Vibration suppression and output constraint of a variable length drilling riser system

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Vibration suppression and output constraint of a variable length drilling riser system Fang Guo a, Yu Liu a,∗, Fei Luo a, Yilin Wu b a School

of Automation Science and Engineering, South China University of Technology, Guangzhou 510640, China of Computer Science, Guangdong University of Education, Guangzhou 510310, China

b Department

Received 7 January 2018; received in revised form 9 July 2018; accepted 9 October 2018 Available online xxx

Abstract In this paper, we mainly concentrate on the control issue of a variable length drilling riser under condition of unknown disturbances and output constraint. The studied flexible drilling riser system with variable length, variable tension, variable speed and restricted boundary output is essentially a nonlinear distributed parameter system. For achieving the vibration suppression and ensuring the boundary output within the constrained range, an appropriate control scheme with output signal barrier is put forward by integrating boundary control method, barrier Lyapunov function with finite-dimensional backstepping technique, where disturbance observer is employed for coping with the boundary disturbance. Moreover, the Lyapunov’s synthetic method is applied for the steadiness research of the studied flexible drilling riser system, and the simulations are presented to display the usefulness of proposed control scheme. © 2018 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

1. Introduction Flexible system, due to its characteristics of light weight, low cost and high efficiency, has been widely used and rapidly developed in the engineering applications, including the manipulators [1,2], risers [3,4], wings [5,6] and so on [7–11]. The flexible drilling riser as an important component part in the drilling system, affects the efficient of the mud transporting process directly [12]. When the drilling riser system begins work, both the external disturbances and the interactions between the accumulational muddy and system devices are ∗

Corresponding author. E-mail address: [email protected] (Y. Liu).

https://doi.org/10.1016/j.jfranklin.2018.10.023 0016-0032/© 2018 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

Please cite this article as: F. Guo, Y. Liu and F. Luo et al., Vibration suppression and output constraint of a variable length drilling riser system, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2018.10.023

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widespread, which will inevitably lead to a certain amplitude of vibration and deformation, and hence degrade the system performance or even cause damage to the equipment. Based on this fact, the vibration control of drilling riser system has become a significant problem, and needs to further study. The flexible drilling riser system is a nonlinear distributed parameter system (DPS), which is expressed by infinite-dimensional equation (PDE) and finite-dimensional equations (ODEs). Recently, many methods and theories have been put forward for DPS by scholars of various countries, including Galerkin’s method and assumed modes method as well as finite element method [13–16], which all rely on the simplification from original infinite-dimensional PDE model to finite-dimensional ODE. All three are effective ways to deal with the vibration problem of DPS, and their greatest strength is simple, but they will cause inaccuracy of system model, resulting in control spillover instability. Instead, boundary control is applied to construct controller based on original PDE [17,18], which differs from the above mentioned control methods. Considering the fact that it could describe system states and infinite dimensional dynamics quite precisely and require relatively few sensors and actuators, boundary control integrated with additional advanced approaches, including adaptive control [19–21], neural network [22–25] and sliding mode control [26–28], has been employed extensively in the control research of flexible systems [29–33]. In [29,30], the axially moving belt systems with S-curve acceleration/deceleration are stabilized via the fusion of boundary control with backstepping technique. In [31], by employing the Lyapunov redesign and active disturbance rejection control, two approaches are proposed for boundary control of plate vibration. In [32], the robust adaptive boundary control is put forward for reducing riser’s vibration and compensating the system parametric uncertainty. In [33], the global stabilization issue is investigated for the two-dimensional marine riser with bending couplings, where the Galerkin’s approximation method is proposed for the research of well-posedness and stability. Despite the research on the control of flexible systems has now made significant progress, the research literature regarding the flexible moving riser system with variable length, variable tension and variable speed is rarely, let alone the literature of integrating boundary control with backstepping technique [34,35] for controlling this kind of system, which inspires us for this paper. Taking into account that excessive vibration of riser will pose the premature fatigue problem, to improve the security and work efficiency in the process of drilling riser moving, it’s necessary to construct controller to ensure riser’s boundary output within the constrained range. The barrier Lyapunov function (BLF) is considered to be a reliable method applied for handling the output constraint, although originally proposed aiming at the ODE, now has been more and more used in PDE and there have been remarkable achievements [36–39]. In [36,37], for the flexible crane systems, boundary controls integrated with BLF are constructed to cope with the restricted problems of the boundary tension and output signals, respectively. In [38], the issue of boundary tension constraint satisfaction is studied for an axially moving string system, where a barrier-based control is put forward for vibration suppression and stability analysis. In [39], the robotic manipulator subject to uncertain dynamics and constrained joint angles is stabilized via the fusion of adaptive neural network control with BLF. Therefore, another innovation of this paper is that the effects of boundary output constraint are taken into account for the variable length drilling riser system, and BLF integrated with backstepping technique and boundary control is employed to design a suitable control scheme aiming at achieving vibration reduction and simultaneously guaranteeing the boundary output within the constrained region. Please cite this article as: F. Guo, Y. Liu and F. Luo et al., Vibration suppression and output constraint of a variable length drilling riser system, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2018.10.023

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Fig. 1. Flexible drilling riser system.

