(0, x) =
∂ 2 u(t, x) ∂u(t, x) −λ 2 ∂x ∂t r √ I I GJ ∂u(t, 1) L Ω(t) − GJ ca ∂x ∂u(t, 0) ∂u(t, 0) a + aF ( ) ∂x ∂t u0 (x), ut (0, x) = u1 (x)
7
(7) (8) (9) (10)
ACCEPTED MANUSCRIPT
where x ∈ (0, 1), t ∈ (0, T ), u0 ∈ K := {u ∈ H 1 (0, 1); u0 (0) = 0}, u1 ∈
95
L2 (0, 1), and Ω(t) ∈ R the control law.
It is obvious that K is a Hilbert space.
CR IP T
The vector space K is equipped with the following scalar product Z 1 1 ∂u (t, x) ∂u2 (t, x) hu1 (t, x), u2 (t, x)iK = dx ∂x ∂x 0
∂u(t,0) T Let us introduce Z = (u(t, x), ∂u(t,x) ∂t , u(t, 1), ∂t ) . Equations (7)-(10) can
Z(0) = Z0
A=
1
0
∂2 ∂x2
−λ
0
I GJ 0 ca hδ1 (x), .i
−ahδ00 (x), .i
0 0
0
0
0
0 0 , H(Z(t)) = 0 0 0 ∂u(t,0) aF ( ∂t ) 0 0
(11)
,
such as δ denotes the Dirac function for which
ED
0 q and f (t) = I L GJ Ω(t) 0
hδ10 (x), u(t, x)i = − ∂u(t,1) and hδ00 (x), u(t, x)i = − ∂u(t,0) ∂x ∂x .
PT
100
√
0
M
where
AN US
be compactly written as ˙ Z(t) = AZ(t) + H(Z(t)) + f (t)
At first, let us consider the problem (11) with H(Z) = 0 and f (t) = 0, accord-
CE
ingly we have the next Theorem. Theorem 1. The operator A generates a C0 semigroup S(t), t ≥ 0 of contrac-
AC
tions on X.
105
u(t, x)
∂u(t,x) ∂t Proof . Let X = { u(t, 1) ∂u(t,0) ∂t
R,
∂u(t,0) ∂t
∈ R}
, u(t, x) ∈ K,
8
∂u(t,x) ∂t
∈ L2 ([0, 1]), u(t, 1) ∈
ACCEPTED MANUSCRIPT
∂u (t,0) ∂t
CR IP T
This vector space X is equipped with the following inner-product u2 (t, x) u1 (t, x) 1 (t,x) ∂u2 (t,x) E D ∂u ∂t ∂u1 (t, x) ∂u2 (t, x) ∂t 1 2 , , iL2 [0,1] 2 X = hu (t, x), u (t, x)iK + h 1 ∂t ∂t u (t, 1) u (t, 1) 1 2 ∂u (t,0) ∂t
hu1 (t, 1), u2 (t, 1)iR + h
+
∂u1 (t, 0) ∂u2 (t, 0) , iR ∂t ∂t
We denote by k . k the norm in X associated to this scalar product.
AN US
Let A : D(A) ⊂ X →X be the linear operator defined by u(t, x) ∂u(t,x) ∂t ∈ X, u ∈ H 2 (0, 1), ∂u ∈ K, ∂u(t,1) = D(A) = { ∂t ∂x u(t, 1) ∂u(t,0) ∂t
∈ R} R, ∂u(t,0) ∂t
u(t, x)
∂u(t,0) ∂t
Moreover
=
∂u(t,x) ∂t ∂ 2 u(t,x) − λ ∂u(t,x) ∂x2 ∂t √ − IcaGJ ∂u(t,1) ∂x a ∂u(t,0) ∂x
ED
∂u(t,x) ∂t A u(t, 1)
M
We have
PT
hAZ, ZiX
= −λ
Z
y1
∂u(t,0) ∂x
u(t, x)
∂u(t,x) ∂t , ∀ u(t, 1) ∂u(t,0) ∂t
= 0, u(t, 1) ∈
∈ D(A)
1
∂u 2 ) dx ≤ 0, ∀ Z ∈ D(A) ∂t 0 (
w1
AC
CE
y2 w2 ∈ X, there exists w = ∈ D(A) It is easy to verify that ∀ y = y3 w3 y4 w4 such that w − Aw = y. Then, D(A) is dense in X and A is closed. Hence,
110
using the Lumer-Phillips theorem (see [32]) A is the infinitesimal generator of a strongly continuous group of isometries S(t), t ≥ 0, on X. Now, we are going to prove the existence and uniqueness of the system (11) with H(Z) and f (t) are different from zero. 9
ACCEPTED MANUSCRIPT
115
Theorem 2. Let f ∈ L1 ([0, T ], X) and Z0 ∈ D(A), then the problem 11 has a unique solution
given by Z(t) = S(t)Z(0) +
Z
\
C 0 ([0, T ], D(A))
CR IP T
Z ∈ C 1 ([0, T ], X)
t
S(t − s)F (Z(s)) + S(t − s)f (s)ds
0
To prove Theorem 2, we need the next lemmas:
120
AN US
Lemma 1. The nonlinear operator H(Z) is locally Lipschitz and dissipative. Proof . Recall that the nonlinear function F given by
µ(
1 L
r
M
GJ ∂u(t,0) I ∂t )
