# Recoil Control of Deepwater-Drilling Riser with Optimal Guaranteed Cost \(H_{\infty}\) Control
## 1 Introduction  
Typhoon, surge, wave, and earthquake generally put forward strict requirements for the safety of engineering structures [1, 2, 3]. For example, ocean current, wind, earthquake, and wave act on the deep-sea-drilling equipment for a long time, potentially leading to economic losses and pipe damage [4, 5]; therefore, the drilling riser is a weak connecting part for deepwater oil and gas exploration between the submarine wellhead and the drilling ship [6, 7, 8]. The components of a deepwater-drilling riser system mainly include the drilling platform, tensioner, split joint, single riser, buoyancy joint, filling valve, lower marine-riser package (LMRP), blowout preventer (BOP), etc. In case of extreme harsh environments or dynamic-positioning failure, emergency disconnection of the LMRP and BOP is required to prevent damage to the drilling vessel [9, 10]. During the disconnection operation, the deepwater-drilling riser will move upward rapidly under the tension of the tensioner and the elastic potential energy, resulting in recoil movements and endangering the safety of the drilling platform; therefore, many scholars have conducted recoil analysis on the riser, see [11, 12, 13, 14, 15, 16, 17], and the references therein. It is known that during riser emergency disconnection, the external load in the recoil process of the deepwater-drilling riser mainly comes from viscous resistance caused by discharging drilling fluid, and more and more research results concerning recoil response have been reported. In [18], the auxiliary program is used to simulate the mud column dischargingprocess without the filling valve. Simulation results show that the auxiliary program and the RESTEN program achieve good agreement in mud kinematic applications. In [11], an one-dimensional finite volume model is proposed to calculate the fluid pressure and velocity. In [19], a slug fluid model is established for seawater injection by the refilled valve. Based on this model, the drilling mud discharge model with a whole fluid column model is developed, where seawater and mud are considered as a whole for force analysis [20]. In [21], the friction force calculation method is introduced for slug fluid model and whole fluid column model. In particular, based on the whole fluid column model and the action force of the fluid on the riser inner wall, the drilling fluid discharge is analyzed in [15], where the drill pipe is also considered. In [22], by comparing the existing drilling mud discharge models, a more accurate three-dimension computational fluid dynamics model is developed. This shows that, to reflect practical response of drilling riser, a more accurate model of the drilling mud discharge process is required. In recent decades, much more research attention mainly focuses on riser recoil analysis, while there are relatively few studies with regard to the riser recoil control of the drilling riser. In [23], a recoil control model of the drilling riser is proposed first, and then the linear quadratic optimal control method is proposed to control the riser recoil displacement. It is found that, under the designed optimal controller, the recoil response of the drilling riser has been reduced effectively. Note that, during the design of the optimal controller for the drilling riser, the effects of the mud discharge on the riser are not considered specifically. In fact, the classic linear quadratic optimal controller design is generally based on the known exact dynamic model, and it is not always effective to cope with uncertain systems subject to external disturbances. In other words, there still exists room for further reduction in the recoil response of the riser, which motivates our study in this paper. Due to the existence of external disturbances and parameter perturbations, the quadratic performance index of the system with an optimal controller may generally not be minimized.