## 5 Results and Discussion
In this work, we assume that the suspension deflection cannot exceed a maximum value, and can be constrained by the mechanical structure \(\left|z_{s}(t)-z_{us}(t)\right|<\beta\). Note that the variable \(\alpha(t)\) is given by the expression \(\alpha(t)=\left(z_{s}(t)-z_{us}(t)\right)^{2}=\left(x_{3}(t)\right)^{2}\). As a result, \(x_{3}(t)\in[-\beta,+\beta]\) and \(\alpha\in[0,\beta 2]\); that is, \(\alpha_{1}=0\) and \(\alpha_{2}=\beta^{2}\), and \(\beta\) is a predefined constant. Based on Equation (39), the faulty discrete state-space representation in Equation (38) can be expressed as a polytopic LPV system:  
\[\left\{ \begin{array}{c}x(k+1)=\sum\
olimits_{r=1}^{2}\rho_{r}(\theta_{k}) (A_{r}x(k)+B_{r}u(k)+D_{r}\omega(k))\\ y_{f}(k)=\sum\
olimits_{r=1}^{2}\rho_{r}(\theta_{k})(C_{r}x(k)+f_{s}(k))\end{array} \right. \tag{40}\]  
where the subscript \(r\) refers to the corresponding vertex of the polytope, and \(\rho_{r}(\theta_{k})\) is the vertex coefficient subject to \(\rho_{r}(\theta_{k})>0\) and \(\sum\limits_{r=1}^{2}\rho_{r}(\theta_{k})=1\). Considering the LPV model as described in Equation (1) with \(N=2\) vertices, the nonlinear system in Equation (31) is described by Equation (35) with the following matrices:  
\[A_{1}=\left[ \begin{array}{cccc}0&1&0&0\\ -\frac{k_{us}}{m_{ss}}&\frac{c_{r}+c_{ss}}{m_{ss}}&\frac{k_{s}}{m_{ss}}&\frac{ c_{s}}{m_{ss}}\\ 0&-1&0&1\\ 0&\frac{c_{s}}{m_{s}}&-\frac{k_{s}}{m_{ss}}&-\frac{c_{s}}{m_{s}}\end{array}  \right],\;A_{2}=\left[ \begin{array}{cccc}0&1&0&0\\ -\frac{k_{us}}{m_{ss}}&\frac{c_{s}+c_{ss}}{m_{ss}}&\frac{k_{s}-N_{s}\beta^{2}} {m_{ss}}&\frac{c_{s}}{m_{ss}}\\ 0&-1&0&1\\ 0&\frac{c_{s}}{m_{s}}&-\frac{k_{s}+N_{s}\beta^{2}}{m_{ss}}&-\frac{c_{s}}{m_{s} }\end{array} \right]\]  
\[B_{1}=B_{2}=\left[ \begin{array}{c}\frac{1}{m_{ss}}\\ 0\\ -\frac{1}{m_{ss}}\end{array} \right],\;D_{1}=D_{2}=\left[ \begin{array}{c}-1\\ \frac{c_{ss}}{m_{ss}}\\ 0\\ 0\end{array} \right]\text{ with }\beta=0.2.\]  
Next, the MPC parameters are selected as follows. The weight matrices \(Q\) and \(R\) of the cost function, in Equation (12), are set to \(115I\) and \(15I\), respectively, where \(I\) is the identity matrix with a proper dimension. A constraint is set on the active control force to limit the power of the hydraulic actuator. Thus, a hard constraint is imposed on the active suspension such that \(\left|u(k)\right|\leq u_{max}\), and \(u_{max}=1500\). In addition, the initial state values of the LPV system are set as follows: \(x(0)=[0,\ 0,\ 0,\ 0,\ 0]\) and for the augmented vector that includes initial fault \(\eta(0)=[0,\ 0,0,0,0.08]\). To test the robustness of the proposed method, the external disturbance, which is considered as a bump road profile, is expressed as \(\omega(k)=0.05\cos(\mathrm{pi}k)\sin(0.26\mathrm{pi}k)\). In the vehicle industry, the sensors are frequently  
The results indicate that the proposed control methodology can robustly improve the ride comfort compared with passive FTC methods. The tire deflection and sprung mass acceleration responses together with their estimated values are plotted in Figures 2 and 3, respectively. The results in Figure 2 show that the tire deflection response has been considerably reduced despite the fact that the system is subject to real-time sensor faults and disturbances. The same comment can be made about the sprung mass acceleration results shown in Figure 3. In addition, the vertical acceleration of the vehicle's vibration, presented in Figure 3, is rapidly reduced by the fault-tolerant predictive control strategy while the actual value has been quickly and accurately estimated. As a result, the system may achieve lower values of body acceleration over time for the proposed active controller, which is better than the passive suspension systems proposed in [21] and, as a consequence, this will ensure better ride comfort. Figure 4 illustrates the suspension deflection, which is bounded by a maximum value of 0.04 m, where the system achieves lower values over time for the closed-loop systems.