is as
GJ ∂u(t, 0) ) I ∂t
ED
q
due to the bit-rock contact
r L 1 GJ ∂u(t, 0) = − T , GJ L I ∂t r r r 1 ∂u(t, 0) L 1 GJ ∂u(t, 0) 1 GJ ∂u(t, 0) = −cb − Wob Rb µ( )sgn( ) GJI ∂t GJ L I ∂t L I ∂t
∂u(t, 0) F ∂t
where µ( L1
∂u(t,0) ∂t
=
γ
µcb + (µsb − µcb )e
1 − υb | L f
√ GJ ∂u(t,0) I
∂t
|
PT
After computing
AC
CE
hH(Z(t)), Z(t)iX
∂u(t, 0) ∂u(t, 0) ) ∂t r∂t 1 ∂u(t, 0) 2 ( ) = −cb GJI ∂t √ GJ ∂u(t,0) γ 1 L − υb | L | I ∂t f − Wob Rb µcb + (µsb − µcb )e GJ r 1 GJ ∂u(t, 0) ∂u(t, 0) × sgn( ) L I ∂t ∂t = F(
Consequently, hH(Z(t)), Z(t)iX ≤ 0 as µsb ≥ µcb . This implies that the operator
125
H(Z) is dissipative. It’s easy to verify that, H is locally Lipschitz. Accordingly, the operator H(Z) is locally Lipschitz and dissipative. 10
ACCEPTED MANUSCRIPT
Lemma 2. For any function f ∈ L1 ([0, T ], X) and any initial condition Z0 ∈ T D(A), the problem (11) has at most one solution in C 1 ([0, T ], X) C 0 ([0, T ], D(A)).
which satisfies the following system ˙ Z(t) = AZ(t) + H(Z(t)) Z(0) = 0
(12)
AN US
Since H(Z) and A are dissipative, we get
CR IP T
T Proof . Suppose Z1 and Z2 are two solutions of (11) in the class C 1 ([0, T ], X) C 0 ([0, T ], D(A)). T Then the difference Z = Z1 −Z2 is an element of C 1 ([0, T ], X) C 0 ([0, T ], D(A))
˙ hZ(t), Z(t)iX = hAZ(t), Z(t)iX + hH(Z(t)), Z(t)iX ≤ 0 130
Then
1 d k Z(t) kX ≤ 0 ⇒k Z(t) kX = 0 2 dt
then this one is unique.
M
Hence, problem (12) has a unique solution Z(t) = 0 for every Z0 ∈ D(A). Thus T proves that Z1 = Z2 and shows that (11) has a solution in C 1 ([0, T ], X) C 0 ([0, T ], D(A)),
135
ED
Proof (Theorem 2). By applying the two Lemmas given above and from results given in (Theorem 4.2 [33], [28, 29, 32]), it is easy to prove that our system
PT
(11) has a unique solution.
AC
CE
2.3. Drillstring modeling and control formulation q aL Let W (t) = cGJ Ω(t) − L1 GJ I ∂t u(t, 1) , we obtain ∂ 2 u(t, x) ∂t2 ∂u(t, 1) ∂x ∂ 2 u(t, 0) ∂t2
=
∂ 2 u(t, x) ∂u(t, x) −λ , x ∈ (0, 1) 2 ∂x ∂t
(13)
=
W (t)
(14)
=
a
∂u(t, 0) ∂u(t, 0) + aF ( ) ∂x ∂t
(15)
To linearize the boundary condition (15), we use the next form (as presented in [12]) u ¯(t, x) =
λwr 2 x − F (wr )x + wr t + u0 2 11
ACCEPTED MANUSCRIPT
∂u ¯(t,x) ∂t .
as a reference trajectory, such that wr = Then the system (13)-(15) becomes ∂ 2 u(t, x) ∂t2 ∂u(t, 1) ∂x ∂ 2 u(t, 0) ∂t2
where b =
∂F ∂w (wr )
and w(t) =
=
∂ 2 u(t, x) ∂u(t, x) −λ 2 ∂x ∂t
=
W (t)
=
a
(16)
CR IP T
140
(17)
∂u(t, 0) ∂u(t, 0) + ab ∂x ∂t
∂u(t,1) ∂t .
(18)
AN US
The local exponential stability of system (16)-(18) is easy to be obtained when we rewrite (16) in this form l r l ∂ ∂ = − λ ∂t ∂x r l 0 q ∂u(t,x) 1 with (r, l) = ( ∂u(t,x) , ) and λ = dL ∂x ∂t IGJ .
M
Introducing the Riemann invariants: y = r − l, z = r + l. Then the system (16)-(18) is transformed into the system (19)-(22)
PT
ED
∂y(t, x) ∂y(t, x) + = ∂t ∂x ∂z(t, x) ∂z(t, x) − = ∂t ∂x z(t, 1) = ∂z(t, 0) ∂y(t, 0) − = ∂t ∂t
λ (z(t, x) − y(t, x)) 2 λ − (z(t, x) − y(t, x)) 2 2W (t) − y(t, 1) a z(t, 0) + y(t, 0) + ab z(t, 0) − y(t, 0) .
