Vibration attenuation control of ocean marine risers with axial-transverse couplings

The target of this paper is designing a boundary controller for vibration suppression of marine risers with coupling mechanisms under environmental loads. Based on energy approach and the equations of axial and transverse motions of the risers are derived. The Lyapunov direct method is employed to formulated the control placed at the riser top-end. Proof of existence and uniqueness of the solutions of the closed-loop system is provided. Stability analysis of the closed-loop system is also included.

INTRODUCTION Due to its physical structure, a riser basically is modeled as a tensioned beam [1], [2], [3], and [4]. In [5], an active boundary control that produces a vibration-free for an Euler-Bernoulli beam system was designed. Similar use of distributed control can be found in [6]. In [7], the authors used differential evolution optimization to search for the best controller model structure and its parameters for beam control problem. The proposed controller is able to suppress the beam?s vibration without knowledge of the system. However, the searching process is conducted within a set of predefined control structures, no proof of the effectiveness of the control was given.
With efforts to make voltage-source converter (VSC) more efficient in handling distributed parameter systems, sliding-mode control (SMC) was given extra flexibility by adding a neural network and fuzzy control in [8]. The author yield a control law in the form of a mass-damper-spring system at the boundary of a moving string. However, difficulties in selecting proper fuzzy membership functions and a slow convergence speed due to online-tuning might be troublesome when applying the aforementioned controls. After accepting that SMC is non-analytical in the sliding surface, in the first control structure, a boundary layer was defined that enabled fuzzy control by taking a switching function and its derivative as inputs while SMC was activated outside this boundary to achieve fast transient responses. A series of papers with applications of SMC to flexible system can be found in [9], [10], and [11]. A second attempt was made to design a fuzzy neural network control (FNNC) that also employed switching variables as its inputs. The proposed FNNC conducted an online-tuning process to regulate fuzzy reasoning to compromise system uncertainties. Both controls resulted in a variation of axially moving string tension as the control action.
In [12], a beam model representing a tensioned riser is investigated, and a boundary controller consisting of the top-end rise information is designed to achieve exponential stability. Krstic tematic approach based on backstepping control for beam-type structure in [13] and [14]. In [15], the authors proposed a control assisted by a disturbance estimator to guarantee asymptotic stability of an Euler-Bernoulli beam system subjected to unknown disturbances. He, et al. In [16], successfully develop a boundary control for a flexible riser with vessel dynamics. In [17], the authors introduce control based on Lyapunov's approach. Through Lyapunov's direct method, the riser's transverse motion under time-varying distributed loads stability is established. A control problem for a coupled nonlinear riser exhibiting longitudinal-transverse couplings is investigated in [3]. Analogous applications to flexible systems are evidenced in [18], [2], and [19]. Since the surface vessel is always control by a dynamic positioning system in practice [20], [21], [22], [23], [24], and [25] the vessel's motions normally are not considered. The paper deals with the vibration control problem for marine risers under environmental disturbances. In addition, the longitudinal-transverse coupling in the riser motion in taken into account. Different from [26], the control is formulated without the assumption of positive tension. Existence, uniqueness, and convergence of the solutions of the closed-loop system is verified in the paper.

MATHEMATICAL FORMULATION
The riser kinetic energy is specified by where u(z, t) is transverse displacements in the X direction and w(z, t) is longitudinal displacement in the Z direction. L denote the riser length, m 0 = ρA is the riser oscillating mass per unit length, A is the riser cross-section area, and ρ represents the mass density of the riser. Assuming that the riser is constrained by constant tension P 0 . The riser potential energy is given as where E is the Young's modulus and I is the second moment of the riser's cross section area. The hydrodynamic forces can be given as [26] where f uD , f wD and f uL , f wL correspond to the distributed damping and external forces. The work done by the hydrodynamic forces acting on the system is calculated as The work done by boundary control is where U u (L, t) and U w (L, t) are the boundary control forces. The total work done on the system is W For the sake of clear presentation, (z, t) is omitted whenever it is applicable. The kinetic energy variation can be written as where δu = δv = δw = 0 at t = t 1 , t 2 have been used. For the riser under consideration, ball joints arranged at both ends ( Figure 1) implying that bending free. In addition, the lower end stationed at the well-head. The riser dynamics is yielded

CONTROL DESIGN
In order to minimize the riser vibration using measured state and applied forces at the top end, we consider the following Lyapunov candidate function Since ∀t ≥ 0 and u(0, t) = w(0, t) = 0, it can be shown that where γ 1 and γ 2 are positive constants, it can be deduced that ❒ ISSN: 1693-6930 The (9) can be lower and upper bounded by and If we select ρ 1 , ρ 2 , γ 1 , and γ 2 such that: where c i , for i = 1 . . . 4, are strictly positive constants. Differentiating (9) and taking (8) into account yieldṡ Since − and noted that −EIu zzz (L, t)+P 0 u z (L, t)+ EA 2 u 3 z (L, t)+EAw z (L, t)u z (L, t) = U u (L, t) and EAw z (L, t)+ EA 2 u 2 z (L, t) = U w (L, t), the boundary controls are designed as follows,  (16) givesV Remark: It is noted that the authors of [26] use the assumption the riser is alsway stretched in order to conclude that L 0 u 2 z w z dt is positive. This is not the case in practice since the riser can be bulked or stretched according to external disturbance. Considering the following term where ∆ con be written as To remove the requirement of positive tension, we use the following property [27] that From (20), the designed parameters are selected such that where c 1 , for i = 5 . . . 9, are strictly positive constants. Applying the upper bound of V in (14), (20) can be written asV where c = min c 5 , c 6 , c 7 , c 8 , ρ1EI m0 , ρ1EA 2m0 , β 1 max m0 Remark: Different from [16], the control design process is carried out in this chapter without any assumptions on boundedness of time and spatial derivatives of the riser system. Equation (26) can be written aṡ where An upper bound of ∆ c can be written as There exists a strictly positive constant ξ such that the following inequality holds From the lower bound of V , it is shown that where Substituting (32)and (33) into (29) giveṡ where and If ξ is picked such thatc = c − ξ ζ is strictly positive, then: Inequality (38) implies that V (t) exponentially converges to nonnegative constant 1 ξ Q. Using Inequality A.2 [26], it can be conclude that all terms |u(z, t)| and |w(z, t)| are bounded and exponentially converge to a non-negative constant defined be the value of external disturbances.

NUMERICAL SIMULATIONS
At this stage, we illustrate the advantages of the proposed control through a set of simulations. The marine riser system parameters are given as in Table 1 The linear current velocity vector in a form of V = [ 1 L s, 0.5 L s, 0] T is employed in numerical simulations. The hydrodynamic forces can be given as [26]. Simulations are carried out without the proposed control and with the control by set k 1 = k 2 = 500. The riser displacements in the X and Z directions for uncontrolled and controlled cases are plotted in Figure 2 and Figure  3, respectively. It can be observed that when the control is activated, displacement magnitudes in all directions (X and Z) are reduced. The reduction in displacement magnitudes illustrates the effectiveness of the proposed control in driving the riser to the vicinity of its equilibrium position. It also can be observed in Figure 4 that the control forces required to drive the risers are reasonable for the riser under consideration.

CONCLUSIONS
The paper copes with minimizing vibration of the marine riser. After deriving the set of equations specifying the riser dynamics, the boundary controller applied at the riser top end is designed thank to Lyapunov's direct method without the assumption of positive tension applied to the riser. The ability in stabilizing the riser at its equilibrium position of the boundary control is validated analytically and illustrated numerically.