## 2 Design of LQ Optimal Controllers
### Design of LQ SOF Controllers  
Generally, we know that all state variables of \(\mathbf{x}\) are difficult or very expensive to measure with sensors in a real vehicle. Moreover, LQR has plenty of gain elements, which are hard to implement in a real vehicle. To resolve those problems, a SOF control has been adopted as feedback structure so far [17; 18; 19; 20; 29; 30]. Unlike the FSF, a SOFC uses available sensor signals in a real vehicle. In this paper, two vectors of available sensor outputs, \(\mathbf{y}_{1}\) and \(\mathbf{y}_{2}\), are derived from the state vector \(\mathbf{x}\) as (18) and (19), respectively. In (18), \(\mathbf{y}_{1}\) represent the suspension displacements, \(d_{f}\) and \(d_{r}\), and the suspension velocities, \(v_{f}\) and \(v_{r}\), of \(S_{f}\) and \(S_{r}\), correspondingly [25; 29; 30]. As shown in (19), the available outputs of \(\mathbf{y}_{2}\) are \(v_{zc}\) and \(\omega_{yc}\)[17; 18; 20]. \[\mathbf{y}_{1}\triangleq \begin{bmatrix}d_{f}\\ v_{f}\\ d_{r}\\ v_{r}\end{bmatrix} =\mathbf{C}_{1}\mathbf{x}= \begin{bmatrix}1&-l_{f}&-1&0&0&0&0 \\ 0&0&0&1&-l_{f}&-1&0\\ 1&l_{r}&0&-1&0&0&0&0\\ 0&0&0&0&1&l_{r}&0&-1\end{bmatrix} \mathbf{x}, \tag{18}\]  
\[\mathbf{y}_{2}\triangleq \begin{bmatrix}\tilde{z}_{c}\\ \hat{\theta}_{c}\end{bmatrix} = \begin{bmatrix}v_{zc}\\ \omega_{yc}\end{bmatrix} =\mathbf{C}_{2}\mathbf{x}= \begin{bmatrix}0&0&0&0&1&0&0 \\ 0&0&0&0&1&0&0\end{bmatrix} \mathbf{x}. \tag{19}\]  
With \(\mathbf{y}_{1}\), the first SOFC, \(\mathbf{u}_{SOF1}\), is proposed as (20) [29; 30]. This is similar to the suspension forces, \(f_{r}\) and \(f_{r}\), as shown in (2). In other words, four gains, \(k_{f1}\), \(k_{f2}\), \(k_{r1}\) and \(k_{r2}\) of (20) correspond to \(k_{gf}\), \(b_{gf}\), \(k_{sr}\) and \(b_{sr}\) of (2), respectively. The first SOFC, \(\mathbf{u}_{SOF1}\), reflects the fact that the control inputs, \(u_{f}\) and \(u_{r}\), are determined by \(S_{f}\) and \(S_{r}\), respectively. This assumes that the vertical motions of FC\({}_{\text{SPMS}}\) and RC\({}_{\text{SPMS}}\) are independent \(\mathbf{K}_{SOF1}\). As shown in (20), \(\mathbf{K}_{SOF1}\) is structured. In other words, \(\mathbf{K}_{SOF1}\) does not have full elements. This makes an optimization procedure difficult when finding \(\mathbf{K}_{SOF1}\) minimizing \(L\). \[\mathbf{u}_{SOF1}\mathbf{=K}_{SOF1}\mathbf{y}_{1}\mathbf{=} \begin{bmatrix}k_{ f1}&k_{f2}&0&0\\ 0&0&k_{r1}&k_{r2}\end{bmatrix} \mathbf{y}_{1}. \tag{20}\]  
The second SOFC is designed with Lotus modal control [18; 29; 30]. The vertical control force, \(F_{zc}\), and the pitch control moment, \(M_{\theta}\), needed to independently control the heave and pitch vibrations of SPMS are calculated with \(\mathbf{y}_{2}\) as (21). As shown in (21), \(F_{zc}\) and \(M_{\theta}\) are obtained from a derivative control. With (8), the relationship between \(F_{zc}\) and \(M_{\theta}\) and \(u_{f}\) and \(u_{r}\) is obtained as (22). From (21) and (22), the second SOFC, \(\mathbf{u}_{SOF2}\), is obtained as (23). This is called Lotus modal control, which converts \(F_{zc}\) and \(M_{\theta}\) into \(u_{f}\) and \(u_{r}\) with the matrix, **G**, as given in (8). As shown in (21), \(\mathbf{K}_{SOF2}\) is structured, similar to \(\mathbf{K}_{SOF1}\). This makes an optimization procedure difficult when finding \(\mathbf{K}_{SOF2}\) minimizing \(L\). \[ \begin{bmatrix}F_{zc}\\ M_{\theta}\end{bmatrix} = \begin{bmatrix}k_{v}\hat{z}_{c}\\ k_{p}\hat{\theta}_{c}\end{bmatrix} = \begin{bmatrix}k_{p}&0\\ 0&k_{p}\end{bmatrix}\begin{bmatrix}\hat{z}_{c}\\ \hat{\theta}_{c}\end{bmatrix} =\mathbf{K}_{SOF2}\mathbf{y}_{2}, \tag{21}\]  
\[ \begin{bmatrix}F_{zc}\\ M_{\theta}\end{bmatrix} =\mathbf{G} \begin{bmatrix}u_{f}\\ u_{r}\end{bmatrix} =\mathbf{G}\mathbf{u}, \tag{22}\]  
\[\mathbf{u}_{SOF2}=\mathbf{G}^{-1} \begin{bmatrix}F_{zc}\\ M_{\theta}\end{bmatrix} =\mathbf{G}^{-1}\mathbf{K}_{SOF2}\mathbf{y}_{2}. \tag{23}\]  
Generally, a SOFC which minimizes \(L\) is called LQ SOF one or LQSOFC.