## 2 Design of LQ Optimal Controllers
### Design of LQ SOF Controllers
In this paper, \(\mathbf{K}_{SOF1}\) and \(\mathbf{K}_{SOF2}\) minimizing \(L\) are called LQSOF1 and LQSOF2, respectively. Up to now, no analytical method has been available to find an optimal \(\mathbf{K}_{SOF1}\) or \(\mathbf{K}_{SOF2}\) that minimize \(L\) and to find a stabilizing \(\mathbf{K}_{SOF1}\) or \(\mathbf{K}_{SOF2}\) for the systems of (14) [47]. To find the gain matrices minimizing \(L\), the optimization problem is constructed as (24), which has been known as non-convex [18; 19; 20; 47; 48; 49]. For this reason, a solution obtained from the optimization cannot be guaranteed to be optimum. In (24), \(\mathbf{K}\) is either \(\mathbf{K}_{SOF1}\) or \(\mathbf{K}_{SOF2}\), and \(\mathbf{A}_{c}\) is the closed-loop system matrix, defined as \(\mathbf{A}+\mathbf{B}_{2}\mathbf{K}_{SOF1}\mathbf{C}_{1}\) and \(\mathbf{A}+\mathbf{B}_{2}\mathbf{G}^{-1}\mathbf{K}_{SOF2}\mathbf{C}_{2}\) for \(\mathbf{u}_{SOF1}\) and \(\mathbf{u}_{SOF2}\), respectively. In (24), for the given \(\mathbf{K}\), the stability of \(\mathbf{A}_{c}\) is tested by max(Real[\(\mathbf{A}_{c}\)]), and then, Lyapunov equation is solved and its solution, \(\mathbf{S}\), is obtained, and then the value of \(L\) is calculated by trace(\(\mathbf{S}\)). To find the optimal solution of (24), the meta-heuristic method, CMA-ES, is selected as an optimizer in this paper [32; 33]. \[ \begin{array}{l}\underset{\mathbf{K}}{\text{mintrace}}(\mathbf{S})\\ s.t.\begin{cases}\mathbf{S}=\mathbf{S}^{T}>\mathbf{0}\\ \max(\text{Real}[\mathbf{A}_{c}])<0\\ \mathbf{A}_{c}^{T}\mathbf{S}+\mathbf{S}\mathbf{A}_{c}+\mathbf{\Pi}+\mathbf{C} ^{T}\mathbf{K}^{T}\Xi^{T}+\Xi\mathbf{K}\mathbf{C}+\mathbf{C}\mathbf{K}^{T} \Theta\mathbf{K}\mathbf{C}=\mathbf{0}\end{cases}.\end{array} \end{array} \tag{24}\]  
In this paper, the SOFC, \(\mathbf{u}_{SOF1}\), uses the signals, \(d_{f}\), \(d_{r}\), \(v_{f}\) and \(v_{r}\), and \(\mathbf{u}_{SOF2}\) uses the signals, \(v_{zc}\) and \(\omega_{yc}\), which are calculated from the accelerometer signals measured at \(\text{FC}_{\text{SPMS}}\), \(\text{RC}_{\text{SPMS}}\), and front and rear wheel centers [17; 18; 19; 20; 50]. As shown in Figure 2, the accelerometers, \(Az_{Sf}\) and \(Az_{Sr}\), are installed on FC\({}_{\text{SPMS}}\) and RC\({}_{\text{SPMS}}\), and \(Az_{Wf}\) and \(Azw_{r}\) are installed on the front and rear wheel centers. Figure 3 shows the calculation procedure for \(d_{f}\), \(d_{r}\), \(v_{f}\), \(v_{r}\), \(v_{zc}\) and \(\omega_{yc}\). The accelerometer signals measured with \(Az_{Sf}\), \(Az_{Sr}\), \(Azw_{f}\) and \(Azw_{r}\) are filtered through high-pass and low-pass filters (HPF and LPF) in order to reduce DC offsets and sensor noises, respectively. Then, the filtered signals pass through the integrator, \(1/s\). As a result, the signals needed for LQSOFCs, \(d_{f}\), \(d_{r}\), \(v_{f}\), \(v_{r}\), \(v_{zc}\) and \(\omega_{yc}\), are obtained. The HPF and LPF can be found in the reference [18]. As an alternative, the suspension displacements, \((d_{f},d_{r})\), can be easily measured with a potentiometer or an encoder in real vehicles [29; 30; 51; 52]. The suspension velocities, \((v_{f},v_{r})\), can be obtained by differentiating the measured \((d_{f},d_{r})\). Figure 3: Block diagram for calculation on the signals for SOFCs.