In contrast with the existing results, the major contributions of this paper include: i. In view of the Lyapunov’s direct method, an appropriate boundary control scheme with output signal barrier is developed to realize the vibration suppression of the drilling riser system and guarantee the output constraint satisfaction by uniting finite-dimensional backstepping technique, BLF and boundary control. ii. The BLF is employed to avoid the constraint violation and disturbance observer is adopted to handle the boundary disturbance, in which the barrier term is included. iii. The stability of the closed-loop drilling riser system is studied and validated by theoretic analysis and simulations. The remainder of the paper is given as below. The system PDE dynamics and preparation knowledge are shown in Section 2. In Section 3, the control scheme with output signal barrier is proposed through a combination of backstepping technique, BLF and boundary control method, which can effectively avoid the constraint violation and guarantee the system stability. Simulations are shown in Section 4 for validating the usefulness of proposed control scheme and the conclusion is drawn in Section 5. 2. Problem statement 2.1. PDE dynamics As depicted in Fig. 1, the drilling riser system with a fixed top boundary can only move vertically. l(t) and L represent the time-varying length and total length of the drilling riser, s represents the independent spatial variable, which satisfies 0 ≤ s ≤ l(t) ≤ L. z(s, t) represents the riser’s vibration displacement at position s for time t in the horizontal direction. d(t), f(s, t) and u(t) represent the time-varying boundary disturbance, spatiotemporally varying distributed disturbance and control force from the hydraulic actuator, respectively. |z(l(t), Please cite this article as: F. Guo, Y. Liu and F. Luo et al., Vibration suppression and output constraint of a variable length drilling riser system, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2018.10.023

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t)| < Q(Q > 0) represents the boundary output constraint. For keeping actuator stationary relative to the drilling riser system, it is connected to the moving trolley, which has the same moving speed v(t ) = l˙(t ) as studied drilling riser system. ˙ = ∂ (∗)/∂ t and (∗) = ∂ (∗)/∂ s are used throughout the paper. Remark 1. For simplicity, (∗) Based on the existing research [12], this paper tries to perfect the mathematical model of drilling riser system by taking the effects of distributed disturbance along the drilling riser and the damping coefficient of tip payload into account, where the governing equation can be updated as ρ[z¨(s, t ) + v˙z (s, t ) + v2 z (s, t ) + 2vz˙ (s, t )] + E I z (s, t ) +c[z˙(s, t ) + vz (s, t )] − T  (s, t )z (s, t ) − T (s, t )z (s, t ) = f (s, t ) and the boundary conditions are ⎧  z (l (t ), t ) = 0 ⎪ ⎪ ⎪ ⎪ ⎨z(0, t ) = 0 z (0, t ) = 0 ⎪ ⎪ ⎪m[z¨(l (t ), t ) + v˙z (l (t ), t ) + vz˙ (l (t ), t )] + T (l (t ), t )z (l (t ), t ) ⎪ ⎩ + ds [z˙(l (t ), t ) + vz (l (t ), t )] − d (t ) − E I z (l (t ), t ) = u(t )

(1)

(2)

where ρ, EI, T(s, t) and c represent uniform mass per unit length, bending stiffness, variable tension and damping coefficient of drilling riser respectively, m and ds represent the mass and damping coefficient of tip payload. 2.2. Preparation knowledge The following lemmas and assumptions are presented for the subsequent work. Lemma 1. Define ν 1 (s, t), ν 2 (s, t) ∈ R as functions on (s, t ) ∈ [0, L] × [0, +∞ ), for δ > 0, the following inequality holds [40] √ ν1 (s, t )ν2 (s, t ) ≤| ( δν1 (s, t ))( √1δ ν2 (s, t )) |≤ δν12 (s, t ) + 1δ ν22 (s, t ) (3) Lemma 2. Define ν(s, t) ∈ R as a function on (s, t ) ∈ [0, L] × [0, +∞ ), if ν(0, t ) = 0 holds, t ∈ [0, +∞ ), the following inequality can be obtained [40] L (4) ν 2 (s, t ) ≤ L 0 [ν  (s, t )]2 ds Assumption 1. It is assumed that there are three constants fm , dm , D ∈ R+ satisfying |f(s, t)| ≤ fm , ∀(s, t ) ∈ [0, L] × [0, +∞ ), |d(t)| ≤ dm , | d˙(t ) |≤ D, ∀t ∈ [0, +∞ ). Assumption 2. For the tension T(s, t), we assume there are six constants Tmin , Tmax , T˙min ,   T˙max , Tmin and Tmax ∈ R+ satisfying 0 ≤ Tmin ≤ T(s, t) ≤ Tmax , 0 ≤ T˙min ≤ T˙ (s, t ) ≤ T˙max and    0 ≤ Tmin ≤ T (s, t ) ≤ Tmax , ∀(s, t ) ∈ [0, L] × [0, +∞ ). 3. Control design Aiming to the control objectives of minimizing the vibration displacement of drilling riser system and simultaneously ensuring the boundary output within the constrained range, an appropriate control scheme is proposed by uniting finite-dimensional backstepping technique, BLF and boundary control method in this section. Moreover, according to Lyapunov’s direct method, the studied drilling riser system is proven to be stable. Please cite this article as: F. Guo, Y. Liu and F. Luo et al., Vibration suppression and output constraint of a variable length drilling riser system, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2018.10.023

