## 3 Reliable Robust \(H_{\infty}\) Controller Design  
Traditionally, most studies mainly consider three requirements, namely, road-holding stability, suspension deflection, and the ride comfort, when designing a control law for suspension systems. Some studies also take energy-saving into consideration. However, few papers have covered the coupling performance of IWMs in suspension design, especially for PMSM. The issues for active suspension system associated with actuator fault tolerance, model parameter uncertainties and coupling effect in IWM based on PMSM have not been well studied in any previous research. Therefore, the reliable robust Ho control law for active suspension of hub-driven EVs is designed. Considering factors mentioned above, the following suspension performances should be taken into account. (1). Maximum actuator force. The active control force provided by the suspension system should be constrained by a threshold due to the limited power of the actuator, that is  
\[\left|u(t)\right|\leq u_{\max} \tag{22}\]  
where \(u_{max}\) is the maximum force of the actuator. (2). Ride comfort. To provide comfort for passengers, a control \(u\) is designed to isolate the sprung mass from the road and magnetic force induced vibration, i.e., to minimize the vertical acceleration \(\tilde{z}_{b}\) in the presence of parameter uncertainties \(m_{b}\) and \(k_{t}\), unknown dynamics \(F_{r}\), and road speed excitation. (3). Suspension deflection. The suspension deflection should not exceed its travel limit to avoid ride comfort degradation and vehicle component damage. i.e.,  
\[\left|z_{s}(t)-z_{b}(t)\right|\leq z_{\max} \tag{23}\]  
where \(z_{max}\) is the maximum deflection of the suspension. (4). Motor unbalance dynamic force. The UEF in IWM promotes the rotor eccentricity which intensifies the bearing wear, shortening the motor life. Furthermore, it aggravates the sprung mass acceleration and provokes cacophony, deteriorating the vehicle comfort significantly. As previously analyzed, the UEF is closely related to the rotor eccentricity. Thus, another primary objective is to minimize the rotor eccentricity when the controller is designed. That is, \(\left|z_{r}-z_{s}\right|\). (5). Road-holding stability. The firm uninterrupted contact of wheels to road should be ensured to make sure the vehicle safety, that is to say, the dynamic load of the tire should not exceed its corresponding static load, i.e.,  
\[k_{t}(z_{r}(t)-z_{g}(t))\leq(m_{b}+m_{s}+m_{r})g \tag{24}\]  
To analyze the performances mentioned above, for the sake of convenience the controlled outputs are defined as  
\[\mathbf{z}_{1}=\left[ \begin{array}{cc}\bar{z}_{b}&\beta(z_{r}-z_{s})\\ \end{array} \right]^{T}\ \mathbf{z}_{2}=\left[ \begin{array}{cc}\gamma\frac{z_{s}(t)- z_{b}(t)}{z_{\max}}&\frac{k_{t}(z_{r}(t)-z_{g}(t))}{(m_{b}+m_{s}+m_{r})g}\\ \end{array} \right]^{T} \tag{25}\]  
Figure 6: (**a**) Electromagnetic suspension system; (**b**) linear motor actuator structure; (**c**) thrust force according to axial direction. Thus, active suspension control system can be described by a state-space equation as:  
\[\left\{ \begin{array}{l}\dot{\mathbf{x}}(t)\textbf{=}\textbf{A}(\mathbf{\lambda}) \mathbf{x}(t)\textbf{+}\textbf{B}_{1}(\mathbf{\lambda})\mathbf{w}(t)\textbf{+} \textbf{B}_{2}(\mathbf{\lambda})u(t)\\ \textbf{z}_{1}\textbf{=}\textbf{C}_{1}(\mathbf{\lambda})\mathbf{x}(t)\textbf{ +}\textbf{D}_{1}(\mathbf{\lambda})u(t)\\ \textbf{z}_{2}\textbf{=}\textbf{C}_{2}(\mathbf{\lambda})\textbf{x}(t)\textbf{ +}\textbf{D}_{2}u(t)\end{array} \right. \tag{26}\]  
where  
\[\textbf{C}_{1}(\mathbf{\lambda})=\left[ \begin{array}{cccccc}-\varepsilon_{s }m_{b}^{-1}&c_{s}m_{b}^{-1}&0&k_{s}m_{b}^{-1}&0&0\\ 0&0&0&\beta&0\end{array} \right]\!\!,\ \ \textbf{D}_{1}(\mathbf{\lambda})=\left[ \begin{array}{ ccc}m_{b}^{-1}&0\end{array} \right]^{T}\!\!,\]  
\[\textbf{C}_{2}(\mathbf{\lambda})=\left[ \begin{array}{cccccc}0&0&0&\gamma z^ {-1}_{max}&0&0\\ 0&0&0&0&k_{t}(m_{b}+m_{s}+m_{r})^{-1}g^{-1}\end{array}