# Recoil Control of Deepwater-Drilling Riser with Optimal Guaranteed Cost \(H_{\infty}\) Control
## 4 Simulation Results
### Effectiveness of the Optimal Guaranteed Cost \(H_{\infty}\) Controller  
To design an optimal guaranteed cost \(H_{\infty}\) controller (OGCHC), the \(H_{\infty}\) performance index level \(\gamma\) is set as 0.55, and the weight matrices \(Q\) and \(R\) in (19) are set as  
\[Q=\text{diag}\{0.01,\ 0.25,\ 0.01,\ 0.25,\ 0.01,\ 0.25\},\ \ R=2.5\times 10^{-12}\]  
Then, by Proposition 2, solving the optimization problem (38) yields an OGCHC with gain matrix \(K\) as  
\[K_{\text{OGCHC}}=10^{5}\times\left[ \begin{array}{cccc}1.0510&4.0249&-1.1883&1. 1332&-0.1114&1.4368\end{array} \right] \tag{42}\]  
For comparison purposes, a classic optimal controller (OC) developed in [23] was designed. Set weight matrices \(Q\) and \(R\) of the linear quadratic performance index as:  
\[Q=10^{4}\times\text{diag}\{1,\ 1,\ 1,\ 1,\ 1,\ 1\},\ R=10^{-7}\]  
Then the gain matrix of an OC is computed as  
\[K_{\text{OC}}=10^{5}\times\left[ \begin{array}{cccc}-3.6435&-4.4226&0.9550&- 1.8504&-0.6362&-2.6558\end{array} \right] \tag{43}\]  
As the OGCHC and OC are applied to the riser system, and the controller constraint (39) is considered at the same time, the curves of the displacement responses of the three mass blocks of the riser and control force are depicted--see Figures 3-6 for the case of constant friction coefficients, and Figures 7-10 for the case of time-varying friction coefficients, respectively. To further compare the two controllers quantitatively, the quadratic performance index of the riser system with OGCHC and OC are computed. Note that the performance index (19) are weight-matrix-dependent, and the designed OGCHC and OC are based on a different weight matrix pair, \((Q,R)\). In this situation, the performance index (19) is modified as two weight-matrix-independent performance indices:  
\[J_{z} =\sqrt{\frac{1}{t_{f}}\int_{0}^{t_{f}}z^{T}(t)z(t)dt} \tag{44}\] \[J_{u} =\sqrt{\frac{1}{t_{f}}\int_{0}^{t_{f}}u^{2}(t)dt} \tag{45}\]  
Clearly, \(J_{z}\) and \(J_{u}\) are state-related and control-related root mean square values of the riser system, respectively. Denote the average displacement peak value of three mass blocks of the riser by \(\bar{d}\). Then, the above indices of the riser system without control, with OC and OGCHC in the two cases of constant and time-varying friction coefficients are computed and listed in Table 3.  
It can be found from Figures 3-10, and Table 3, that the controllers OGCHC and OC can reduce the recoil response of the riser system efficiently; however, the OGCHC is more economical than the OC. In fact, the average displacement amplitudes of the three mass blocks of the riser are almost the same, while the control cost by the OGCHC is less than that by the OC. In addition, the reduction in the average displacement peak value of the riser under OGCHC is larger than that under OC. Figure 4: Displacement 2 of the riser without control, with OC and OGCHC under case I.  
Figure 3: Displacement 1 of the riser without control, with OC and OGCHC under case I. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Cases** & **Controllers** & \(\mathbf{J_{\mathrm{z}}}\) **(m)** & \(\mathbf{\hat{d}}\) **(m)** & \(\mathbf{J_{\mathrm{u}}}\)**(10\({}^{4}\) N)** \\ \hline \multirow{3}{*}{I} & No control & 0.2362 & 6.2985 & \(-\) \\  & OC [23] & 0.1888 & 4.3066 & 1.0859 \\  & OGCHC & 0.1912 & 3.8065 & 1.0625 \\ \hline \multirow{3}{*}{II} & No control & 0.3302 & 4.9055 & \(-\) \\  & OC [23] & 0.3001 & 3.6741 & 0.9758 \\ \cline{1-1}  & OGCHC & 0.3049 & 2.8722 & 0.8956 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Performance indices of the riser without control, with OC and OGCHC. Figure 5: Displacement 3 of the riser without control, with OC and OGCHC under case I.  
Figure 6: Control force with OC and OGCHC under case I.  
Figure 7: Displacement 1 of the riser without control, with OC and OGCHC under case II. Figure 8: Displacement 2 of the riser without control, with OC and OGCHC under case II. Figure 10: Control force with OC and OGCHC under case II.