(19) (20) (21) (22)
CE
Now, we are going to construct changes of variables for the 2 × 2 PDE states
AC
(19) -(22), which are facilitate our analysis. Hence, we introduce the notations λ
λ
φ(x) = e 2 x , β(x) = e− 2 x
and the new states v1 (t, x) = φ(x)y(t, x),
v2 (t, x) = β(x)z(t, x).
12
ACCEPTED MANUSCRIPT
Then the system (19)-(22) is transformed in the next form: =
0
=
v2 (t, 1)
=
∂v2 (t, 0) ∂v1 (t, 0) − ∂t ∂t 145
∂v1 (t, x) ∂v1 (t, x) λφ + − v2 (t, x) ∂t ∂x 2β ∂v2 (t, x) ∂v2 (t, x) λ β − − v1 (t, x) ∂t ∂x 2φ β(1) 2β(1)W (t) − v1 (t, 1) φ(1)
(23) (24)
CR IP T
0
(25)
= a(v2 (t, 0) + v1 (t, 0)) + ab(v2 (t, 0) − v1 (t, 0)) (26)
After transformation of the damped wave equation into a 2 × 2, first order, lin-
AN US
ear hyperbolic PDE, we propose a stabilizing boundary control scheme for the linearized system.
As suggested by the boundary conditions (25)-(26) for the drilling process, the objectives is to find a feedback control law W (t) that locally-exponentially sta150
bilizes the zero equilibrium of system (23)-(24).
M
3. BOUNDARY CONTROL DESIGN
This section shows the importance of backstepping transformations, well-
ED
posedness propriety of kernel equations, and the Lyapnov theory, providing a useful analysis for stability of oilwell drilling system. The backstepping tech155
niques is to eliminate the effects destabilizing terms that appear throughout the
PT
domain while the control is acting only from the boundary. The backstepping transformation as also known Volterra transformation has a lower triangular
CE
structure. The terms of instability in (23)-(25) are
λφ 2β v2
and
λβ 2 φ v1 ,
our main
objective is to eliminate these both terms to show the stablity of the target system.
AC
160
13
ACCEPTED MANUSCRIPT
3.1. Target system stability Our objectif is to design the control W (t) such that the transformed system (23)-(26) maps the next target system ∂χ(t, x) ∂x ∂ζ(t, x) = ∂x = 0
CR IP T
∂χ(t, x) ∂t ∂ζ(t, x) ∂t ζ(t, 1) ∂ζ(t, 0) ∂χ(t, 0) − ∂t ∂t
= −
(27) (28)
(29) = a ζ(t, 0) + χ(t, 0) − ab2 ζ(t, 0) − χ(t, 0) . (30)
ζ(0, x) = ζ0 (x), we get
ζ(t, x) =
AN US
Computing ζ by the method of characteristics such that the initial condition ζ0 (t + x) 0
0 ≤ t + x < 1, t + x ≥ 1.
The necessary relations between (χ, ζ) and (v1 , v2 ) will be identified in subsection 3.2.
M
165
Theorem 3. Consider system (27)-(30) with initial conditions (χ0 , ζ0 ) ∈ L2 ([0, 1]).
ED
Then the zero equilibrium of (27)-(30) is locally exponentially stable in the L2 sense. Further ζ go to zero in finite time.
PT
Proof . Let us introduce the next Lyapunov function Z 1 2 V (t) = M e−ηx χ2 (t, x) + N eηx ζ 2 (t, x)dx + ζ(t, 0) − χ(t, 0) ,
CE
0
where M , N and η are positive constants. N ≥ 2a ≥ M .
Differentiating V with respect to time, integrating by parts and using the bound-
AC
170
Choosing M and N such that
14
ACCEPTED MANUSCRIPT
CR IP T
ary conditions (29)-(30), we get Z 1 ∂χ ∂ζ ∂ζ ∂χ ˙ 2M χ e−ηx + 2N ζ eηx dx + 2( (t, 0) − V (t) = (t, 0))(ζ(t, 0) − χ(t, 0)) ∂t ∂t ∂t ∂t 0 Z 1 ηM e−ηx χ2 (t, x)dx + [N eηx ζ 2 (t, x)]10 = −[M e−ηx χ2 (t, x)]10 − 0 h i + 2 a(ζ(t, 0) + χ(t, 0)) − ab2 (ζ(t, 0) − χ(t, 0)) (ζ(t, 0) − χ(t, 0)) Z 1 N eηx ζ 2 (t, x)dx − η 0
= −M e−η χ2 (t, 1) − (2a − M )χ2 (t, 0) − (N − 2a)ζ 2 (t, 0) ≤
− min(η, 2ab2 )V (t).
