# Recoil Control of Deepwater-Drilling Riser with Optimal Guaranteed Cost \(H_{\infty}\) Control
## 2 Problem Formulation
### State Space Model of the Riser System and Active-Recoil Control Problem  
Let  
\[\left\{\begin{array}{l}z_{1}(t)=x_{1}(t),\ z_{2}(t)=\dot{x}_{1}(t),\ z_{3}(t)= x_{2}(t)\\ z_{4}(t)=\dot{x}_{2}(t),\ z_{5}(t)=x_{3}(t),\ z_{6}(t)=\dot{x}_{3}(t)\end{array}\right. \tag{10}\]  
and denote \(z(t):=[z_{1}(t)\ z_{2}(t)\ z_{3}(t)\ z_{4}(t)\ z_{5}(t)\ z_{6}(t)]^{T}\). Then, the dynamic Equation (6) can be written as  
\[\dot{z}(t)=Az(t)+Bu(t)+D_{1}w(t)+\bar{D}_{2}h_{1}(t),\ \ z(0)=z_{0} \tag{11}\]  
where \(z_{0}\) is the system initial value as  
\[z_{0}=[0\ \ 0\ \ -x_{0}\ \ 0\ \ -x_{t}\ \ \ 0]^{T} \tag{12}\]  
with \(x_{t}=(f_{t}-m_{1}g-m_{2}g+F_{b_{1}}+F_{b_{2}})/k_{3}\), and  
\[A=\left[\begin{array}{cccc}0&1&0&0&0\\ -\frac{k_{1}+k_{2}}{m_{1}}&-\frac{c_{1}+c_{2}}{m_{1}}&\frac{k_{2}}{m_{1}}& \frac{c_{2}}{m_{1}}&0&0\\ 0&0&0&1&0&0\\ \frac{k_{2}}{m_{2}}&\frac{c_{2}}{m_{2}}&-\frac{k_{2}+k_{3}}{m_{2}}&-\frac{c_{ 2}+c_{3}}{m_{2}}&\frac{k_{3}}{m_{2}}&\frac{c_{3}}{m_{2}}\\ 0&0&0&0&0&1\\ 0&0&\frac{k_{3}}{m_{3}}&\frac{c_{3}}{m_{3}}&-\frac{k_{3}}{m_{3}}&-\frac{c_{3}} {m_{3}}\\ \end{array}\right],\ B=\left[\begin{array}{c}0\\ -\frac{1}{m_{1}}\\ 0\\ 0\\ 0\\ 0\end{array}\right]\]  
\[D_{1}=\left[\begin{array}{c}0\\ \frac{k_{1}}{m_{1}}\\ 0\\ 0\\ 0\\ 0\end{array}\right],\ \bar{D}_{2}=\left[\begin{array}{cccc}0&0&0\\ -\frac{1}{m_{1}}&0&0\\ 0&0&0\\ 0&-\frac{1}{m_{2}}&0\\ 0&0&0\\ 0&0&-\frac{1}{m_{3}}\end{array}\right],\ h_{1}(t)=\left[\begin{array}{c}F_{1} -f_{t}\\ F_{2}+k_{3}x_{0}\\ F_{3}-k_{3}x_{0}\end{array}\right]\]  
Note that the equilibrium state of the system is given by  
\[\left\{\begin{array}{l}z_{e_{1}}=\frac{\kappa}{k_{1}}\\ z_{e_{3}}=\frac{\kappa}{k_{1}}-\frac{m_{2}g+m_{3}g-F_{b_{2}}-F_{b_{3}}}{k_{2}} \\ z_{e_{5}}=\frac{\kappa}{k_{1}}+\frac{\kappa}{k_{2}}-\frac{m_{3}g-F_{b_{3}}}{k _{3}}\\ z_{e_{2}}=z_{e4}=z_{e6}=0\end{array}\right. \tag{13}\]  
where \(\kappa=f_{t}-m_{1}g-m_{2}g-m_{3}g+F_{b_{1}}+F_{b_{2}}+F_{b_{3}}\).  
Let  
\[\dot{z}_{i}(t)=z_{i}(t)-z_{e_{i}},\ i=1,2,\cdots,6. \tag{14}\]  
Then from (10)-(14), one yields an incremental equation of the system as  
\[\dot{z}(t)=A\dot{z}(t)+Bu(t)+D_{1}w(t)+D_{2}h(t),\ \ \dot{z}(0)=\dot{z}_{0} \tag{15}\]  
where  
\[D_{2}=\left[\begin{array}{cccc}0&-\frac{1}{m_{1}}&0&-\frac{1}{m_{2}}&0&- \frac{1}{m_{3}}\end{array}\right]^{T},\ \ h(t)=\frac{F_{w}+F_{m}}{3}\]Let  
\[\hat{h}(t)=[w^{T}(t),\;h^{T}(t)]^{T},\;\;D=[D_{1}\;\;D_{2}] \tag{16}\]  
Then, the system Equation (15) can be written as  
\[\hat{z}(t)=A\bar{z}(t)+Bu(t)+D\hat{h}(t),\;\;\bar{z}(0)=\bar{z}_{0} \tag{17}\]  
The output equation of the system is given as  
\[\eta(t)=C_{1}\bar{z}(t)+E_{1}\hat{h}(t) \tag{18}\]  
where \(C_{1}\) and \(E_{1}=[E_{w}\;\;E_{h}]\) are given matrices with appropriate dimensions.  
Introducing the following quadratic performance index as  
\[J=\int_{0}^{\infty}\Bigl{[}\bar{z}^{T}(t)Q\bar{z}(t)+u^{T}(t)Ru(t)\Bigr{]}dt \tag{19}\]  
where the weight matrices \(Q=Q^{T}\geq 0\) and \(R=R^{T}>0\).  
A state feedback control law is designed as  
\[u(t)=K\bar{z}(t) \tag{20}\]  
where \(K\) is a \(1\times 6\) gain matrix to be designed.  
From (17) and (20), one yields the closed-loop system as  
\[\hat{z}(t)=(A+BK)\bar{z}(t)+D\hat{h}(t),\;\;\bar{z}(0)=\bar{z}_{0} \tag{21}\]  
The aim of this paper is to design the optimal guaranteed cost \(H_{\infty}\) recoil controller (20) such that  
* The closed-loop system (21) with \(w(t)=0\) and \(h(t)=0\) is asymptotically stable, and \(J\leq J^{*}\), \(J^{*}\) is the upper bounds of the performance;
* The \(H_{\infty}\) performance index \[\|\eta(t)\|\leq\gamma\|\hat{h}(t)\|\] (22) of the closed-loop system can be guaranteed for a given \(\gamma>0\).