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3.1. Boundary control scheme design On account of backstepping technique, the dynamics Eqs. (1) and (2) of studied drilling riser system is transformed as ⎧ ρ[z¨(s, t ) + v˙z (s, t ) + v2 z (s, t ) + 2vz˙ (s, t )] + E I z (s, t ) ⎪ ⎪ ⎪ ⎪ +c[z˙(s, t ) + vz (s, t )] − T  (s, t )z (s, t ) − T (s, t )z (s, t ) = f (s, t ) ⎪ ⎪ ⎪  ⎪ z (l (t ), t ) = 0 ⎪ ⎪ ⎪ ⎨z(0, t ) = 0 (5) z (0, t ) = 0 ⎪ ⎪ ⎪ y (t ) = z(l (t ) , t ) 1 ⎪ ⎪ ⎪ ⎪ y˙1 (t ) = y2 (t ) ⎪ ⎪   ⎪ ⎪ ⎩y˙ (t ) = E I z (l (t ), t ) − ds y2 (t ) + u(t ) − T (l (t ), t )z (l (t ), t ) + d (t ) + vz˙ (l (t ), t ) 2 m 3.1.1. Step One First, we use γ (t) to express the virtual control of y2 (t) and then the corresponding error is defined as e(t ) = y2 (t ) − γ (t )

(6)

The following Lyapunov candidate function is chosen Fx (t ) = Fa (t ) + Fb (t )

(7)

where energy term Fa (t) is Fa (t ) =

  ψE I l (t )  ψ l (t ) [z (s, t )]2 ds + T (s, t )[z (s, t )]2 ds 2 2 0 0  ψρ l (t ) + [z˙(s, t ) + vz (s, t )]2 ds 2 0

(8)

and small crossing term Fb (t) is 

l (t )

Fb (t ) = ς ρ

sz (s, t )[z˙(s, t ) + vz (s, t )]ds

(9)

0

where ψ, ς > 0. Lemma 3. The Lyapunov candidate function Fx (t) Eq. (7) is positive definite and there exist constants β 1 and β2 ∈ R+ such that the inequality below holds 0 < β1 Fa (t ) ≤ Fx (t ) ≤ β2 Fa (t )

(10)

Proof. See Appendix A. Differentiating Eq. (7) yields F˙x (t ) = F˙a (t ) + F˙b (t )

(11)

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Differentiating Fa (t) and substituting Eq. (1), then integrating by parts yield  ψ l (t ) F˙a (t ) = ψ[T (l (t ), t )z (l (t ), t ) − E I z (l (t ), t )]y2 (t ) + [T˙ (s, t ) 2 0  l (t ) + vT  (s, t )][z (s, t )]2 ds − ψE I v[z (0, t )]2 − ψc [vz (s, t ) 0



l (t )

+ z˙(s, t )]2 ds + ψ

f (s, t )[vz (s, t ) + z˙(s, t )]ds

(12)

0

For Fb (t), similarly, we can obtain F˙b (t ) =

 ς ρl (t ) 2 ς T (l (t ), t )l (t )  ς ρ l (t ) 2 y2 (t ) + [z (l (t ), t )]2 − z˙ (s, t )ds 2 2 2 0  3ς E I l (t )  − ς E I l (t )z (l (t ), t )z (l (t ), t ) − [z (s, t )]2 ds 2 0  1 l (t ) − [2ς csv + ς T (s, t ) − ς sT  (s, t ) − ς ρv2 ][z (s, t )]2 ds 2 0  l (t )  l (t ) − ςc sz (s, t )z˙(s, t )ds + ς sz (s, t ) f (s, t )ds 0

(13)

0

Substituting Eqs. (12) and (13) into Eq. (11) and combining with Eq. (6), we derive ς ρl (t ) F˙x (t ) = ψ[−E I z (l (t ), t ) + T (l (t ), t )z (l (t ), t )][e(t ) + γ (t )] + 2  l (t ) ς l (t ) T (l (t ) , t ) ς ρ × [e(t ) + γ (t )]2 + [z (l (t ), t )]2 − z˙2 (s, t )ds 2 2 0  l (t ) − ς E I l (t )z (l (t ), t )z (l (t ), t ) + ς sz (s, t ) f (s, t )ds 0



l (t )

− ψc −

1 2



0 l (t )

[vz (s, t ) + z˙(s, t )]2 ds −

3ς E I 2



l (t )

[z (s, t )]2 ds

0

[2ς csv + ς T (s, t ) − ς sT  (s, t ) − ς ρv2 − ψ T˙ (s, t )

0 





l (t )

− ψvT (s, t )][z (s, t )] ds − ς c 

2



− ψE I v[z (0, t )] + ψ 2

sz (s, t )z˙(s, t )ds

0 l (t )

f (s, t )[vz (s, t ) + z˙(s, t )]ds

(14)

0

 Remark 2. It can be seen from Eq. (14) that there exist an indeterminate term −ς E I l (t )z (l (t ), t )z (l (t ), t ) and a positive term ςl (t )T2(l (t ),t ) [z (l (t ), t )]2 , which make the system stability cannot be achieved. Thus, the virtual control γ (t) is designed to generate the cross term ς EIl(t)z   (l(t), t)z (l(t), t) and negative definite term −[z (l (t ), t )]2 and thus to stabilize them. From (14), we design the virtual control γ (t) as ς γ (t ) = − l (t )z (l (t ), t ) (15) ψ Please cite this article as: F. Guo, Y. Liu and F. Luo et al., Vibration suppression and output constraint of a variable length drilling riser system, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2018.10.023

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According to Eqs. (14) and (15), we get F˙x (t ) =

ς ρl (t ) 2 e (t ) + [−ψE I z (l (t ), t ) + ψT (l (t ), t )z (l (t ), t ) 2  ς l (t )T (l (t ), t ) ς 3 ρl 3 (t ) ς 2 ρl 2 (t )  − z (l (t ), t )]e(t ) − − ψ 2 2ψ 2  l (t )  l (t ) ςρ z˙2 (s, t )ds + ς sz (s, t ) f (s, t )ds × [z (l (t ), t )]2 − 2 0 0  l (t )  3ς E I l (t )  − ψc [vz (s, t ) + z˙(s, t )]2 ds − [z (s, t )]2 ds 2 0 0  1 l (t ) − [2ς csv + ς T (s, t ) − ς sT  (s, t ) − ς ρv2 − ψ T˙ (s, t ) 2 0  l (t ) − ψvT  (s, t )][z (s, t )]2 ds − ς c sz (s, t )z˙(s, t )ds − ψE I v[z (0, t )]2 + ψ

0



l (t )

f (s, t )[vz (s, t ) + z˙(s, t )]ds

(16)

0

3.1.2. Step Two In this step, e(t) can be stabilized at a small neighborhood of zero through designing appropriate controller u(t). Differentiating e(t) and combining the last equation of (5), (6) with Eq. (15), we can obtain e˙(t ) =

E I z (l (t ), t ) − ds y2 (t ) + u(t ) − T (l (t ), t )z (l (t ), t ) + d (t ) ς + v(t )z (l (t ), t ) m ψ  ς + l (t ) + v(t ) z˙ (l (t ), t ) ψ

(17)

The following Lyapunov candidate function is chosen Fz (t ) = Fx (t ) +

me2 (t ) 2Q2 ln 2 2 Q − y12 (t )

(18)

Differentiating Eq. (18) and combining with Eqs. (16) and (17) result in F˙z (t ) =

ς ρl (t ) 2 e (t ) + [ψT (l (t ), t )z (l (t ), t ) − ψE I z (l (t ), t ) 2 ς 2 ρl 2 (t )  − z (l (t ), t )]e(t ) + {u(t ) + E I z (l (t ), t ) ψ  mς + d (t ) − T (l (t ), t )z (l (t ), t ) + mv + l (t ) z˙ (l (t ), t ) ψ mς 2Q2 − ds y2 (t ) + v(t )z (l (t ), t )}e(t ) ln 2 ψ Q − y12 (t )  me2 (t )y1 (t )y2 (t ) ς T (l (t ), t )l (t ) ς 3 ρl 3 (t )  [z (l (t ), t )]2 + − − 2 2ψ 2 Q2 − y12 (t )   l (t ) ς ρ l (t ) 2 − z˙ (s, t )ds − ψc [vz (s, t ) + z˙(s, t )]2 ds 2 0 0

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  l (t ) 3ς E I l (t )  2 − [z (s, t )] ds + ς sz (s, t ) f (s, t )ds 2 0 0  1 l (t ) − [2ς csv + ς T (s, t ) − ς sT  (s, t ) − ς ρv2 − ψ T˙ (s, t ) 2 0  l (t )   2 − ψvT (s, t )][z (s, t )] ds − ς c sz (s, t )z˙(s, t )ds 0



l (t )

+ψ

f (s, t )[vz (s, t ) + z˙(s, t )]ds − ψE I v[z (0, t )]2

(19)

0

From Eq. (19), we design u(t) as u(t ) = −E I z (l (t ), t ) + ds y2 (t ) + T (l (t ), t )z (l (t ), t )  mς mς − mv + l (t ) z˙ (l (t ), t ) − v(t )z (l (t ), t ) − k1 e(t ) − d (t ) ψ ψ  2 2 1 (t )y2 (t ) ψT (l (t ), t ) − ς ρlψ (t ) z (l (t ), t ) + me(tQ2)y−y + k2 e(t ) − ψE I z (l (t ), t ) 2 1 (t ) − 2Q2 ln Q2 −y 2 (t )

(20)

1

where k1 , k2 > 0, and the estimate d (t ) is d (t ) = (t ) + k3 my2 (t )

(21)

where k3 > 0 and the dynamics of ϱ(t) is designed as

˙ (t ) = −k3 (t ) − k3 [E I z (l (t ), t ) + u(t ) − T (l , t )z (l (t ), t ) 2Q2 − ds y2 (t ) + mvz˙ (l (t ), t ) + k3 my2 (t )] + e(t ) ln 2 Q − y12 (t )

(22)

Define the corresponding estimation error as d (t ) = d (t ) − d (t )

(23)

