# Vibration Control of Multi-Modular VLFS in Random Sea Based on Stiffness-Adjustable Connectors
## 3 Vibration Control Method Based on the Stiffness-Adjustable Connector
### Feedback Control Gains and the Parameters of Hydraulic-Actuators  
The control flow chart is shown in Figure 4. With the wave parameters, the feedback gains \(\left[\lambda_{1}^{q,p},\lambda_{2}^{q,p},\cdots,\lambda_{N}^{q,p}\right]\) are obtained by using Sequential Quadratic Programming (SQP), where the optimization goal is the minimization of the 1-in-1000 amplitude of the extreme responses of all modules in the intended directions. Using the feedback gains and the VLFS states, the outputs of the actuators can be obtained, which are the forces imposed on the VLFS to suppress the vibration of the system. Substituting Equation (23), Equation (16), Equation (12), and Equation (11) into Equation (1), the mathematical model of the VLFS can be rewritten in the simple form  
\[\mathbf{M}\mathbf{\ddot{x}}+\mathbf{C}\mathbf{\dot{x}}+\mathbf{K}\mathbf{x}= \mathbf{\ddot{f}}_{i,w}e^{-i\omega t} \tag{24}\]  
where the matrices are  
\[\mathbf{M} =\left[ \begin{array}{cccc}\mathbf{M}_{1}+\Delta\mathbf{M}_{1,1}& \Delta\mathbf{M}_{1,2}&\cdots&\Delta\mathbf{M}_{1,N}\\ \Delta\mathbf{M}_{2,1}&\mathbf{M}_{2}+\Delta\mathbf{M}_{2,2}&\cdots&\Delta \mathbf{M}_{2,N}\\ \vdots&\vdots&\ddots&\vdots\\ \Delta\mathbf{M}_{N,1}&\Delta\mathbf{M}_{N,2}&\cdots&\mathbf{M}_{N}+\Delta \mathbf{M}_{N,N}\end{array} \right]\] \[\mathbf{C} =\left[ \begin{array}{cccc}\Delta\mathbf{C}_{1,1}&\Delta \mathbf{C}_{1,2}&\cdots&\Delta\mathbf{C}_{1,N}\\ \Delta\mathbf{C}_{2,1}&\Delta\mathbf{C}_{2,2}&\cdots&\Delta\mathbf{C}_{2,N}\\ \vdots&\vdots&\ddots&\vdots\\ \Delta\mathbf{C}_{N,1}&\Delta\mathbf{C}_{N,2}&\cdots&\Delta\mathbf{C}_{N,N} \end{array} \right]\] \[\mathbf{K} =\left[ \begin{array}{cccc}\mathbf{R}_{1}&&&\\ &\mathbf{R}_{2}&&\\ &&\ddots&\\ &&\mathbf{R}_{N}\end{array} \right]+\left[ \begin{array}{cccc}\mathbf{K}_{M_{ 1}}&&&\\ &\mathbf{K}_{M_{2}}&&\\ &&\ddots&\\ &&\mathbf{K}_{M_{N,N}}\end{array} \right]+\sum_{q=1}^{N_{\mathrm{C}}}\left[ \begin{array}{cccc}\mathbf{\ddot{H}}^{1,1}&&&\\ &-\mathbf{\ddot{H}}^{1,1}+\mathbf{\ddot{H}}^{2}&\\ &\vdots&\\ -\mathbf{H}^{N,-2}+\mathbf{H}^{N,-1}&\\ &-\mathbf{\ddot{H}}^{N,-1}&\end{array} \right] \tag{25}\] \[+\sum_{q=1}^{N_{\mathrm{C}}}\left[\begin{array}{cccc}\left( \mathbf{W}_{1}^{q\pm 1}\right)^{\mathrm{T}}\mathbf{K}_{\mathbf{c}}\mathbf{W}_{1}^{q\pm 1}&- \left(\mathbf{W}_{1}^{q\pm 1}\right)^{\mathrm{T}}\mathbf{K}_{\mathbf{c}} \mathbf{W}_{2}^{q\pm 1}&\mathbf{0}&\mathbf{0}&\mathbf{0}\\ -\left(\mathbf{W}_{2}^{q\pm 1}\right)^{\mathrm{T}}\mathbf{K}_{\mathbf{c}} \mathbf{W}_{1}^{q\pm 1}&\left(\mathbf{W}_{2}^{q\pm 1}\right)^{\mathrm{T}} \mathbf{K}_{\mathbf{c}}\mathbf{W}_{2}^{q\pm 1}&-\left(\mathbf{W}_{1}^{q\pm 2}\right)^{ \mathrm{T}}\mathbf{K}_{\mathbf{c}}\mathbf{W}_{2}^{q\pm 2}&\mathbf{0}&\mathbf{0}\\ \ddots&\ddots&\ddots&\\ \mathbf{0}&\mathbf{0}&-\left(\mathbf{W}_{2}^{q,N-2}\right)^{\mathrm{T}}\mathbf{K }_{\mathbf{c}}\mathbf{W}_{1}^{q\pm 1}& \begin{array}{cccc}\left(\mathbf{W}_{1}^{q\pm 1} \right)^{\mathrm{T}}\mathbf{K}_{\mathbf{c}}\mathbf{W}_{1}^{q\pm 1}\\ +\left(\mathbf{W}_{2}^{q\pm 2}\right)^{\mathrm{T}}\mathbf{K}_{\mathbf{c}} \mathbf{W}_{2}^{q\pm 1-2}&-\left(\mathbf{W}_{1}^{q\pm 1}\right)^{\mathrm{T}}\mathbf{K}_{ \mathbf{c}}\mathbf{W}_{2}^{q\pm 1-1}\end{array} \right]\]  
From the matrix \(\mathbf{K}\) in Equation (25), we can see that the control effects are actually converted to a stiffness matrix expression. In view of the motion equation of the VLFS in Equation (24), the actuators in the connectors actually change the system stiffness and  
Figure 4: The flow chart of the control process. are adjustable according to sea conditions. This is why we call the novel connectors stiffness-adjustable connectors.