# An Observer-Based Active Fault Tolerant Controller for Vehicle Suspension System
## 2 System Modelling and Problem Statement
### Problem Statement of Active Fault-Tolerant Control
\right]\text{,}\]  
\[\mathbf{C}_{e}=\left[ \begin{array}{cc}\mathbf{C}_{1}&\mathbf{D}_{1f}\mathbf{C}_{wf}\end{array}  \right]\text{, }\mathbf{C}_{ef}=\left[ \begin{array}{cc}\mathbf{0}&\mathbf{C}_{wf}\end{array} \right] \text{. }\]  
It is assumed that in the augmented system (9), (\(\mathbf{A}_{ef}\), \(\mathbf{B}_{ef}\)) is controllable, and (\(\mathbf{C}_{ef}\), \(\mathbf{A}_{ef}\)) is measurable, thus an augmented observer can be constructed to detect the system state and fault signal information after the system faults occur, and the full-order state observer of the system (9) is then expressed as:  
\[\left\{ \begin{array}{l}\dot{\mathbf{\xi}}(t)=\mathbf{A}_{ef}\xi(t)+\mathbf{B}_{ef}u(t)+ \mathbf{L}(\mathbf{y}(t)-\mathbf{C}_{e}\mathbf{\xi}(t))\\ \mathbf{f}(t)=\mathbf{C}_{ef}\mathbf{\xi}(t)\end{array} \right. \tag{10}\]  
where \(\mathbf{\xi}(t)\in R^{n+n_{f}}\) and \(\mathbf{L}\) are the state vector and gain matrix for the augmented observer, respectively. Let  
\[\mathbf{e}(t)=\mathbf{\xi}(t)-\mathbf{\xi}(t) \tag{11}\]  
\[\widehat{f}(t)=\mathbf{f}(t)-\widehat{f}(t) \tag{12}\]  
From (9) and (10), we have  
\[\left\{ \begin{array}{l}\dot{\mathbf{\epsilon}}(t)=(\mathbf{A}_{ef}-\mathbf{L}\mathbf{C}_{e} )\mathbf{e}(t)+\mathbf{F}_{e}\mathbf{w}(t)\\ \widetilde{f}(t)=\mathbf{C}_{ef}\mathbf{e}(t)\end{array} \right. \tag{13}\]  
Synthesizing (8) and (13) gives the active fault-tolerant control system of half-vehicle suspension as follows:  
\[\left\{ \begin{array}{l}\dot{\mathbf{x}}(t)=\mathbf{A}\mathbf{x}(t)+\mathbf{B}\mathbf{u}_{N}(t) +\mathbf{B}_{f}\mathbf{C}_{ef}\mathbf{e}(t)+\mathbf{B}_{w}\mathbf{w}(t)\\ \dot{\mathbf{e}}(t)=(\mathbf{A}_{ef}-\mathbf{L}\mathbf{C}_{e})\mathbf{e}(t)+\mathbf{F}_{e}\mathbf{w}(t)\\ \mathbf{z}(t)=\mathbf{C}_{2}\mathbf{x}(t)+\mathbf{D}_{2}\mathbf{u}_{N}(t)+\mathbf{D}_{2f}\mathbf{C}_{ef}\bm {e}(t)\end{array} \right. \tag{14}\]  
where \(\mathbf{B}_{w}=[\mathbf{B}_{d}\ \mathbf{0}]\). Through establishing the augmented system (14), the design of active FTC control scheme can be transformed into solving the gain vector \(\mathbf{L}\) of the fault observer and the control law \(\mathbf{u}_{N}(t)\) of \(H_{\infty}\) output-feedback controller by satisfying the following performance conditions:  
(a) The closed-loop system (14) is asymptotically stable. (b) Under zero initial conditions, \(\forall\mathbf{w}(t)\in L_{2}[0,+\infty)\), supposing that the transfer function from the disturbance input \(\mathbf{w}(t)\) to the fault error \(\widehat{f}(t)\) is \(T_{fw}(\mathbf{s})\), solving \(\mathbf{L}\) makes the system (13) satisfying \(H_{\infty}\) performance index \(|\ 1\,T_{\widehat{f}w}(\mathbf{s})\ |\ |<\beta\). (c) Under zero initial conditions, \(\forall\mathbf{w}(t)\in L_{2}[0,+\infty)\), supposing that the transfer function from the disturbance input \(\mathbf{w}(t)\) to the control output \(\mathbf{z}(t)\) is \(T_{zw}(\mathbf{s})\), solving \(\mathbf{u}_{N}(t)\)=\(\mathbf{K}(\mathbf{s})\mathbf{y}(t)\) makes the system (13) satisfying \(|\ 1\,T_{zw}(\mathbf{s})\ |\ |_{\infty}\)\(<\lambda\). To facilitate the following descriptions and our fault-tolerant controller design, Lemma 1 was introduced. Lemma 1 ([32]): _Consider the following linear time-invariant system_  
\[\left\{ \begin{array}{l}\mathbf{x}(t)=\mathbf{A}\mathbf{x}(t)+\mathbf{B}\mathbf{w}(t)\\ \mathbf{z}(t)=\mathbf{C}\mathbf{x}(t)+\mathbf{D}\mathbf{w}(t)\end{array} \right. \tag{15}\]_Given a positive constant \(\gamma\), if there exists a positive definite matrix \(\mathbf{P}>0\) satisfying_  
\[\left[ \begin{array}{ccc}\mathbf{A}^{T}\mathbf{P}+\mathbf{P}\mathbf{A}&\mathbf{ P}\mathbf{B}&\mathbf{C}^{T}\\ *&-\gamma^{2}\mathbf{I}&\mathbf{D}^{T}\\ *&*&-\mathbf{I}\end{array} \right] \tag{16}\]  
_Thus, the system (15) will reach the asymptotical stability and the inequality \(\mid\mid\mathbf{z}(t)\mid\mid_{2}<\gamma\mid\mid\mathbf{w}(t)\mid\mid\mid_{2}\) holds under zero initial condition._