## 3 Design of Parameter Adaptive Controllers  
In this paper, PAC is designed with extended Kalman filter. PAC is based on the structure of SOFCs, as given in (20) and (23) [38]. In the equations of motions of SPMS given in (3), if the front and rear suspension forces, \(f_{f}\) and \(f_{r}\), become zero, then there will be no heave and pitch motions of SPMS. As a result, an ideal ride comfort can be achieved due to zero heave acceleration. To go into more detail, if (25) is satisfied for \(f_{f}\) and \(f_{r}\), then (26) is achieved. From (25), the control inputs are obtained as (27). If the control inputs in (27) are applied, then the heave and pitch angular accelerations of the SPMS are not generated as a result [34; 35; 36; 37; 38]. Equation (27) is converted into the vector-matrix form of (28). Equation (28) is identical to \(\mathbf{u}_{SOF1}\) in (20), where \(k_{g}\), \(b_{g}\), \(k_{sr}\) and \(b_{sr}\) of (28) correspond to \(k_{f1}\), \(k_{f2}\), \(k_{r1}\) and \(k_{r2}\), respectively. In this paper, four gains in (20) are adaptively updated by RLS-equivalent EKF [39; 40; 41; 42; 43; 44; 45]. Similarly, the two gains of (21) are adaptively updated by RLS-equivalent EKF for the purpose of satisfying (26). \[ \begin{cases}f_{f}=-k_{sf}d_{f}-b_{sf}v_{f}+u_{f}=0\\ f_{r}=-k_{sr}d_{r}-b_{sr}v_{r}+u_{r}=0\end{cases} , \tag{25}\]  
\[ \begin{cases}m_{s}\tilde{z}_{c}=f_{f}+f_{r}=0\\ l_{g}\tilde{\theta}_{c}=-l_{f}f_{f}+l_{r}f_{r}=0\end{cases} , \tag{26}\]  
\[ \begin{cases}u_{f}=k_{sf}d_{f}+b_{sf}v_{f}\\ u_{r}=k_{sr}d_{r}+b_{sr}v_{r}\end{cases} , \tag{27}\]  
\[\mathbf{u}= \begin{bmatrix}u_{f}\\ u_{r}\end{bmatrix} = \begin{bmatrix}k_{sf}&b_{sf}&0&0\\ 0&0&k_{sr}&b_{sr}\end{bmatrix}\begin{bmatrix}d_{f}\\ v_{f}\\ d_{r}\\ v_{r}\end{bmatrix} . \tag{28}\]  
Generally, it has been shown that RLS is equivalent to EKF [38]. To adaptively update the gains of the SOFCs, EKF is selected [39; 40; 41; 42; 43; 44; 45]. The parameter vector of RLS is equivalent to the state vector of EKF. The input vector of RLS is equivalent to the output matrix of EKF. By combining (25) and (26), (29) is obtained. From (29), the state vector and output matrix of EKF for \(\mathbf{u}_{SOF1}\) are obtained as (30). From (3) and (21), (31) is obtained. With (31), the state vector and output matrix of EKF for \(\mathbf{u}_{SOF2}\) are obtained as (32). The vector of the desired outputs, \(\mathbf{d}_{p}(k)\), is obtained as (33) from (26). As shown in (30) and (32), the state vector, \(\mathbf{x}_{p}\), acts as a gain of SOFCs. As shown in (30), (32) and (33), EKF is run in discrete-time domain. When discretizing the state-space equation or transfer function, the sampling period was set to 1 ms in this paper. \[ \begin{cases}m_{s}\tilde{z}_{c}=-k_{sf}d_{f}-b_{sf}v_{f}-k_{sr}d_{r}-b_{sr}v_{ r}+u_{f}+u_{r}\\ l_{g}\tilde{\theta}_{c}=-l_{f}\left(-k_{sf}d_{f}-b_{sf}v_{f}+u_{f}\right)+l_{r} (-k_{sr}d_{r}-b_{sr}v_{r}+u_{r})\end{cases} , \tag{29}\]  
\[\mathbf{x}_{p}(k)\triangleq \begin{bmatrix}k_{sf}(k)\\ b_{sf}(k)\\ k_{sr}(k)\\ b_{sr}(k)\end{bmatrix} ,\,\mathbf{C}_{p}\triangleq \begin{bmatrix}-d_{f}(k)&-v_ {f}(k)&-d_{r}(k)&-v_{r}(k)\\ l_{f}d_{f}(k)&l_{f}v_{f}(k)&-l_{r}d_{r}(k)&-l_{r}v_{r}(k)\end{bmatrix} , \tag{30}\]  
\[ \begin{bmatrix}m_{s}\tilde{z}_{c}\\ l_{y}\tilde{\theta}_{c}\end{bmatrix} = \begin{bmatrix}k_{v}&0\\ 0&k_{p}\end{bmatrix}\begin{bmatrix}\tilde{z}_{c}\\ \tilde{\theta}_{c}\end{bmatrix} , \tag{31}\]  
\[\mathbf{x}_{p}(k)\triangleq \begin{bmatrix}k_{v}(k)\\ k_{p}(k)\end{bmatrix} ,\,\mathbf{C}_{p}\triangleq \begin{bmatrix}\tilde{z}_{c} &0\\ 0&\hat{\theta}_{c}(k)\end{bmatrix} , \tag{32}\]\[\mathbf{d}_{p}(k)= \begin{bmatrix}m_{s}\tilde{z}_{c}(k)+u_{f}(k-1)+u_{r}(k-1)\\ I_{y}\tilde{\varrho}_{c}(k)-l_{f}u_{f}(k-1)+l_{r}u_{r}(k-1)\end{bmatrix} . \tag{33}\]  
The state-space and output equations of EKF are given as (34) [39; 40; 41; 42; 43; 44; 45]. In (34), \(\boldsymbol{\chi}\) and \(\boldsymbol{\sigma}\) are the vectors of the white-noise processes of system and measurement, respectively.