Differentiating Eq. (23) and combining the last equation of (5), (21) and Eq. (22) yield ˙ (t ) = d˙(t ) − k d (t ) − e(t ) ln d 3

Q2

2Q2 − y12 (t )

(24)

Substituting Eq. (20) into Eq. (19) results in 2Q2 2Q2

(t )e(t ) ln + d Q2 − y12 (t ) Q2 − y12 (t )   ς ρl (t ) 2 ς T (l (t ), t )l (t ) ς 3 ρl 3 (t )  e (t ) − [z (l (t ), t )]2 − k2 − − 2 2 2ψ 2   l (t ) ς ρ l (t ) 2 − z˙ (s, t )ds − ψc [vz (s, t ) + z˙(s, t )]2 ds 2 0 0   l (t ) 3ς E I l (t )  − [z (s, t )]2 ds + ς sz (s, t ) f (s, t )ds 2 0 0

F˙z (t ) = −k1 e2 (t ) ln

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(a) z(l(t),t) (m)

1.5 1 0.5 0 0

(b) z(l(t),t) (m)

9

50

100

150 Time (s)

200

250

300

50

100

150 Time (s)

200

250

300

50

100

150 Time (s)

200

250

300

0.1

0.075 0.05 0.025 0 0

(c) z(l(t),t) (m)

0.75 0.5 0.25 0 0

Fig. 2. Output z(l(t), t) of the drilling riser system under boundary disturbance d (t ) = 1 + sin (πt ) + sin (2πt ) + sin (3πt ) and moving speed v(t ) = 0.007 + 0.00002t. (a) Without control, (b) With the proposed control (28), (c) With PD control (29).

−

1 2



l (t )

[2ς csv + ς T (s, t ) − ς sT  (s, t ) − ς ρv2 − ψ T˙ (s, t )

0

− ψvT  (s, t )][z (s, t )]2 ds − ς c − ψE I v[z (0, t )]2 + ψ



l (t )

sz (s, t )z˙(s, t )ds

0



l (t )

f (s, t )[vz (s, t ) + z˙(s, t )]ds

(25)

0

Lemma 4. The following Lyapunov candidate function is considered 1 F (t ) = Fz (t ) + d 2 (t ) 2

(26)

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(a) z(l(t)/2,t) (m)

0.6 0.4 0.2 0 0

(b)

150 Time (s)

200

250

300

50

100

150 Time (s)

200

250

300

50

100

150 Time (s)

200

250

300

0.2

z(l(t)/2,t) (m) z(l(t)/2,t) (m)

100

0.3

0.1 0 −0.1 0

(c)

50

0.45 0.3 0.15 0 0

Fig. 3. Output z(l(t)/2, t) of the drilling riser system under boundary disturbance d (t ) = 1 + sin (πt ) + sin (2πt ) + sin (3πt ) and moving speed v(t ) = 0.007 + 0.00002t. (a) Without control, (b) With the proposed control (28), (c) With PD control (29).

it can be concluded that there exist constants ϑ and ε ∈ R+ such that the inequality below holds F˙ (t ) ≤ −ϑF (t ) + ε

(27)

Proof. See Appendix B. Combining Eq. (6) with Eq. (15), the controller (20) can be updated as u(t ) = −d (t ) − (k1 − ds )z˙(l (t ), t ) + [ds v(t ) − k1 v(t ) + T (l (t ), t )  mς mς ς − v(t ) − l (t )k1 ]z (l (t ), t ) − E I z (l (t ), t ) − mv + l (t ) ψ ψ ψ ς   m[ z ˙ (l (t ) , t ) + vz (l (t ) , t ) + l (t ) z (l (t ) , t )] z(l (t ), t )[z˙(l (t ), t ) + vz (l (t ), t )] ψ  × z˙ (l (t ), t ) − 2Q2 {Q2 − [z(l (t ), t )]2 } ln Q2 −[z(l (t ),t )]2 Please cite this article as: F. Guo, Y. Liu and F. Luo et al., Vibration suppression and output constraint of a variable length drilling riser system, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2018.10.023

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100 0

˜( ) (N)

−100 −200 −300 −400 −500 0

50

100

150 Time (s)

200

250

300

Fig. 4. Disturbance estimation error d (t ).

−

[ψT (l (t ), t ) −

ς 2 ρl 2 (t ) ψ

+ k2 v(t ) +

ς l (t )k2 ]z (l (t ), t ) ψ 2Q2 ln Q2 −[z(l (t ),t )]2

− ψE I z (l (t ), t ) + k2 z˙(l (t ), t ) (28) 

