## 3 Analysis of Fault Model by Using Convex Optimization
### Treatment of the Fault Model Uncertainty  
Since this work considers only the gradual faults in NT-DRAS, it is reasonable to assume that the fault signature in parameter drift form is limited. Hence, upper bounds \(\left|\mathbf{\Delta}\boldsymbol{A}\right|_{\text{bound}}\) and \(\left|\boldsymbol{\Delta}\boldsymbol{B}\right|_{\text{bound}}\) for the uncertainty matrices \(\boldsymbol{\Delta}\boldsymbol{A}\) and \(\boldsymbol{\Delta}\boldsymbol{B}\) can be defined. To describe the system with uncertainty using defined upper bounds, two other time independent factors \(\rho_{\text{A}}\) and \(\rho_{\text{B}}\) are introduced, both of which are random variables from the interval \([-1,\ \ +1]\). Finally, another form of the original system with gradual faults can be obtained as:  
\[\left\{ \begin{array}{l}\dot{\boldsymbol{x}}(t)=(\boldsymbol{A}+\rho_{\text{A }}|\boldsymbol{\Delta}\boldsymbol{A}|_{\text{bound}})\boldsymbol{x}(t)+( \boldsymbol{B}+\rho_{\text{B}}|\boldsymbol{\Delta}\boldsymbol{B}|_{\text{ bound}})\boldsymbol{u}(t)+\boldsymbol{G}w(t)\\ \boldsymbol{y}(t)=\boldsymbol{C}\boldsymbol{x}(t)\end{array} \right. \tag{13}\]  
where  
\[\left\{ \begin{array}{l}\left|\boldsymbol{\Delta}\boldsymbol{A}\right|_{ \text{bound}}=\left[\begin{array}{cccc}0&0&0&0\\ 0&\left|\Delta a_{22}\right|_{\text{bound}}&0&0\\ 0&\left|\Delta a_{32}\right|_{\text{bound}}&\left|\Delta a_{33}\right|_{\text{ bound}}&\left|\Delta a_{34}\right|_{\text{bound}}\\ 0&0&0&0\end{array}\right]\\ \left|\boldsymbol{\Delta}\boldsymbol{B}\right|_{\text{bound}}=\left[\begin{array} []{c}0\\ 0\\ 0\\ 0\\ |\Delta b_{41}|_{\text{bound}}\end{array}\right]\end{array} \right. \tag{14}\]  
To form a closed-loop system, the state feedback \(\boldsymbol{K}_{\boldsymbol{x}}(t)\) was usually chosen as a part of the system input \(\boldsymbol{u}(t)\), where \(\boldsymbol{K}_{\boldsymbol{x}}\) is state feedback control gain. Since gradual faults can affect the pole position, and assuming that the gradual faults are limited, the following assumption is made as a basis of proof and following conclusions. **Assumption 1**. : _The gradual faults are assumed to be limited and their effects on the system can be expressed in the form of the closed-loop system eigenvalues, where the real parts of the eigenvalues of the fault system change according to the following inequality relationship with respect to the fault-free system_  
\[\left\{ \begin{array}{l}\left|\text{Re}\lambda_{i}(\left|\boldsymbol{\Delta }\boldsymbol{A}\right|_{\text{bound}})\right|<\left|\text{Re}\lambda_{i}( \boldsymbol{A})\right|\\ \left|\text{Re}\lambda_{i}(\left|\boldsymbol{\Delta}\boldsymbol{B}\right|_{ \text{bound}}\boldsymbol{K}_{\boldsymbol{x}})\right|<\left|\text{Re}\lambda_{i} (\boldsymbol{B}\boldsymbol{K}_{\boldsymbol{x}})\right|\end{array} \right. \tag{15}\]  
In order to determine the controller gains, based on the upper bounds of the uncertainty matrices, the deterministic model needs to be obtained. Here, the extremum of \begin{table}
\begin{tabular}{c c c} \hline \hline
**Number** & **Illustration** & **The Corresponding Coefficient** \\ \hline Fault-1 & Hydraulic cylinder leakage & leakage coefficient \(\text{C}_{\text{ehal}}\) \\ Fault-2 & Motion damping increasing & damping coefficient \(\text{B}_{\text{eha}}\) \\ Fault-3 & Oil deterioration & bulk modulus \(\text{E}_{\text{eha}}\) \\ Fault-4 & Resistance increasing & motor armature resistance \(\text{R}_{\text{e}}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Fault types and corresponding coefficient. \(\rho_{\text{A}}\) and \(\rho_{\text{B}}\) are considered, resulting in four critical subsystems as the boundary condition models to be used by the LMI technique. \[\left\{ \begin{array}{l}\dot{\mathbf{x}}(t)=(\mathbf{A}\pm|\mathbf{\Delta}\mathbf{A}|_{\text{ bound}})\mathbf{x}(t)+(\mathbf{B}\pm|\mathbf{\Delta}\mathbf{B}|_{\text{bound}})\mathbf{u}(t)+\mathbf{G}w(t)\\ \mathbf{y}(t)=\mathbf{\text{C}}\mathbf{x}(t)\end{array} \right.