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Vibration Control of Nonlinear Three‐dimensional Marine Riser Model with Output Constraints

: A typical three-dimensional flexible marine riser system described as a distributed parameter system with several partial differential equations and ordinary differential equations is taken into account in this article, we are aiming at limiting the displacement of riser within restricted range. Appropriate boundary control laws by integrating backstepping technique with barrier Lyapunov functions are put forwarded to suppress the vibration of flexible riser under the external disturbances. With the designed boundary control, Lyapunov synthesis method is used to prove the stability of the closed-loop system without any discretization or simplification of the dynamic in the time and space when the initial conditions meet the requirements. Furthermore, in order to illustrate the effectiveness of proposed control laws, numerical simulation studies are carried.


Introduction
The flexible structures are widely used in various fields such as manipulators [1][2], wings [3][4] and other engineering applications due to their light weight, low cost and high efficiency. Marine risers are modeled as typical flexible structures [5][6][7]. In the process of deep-water drilling operations, marine risers play an increasingly critical role, which is used to isolated the internal drilling fluid from the external seawater, establish drilling fluid circulation channels and transport oil and gas raw materials and other undersea economic resources to production lines. However, it is highly likely to cause vibration and deformation influenced by the surrounding disturbance, which can not only cause premature vulnerability problems, shorten its service life, but also require expensive maintenance [8][9].
Motivated by this, an increasingly number of scholars devoted themselves to the studies on vibration control of marine riser and many control techniques have been presented. Such as finite element method, modal analysis, boundary control and other control methods. The flexible marine riser system can be considered as an infinite dimension model, when the high-frequency parts of model are deleted, the system will appear "spillover" phenomenon, and boundary control can solve the phenomenon well. Therefore boundary control is more suitable than other methods for infinite dimension model. In the existing corresponding works, the riser system is modeled as a Euler-Bernoulli beam because the ratio of its diameter to length is very small, and the dynamic behavior of it is described as a distributed parameter system composed of partial differential equations (PDEs) and ordinary differential equations (ODEs). For control problems of DPS [10]- [12], one of the inevitable challenges is how to use finite sensors to achieve the goal of controlling an infinite-dimensional model. This challenge has been overcome by boundary control, the designed boundary control can decrease the vibration of marine riser [13]- [15].In [16], an adaptive feedback controller based on boundary control is proposed to stabilizing the riser system, the high-gain observer and the robust strategy are used to handle the uncertainty of the system. Besides, combine boundary control with backstepping technique and adaptive observer to tackled with the uncertain parameter in system, the actuators can obtain better control performance which can be seen in [17].
Despite the researches on control of flexible riser have acquired desired results, they only focus on the transverse vibration in two-dimensional space. In fact, the marine riser produces more than just transverse vibration under the complex environmental interference. To reduce the lateral and longitudinal vibration of a coupled nonlinear flexible riser, two actuators are set on the top of the riser through designing proper boundary control in [18]. The barrier Lyapunov function (BLF) has been increasingly used in partial differential equations, although it was originally proposed to deal with ordinary differential equations and many remarkable results about BLF have been achieved. By combining backstepping and Lyapunov method, boundary control with observer and barrier Lyapunov function is introduced to solve the output constraint problem of flexible riser system [19]. In work [20], a controller with integral-barrier Lyapunov function and boundary control is designed to suppress the riser's vibration with a top tension constraint. To handle with top-level tension and input saturation constraint, boundary control with barrier term and auxiliary system is exploited in [21].
In order to make further study, the flexible riser system is studied in three-dimensional space. On the basis of integral-barrier Lyapunov function, the objective of reducing the vibration of riser under the external nonlinear disturbances has been accomplished, three actuators are equipped at the top of the riser and boundary control makes sure the joint angles in the limited range at [22]. Inspired by this work, the author proposed different control laws aimed at limiting the amplitude at the top of the riser within constrained range and suppressing the vibration of riser in three-dimensional space.

Boundary control integrated with backstepping technique and barrier Lyapunov function is employed.
Compared with the existing works, the main contributions of this paper are summarized as follows: (i)The flexible riser system under the external disturbance in three-dimensional space is described as a distributed parameters system which increases the accuracy of establishing model and the difficulties of the control designing greatly.
(ii)Based on the backstepping technique and barrier Lyapunov function, boundary control with barrier terms is proposed to guarantee the stability of the system and achieve the constraint of the top displacement of riser system.
(iii)By comparing the proposed control with the traditional PD control via numerical simulations, it can be concluded that regardless of whether there is distributed interference or not, the performance of proposed control is better than that of PD control .

Dynamics and preliminaries
A typical marine riser in three-dimensional is shown in Figure 1. The controllers are implemented from three actuators installed at the top boundary of the riser. We ignore the effect of gravity and make t s, be independent spatial and time variables. For clarity, the notations, t  are introduced throughout the whole paper.