AN US
− 2ab2 (ζ(t, 0) − χ(t, 0))2 − ηV (t)
This proves local exponential stability of (χ, ζ). Finally from (28) and (29),
M
basically, the solution of ζ satisfies ζ0 (t + x) ζ(t, x) = 0
0 ≤ t + x < 1, t + x ≥ 1,
ED
which implies that ζ tends to zero in finite-time. 3.2. Backstepping techniques and control law In order, to convert the original system (23)-(26) into the target system
175
PT
(27)-(30), we consider the next backstepping transformation (more details about
AC
CE
backstepping transformation and kernel equations in [25]): Z x Z x χ(t, x) = v1 (t, x) − Auu (x, ξ)v1 (t, ξ)dξ − Auv (x, ξ)v2 (t, ξ)dξ 0
0
1 + (b + b2 )e−abx (v2 (t, 0) − v1 (t, 0)). 2 Z x Z vu ζ(t, x) = v2 (t, x) − A (x, ξ)v1 (t, ξ)dξ − 0
+
λφ(x) 2β(x) ,
1 (b + b2 )eabx (v2 (t, 0) − v1 (t, 0)). 2
ε2 (x) =
λ β(x) 2 φ(x) ,
Avv (x, ξ)v2 (t, ξ)dξ
0
Let consider the notations: ε1 (x) =
(31) x
λ is defined above,
15
(32) v1 (t, x) , E(t, x) = v2 (t, x)
180
χ(t, x) Auu (x, ξ) Auv (x, ξ) , A(x, ξ) = , δ(t, x) = ζ(t, x) Avu (x, ξ) Avv (x, ξ) 1 2 −abx (b + b )e (v (t, 0) − v (t, 0)) 2 1 . and Φ(t, x) = 2 1 2 abx (b + b )e (v (t, 0) − v (t, 0)) 2 1 2
CR IP T
ACCEPTED MANUSCRIPT
In compact form, using (31) and (32) we have Z x δ(t, x) = E(t, x) − A(x, ξ)E(t, ξ)dξ + Φ(t, x). 0
(33)
Plugging (31)-(32) into (27)-(28), integrating by parts and using the boundary conditions, we obtain the next kernel equations
AN US
185
∂Auu (x, ξ) ∂Auu (x, ξ) − − ε2 (ξ)Auv (x, ξ) = 0 ∂x ∂ξ ∂Auv (x, ξ) ∂Auv (x, ξ) + − ε1 (ξ)Auu (x, ξ) = 0 − ∂x ∂ξ ∂Avu (x, ξ) ∂Avu (x, ξ) − − ε2 (ξ)Avv (x, ξ) = 0 ∂x ∂ξ ∂Avv (x, ξ) ∂Avv (x, ξ) + − ε1 (ξ)Avu (x, ξ) = 0 ∂x ∂ξ
(34) (35) (36) (37)
M
−
with the next boundary conditions
ED
−Auv (x, 0) = =
Avu (x, x)
=
PT
Auv (x, x)
1 (b + b2 )ae−abx 2
1 ε1 (x) 2 1 − ε2 (x) 2 Avu (x, 0) =
(38) (39) (40)
1 (b + b2 )aeabx . 2
(41)
CE
−Avv (x, 0) =
Auu (x, 0) =
The kernel of the backstepping transformation also known as Volterra transformation satisfies an interesting system of PDE which is easily solvable. Then,
AC
this equations are defined on the triangular domain ∆ = {(x, ξ) ∈ R2 : 0 ≤ ξ ≤
190
x ≤ 1}. By Theorem 5 (see the Appendix), we show the existence, uniqueness
and continuity of the solution for system (34)-(37) with boundary conditions (38)-(41). In order, to ensure that the closed-loop and the target systems have equivalent stability properties, the backstepping transformations (31) and (32) have to be 16
ACCEPTED MANUSCRIPT
195
invertible. Then, we study the invertibility of the transformation (31) and (32), we look for a transformation of the target system (27)- (28) into the closed-loop
x
Lχχ (x, ξ)χ(t, ξ)dξ +
v2 (t, x)
b + b2 −abx e (ζ(t, 0) − χ(t, 0)). 2 Z x
Lζχ (x, ξ)χ(t, ξ)dξ +
= ζ(t, x) +
0
b + b2 abx e (ζ(t, 0) − χ(t, 0)). 2
Denoting 200
Lχζ (x, ξ)ζ(t, ξ)dξ
(42)
Z
0
x
Lζζ (x, ξ)ζ(t, ξ)dξ
(43)
AN US
−
x
0
0
−
Z
CR IP T
system (23) - (26) as follows Z v1 (t, x) = χ(t, x) +
