Output feedback control of a flexible marine riser with the top tension constraint

Abstract

In this paper, for the flexible marine riser system with the top tension constraint, the output feedback controller is developed to suppress the vibration of the riser. The dynamic behavior of the flexible riser is expressed by the partial differential equation (PDE). Firstly, a high-gain observer is designed to estimate unmeasurable system states. Secondly, the top tension of the riser is constrained with the aid of the logarithmic barrier Lyapunov function, and the output feedback boundary controller is developed using the direct Lyapunov method. Through rigorous mathematical analysis, the consistently bounded stability of the closed-loop system is achieved. Finally, the simulation is carried out to show that the developed control can stabilize the system with good performance under proper selection of design parameters.

Introduction

Over recent years, oil and gas production activities in deep sea development have increased dramatically, and more and more oil and gas resources are being discovered in deep sea. Regardless of the type of floating device used for processing, the marine riser is always an important equipment unit of the marine infrastructure [1], [2]. The flexible marine riser is the connected pipeline from the subsea oil and gas wellhead to the vessel, which has a significant function in the development of marine oil and gas [3]. As the development of deep water resources, the marine riser faces increasingly harsh environment with risers subjected to wind, wave, ocean current, ice and earthquake and other loads [4]. These external disturbances lead to undesired vibration and deformation of the riser, and also increase the top tension. If the top tension is over the limit, it will cause the riser to break. The oil and gas leakage from the damaged area will lead to marine environment pollution [5], [6]. Therefore, the top tension of the riser needs to be limited to a constraint range during design and production operations. Meanwhile, in order to ensure the production safety during the working of flexible risers, appropriate control forces need to be applied to suppress the vibration of the riser [7], [8], [9].

In view of the small ratio of the diameter to the length of the marine riser, the flexible marine riser system is commonly modeled as an Euler–Bernoulli beam structure [10]. Mathematically, the dynamics of the flexible marine riser is modeled by the distributed parameter system (DPS) [11], [12], [13]. For the DPS, the traditional control approach is the assumed mode method. The method is suitable for control design based on truncated finite-dimensional models, and will produce control overflow instability if applied to infinite-dimensional systems. The distributed control has better control effect compared to the mode method [14]. But the number of actuators and controllers will be increased by applying the distributed control, which will lower its practicality and become more unimplementable [2]. Compared with the former two approaches, the boundary control is simple in structure, which can be applied to infinite dimensional systems, and requires few actuators. Hence, the boundary control is considered to be a economical, efficient and easy implement approach for stabilizing the flexible riser system [15]. The Prof. Krstic has made pioneering contributions in the areas of distributed parameter system control and PDE boundary control [16], [17], [18], [19], [20]. Among them, the book of [16] introduced in detail boundary control, PDE backstepping method, observer design for hyperbolic PDEs, etc., which became an important reference book in the field of PDE control.

Numerous achievements in boundary control of flexible structures were already made within recent years [6], [21], [22], [23], [24], [25], [26]. In [6], a boundary controller was developed for a riser system having time-varying internal fluid. The output regulation problem of boundary feedback control for the nonlinear PDE processes was studied in [21]. [22] introduced new methods for the design of feedback controllers for infinite dimensional systems from boundary control. The boundary control scheme was developed to design active constrained layer damping (ACLD) to deal with the beam vibrations in [24]. For the flexible 3D riser system, the authors designed a boundary controller in [25], in which faults were compensated using the Nussbaum function. Additionally, a control method combining boundary control with iterative learning control was utilized in [26], to restrain the vibration of a riser involving periodic external disturbances. To be noted, the above results fail to consider the constraint problem. Nevertheless, almost all practical systems are subject to various forms of constraints. Violation of these constraints degrades the performance of the system and causes system instability. Consequently, it is meaningful and essential to consider constraints in the controller design process.

Nowadays, the barrier Lyapunov function (BLF) approach is widely used to solve constraint problems [15], [17], [27], [28], [29], [30], [31], [32]. The BLF is a Lyapunov function which tends to infinity when the constrained variable in the BLF is close to a certain limit. The analysis in the closed-loop system ensures that the BLF is bounded, which in turn ensures that the constraint cannot be violated [27]. In [28], a novel time-varying integral BLF was introduced in the adaptive control of time-varying full-state constrained nonlinear systems. Two adaptive control methods were constructed for nonlinear systems, employing the Nussbaum gain technique and the logarithmic BLF to deal with unknown control directions and full-state constraint problem [29]. In [15], with the help of BLF, a boundary control scheme was proposed to restrain the vibration of the riser, while ensuring that the top tension does not violate the constraint. The vibration problem of variable-length drilling riser with output constraints was studied in [17], where a proper control strategy was proposed through the combination of BLF method and the backstepping technique. In above research results, all signals were assumed to be measurable, and the case of non-measurable signals was ignored, which will lead to the inability to control the system accurately.

Actually, the noise of the sensor is unavoidable. So for system states obtained by the backward difference method, the noise error will be further amplified during the differential calculation, which will affect the performance of the controller. For this problem, state observers usually need to be introduced to solve it [18], [19], [20], [33], [34], [35], [36], [37], [38], [39], [40], [41], [42], [43]. In [33], the linear Volterra transformation was used to transform the observer error system into a heat equation, and an exponentially convergent infinite dimensional observer is proposed. From papers such as [34], [35], the high-gain observer have gradually evolved into an essential subject in the estimation of states for nonlinear systems. For [18], the stability for the nonlinear delay system was investigated based on approximate predictors and high-gain observers, and the input-to-state stability for the global Lipschitz system over the existence of disturbances was proved. In [36], [37], [38], [39], output feedback controllers were proposed to restrain the vibration of flexible structures, with the help of backstepping approach and high gain observation techniques. The belt system with saturation constraint is studied in [40], and the authors designed a state feedback control scheme and an output feedback control scheme, respectively. Although the boundary control of riser systems to suppress vibration has made great progress, the researches on the control of the marine riser with the tension constraint that consider unmeasurable states is limited. The restriction of top tension and the unmeasured states of the system will add to the difficulty in controlling the riser system.

Motivated by the above observations, the present paper investigates the flexible marine riser system, considering not only the top tension constraint of the riser, but unmeasured states of the system. Adopting the Lyapunov direct method, an output feedback boundary controller is presented for application to the riser top boundary, so as to guarantee a good vibration damping effect of the riser system. Based on the designed controller, the uniformly bounded stability of the closed-loop system is demonstrated by mathematical evolution and combining with Lyapunov stability theory. In comparison to previous findings, the major contributions of this article are highlighted below:

• The logarithmic barrier Lyapunov function is employed to guarantee that the top tension constraint $\left|T\left(\ell ,t\right)\right|\le T_M$ is not violated. This is the first implementation under the unmeasured states.

• These literature [6], [11], [15], [23] study the system states of risers that are measurable. However, in practical riser systems, we cannot ignore the effect of measurement noise. Compared to them, this paper studies the case of unmeasured boundary states, which are estimated using the high-gain observer.