# Recoil Control of Deepwater-Drilling Riser with Optimal Guaranteed Cost \(H_{\infty}\) Control
## 3 Design of Optimal Guaranteed Cost \(H_{\infty}\) Recoil Controller
### The Existence Condition of the Optimal Guaranteed Cost \(H_{\infty}\) Recoil Controller  
To analyze the asymptotic stability of the closed-loop system (21), consider the following Lyapunov function as  
\[V(\bar{z}(t))=\bar{z}^{T}(t)P\bar{z}(t) \tag{23}\]  
where \(P\) is a \(6\times 6\) positive definite matrix. Then, the time derivative of \(V(\bar{z}(t))\) along system (21) is given as  
\[\dot{V}(\bar{z}(t))=\bar{z}^{T}(t)\Pi\bar{z}(t)+2\bar{z}^{T}(t)PD\hat{h}(t) \tag{24}\]  
where  
\[\Pi=A^{T}P+K^{T}B^{T}P+PA+PBK \tag{25}\]To prove asymptotic stability of the closed-loop system (21), set \(\hat{h}(t)=0\). Then from (24), one obtains  
\[\dot{V}(\tilde{z}(t))=\tilde{z}^{T}(t)(Y-Q-K^{T}RK)\tilde{z}(t) \tag{26}\]  
where \(Y=\Pi+Q+K^{T}RK\).  
Note that  
\[\dot{V}(\tilde{z}(t))\leq z^{T}(t)Y\tilde{z}(t)-\lambda_{min}(Q+K^{T}RK)\| \tilde{z}(t)\|^{2} \tag{27}\]  
where \(\lambda_{min}\) represents the minimum eigenvalue of the matrix.  
To guarantee the asymptotic stability of the system, we require \(Y<0\). In fact, by the Schur Complement, it is equivalent to the following matrix inequality as  
\[\left[\begin{array}{ccc}\Pi&K^{T}R&Q\\ *&-R&0\\ *&*&-Q\end{array}\right]<0 \tag{28}\]  
If the matrix inequality (28) holds, then there exists a sufficiently small positive scalar \(c\) such that \(\dot{V}(\tilde{z}(t))<-c\|\tilde{z}(t)\|^{2},\tilde{z}(t)\
eq 0\), which indicates that riser system (17) is asymptotically stable.  
Integrating both sides of (26) from \(0\) to \(\infty\), one obtains  
\[\int_{0}^{\infty}\tilde{z}^{T}(t)(Q+K^{T}RK)\tilde{z}(t)dt<V(0) \tag{29}\]  
That is to say, the quadratic performance index \(J\) satisfies \(J<\tilde{z}^{T}(0)P\tilde{z}(0)\).  
Then, we try to find the condition guaranteeing the \(H_{\infty}\) performance index (22) of the riser system. Denote  
\[\beta(t)=[\tilde{z}^{T}(t)\ \ \hat{h}^{T}(t)]^{T} \tag{30}\]  
Then, it follows from (24) and (18) that  
\[\dot{V}(\tilde{z}(t))+\eta^{T}(t)\eta(t)-\gamma^{2}\hat{h}^{T}(t)\hat{h}(t) \leq\beta^{T}(t)\Psi\beta(t) \tag{31}\]  
where  
\[\Psi=\left[\begin{array}{ccc}Y+C_{1}^{T}C_{1}&PD+C_{1}^{T}E_{1}\\ *&-\gamma^{2}I+E_{1}^{T}E_{1}\end{array}\right] \tag{32}\]  
Note that \(\Psi<0\) is equivalent to the following matrix inequality as  
\[\left[\begin{array}{cccc}\Pi&PD&K^{T}R&Q&C_{1}^{T}\\ *&-\gamma^{2}I&0&0&E_{1}^{T}\\ *&*&-R&0&0\\ *&*&*&-Q&0\\ *&*&*&*&-I\end{array}\right]<0 \tag{33}\]  
If the above inequality holds, then from (31), one yields  
\[\dot{V}(\tilde{z}(t))+\eta^{T}(t)\eta(t)-\gamma^{2}\hat{h}^{T}(t)\hat{h}(t)<0 \tag{34}\]  
Further, integrating both sides of (34) and noticing the zero initial state condition of the system yields the \(H_{\infty}\) performance index inequality (22) directly.  
Note that the inequality (33) implies that (28) holds, which indicates if the inequality (33) holds, then the closed-loop system (21) with \(\hat{h}(t)=0\) is asymptotically stable, the quadratic performance index (19) is bounded, and the \(H_{\infty}\) performance (22) is guaranteed for the friction force \(\hat{h}(t)\).  
To obtain the minimum upper bound of the quadratic performance index \(J\), introducing an upper bound \(\mu>0\), i.e., \(\hat{z}^{T}(0)P\hat{z}(0)<\mu\), which is equivalent to the one as  
\[\left[\begin{array}{cc}-\mu&z^{T}(0)\\ *&-P^{-1}\end{array}\right]<0 \tag{35}\]  
Now, we state a proposition which provides the existence of the optimal guaranteed cost \(H_{\infty}\) recoil controller for the riser system.  
Proposition 1: _For a given scalar \(\gamma>0\), matrices \(Q\geq 0\), \(R>0\), and the initial state value \(\hat{z}(0)\), if there exist \(6\times 6\) matrix \(P>0\) and \(1\times 6\) matrix \(K\) such that the inequalities (33) and (35) hold, then the optimal guaranteed cost \(H_{\infty}\) recoil controller (20) for the riser system (6) can be designed, and the gain matrix \(K\) of the controller is solvable._