Lχχ (x, ξ) Lχζ (x, ξ) − 12 (b + b2 )e−abx (ζ(t, 0) − χ(t, 0)) and Λ(t, x) = . L(x, ξ) = Lζχ (x, ξ) Lζζ (x, ξ) − 21 (b + b2 )eabx (ζ(t, 0) − χ(t, 0)) We rewritten (42)-(43) in this form Z x E(t, x) = δ(t, x) + L(x, ξ)δ(t, ξ)dξ + Λ(t, x). (44)
M
0
Plugging (42)-(43) into (23)-(26), integrating by parts and using the boundary conditions we find the next kernel equations:
CE
PT
ED
∂Lχχ (x, ξ) ∂Lχχ (x, ξ) + − ε1 (x)Lζχ (x, ξ) = 0 ∂ξ ∂x ∂Lχζ (x, ξ) ∂Lχζ (x, ξ) − + − ε1 (x)Lζζ (x, ξ) = 0 ∂ξ ∂x ∂Lζχ (x, ξ) ∂Lζχ (x, ξ) − − ε2 (x)Lχχ (x, ξ) = 0 ∂ξ ∂x ∂Lζζ (x, ξ) ∂Lζζ (x, ξ) − − ε2 (x)Lχζ (x, ξ) = 0 − ∂ξ ∂x
(45) (46) (47) (48)
AC
with the next boundary conditions Lχχ (x, 0) = Lχζ (x, x) = Lζχ (x, x) = Lζζ (x, 0) =
Lχζ (x, 0) + a(b + b2 )e−abx ε1 (x) 2 ε2 (x) − 2 Lζχ (x, 0) − a(b + b2 )eabx , 17
(49) (50) (51) (52)
ACCEPTED MANUSCRIPT
To prove existence, uniqueness and continuity of the system (45)-(48), with 205
boundary conditions (49)-(52) we use the Theorem 5 (see the Appendix). By plugging the transformation (32) into (29), the control law is obtained. So,
ζ(t, 1)
= v2 (t, 1) − +
Z
0
1
Avu (1, ξ)v1 (t, ξ)dξ −
1 (b + b2 )eab (v2 (t, 0) − v1 (t, 0)) = 0. 2
Hence
+
0
1
Avv (1, ξ)v2 (t, ξ)dξ
Z 1 Z 1 β(1) v1 (t, 1) − Avu (1, ξ)v1 (t, ξ)dξ − Avv (1, ξ)v2 (t, ξ)dξ φ(1) 0 0 1 (b + b2 )eab (v2 (t, 0) − v1 (t, 0)) = 0. 2
AN US
2β(1)W (t) −
Z
CR IP T
plugging (32) into (29), and using (25) we get
M
From this equality, the control law can be easily computed, which is given by: Z Z 1 1h 1 1 1 vu W (t) = v1 (t, 1) + A (1, ξ)v1 (t, ξ)dξ + Avv (1, ξ)v2 (t, ξ)dξ 2 φ(1) β(1) 0 0 i 1 (b + b2 )eab (v2 (t, 0) − v1 (t, 0)) . (53) − 2
ED
210
Theorem 4. Consider system (23)-(26), with initial conditions (v10 , v20 ) and the control law (53) where the kernels Avu and Avv are obtained from (34)-(41)
PT
with Λ(t, x), Φ(t, x) ∈ L2 ([0, 1]). Then the equilibrium (v1 , v2 ) = (0, 0) is locally
exponentially stable in the L2 sense. Further, the equilibrium v2 vanishes to zero in finite time.
CE
215
AC
Proof . Recall that E(t, x) = δ(t, x) +
Z
x
L(x, ξ)δ(t, ξ)dξ + Λ(t, x),
0
and δ(t, x)
=
E(t, x) −
Z
x
A(x, ξ)E(t, ξ)dξ + Φ(t, x).
0
18
ACCEPTED MANUSCRIPT
By Theorem 5 (see the Appendix) the kernel A and L are continuous. Consequently, Z . L(., ξ)δ(t, ξ)dξ ||δ(t, .)||L2 ([0,1]) + kΛ(t, .)kL2 ([0,1]) +
≤
0
≤
||δ(t, .)||L2 ([0,1]) + kΛ(t, .)kL2 ([0,1]) + |||L|||∞ ||δ(t, .)||L2 ([0,1])
≤ C1 (t)||δ(t, .)||L2 ([0,1])
where C1 (t) = Besides
n kΛ(t,.)k
L2 ([0,1])
||δ(t,.)||L2 ([0,1])
||δ(t, .)||L2 ([0,1])
o + 1 + |||L|||∞ .
Z . ||E(t, .)||L2 ([0,1]) + kΦ(t, .)kL2 ([0,1]) + A(., ξ)E(t, ξ)dξ
≤
AN US
220
L2 ([0,1])
CR IP T
||E(t, .)||L2 ([0,1])
0
≤
||E(t, .)||L2 ([0,1]) + kΦ(t, .)kL2 ([0,1]) + |||A|||∞ ||E(t, .)||L2 ([0,1])
≤ C2 (t)||E(t, .)||L2 ([0,1])
where C2 (t) = If t = 0 then
n kΦ(t,.)k
L2 ([0,1])
||E(t,.)||L2 ([0,1])
o + 1 + |||A|||∞ .
M
||δ(0, .)||L2 ([0,1]) ≤ C2 (0)||E(0, .)||L2 ([0,1]) , n kΦ(0,.)k o 2 such that C2 (0) = ||E(0,.)||L 2([0,1]) + 1 + |||A|||∞ . L ([0,1])
ED
By Theorem 3, there exists c > 0 and κ > 0 such that
||δ(t, .)||L2 ([0,1]) ≤ c||δ(0, .)||L2 ([0,1]) e−κt .