Dynamic Analysis
The kinetic energy of the system Ek can be represented as: Where the M denotes the mass of the vessel, ) ,  The potential energy of the system E p can be represented as: Where EI is the bending stiffness of the riser and T is the top tension of the riser. EA is the axial stiffness of the riser. on the tip payload is given as: The virtual work done by boundary control force to suppress the vibration and prevent the top amplitude from the constraint can be represented as: Therefore, the total virtual work ) (t W  done on the system can be described as: ， and the boundary conditions are as follows：

Preliminaries
Definition 1 [25]: A barrier Lyapunov function (BLF) which is a scalar function on an open region D containing the origin, which is continuous , positive definite, has continuous first-order partial derivatives at every point of D ,has the property   ) (x V as x approaches the boundary of D and satisfies 0 , and a positive constant b .

Lemma 1[26]: Let
  ,the following inequalities hold: represent the constraints of three directions.

Lemma 3 [25]: Let
continuous in t and locally Lipschitz in ,continuously differentiable and positive definite in their respective domains, such that: ,in the rest of paper.
Property 1 [27]: If kinetic energy given by equation (1) of the system (6)- (14) is bounded, then the Property 2 [27]: If potential energy given by equation (2) of the system (6)- (14) is bounded, then the Assumption 1 [22]: For the known distributed ocean disturbances and the unknown boundary ,these are reasonable assumptions, because the disturbances have finite and bounded energy. Furthermore, the knowledge of exact values of the disturbances are not required.

Boundary Control Design
The control objective is to suppress the vibration of the riser and stabilize the riser system at constrained range in three-dimensional space, even though the presence of distributed disturbance and boundary disturbance. The control inputs are proposed to stabilize the closed-loop system and make the For the given riser system, we propose the following control laws ： To use the backstepping technique conveniently, the PDE dynamics (6)-(14) of considered riser system can be transformed as: Step one ： to express the virtual control of Consider the Lyapunov function candidate as： Where 1 V is energy term, which is designed on account of the kinetic and potential energies of riser system. The small crossing term 2 V is designed to promote stability analysis. 3 V is associated with the state of the system.
Differentiating (30) and combining the riser model in three-dimension (26) with (27) From the (35), the virtual controls can be designed as: Take the inequality (15)    Step two: In this step, the controllers are designed to regulate the errors Choose the Lyapunov function candidate b V as: Differentiating (43) and (44):

Stability Analysis.
Theorem 1: For the considered riser system (6)- (14), under the proposed controls (20) .the states of the system will eventually converge to the compact 2  defined by:  (16), (34) and (47), we obtain Rearranging the terms of above three inequalities, we can obtain ,Consequently, we know that

Remark 4:
According to the property 1 and property 2, we can draw a conclusion that And we can conclude that control inputs

Simulation
In this part, to verify the effectiveness and practicability of the proposed control laws (20)-(22), the finite difference method (FDM) [32]- [35] is chosen to simulate the performance of system. The corresponding initial conditions of the system are given as: Reader can refer to [8] for the detailed parameters of distributed disturbance on the riser. In this paper, the distributed disturbance cause effects in all three directions, therefore, we assume that The performance of proposed control is shown in Figure 5. The top displacement of riser with proposed control is shown in figure 6. Where the PD control parameters are selected as:  Figure 3. Displacement of the riser without distributed disturbance with PD control. (c) Figure 6. Top displacement of the riser without distributed disturbance with proposed control.
(iii): When there exist distributed disturbance, the responses of riser system with above PD control and parameters are shown in Figure 7. Top displacement of the riser with distributed disturbance under PD control is depicted in Figure 8. The performance of proposed control with above control parameters is depicted in Figure 9. Top displacement of the riser with distributed disturbance under proposed control is depicted in Figure 10. range than PD control in all directions, which can be seen in Figure 11. In addition, we compare the performances of the proposed control with distributed interference and without distributed disturbance, the results are shown in Figure 12. In order to show the simulation results more intuitively, we list the following table: Table 1 Comparison of simulation results Remark 6: From above table, we can find that it is easy to violate the requirements of the American Petroleum Institute which is that the average value of the upper and lower joint angle keep within 2ﾟ and the maximum non drilling angle should be limited to 4 ﾟ [14] when there is no control. As we can know from above simulation results, both PD control and proposed control can reduce the amplitudes of vibration in three directions. Obviously, the performance of proposed control are far better than that of PD control. Whether there is distributed disturbance or not, the vibration of three directions will stay in a very small range with proposed control while PD control is only to reduce the amplitude of vibration a little. At the same time, we can know that the distributed disturbance decrease the control effects on riser.  control. Therefore, how to overcome the distributed disturbance will be a meaningful study in the future.

Appendix A. proof of lemma 3
Proof. Applying the inequality (15) and equality (31) yields: Then we can obtain: Then we further yield: Combining with the Lyapunov function (30), we obtain: And the designed parameters are selected to satisfy the following conditions: