# Recoil Control of Deepwater-Drilling Riser with Optimal Guaranteed Cost \(H_{\infty}\) Control
## 2 Problem Formulation
### Dynamic Model of a Riser-Tension System  
Notice that the tension system uses the hydraulic cylinder to drive the riser directly, and provides the required tension force for the riser. The tension force generated by a single tensioner can be calculated by [23]  
\[N_{t}(t)=f_{t}-k_{1}x_{p}-u(t) \tag{4}\]  
where  
\[f_{t}=P_{H_{0}}A_{r}-P_{L_{0}}A_{p},\ \ k_{1}=\frac{eP_{H_{0}}A_{r}^{2}}{V_{H_{ 0}}}+\frac{eP_{L_{0}}A_{p}^{2}}{V_{L_{0}}} \tag{5}\]  
with \(f_{t}\) representing the top tension force of the initial disconnection moment, N; \(P_{H_{0}}\) and \(P_{L_{0}}\) are the pressure of the high pressure air vessel and low pressure nitrogen vessel at the initial time, respectively, Pa; \(A_{r}\) and \(A_{p}\) are the area of piston rod side and piston rod-less side, respectively, m\({}^{2}\); \(k_{1}\) is the spring stiffness of tensioner, N/m; \(\epsilon\) is the gas constant, \(e\in[1.0,\ 1.4]\); \(V_{H_{0}}\) and \(V_{L_{0}}\) are the volume of the high pressure gas and low pressure nitrogen vessel at the initial time, respectively, m\({}^{3}\); \(x_{p}\) is the displacement of piston relative to hydraulic cylinder at time \(t\), m; \(u(t)\) is the pressure drop. Now, we aim to present a dynamic model of the riser-tension recoil control system. Similar to [15; 23], the system can be simplified as a three-element mass-damper-spring model as seen in Figure 1, where \(m_{1}\), \(m_{2}\), and \(m_{3}\) are the masses of three blocks of the whole riser, \(x_{1}\), \(x_{2}\), and \(x_{3}\) are displacements of three mass blocks of the riser, \(c_{1}\), \(c_{2}\), and \(c_{3}\) are the damping on the three mass blocks, \(k_{2}\) and \(k_{3}\) are equivalent spring stiffness of riser system with \(k_{2}=k_{3}=EA_{e}/L\). \(E\) is the elastic modulus of riser, \(A_{e}\) is the cross-sectional area of riser, and \(L\) is the effective length of the riser. To present dynamic equations of the riser, the following assumptions are required [15; 23]. **Assumption 1**. : _The upward direction of force and displacement is positive, and the coordinate origin of riser system is located at the position of respective initial disconnection moment._  
**Assumption 2**. : _The influence of buoyancy modules on the riser stiffness is ignored, the drilling pipe and the riser are supposed to be concentric. The viscous resistance of drilling mud discharge is evenly distributed along the axial direction of the riser._  
Based on the Assumptions aforesaid, by [PERSON]'s second law, the dynamic equation of riser-tension control system can be expressed as  
\[\left\{ \begin{array}{l}m_{1}\ddot{x}_{1}=-k_{2}(x_{1}-x_{2})-c_{1}\dot{x}_{1 }-c_{2}(\dot{x}_{1}-\dot{x}_{2})+N_{t}-F_{1}\\ m_{2}\ddot{x}_{2}=k_{2}(x_{1}-x_{2})+c_{2}(\dot{x}_{1}-\dot{x}_{2})-c_{3}(\dot {x}_{2}-\dot{x}_{3})-k_{3}(x_{2}+x_{0}-x_{3})-F_{2}\\ m_{3}\ddot{x}_{3}=k_{3}(x_{2}+x_{0}-x_{3})+c_{3}(\dot{x}_{2}-\dot{x}_{3})-F_{3 }\end{array} \right. \tag{6}\]  
where \(x_{0}=(f_{t}-m_{1}g+F_{b_{1}})/k_{2}\), and  
\[\left\{ \begin{array}{l}F_{1}=F_{w_{1}}+F_{m_{1}}-F_{b_{1}}+m_{1}g\\ F_{2}=F_{w_{2}}+F_{m_{2}}-F_{b_{2}}+m_{2}g\\ F_{3}=F_{w_{3}}+F_{m_{3}}-F_{b_{3}}+m_{3}g\end{array} \right. \tag{7}\]  
with \(F_{w_{i}}=F_{w}/3\), \(F_{m_{i}}=F_{m}/3\), \(i=1,2,3\), and \(F_{b_{1}}\), \(F_{b_{2}}\), \(F_{b_{3}}\) representing the buoyancy during recoil of the riser. The mass of the three blocks can be computed as  
\[\left\{ \begin{array}{l}m_{1}=M_{\text{riser}}/3+M_{\text{slip}}\\ m_{2}=M_{\text{riser}}/3\\ m_{3}=M_{\text{riser}}/3+M_{\text{LMRP}}\end{array} \right. \tag{8}\]  
where \(M_{\text{riser}}=N_{b}M_{\text{buoyancy}}\) is the dry weight of the riser, kg; \(N_{b}\) is the number of buoyancy joint; \(M_{\text{buoyancy}}\) is the dry weight of the buoyancy joint, kg; \(M_{\text{slip}}\) is the dry weight of slip joint outer cylinder, kg; \(M_{\text{LMRP}}\) is the dry weight of LMRP, kg. Figure 1: A riser-tension recoil control system model. Adapted with permission from [23; 31]. 2020, 2021, Elsevier.