Remark 3. In the light of the Lyapunov’s direct method for stability, the Lyapunov candidate function F(t) is required to be a positive definite function and its derivative satisfies F˙ (t ) ≤ −ϑF (t ) + ε. The energy term Fa (t) is designed to ensure the system energy convergence, 2 2Q2 which is relevant to the system mechanical energy. The auxiliary term me2(t ) ln Q2 −y is 2 1 (t ) designed to meet constraint need |z(l(t), t)| < Q, which is relevant to the kinetic energy of the tip payload. The error term 21 d 2 (t ) is designed to cope with boundary disturbance d(t) and the crossing term Fb (t) is designed to enhance the steadiness research process. In combination with the system dynamics Eqs. (1) and (2), both the Lyapunov candidate function and controller are constantly revised and improved until the appropriate F(t) and u(t) are derived. Remark 4. Unlike existing relevant literatures [3,4,12,32,33] on vibration control for the riser systems, this paper not only thinks about vibration abatement of a variable length drilling riser, but also takes account of output constraint problem. Through the integrated use of boundary control, finite-dimensional backstepping technique and BLF, a boundary control scheme with output signal barrier is developed aiming at realizing the vibration abatement and constraint satisfaction, in which BLF is proposed to ensure the boundary output within the constrained range and the disturbance observer containing the barrier term is designed to counteract boundary disturbance d(t). It is worth stressing that the proposed control scheme in this paper can effectively eliminate chattering, which always appears in sliding mode control and caused by discontinuous switch character [41,42]. Please cite this article as: F. Guo, Y. Liu and F. Luo et al., Vibration suppression and output constraint of a variable length drilling riser system, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2018.10.023

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(a) z(l(t),t) (m)

1.5 1 0.5 0 0

z(l(t),t) (m)

(b)

50

100

150 Time (s)

200

250

300

50

100

150 Time (s)

200

250

300

50

100

150 Time (s)

200

250

300

0.1 0.05 0

−0.05 −0.1 0

z(l(t),t) (m)

(c)

0.75 0.5 0.25 0 0

Fig. 5. Output z(l(t), t) of the drilling riser system under boundary disturbance d1 (t ) = 2[sin (0.5πt ) + sin (2πt )] and moving speed v1 (t ) = 0.005 + 0.00006t. (a) Without control, (b) With the proposed control (28), (c) With PD control (29).

3.2. Stability analysis Theorem 1. For the studied variable length drilling riser system (1) and (2), under proposed boundary control scheme (28), Assumptions (1) and (2) and bounded initial conditions, when appropriate parameters ς , ψ, k1 ∼ k3 and δ 1 ∼ δ 4 are selected to satisfy constraints (B.3), there is a conclusion that the system signal z(s, t) is uniformly ultimately bounded. Proof. See Appendix C.  Remark 5. Combining Eq. (B.2) withEq. (B.5), quite evidently, increasing k3 will cause a 2Lε relatively larger ϑ, then the value of ψTmin in Eq. (C.5) will be decreased, which will α1 ϑ ultimately improve the control effect. But the increase of k3 will also cause high gain control. Based on this, it is of fundamental importance to select suitable control parameters in practice. Please cite this article as: F. Guo, Y. Liu and F. Luo et al., Vibration suppression and output constraint of a variable length drilling riser system, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2018.10.023

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(a) z(l(t)/2,t) (m)

0.6 0.4 0.2 0 0

(b) z(l(t)/2,t) (m) z(l(t)/2,t) (m)

50

100

150 Time (s)

200

250

300

50

100

150 Time (s)

200

250

300

50

100

150 Time (s)

200

250

300

0.3 0.2 0.1 0 0

(c)

13

0.45 0.3 0.15 0 0

Fig. 6. Output z(l(t)/2, t) of the drilling riser system under boundary disturbance d1 (t ) = 2[sin (0.5πt ) + sin (2πt )] and moving speed v1 (t ) = 0.005 + 0.00006t. (a) Without control, (b) With the proposed control (28), (c) With PD control (29).

4. Simulation analysis The performance of studied drilling riser system with proposed control scheme (28) is simulated using the finite difference method (FDM) [43,44], where the space and time step sizes are selected as s = 3 m and t = 3 × 10−3 s, respectively. It should be emphasised that the sizes of space and time step will directly affect the FDM’s stability and simulation precision. The system parameters are ρ = 500 kg/m, m = 75 kg, L = 300 m, c = 2.0 Ns/m2 , E I = 1.5 × 107 Nm2 , ds = 1.0 Ns/m, Q = 0.1 m, v(t ) = 0.007 + 0.00002t and T (s, t ) = s [m + ρ(l (t ) − s)](g − v˙). The initial conditions are z(s, 0) = 5000 and z˙(s, 0) = 0. The external environmental disturbance is given as d (t ) = 1 + sin (πt ) + sin (2πt ) + sin (3πt ), and distributed ocean current disturbance f(s, t) is given in [32]. For analysing and validating the control performance, the following three different cases are simulated for the studied drilling riser system: Please cite this article as: F. Guo, Y. Liu and F. Luo et al., Vibration suppression and output constraint of a variable length drilling riser system, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2018.10.023

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100 0

˜1( ) (N)

−100 −200 −300 −400 −500 0

50

100

150 Time (s)

200

250

300

Fig. 7. Disturbance estimation error d 1 (t ).