PT
Hence, there exists a constant C > 0 and κ > 0 such that
CE
||E(t, .)||L2 ([0,1]) ≤ C||E(0, .)||L2 ([0,1]) e−κt .
It is straightforward E(t, x) = (v1 (t, x), v2 (t, x)) tends local exponentially to zero. Finally, we know that as ζ go to zero in finite time, then also v2 go to zero in
AC
finite time. This ends the proof.
225
L2 ([0,1])
4. SIMULATION Controlling drillstring vibrations requires a good understanding of the bottom hole assembly (BHA) dynamics and its interactions with the drillstring. 19
ACCEPTED MANUSCRIPT
The drillstring vibrations can be self-excited and induced by nonlinear dynamic interactions between drilled formation and drill bit, and between drillstring 230
and borehole which are often characterized by erratic patterns and behavior.
CR IP T
Uncontrolled vibrations can decrease the rate of penetration, and accordingly increase the well cost. Furthermore, excessive oscillations can interfere with
measurement while drilling tools or even cause their damage. They can also cause a significant waste of drilling energy. Vibrations often induce well bore 235
instabilities that can worsen the condition of the well and reduce the directional control. In the last years, the dynamics of drilling has been studied numerically
AN US
and experimentally but in most researches local problems, for example, torsional vibrations or stick-slip have been focussed [34].
In this section, simulation results of the torsional vibrations model are presented. 240
The strategy developed in section 2 and 3 is applied to find the synthesis of a stabilizing controller ensuring the suppression of the torsional vibrations and consequently the reduction of the bit bouncing phenomenon. As the drillstring
M
system (1)-(3) is equivalent to system of study (16)-(19), it is important to test the effectiveness of the boundary control law for the local exponential stability
ED
of system (16)-(19). The next physical parameters are used in simulation: Variable
Value
Description
L
2000 m
Drillstring length
PT
245
I
0.095 kg.m
Inertia per unit length
Ib
311 kg.m2
Inertia at the drillstring bottom 4
1.19.10 m
ca
2000 N m.s.rad−1
CE
J
d
AC
5
0.09 kg.m.s
−1
Geometrical moment of inertia Sliding torque coefficient Drillstring damping
Table 1: Different physical parameters
As shown by figures 2 and 3, the states v1 (t, x) and v2 (t, x) reach zero local.
20
AN US
CR IP T
ACCEPTED MANUSCRIPT
AC
CE
PT
ED
M
Figure 2: Stabilization of the state v1 (t, x).
Figure 3: Stabilization of the state v2 (t, x).
21
ACCEPTED MANUSCRIPT
The simulation results of (Fig. 2)-(Fig. 3) show that the stick-slip vibrations are reduced by means of the application of the control law (53) depending on the angular velocity at the bottom extremity.
CR IP T
250
Conclusion
In this paper, the oilwell drilling system described by a damped wave equa-
tion was transformed into 2 × 2 first-order linear hyperbolic PDEs. The gen-
erally neglected damping term is considered here, and the boundary conditions 255
are defined for the two first-order hyperbolic PDEs. Basically this makes our
AN US
stabilizing problem different. The fundamental tools such that the backstepping technique, the well-posedness propriety of the kernel equations, and the Lyapunov theory were shown their importance in the transformation procedure and
the control law design. The oilwell drilling system is shown locally exponentially 260
stable from the stick-slip suppression. Future investigations should focus on the
M
performances of the controllers. It is important to evaluate their robustness to changes in operating conditions, such as variations of the weight-on-bit Wob . This, along with the design of an observer for the backstepping controller, will
APPENDIX
PT
265
ED
be topic of future works.
The studying of the existence, uniqueness and continuity of the solution
CE
to systems (34)-(37), (45)-(48), with boundary conditions respectively (38)(41), (49)-(52), are similar. Because both systems have similar structures. We consider the ”generalized Goursat” problem in which the direct kernel (34)(37) with boundary conditions (38)-(41) and the inverse kernel equations (38)-
AC
270
(41) with boundary conditions, (49)-(52), are a particular case. Defining for
22
ACCEPTED MANUSCRIPT
i = 1, .., 4 the next system equations: ∂G1 (x, ξ) ∂G1 (x, ξ) + c1 (ξ) ∂x ∂ξ 2
c1 (x) c2 (x)
4 X
ε1i (x, ξ)Gi (x, ξ) + g1 (x, ξ), (54)
i=1
2
∂G (x, ξ) ∂G (x, ξ) − c2 (ξ) ∂x ∂ξ
=
∂G3 (x, ξ) ∂G3 (x, ξ) − c1 (ξ) ∂x ∂ξ
=
4 X i=1
4 X i=1
4
4
c2 (x)
=
∂G (x, ξ) ∂G (x, ξ) + c2 (ξ) ∂x ∂ξ
=
4 X i=1
ε3i (x, ξ)Gi (x, ξ) + g3 (x, ξ) (56)
ε4i (x, ξ)Gi (x, ξ) + g4 (x, ξ), (57)
AN US
with the next boundary conditions
ε2i (x, ξ)Gi (x, ξ) + g2 (x, ξ), (55)
CR IP T
c1 (x)
G1 (x, 0)
= q1 (x)G2 (x, 0) + h1 (x),
(58)
G2 (x, x)
= h2 (x),
(59)
G3 (x, x)
= h3 (x),
(60)
G4 (x, 0)
=
q2 (x)G3 (x, 0) + h4 (x).
(61)
275
M
This system is defined on the triangular domain ∆ = {(x, ξ) : 0 ≤ ξ ≤ x ≤ 1}. We use the method of characteristics to prove the existence and uniqueness of
ED
the equations. Hence, to prove the existence and uniqueness we prefer to use the next theorem:
Theorem 5. [35] Consider system (54)-(57), with boundary conditions (58)C N ([0, 1]). Then there exists a unique C N (∆) solution Gi for i = 1, 2, 3, 4.
CE
280
PT
(61) such that hi , qi ∈ C N ([0, 1]), gi , εij ∈ C N (∆), i, j = 1, 2, 3, 4 and c1 , c2 ∈
References
AC
References
[1] J. D. Jansen, Nonlinear dynamics of oilwell drillstrings, Delft University of
285
Technology, 1993. doi:10.1007/s10596-016-9576-1.44.
[2] J. J. Bailey, I. Finnie, An analytical study of drill-string vibration, Journal of Engineering for Industry 82 (2) (1960) 122–127. doi:10.1115/1. 3663017. 23
ACCEPTED MANUSCRIPT
[3] S. Toumi, L. Beji, R. Mlayeh, A. Abichou, Stability analysis of coupled torsional vibration and pressure in oilwell drillstring system, International Journal of Control (2017)doi:10.1080/00207179.2016.1278269.
290
CR IP T
[4] S. Dwars, Recent advances in soft torque rotary systems, Society of Petroleum Engineers (2015)doi:http://dx.doi.org/10.2118/ 173037-MS.
[5] X. Liu, N. Vlajic, X. Long, G. Meng, B. Balachandran, State-dependent
delay influenced drill-string oscillations and stability analysis, Journal of
295
AN US
Vibration and Acoustics 136 (5) (2014) 051008. doi:10.1115/1.4027958. [6] B. Saldivar, S. Mondi´e, J. J. Loiseau, V. Rasvan, Stick-slip oscillations in
oillwell drilstrings: distributed parameter and neutral type retarded model approaches, IFAC Proceedings Volumes 44 (1) (2011) 284–289. doi:10. 3182/20110828-6-IT-1002.00084.
300
M
[7] B. Saldivar, T. Kn¨ uppel, F. Woittennek, I. Boussaada, H. Mounier, S.I. Niculescu, Flatness-based control of torsional-axial coupled drilling vidoi:
ED
brations, IFAC Proceedings Volumes 47 (3) (2014) 7324–7329. 10.3182/20140824-6-ZA-1003.02205. 305
[8] S. Toumi, L. Beji, R. Mlayeh, A. Abichou, Stabilization of stick-slip os-
PT
cillations integrating fluid injection in oilwell drillstring system, European Control Conference (2016) 352–357doi:10.1109/ECC.2016.7810310.
CE
[9] B. Saldivar, I. Boussaada, H. Mounier, S. Niculescu, Analysis and Control of Oilwell Drilling Vibrations, A Time-Delay Systems Approach, Advances in Industrial Control. Springer, New York, 2015. doi:10.1007/
AC
310
978-3-319-15747-4.
[10] S. Toumi, R. Mlayeh, L. Beji, A. Abichou, Stability analysis of oilwell drilling torsional vibrations, Control and Automation (MED), 24th Mediterranean Conference (2016) 677–682doi:10.1109/MED.2016.
315
7535945. 24
ACCEPTED MANUSCRIPT
[11] T. Vromen, Control of stick-slip vibrations in drilling systems Eindhoven, Technische Universiteit Eindhoven, 2015. [12] C. Sagert, F. D. Meglio, M. Krstic, P. Rouchon, Backstepping and
CR IP T
flatness approaches for stabilization of the stick-slip phenomenon for
drilling, IFAC Proceedings Volumes 46 (2) (2013) 779–784. doi:10.3182/
320
20130204-3-FR-2033.00126.
[13] X. Liu, N. Vlajic, X. Long, G. Meng, B. Balachandran, Coupled axialtorsional dynamics in rotary drilling with state dependent delay: Stability
AN US
and control, Nonlinear Dynamics 78 (3) (2014) 1891–1906. doi:10.1007/ s11071-014-1567-y.
325
[14] X. Liu, N. Vlajic, X. Long, G. Meng, Nonlinear oscillations of a flexible rotor with a drill bit: Stick-slip and delay effects, Nonlinear Dynamics 72 (1-2) (2013) 61–77. doi:10.1007/s11071-012-0690-x.
M
[15] E. Kreuzer, L. Krumm, M.-A. Pick, E. Solowjow, M. Steidl, Active vibration isolation via decomposition of traveling waves, 11th International Con-
330
ED
ference on Vibration Problems, Lisbon, Portugal (9–12 September 2013). [16] P. Rouchon, Flatness and stick-slip stabilization, Technical Report 492. France: Ecole des Mines de Paris (1998).
