## 4 Control Design  
A general control architecture for extracting energy is shown in Figure 4, where it is assumed the instantaneous displacement states, wave elevation \(\eta\), and the reference displacement states, \(\xi_{r}\), \(\xi_{r}\) and \(\xi_{r}\) are known. A variety of control laws could be used with this architecture, where we will consider feedback linearization with a sinusoidal reference to illustrate a model-based approach. It should be noted that this results in the cancellation of all the nonlinear terms in Equation (13). Such cancellation is avoided by  
Figure 3: Simulated steady state results of a model without \(F_{f_{k}d}\). The upper plot is \(F_{b}=F_{f_{k}s}+F_{g}\) force and the lower plot is the buoy displacement.  
Figure 2: Simulated steady state results comparing the three noninertial terms of Equation (11): \(F_{b}=F_{f_{k}s}+F_{g}\), \(F_{f_{k}d}\), \(b_{c}^{\dagger}\) and their sum (upper plot) and the buoy displacement and wave elevation (lower plot) for a deep water wave with \(T=18\) s and the buoy parameters shown in Table 1.  
including these terms in the reference command leading to a control solution that does not require a reference and relies on exploiting the buoy's nonlinear response to extract energy.  
The error states are defined by Equation (15)  
\[\begin{split} e_{1}&=\zeta-\zeta_{r}\\ e_{2}&=\dot{\zeta}-\dot{\zeta}_{r}\end{split} \tag{15}\]  
and then differentiated to form Equation (16).  
\[\begin{split}\dot{e}_{1}&=e_{2}\\ \dot{e}_{2}&=\dot{\zeta}-\dot{\zeta}_{r}\end{split} \tag{16}\]  
Equation (11) can be solved for \(\zeta\)  
\[\begin{split}\dot{\zeta}=M^{-1}\Big{(}F_{c}-& b\dot{\zeta}+\frac{1}{3}\pi\rho g\Re^{2}(\eta-\zeta)^{3}\\ &+\frac{2\pi}{\chi^{2}}\rho g\Re^{2}\eta\Big{[}2-e^{-\chi h}-e^{- \chi(\eta-\zeta)}-\chi(h+\eta-\zeta)\Big{]}\Big{)}\end{split} \tag{17}\]  
and substituted into \(\dot{e}_{2}\) of Equation (16) to create Equation (18)  
\[\dot{e}_{2}=\
u-\frac{b}{M}\dot{e}_{2} \tag{18}\]  
where \(\
u\) is:  
\[\begin{split}\
u=M^{-1}\Big{(}F_{c}-& b\dot{\zeta} _{r}+\frac{1}{3}\pi\rho g\Re^{2}(\eta-\zeta)^{3}\\ &+\frac{2\pi}{\chi^{2}}\rho g\Re^{2}\eta\Big{[}2-e^{-\chi h}-e^{- \chi(\eta-\zeta)}-\chi(h+\eta-\zeta)\Big{]}\Big{)}-\zeta_{r}\end{split} \tag{19}\]  
and the error state system of Equation (16) is written compactly in Equation (20).  
\[\begin{split}\dot{e}_{1}&=e_{2}\\ \dot{e}_{2}&=-\frac{b}{M}\dot{e}_{2}+\
u\end{split} \tag{20}\]  
State feedback control  
\[\
u=-\vec{k}^{\,T}\vec{\varepsilon} \tag{21}\]  
is applied to \(\
u\) and the control law is Equation (22)  
\[\begin{split} F_{c}=-M\vec{k}^{\,T}\vec{\varepsilon}+M\ddot{ \zeta}_{r}+& b\dot{\zeta}_{r}-\frac{1}{3}\pi\rho g\Re^{2}(\eta- \zeta)^{3}\\ &-\frac{2\pi}{\chi^{2}}\rho g\Re^{2}\eta\Big{[}2-e^{-\chi h}-e^{ -\chi(\eta-\zeta)}-\chi(h+\eta-\zeta)\Big{]}\end{split} \tag{22}\]  
which is simply Equation (23).  
\[F_{c}=-M\vec{k}^{\,T}\vec{\varepsilon}+M\ddot{\zeta}_{r}+b\dot{\zeta}_{r}-F_{ b}-F_{fk,d} \tag{23}\]  
Figure 4: Control architecture illustrating its access to wave elevation, buoy states, and reference states.  
A variety of methods can be used to create the gain vector \(\vec{k}\), where the infinite horizon, linear quadratic regulator approach is used, which minimizes the cost function of Equation (24) [41]  
\[J=\int_{0}^{\infty}(\vec{e}^{T}Q\,\vec{e}+\
u^{T}Rv)dt \tag{24}\]  
where in our case  
\[Q=\begin{bmatrix}10&0\\ 0&1\end{bmatrix}\quad\text{and}\quad R=1 \tag{25}\]  
yielding \(\vec{k}^{T}=\begin{bmatrix}3.16&2.53\end{bmatrix}\). The effect of varying the \(Q_{11}\) gain value, nominally set to 10, will be examined in Section 6.3.  
The control law shown in the block diagram of Figure 4 can be summarized by Equation (23), the error vector of Equation (15), the hourglass buoy parameters of Table 1, the Froude-Krylov terms of Equation (8) and the gains of \(\vec{k}\) above. It requires the reference trajectory and measurements of the wave elevation and the buoy displacement states.