(1) Without control: The deflections of endpoint and midpoint of the studied system under no control, namely, u(t ) = 0, are respectively depicted in Figs. 2(a) and 3(a). (2) With the proposed control (28): When the proposed control scheme (28) acts on studied drilling riser system, the corresponding deflections of the endpoint and midpoint are respectively depicted in Figs. 2(b) and 3(b), where the parameters are selected as ς = 10−4 , ψ = 0.1, k1 = 5 × 103 , k2 = 2 × 104 and k3 = 0.1. (3) With PD control: When the following PD control (29) acts on studied drilling riser system, the deflections of the endpoint and midpoint are respectively depicted in Figs. 2(c) and 3(c), where the parameters are selected as ka = 20 and kb = 500.

u1 (t ) = −ka z˙(l (t ), t ) − kb z(l (t ), t )

(29)

To further show the robustness of proposed control scheme (28), we choose a different set of boundary disturbance and moving speed, i.e., d1 (t ) = 2[sin (0.5πt ) + sin (2πt )] and v1 (t ) = 0.005 + 0.00006t, the corresponding deflections of the endpoint and midpoint of studied drilling riser system under the above three cases are respectively displayed in Figs. 5 and 6, and the time histories of the disturbance estimation errors d (t ) and d 1 (t ) are respectively given in Figs. 4 and 7. As shown in Figs. 2 and 3, there exists a large vibration deflection for the uncontrolled drilling riser system. Under the action of proposed control scheme (28) and PD control (29), the deflections of the endpoint and midpoint of the studied system are all greatly reduced. However, it’s clear that proposed control scheme (28) shows a better control performance compared with PD control (29), besides, only under the proposed control scheme (28) can the deflections of system endpoint be guaranteed within the constrained region, i.e., |z(l (t ), t )| < Q, Q = 0.1 m. Moreover, combining with Figs. 5 and 6, through the comparison of control performance for different boundary disturbance and moving speed, the validity and robustness of proposed boundary control scheme (28) can also be further proved. As shown in Figs. 4 Please cite this article as: F. Guo, Y. Liu and F. Luo et al., Vibration suppression and output constraint of a variable length drilling riser system, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2018.10.023

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and 7, the estimation errors of boundary disturbances d (t ) and d 1 (t ) can stabilize near zero as time goes by. All in all, the proposed control scheme (28) can realize the control objectives of vibration abatement and constraint satisfaction. 5. Conclusion This paper discussed the vibration control and constraint issues of a drilling riser system with variable length, variable tension, variable speed and restricted output. Through a combination of backstepping technique, boundary control method and BLF, an appropriate boundary control scheme with output signal barrier was developed, where the disturbance observer was employed for handling boundary disturbance, and BLF was adopted for avoiding constraint violation. Moreover, simulation results were given for demonstrating the usefulness of proposed control scheme. In the future work, we will focus on the practical experiment investigation to further demonstrate the effectiveness of the proposed control scheme. Acknowledgements This work was supported by the National Natural Science Foundation of China (61203060); the Science and Technology Planning Project of Guangdong Province (2017B090910006, 2016A010106007, 2017B010117007, 2017B090910011); the Fundamental Research Funds for Central Universities of SCUT (2017ZD058); the Opening Foundation of Key Researcfh and Development Program of Guangdong province in 2018-2019. Appendix A. Proof of Lemma 3 Proof. Applying Lemma 1 to Fb (t) leads to   ς ρL l (t ) ς ρL l (t )  | Fb (t ) |≤ [z (s, t )]2 d s + [z˙(s, t ) + vz (s, t )]2 d s ≤ βFa (t ) 2 0 2 0

(A.1)

where β = ψ minςρL . (Tmin ,ρ) Rearranging inequality Eq. (A.1) yields −βFa (t ) ≤ Fb ≤ βFa (t )

(A.2)

Choosing ς and ψ appropriately for ensuring 0 < β < 1, combining Eq. (A.1), we have ς< 

ψ min (Tmin , ρ) ρL

(A.3)

Defining β1 = 1 − β and β2 = 1 + β and combining Eq. (A.3) yield β1 = 1 − β2 = 1 +

ςρL ψ min (Tmin ,ρ) ςρL ψ min (Tmin ,ρ)

>0 >1

(A.4)

Combining Eqs. (7), (A.2) with Eq. (A.4) yields 0 < β1 Fa (t ) ≤ Fx (t ) ≤ β2 Fa (t )

(A.5)

which completes the proof.  Please cite this article as: F. Guo, Y. Liu and F. Luo et al., Vibration suppression and output constraint of a variable length drilling riser system, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2018.10.023

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Appendix B. Proof of Lemma 4 Proof. Taking the derivative of F(t) leads to ˙ (t ) F˙ (t ) = F˙z (t ) + d (t )d

(B.1)

Substituting Eqs. (24) and (25) into Eq. (B.1) and combining Lemmas 1 and 2 yield  2Q2 ς ρl (t ) 2 e (t ) − (k3 − δ1 )d 2 (t ) F˙ (t ) ≤ −k1 e2 (t ) ln 2 − k − 2 2 Q − y12 (t )  ς T (l (t ), t )l (t ) ς 3 ρl 3 (t )   2 [z (l (t ), t )]2 − ψE I v[z (0, t )] − − 2 2ψ 2    l (t ) ςρ ς cL  2 − ψ (c − δ3 ) [vz (s, t ) + z˙(s, t )] ds − − 2 δ2 0  l (t )  l (t )  l (t ) ς E I 3 1 z˙2 (s, t )ds − [z (s, t )]2 ds − [2ς csv × 2 2 0 0 0 + ς T (s, t ) − ς sT  (s, t ) − ς ρv2 − ψ T˙ (s, t ) − ψvT  (s, t )   ψ 1 ˙2 ςL  2 − 2ς cLδ2 − 2ς Lδ4 ][z (s, t )] d s + d (t ) + + δ1 δ3 δ4  l (t ) f 2 (s, t )ds × 0  me2 (t ) 2Q2 1 2 ≤ −ϑ1 Fa (t ) + ln 2 + d (t ) + ε 2 Q − y12 (t ) 2 (B.2) 

