## 2 Design of LQ Optimal Controllers  
### Half-Car Model and Derivation of State-Space Equation  
When designing LQ optimal controllers such as LQR and LQSOFC, a linear SSEQN is needed. Generally, a linear SSEQN is derived from a linear vehicle model. To describe the heave and pitch motions of a SPMS and the vertical motions of unsprung masses (USPMSs), a HCM is selected as a vehicle one in this paper [17; 19; 20]. Figure 2 shows the HCM with linear springs and dampers and its parameters and variables. In Figure 2, \(S_{f}\) and \(S_{r}\) stand for the front and rear suspensions, respectively. \(u_{f}\) and \(u_{r}\) are the control input or the vertical control force generated by an active actuator in \(S_{f}\) and \(S_{r}\). The parameters in the HCM are described in Nomenclature. Let denote the front and rear corners of the SPMS as FC\({}_{\text{SPMS}}\) and RC\({}_{\text{SPMS}}\), respectively. In Figure 2, \(z_{sf}\) and \(z_{sr}\) are the vertical displacements of FC\({}_{\text{SPMS}}\) and RC\({}_{\text{SPMS}}\), respectively. In Figure 2, the suspension displacements, suspension velocities and tire deflections of \(S_{f}\) and \(S_{r}\) are defined as (1), where \(d\), \(v\) and \(w\) stand for the suspension displacement, the suspension velocity and the tire deflection, respectively. With those definitions, the suspension forces at \(S_{f}\) and \(S_{r}\) are derived as (2). As shown in (2), the suspension forces are calculated with \(d\) and \(v\) at \(S_{f}\) and \(S_{r}\). With (2), the equations of heave and pitch motions of the SPMS, \(m_{s}\) and the vertical motions of USPMSs, \(m_{uf}\) and \(m_{ur}\), are obtained as (3). From the spatial relationship between the center of gravity (C.G.) and FC\({}_{\text{SPMS}}/\text{RC}_{\text{SPMS}}\), and the approximation, \(\sin\!\theta_{c}\approx\theta_{c}\), \(z_{sf}\) and \(z_{sr}\), the vertical displacements at FC\({}_{\text{SPMS}}\) and RC\({}_{\text{SPMS}}\), are obtained as (4) from \(z_{c}\) and \(\theta_{c}\). The value of (4) is converted into the matrix-vector form of (5). New vectors and matrices are defined as (6), where \(\mathbf{x}\), \(\mathbf{w}\) and \(\mathbf{u}\) are the vectors of state variables, disturbances and control inputs, respectively. With these definitions in (6), the suspension force of (2) is represented by (7). With (6) and (7), the equation of the sprung mass given in (3) is converted into (8), and the vertical displacements of (5) are converted into (9). With (7) and (9), (8) is converted into (10). A new matrix-vector Equation (11) is derived from (10). With the new matrices defined in (11) and the vector \(\mathbf{z}\) defined in (6), (12) is obtained. By defining new matrices \(\mathbf{A}\), \(\mathbf{B}_{1}\) and \(\mathbf{B}_{2}\) as given in (13), the SSEQN of the HCM is obtained as (14). \[ \begin{cases}d_{f}\triangleq z_{sf}-z_{uf}\\ d_{r}\triangleq z_{sr}-z_{ur}\end{cases}\begin{cases}v_{f}\triangleq\dot{z}_{ sf}-\dot{z}_{uf}\\ v_{r}\triangleq\dot{z}_{sr}-z_{ur}\end{cases}\begin{cases}w_{f}\triangleq z _{uf}-z_{rf}\\ w_{r}\triangleq z_{ur}-z_{rr}\end{cases} , \tag{1}\]  
\[ \begin{cases}f_{f}=-k_{sf}\Big{(}z_{sf}-z_{uf}\Big{)}-b_{sf}\Big{(}\dot{z}_{ sf}-\dot{z}_{uf}\Big{)}+u_{f}=-k_{sf}d_{f}-b_{sf}v_{f}+u_{f}\\ f_{r}=-k_{sr}(z_{sr}-z_{ur})-b_{sr}(\dot{z}_{sr}-\dot{z}_{ur})+u_{r}=-k_{sr}d_{ r}-b_{sr}v_{r}+u_{r}\end{cases} , \tag{2}\]  
Figure 2: Half-car model with front and rear suspensions. \[ \begin{cases}m_{s}\ddot{z}_{c}=f_{f}+f_{r}\\ I_{y}\ddot{\theta}_{c}=-l_{f}f_{f}+l_{r}f_{r}\end{cases}\begin{cases}m_{u}\ddot{z }_{uf}=-f_{f}-k_{tf}w_{f}\\ m_{ur}\ddot{z}_{ur}=-f_{r}-k_{tr}w_{r}\end{cases} , \tag{3}\]  
\[ \begin{cases}z_{sf}=z_{c}-l_{f}\sin\theta_{c}\simeq z_{c}-l_{f}\theta_{c}\\ z_{sr}=z_{c}+l_{r}\sin\theta_{c}\simeq z_{c}+l_{r}\theta_{c}\end{cases} , \tag{4}\]  
\[ \begin{bmatrix}z_{sf}\\ z_{sr}\end{bmatrix} = \begin{bmatrix}1&-l_{f}\\ 1&l_{r}\end{bmatrix}\begin{bmatrix}z_{c}\\ \theta_{c}\end{bmatrix} , \tag{5}\]  
\[