# An Observer-Based Active Fault Tolerant Controller for Vehicle Suspension System
## 3 Robust \(H_{\infty}\) Fault-Tolerant Controller Design
### The Design of \(H_{\infty}\) Weighted Output Feedback Controller  
Because it is easy to measure the output variable \(y(t)\) with an explicit physical property, the weighed \(H_{\infty}\) output feedback controller is developed in this paper to improve the vehicle dynamics performances, the control block of which is shown in Figure 3, where \(\widetilde{w}(t)=\left[d(t)\widetilde{f}(t)\right]^{T}\) is the disturbance input, \(\mathbf{S}_{w}\) is the weighted coefficient matrix of \(\widetilde{w}(t)\), \(\mathbf{G}(s)\) is the transfer function of the closed-loop system (14), \(\mathbf{K}(s)\) is the weighted \(H_{\infty}\) output feedback controller to be designed, \(\mathbf{W}_{\rm z}\) and \(\mathbf{S}_{\rm z}\) are the weighted transfer function matrix and coefficient matrix of control output \(\mathbf{z}(t)\), respectively, \(\mathbf{u}_{N}(t)=\mathbf{K}(s)\mathbf{y}(t)\) is the weighted \(H_{\infty}\) output feedback control law satisfying [33]:  
\[\left\{ \begin{array}{l}\dot{\mathbf{x}}_{c}(t)=\mathbf{A}_{c}\mathbf{x}_{c}(t)+\mathbf{B}_{ c}\mathbf{y}(t)\\ \mathbf{u}_{N}(t)=\mathbf{C}_{c}\mathbf{x}_{c}(t)+\mathbf{D}_{c}\mathbf{y}(t)\end{array} \right. \tag{21}\]  
In Equation (21), \(\mathbf{x}_{c}\in R^{n_{c}}\) is the state vector of the desirable \(H_{\infty}\) weighted output feedback controller, and \(\mathbf{A}_{c}\), \(\mathbf{B}_{c}\), \(\mathbf{C}_{c}\) and \(\mathbf{D}_{c}\) are the parameter matrices of the designed controller. Substituting (21) to (14), we can obtain the active fault-tolerant control system of the half-vehicle suspension system as [33]:  
\[\left\{ \begin{array}{l}\dot{\mathbf{x}}_{cl}(t)=\mathbf{A}_{cl}\mathbf{x}_{cl}(t)+\mathbf{B}_ {cl}\widetilde{w}(t)\\ \mathbf{z}(t)=\mathbf{C}_{cl}\mathbf{x}_{cl}(t)+\mathbf{D}_{cl}\widetilde{w}(t)\end{array} \right. \tag{22}\]  
where \(\mathbf{x}_{cl}(t){=}\left[\mathbf{x}(t),\ \mathbf{x}_{c}(t)\right]^{T}\) is the state vector of the augmented system (22), and the corresponding coefficient matrices are as follows:  
\[\mathbf{A}_{cl}=\left[ \begin{array}{cc}\mathbf{A}+\mathbf{B}\mathbf{D}_{c}\mathbf{C}_{1}&\mathbf{B} \mathbf{C}_{c}\\ \mathbf{B}_{c}\mathbf{C}_{2}&\mathbf{A}_{c}\end{array} \right],\ \mathbf{B}_{cl}=\left[ \begin{array}{c}\mathbf{B}_{ \widetilde{w}}\\ \mathbf{0}\end{array} \right],\]  
\[\mathbf{C}_{cl}=\left[ \begin{array}{cc}\mathbf{C}_{2}+\mathbf{D}_{2}\mathbf{D}_{c}\mathbf{C}_{1 }&\mathbf{D}_{2}\mathbf{C}_{c}\end{array} \right],\ \mathbf{D}_{cl}=[\mathbf{0}].\]  
Now, the objective of the controller design is converted into solving the coefficient matrix \(\mathbf{A}_{cl}\), \(\mathbf{B}_{cl}\), \(\mathbf{C}_{cl}\), and \(\mathbf{D}_{cl}\) to guarantee the asymptotic stability of the closed-loop system (22), and the \(H_{\infty}\) norm \(\mid\!\!\mid\!T({\rm s})\!\mid\!\mid_{\infty}\) of the system (22) is bounded by \(\lambda\). It was noted that the parameter matrices \(\mathbf{A}_{c}\), \(\mathbf{B}_{c}\), \(\mathbf{C}_{c}\) and \(\mathbf{D}_{c}\) could be obtained by using the MATLAB internal function hinflmi (2017a, The MathWorks, Inc., Natick, MA, USA) to obtain the control law \(\mathbf{u}_{N}(t)\). Subsequently, substituting \(\mathbf{A}_{c}\), \(\mathbf{B}_{c}\), \(\mathbf{C}_{c}\) and \(\mathbf{D}_{c}\) into \(\mathbf{A}_{cl}\), \(\mathbf{B}_{cl}\), \(\mathbf{C}_{cl}\) and \(\mathbf{D}_{cl}\) could yield to the active fault-tolerant control system (18). According to ISO2631-1:1997 (E) criteria, the human body is more sensitive to the vertical vibrations of 4-12.5 Hz and the horizontal vibrations of 1-2 Hz. In order to improve the ride comfort in these two specific frequency ranges, the weight functions shown in (23) and (24) were selected to achieve the corresponding frequency attenuation [33]. \[W_{1}=\frac{s^{2}+314.2s+987}{s^{2}+43.98s+987} \tag{23}\]  
\[W_{2}=\frac{s^{2}+50.27s+25.72}{s^{2}+7.037s+25.72} \tag{24}\]  
Figure 3: The diagram of output feedback \(H_{\infty}\) controller for the active suspension system.