˙



T −ψvT −2ς cLδ2 −2ς Lδ4 where δ 1 ∼ δ 4 are all positive constants, ϑ1 = min[ 2ς cLv+ς Tmin −ς LT −ς ρvψT−ψ , max  l (t ) 2 ψ 3ς 2(c−δ3 ) 2k1 ςL 1 ˙2 , ρ , m , 2(k3 − δ1 )] > 0 and ε = max[ δ1 d (t ) + ( δ3 + δ4 ) 0 f (s, t )ds] < +∞. Beψ sides, combining with Eq. (A.3), the constants ς , ψ, k1 ∼ k3 and δ 1 ∼ δ 4 are selected to satisfy the following conditions ⎧ (Tmin ,ρ) ς < ψ minρL ⎪ ⎪ ⎪ ⎨k − ςρL ≥ 0 2 2 3 3 (B.3) ςL ⎪ T − ς2ψρL2 ≥ 0 min ⎪ 2 ⎪ ⎩ ςρ − ςcL ≥0 2 δ2

According to Eqs. (10), (18) and (26), we can obtain  me2 (t ) 2Q2 1 2 0 < α1 Fa (t ) + ln 2 + d (t ) ≤ F (t ) 2 Q − y12 (t ) 2  me2 (t ) 2Q2 1 2 ≤ α2 Fa (t ) + ln 2 + d (t ) 2 Q − y12 (t ) 2

2

(B.4)

where α1 = min (β1 , 1) and α2 = max (β2 , 1). Combining Eq. (B.2) with Eq. (B.4) leads to F˙ (t ) ≤ −ϑF (t ) + ε

(B.5)

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where ϑ = ϑα21 . Therefore, Lemma 4 is proved completely.  Appendix C. Proof of Theorem 1 Proof. Multiplying Eq. (27) by eϑt yields F˙ (t )eϑt ≤ −ϑF (t )eϑt + εeϑt Integrating (C.1), we can obtain  ε ε ε −ϑt F (t ) ≤ F (0) − e + ≤ F (0)e−ϑt + ϑ ϑ ϑ

(C.1)

(C.2)

According to Lemma 2 and Assumption 2, combining Eq. (8) with Eq. (B.4) yields  ψTmin 2 ψ l (t ) z (s, t ) ≤ T (s, t )[z (s, t )]2 ds ≤ Fa (t ) 2L 2 0 me2 (t ) 2Q2 1 1 ≤ Fa (t ) + ln 2 + d 2 (t ) ≤ F (t ) (C.3) 2 2 α1 Q − y1 (t ) 2 Substituting Eq. (C.3) into Eq. (C.2), we can obtain  2L  ε | z(s, t ) |≤ F (0)e−ϑt + ψTmin α1 ϑ ∀(s, t ) ∈ [0, L] × [0, +∞ ). Thus, we can further obtain   2L ε 2Lε lim | z(s, t ) |≤ lim [F (0)e−ϑt + ] = t→∞ t→∞ ψTmin α1 ϑ ψTmin α1 ϑ

(C.4)

(C.5)

∀s ∈ [0, L], which completes the proof.  References [1] C.Y. Sun, W. He, J. Hong, Neural network control of a flexible robotic manipulator using the lumped spring-mass model, IEEE Trans. Syst. Man Cybern. Syst. 47 (8) (2017) 1863–1874. [2] X. Liu, C.G. Yang, Z.G. Chen, M. Wang, C.Y. Su, Neuro-adaptive observer based control of flexible joint robot, Neurocomputing 275 (2018) 73–82. [3] B.V.E. How, S.S. Ge, Y.S. Choo, Active control of flexible marine risers, J. Sound Vib. 320 (4–5) (2009) 758–776. [4] K.D. Do, J. Pan, Boundary control of transverse motion of marine risers with actuator dynamics, J. Sound Vib. 318 (4) (2008) 768–791. [5] A.A. Paranjape, J. Guan, S.J. Chung, M. Krstic, PDE boundary control for flexible articulated wings on a robotic aircraft, IEEE Trans. Robot. 29 (3) (2013) 625–640. [6] W. He, S. Zhang, Control design for nonlinear flexible wings of a robotic aircraft, IEEE Trans. Control Syst. Technol. 25 (1) (2017) 351–357. [7] Y. Liu, Z.J. Zhao, W. He, Boundary control of an axially moving system with high acceleration/deceleration and disturbance observer, J. Frankl. Inst. 354 (7) (2017) 2905–2923. [8] Z.J. Liu, J.K. Liu, W. He, Modeling and vibration control of a flexible aerial refueling hose with variable lengths and input constraint, Automatica 77 (2017) 302–310. Please cite this article as: F. Guo, Y. Liu and F. Luo et al., Vibration suppression and output constraint of a variable length drilling riser system, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2018.10.023

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