## 2 System Modelling and Problem Formulation
### Active Suspension System Modelling  
The sprung mass \(m_{b}\) includes the driver mass and passenger mass, so it usually changes with the number of drivers and passengers. The tire stiffness \(k_{t}\) usually varies due to tire pressure. In this study, it is supposed that the sprung mass and the tire stiffness reside in an interval as follows:  
\[m_{b}=m_{b0}(1+\lambda_{1})k_{t}=k_{t0}(1+\lambda_{2}) \tag{19}\]  
where \(m_{b0}\) and \(k_{t0}\) are the nominal sprung mass and the nominal tire stiffness, respectively. \(|\lambda_{1}|\leq\lambda_{10}\) and \(|\lambda_{2}|\leq\lambda_{20}\) are used to exhibit the interval that the sprung mass and the tire stiffness are located within. Furthermore, \(m_{bl}=m_{b0}(1+\lambda_{1})\big{|}_{\lambda 1=-\lambda 10}\) and \(k_{tl}=k_{t0}(1+\lambda_{2})\big{|}_{\lambda 2=-\lambda 20}\) are, respectively, expressed as the lower bounds of the sprung mass and the tire stiffness. Similarly, \(m_{bu}=m_{b0}(1+\lambda_{1})\big{|}_{\lambda 1=\lambda 10}\) and \(k_{tu}=k_{t0}(1+\lambda_{2})\big{|}_{\lambda 2=\lambda 20}\) are expressed as the corresponding upper bounds. Therefore, the uncertain parameter vector can be described as \(\mathbf{\lambda}=[\lambda_{1},\lambda_{2}]\).  
According to Equation (11), and defining vehicle state vector as  
\[\mathbf{x}(t)=\Big{[}\begin{array}{cccc}\dot{z}_{b}&\dot{z}_{s}&\dot{z}_{r}& z_{s}-z_{b}&z_{r}-z_{s}&z_{g}-z_{r}\end{array}\Big{]}^{T} \tag{20}\]  
the dynamic model of the vehicle can be described by the following state-space equations.  
\[\dot{\mathbf{x}}(t)\textbf{=A}(\mathbf{\lambda})\mathbf{x}(t)\textbf{+B}_{1}(\bm {\lambda})\mathbf{w}(t)\textbf{+B}_{2}(\mathbf{\lambda})\mathbf{\xi}\mathbf{u}(t) \tag{21}\]  
where  
\[\mathbf{A}(\mathbf{\lambda})=\left[\begin{array}{cccc}-c_{s}m_{b}^{-1}&c_{s}m_{ b}^{-1}&0&k_{s}m_{b}^{-1}&0&0\\ c_{s}m_{s}^{-1}&-c_{s}m_{s}^{-1}&0&-k_{s}m_{s}^{-1}&k_{m}m_{s}^{-1}&0\\ 0&0&-c_{t}m_{r}^{-1}&0&-k_{m}m_{r}^{-1}&k_{t}m_{r}^{-1}\\ -1&1&0&0&0&0\\ 0&-1&1&0&0&0\\ 0&0&-1&0&0&0\\ \end{array}\right]\!,\ \mathbf{B}_{1}(\mathbf{\lambda})=\left[\begin{array}{cccc}0&0&c_{t}m_{r}^{-1 }&0&0&1\\ 0&m_{s}^{-1}&-m_{r}^{-1}&0&0&0\\ \end{array}\right]\!\!^{T}\!,\] \[\mathbf{B}_{2}(\mathbf{\lambda})=\left[\begin{array}{cccc}m_{b}^{-1}& -m_{s}^{-1}&0&0&0&0\\ \end{array}\right]\!\!^{T}\!,\ \mathbf{w}(t)=\left[\begin{array}{cccc}\dot{z}_{g}&F_{r} \end{array}\right]\!\!^{T}\!,\]  
where \(\mathbf{\lambda}\) represents uncertain parameter vector and \(\mathbf{u}(t)\) is the actuator force produced by active suspension. As shown in Figure 5(a), the active suspension system consists of an electro-magnetic actuator and a mechanical spring. Fail safe passive damping is provided by means of eddy-currents. The linear motor actuator is a tubular slotted three-phase permanent magnet actuator with a peak force of 2000 N. The actuator is shown as Figure 5(b), where \(\tau_{p}\) represents the PM pole pitch; \(R_{s}\) represents the coils outer radius; \(\tau_{cp}\) and \(\tau_{sp}\) represent the coil and slot pitches, respectively; \(\tau_{mz}\) and \(\tau_{mr}\) denote the pitches of axially and radially magnetized PMs, respectively; \(\tau_{s}\) represents the slot width; and \(R_{0}\) and \(R_{m}\) represent the inner and outer radii of the PMs. The details can be found in the literature [51; 52]. The thrust force according to axial direction is shown in Figure 5(c) (detailed calculation of the thrust force can be found in literature [52]).