PT
[17] S. Toumi, L. Beji, R. Mlayeh, A. Abichou, Boundary observer design for hyperbolic pde in rotary drilling systems, IEEE 55th Conference on
335
CE
Decision and Control (CDC), Las Vegas, USA (2016) 2128–2133doi: 10.1109/CDC.2016.7798578.
AC
[18] A. Smyshlyaev, E. Cerpa, M. Krstic, Boundary stabilization of a 1-d wave
340
equation with in-domain antidamping, SIAM Journal on Control and Optimization 48 (6) (2010) 4014–4031. doi:10.1137/080742646.
[19] A. Diagne, G. Bastin, J. M. Coron, Lyapunov exponential stability of linear hyperbolic systems of balance laws, IFAC Proceedings Volumes 44 (1) (2011) 13320–13325. doi:10.3182/20110828-6-IT-1002.01506. 25
ACCEPTED MANUSCRIPT
[20] F. D. Meglio, R. Vazquez, M. Krstic, N. Petit, Backstepping stabilization of an underactuated 3× 3 linear hyperbolic system of fluid flow equations,
345
American Control Conference (ACC) (2012) 3365–3370doi:10.1.1.400.
CR IP T
2886. [21] J. M. Coron, R. Vazquez, M. Krstic, G. Bastin, Local exponential h2
stabilization of a 2×2 quasilinear hyperbolic system using backstepping, SIAM Journal on Control and Optimization 51 (3) (2013) 2005–2035.
350
doi:10.1137/120875739.
AN US
[22] C.-Z. Xu, G. Sallet, Exponential stability and transfer functions of processes
governed by symmetric hyperbolic systems, ESAIM: Control, Optimisation and Calculus of Variations 7 (2002) 421–442. doi:10.1051/cocv:2002062. 355
[23] J. Rauch, M. E. Taylor, Exponential decay of solutions to hyperbolic equations in bounded domains, Indiana university Mathematics journal 21 (1)
M
(1974) 79–86. doi:238228429.
[24] F. D. Meglio, R. Vazquez, M. Krstic, Stabilization of a system of cou-
ED
pled first-order hyperbolic linear pdes with a single boundary input, IEEE Transactions on Automatic Control 58 (12) (2013) 3097–3111.
360
doi:
10.1109/TAC.2013.2274723.
PT
[25] M. Krstic, A. Smyshlyaev, Backstepping boundary control for first-order hyperbolic pdes and application to systems with actuator and sensor de-
CE
lays, Systems & Control Letters 57 (9) (2008) 750–758. doi:10.1016/j. sysconle.2008.02.005.
365
AC
[26] J. M. Coron, B. d’Andrea Novel, G. Bastin, A strict lyapunov function
370
for boundary control of hyperbolic systems of conservation laws, IEEE transactions on automatic control 52 (1) (2007) 2–11. doi:10.1109/TAC. 2006.887903.
[27] R. Vazquez, M. Krstic, J. M. Coron, Backstepping boundary stabiliza-
26
ACCEPTED MANUSCRIPT
tion and state estimation of a 2× 2 linear hyperbolic system (2011) 4937– 4942doi:10.1109/CDC.2011.6160338. [28] T. D. Nguyen, O. Egeland, Stabilization of towed cables, Decision and
CR IP T
Control, 2004. CDC. 43rd IEEE Conference on 5 (2004) 5059–5064. doi: 10.1109/CDC.2004.1429609.
375
[29] A. Pazy, Semigroups of linear operators and applications to partial differential equations, Vol. 44, Springer New York, 1983. 978-1-4612-5561-1.
doi:10.1007/
AN US
[30] N. Challamel, Rock destruction effect on the stability of a drilling structure,
Journal of Sound and Vibration 233 (2) (2000) 235–254. doi:10.1006/
380
jsvi.1999.2811.
[31] E. Navarro-L´ opez, R. Su´ arez, Practical approach to modelling and controlling stick-slip oscillations in oilwell drillstrings, Control Applications. Pro-
M
ceedings of the 2004 IEEE International Conference 2 (2004) 1454–1460. doi:10.1109/CCA.2004.1387580.
385
ED
[32] J. M. Coron, Control and nonlinearity, no. 136, American Mathematical Soc, 2007. doi:1098765432112111009.
PT
[33] M. Slemrod, Feedback stabilization of a linear control system in hilbert space with ana priori bounded control, Mathematics of Control, Signals and Systems 2 (3) (1989) 265–285. doi:10.1007/BF02551387.
CE
390
[34] N. Mihajlovic, A. van Veggel, N. van de Wouw, H. Nijmeijer, Analysis of
AC
friction induced limit cycling in an experimental drill-string system, Journal
395
of dynamic systems, measurement, and control 126 (4) (2004) 709. doi: 10.1115/1.1850535.
[35] R. Vazquez, J. M. Coron, M. Krstic, G. Bastin, Local exponential h2 stabilization of a 2× 2 quasilinear hyperbolic system using backstepping (2011) 1329–1334doi:10.1109/CDC.2011.6161075.
27