# An Observer-Based Active Fault Tolerant Controller for Vehicle Suspension System
## 2 System Modelling and Problem Statement  
### Generalized Fault Model of Active Suspension System  
A half-vehicle suspension model with four DOFs (degree-of-freedoms) is shown in Figure 1, in which, \(m_{\rm s}\) an \(I_{\rm y}\) denote the sprung mass and its moment of inertia, \(z_{\rm s}\) and \(\varphi\) denote the vertical and pitch angular displacement at the center gravity (CG) of the vehicle body, respectively; \(a\) and \(b\) denote the horizontal distances from the CG of the vehicle body to the front and rear axles, respectively; \(m_{\rm uf}\) and \(m_{\rm ur}\) denote the unsprung mass of the front and rear suspension, respectively; \(c_{\rm f}\) and \(c_{\rm r}\) represent the damping coefficients of the front and rear suspension, respectively; \(k_{\rm f}\) and \(k_{\rm r}\) represent the stiffness coefficients of the front and rear suspension, respectively; \(z_{\rm si}\), \(z_{\rm ui}\), and \(z_{\rm ri}\) (\(i=\rm f,\,r\), stands for the front and rear suspension) denote the suspension vertical displacements of the sprung and unsprung mass, and the road disturbance inputs to the front and rear wheels, respectively; \(u_{\rm f}\) and \(u_{\rm r}\) represent the control forces of the front and rear suspension, respectively. Ignoring the damping characteristics of tires, \(k_{\rm tf}\) and \(k_{\rm tr}\) denote the tire stiffness coefficients of the front and rear wheels, respectively. It was assumed that the front and rear wheels had the same road excitation signals, thus the vertical displacements of the front and rear suspension sprung masses had the following relationship with a smaller pitch angular \(\varphi\) as [28]  
\[\left\{ \begin{array}{l}z_{\rm sf}\cong z_{\rm s}-a\sin\varphi\approx z_{\rm s }-a\varphi\\ z_{\rm sr}\cong z_{\rm s}+b\sin\varphi\approx z_{\rm s}+b\varphi\end{array} \right. \tag{1}\]  
Based on [PERSON]'s second law, the dynamics equations of the half-vehicle suspension system was written as follows:  
\[\left\{ \begin{array}{l}m_{\rm s}\ddot{z}_{\rm s}=-k_{\rm f}(z_{\rm sf}-z_{ \rm uf})-c_{\rm f}(\dot{z}_{\rm sf}-\dot{z}_{\rm uf})-k_{\rm r}(z_{\rm sr}-z_{ \rm ur})-c_{\rm r}(\dot{z}_{\rm sr}-\dot{z}_{\rm ur})+u_{\rm f}+u_{\rm r}\\ l_{\rm y}\ddot{\varphi}=-ak_{\rm f}(z_{\rm sf}-z_{\rm uf})-ac_{\rm f}(\dot{z}_{ \rm sf}-\dot{z}_{\rm uf})+bk_{\rm r}(z_{\rm sr}-z_{\rm ur})+bc_{\rm r}(\dot{z} _{\rm sr}-\dot{z}_{\rm ur})+au_{\rm f}-bu_{\rm r}\\ m_{\rm uf}\ddot{z}_{\rm uf}=k_{\rm f}(z_{\rm sf}-z_{\rm uf})+c_{\rm f}(\dot{z} _{\rm sf}-\dot{z}_{\rm uf})-k_{\rm tf}(z_{\rm uf}-z_{\rm rf})-u_{\rm f}\\ m_{\rm ur}\ddot{z}_{\rm ur}=k_{\rm r}(z_{\rm sr}-z_{\rm ur})+c_{\rm r}(\dot{z} _{\rm sr}-\dot{z}_{\rm ur})-k_{\rm rr}(z_{\rm ur}-z_{\rm rr})-u_{\rm r}\end{array} \right. \tag{2}\]  
Define the state vector as \(\mathbf{x}(t)=\left[z_{\rm s},\dot{z}_{\rm s},\varphi,\dot{\varphi},z_{\rm ur},z_{ \rm ur},\dot{z}_{\rm uf},\dot{z}_{\rm ur}\right]^{T}\), the disturbance input as \(\mathbf{d}(t)=[z_{\rm rf},z_{\rm rr}]^{T}\), the control input as \(\mathbf{u}(t)=[u_{\rm f},\,u_{\rm r}]^{T}\), the control output as \(\mathbf{z}(t)=\left[\ddot{z}_{\rm s},\,\ddot{\varphi},\,z_{\rm sf}-z_{\rm uf}, \,z_{\rm sr}-z_{\rm ur},\,u_{\rm f},\,u_{\rm r}\right]^{T}\), and the measured output as \(\mathbf{y}(t)=\left[\dot{z}_{\rm s},\,\dot{\varphi},\,z_{\rm sf}-z_{\rm uf},\,z_{ \rm sr}-z_{\rm ur}\right]^{T}\), wherein \(\ddot{z}_{\rm s}\) and \(\ddot{\varphi}\) represent the vertical and pitch angular accelerations of the vehicle body, respectively;  
Figure 1: Half-vehicle active suspension model. \(\dot{z}_{\rm s}\) and \(\dot{\varphi}\) represent the vertical and pitch angular velocities of the vehicle body, respectively; \(z_{\rm sf}-z_{\rm uf}\) and \(z_{\rm sr}-z_{\rm ur}\) represent the dynamic deflections of the front and rear suspension, respectively. Then, the system state-space equation of half-vehicle suspension was given by  
\[\left\{