# An Observer-Based Active Fault Tolerant Controller for Vehicle Suspension System
## 2 System Modelling and Problem Statement
### Problem Statement of Active Fault-Tolerant Control  
For the active suspension system in (6), our main objective was to develop a fault-tolerant controller to improve the vehicle dynamics performance in the presence of the actuator faults and the external road disturbance. The proposed control scheme for this active suspension system is shown in Figure 2.  
As shown in Figure 2, when the system (6) has no fault, that is, \(\mathbf{f}(t)=0\), the \(H_{\infty}\) output feedback controller works as the proposed active FTC scheme, which can guarantee a better control performance of vehicle suspension system. When there are some systematic faults in (6), i.e., \(\mathbf{f}(t)\
eq 0\), the proposed active FTC scheme is composed of the \(H_{\infty}\) output feedback controller and \(H_{\infty}\) observer-based fault compensation controller, in which the fault compensation controller is employed to observe and estimate the fault information and further, to design the fault compensative control law to eliminate the effects of the faults on vehicle suspension system performances. At the same time, the \(H_{\infty}\) output feedback controller starts to work so as to ensure that the closed-loop system is stable and robust to the external disturbance inputs. In this control scheme, both of the fault compensation controller and robust \(H_{\infty}\) output feedback controller work together to achieve the active fault-tolerant control for active suspension system (6). To sum up, the designed active fault-tolerant controller is formed as:  
\[\mathbf{u}(t)=\mathbf{u}_{N}(t)+\mathbf{u}_{C}(t) \tag{7}\]  
where \(\mathbf{u}_{N}(t)\) is the control law of \(H_{\infty}\) output-feedback controller, and \(\mathbf{u}_{C}(t)\) is the control law of fault compensation controller. Substituting (7) into (6) gives the closed-loop system as:  
\[\left\{ \begin{array}{l}\dot{\mathbf{x}}(t)=\mathbf{A}\mathbf{x}(t)+\mathbf{B}\mathbf{u}_{N}(t)+ \mathbf{B}\mathbf{u}_{C}(t)+\mathbf{B}_{d}\mathbf{d}(t)+\mathbf{B}_{f}\mathbf{f}(t)\\ \mathbf{y}(t)=\mathbf{C}_{1}\mathbf{x}(t)\\ \mathbf{z}(t)=\mathbf{C}_{2}\mathbf{x}(t)+\mathbf{D}_{2}\mathbf{u}_{N}(t)+\mathbf{D}_{2}\mathbf{u}_{C}(t)+ \mathbf{D}_{2}\mathbf{f}\mathbf{f}(t)\end{array} \right. \tag{8}\]  
Therefore, the control objective in this article can be defined as: For the system (8), one can design \(\mathbf{u}_{C}(t)=-\mathbf{M}\mathbf{f}(t)\) to make \(\mathbf{B}\mathbf{u}_{C}(t)+\mathbf{B}_{f}\mathbf{f}(t)=\mathbf{0}\), and \(\mathbf{D}_{2}\mathbf{u}_{C}(t)+\mathbf{D}_{2f}\mathbf{f}(t)=\mathbf{0}\), thus to compensate for the performance penalties caused by the actuator faults. Moreover, the control law \(\mathbf{u}_{N}(t)=\mathbf{K}(s)\mathbf{y}(t)\) is designed to ensure the asymptotic stability of the closed-loop system (8). It should be noted here that \(\mathbf{\hat{f}}(t)\) is the estimation of \(\mathbf{f}(t)\). To fulfill the fault estimation and compensation, substituting (4) into (6) yields the following augmented system as:  
\[\left\{ \begin{array}{l}\dot{\mathbf{\zeta}}(t)=\mathbf{A}_{ef}\mathbf{\zeta}(t)+\mathbf{B}_ {ef}\mathbf{u}(t)+\mathbf{F}_{\varepsilon}\mathbf{w}(t)\\ \mathbf{y}(t)=\mathbf{C}_{\varepsilon}\mathbf{\zeta}(t)\\ \mathbf{f}(t)=\mathbf{C}_{ef}\mathbf{\zeta}(t)\end{array} \right. \tag{9}\]  
where  
\[\mathbf{\zeta}(t)=\left[ \begin{array}{c}\mathbf{x}(t)\\ \mathbf{x}_{f}(t)\end{array} \right]\text{, }\mathbf{w}(t)=\left[ \begin{array}{c}\mathbf{d}(t)\\ \mathbf{\delta}(t)\end{array} \right]\text{,}\]  
Figure 2: The diagram of the active fault tolerant controller. \[\mathbf{A}_{ef}=\left[ \begin{array}{cc}\mathbf{A}&B_{f}\mathbf{C}_{wf}\\ \mathbf{0}&\mathbf{A}_{wf}\end{array} \right]\text{, }\mathbf{B}_{ef}=\left[ \begin{array}{c}\mathbf{B}\\ \mathbf{0}\end{array} \right]\text{,}\]  
\[\mathbf{F}_{e}=\left[ \begin{array}{cc}\mathbf{F}_{1e}&\mathbf{F}_{2e}\end{array} \right] \text{, }\mathbf{F}_{1e}=\left[ \begin{array}{c}\mathbf{B}_{d}\\ \mathbf{0}\end{array} \right]\mathbf{F}_{2e}=\left[ \begin{array}{c}\mathbf{0}\\ \mathbf{B}_{wf}\end{array}