## 2 System Statement
### Modeling of Mode-1 under Gradual Fault Conditions
#### 2.2.2 Gradual Fault Mechanism-Based System Model  
The gradual faults of EHA can be classified in the above categories shown in Table 2, where the faults correspond with relative parameter drift: Fault-1 represents that leakage coefficient \(\rm C_{ehal}\) changes due to the hydraulic cylinder leakage, Fault-2 represents that damping coefficient \(\rm B_{eha}\) changes due to the increasing motion damping, Fault-3 represents that bulk modulus \(\rm E_{eha}\) changes due to oil deteriorates caused by the air in oil, and Fault-4 represents that \(\rm R_{e}\) changes due to its increasing caused by heat production. These types of gradual faults are difficult to detect and yet result in the system performance degradation. The relative parameters drift with the gradual fault degrees within limited ranges. Though similar to the system model uncertainty in fault-free conditions, the gradual faults cause more severe parameter drift, which can be regarded as gradual faults caused by the system uncertainty. In order to model these faults in NT-DRAS, uncertainty matrices are used to describe the parameter uncertainty caused by these faults. The NT-DRAS model of the mode-1 with gradual faults can be represented in state-space form as:  
\[\left\{\begin{array}{l}\dot{\mathbf{x}}(t)=(\mathbf{A}+\mathbf{\Delta A})\mathbf{x}(t)+(\mathbf{ B}+\mathbf{\Delta B})\mathbf{u}(t)+\mathbf{G}w(t)\\ \mathbf{y}(t)=\mathbf{C}\mathbf{x}(t)\end{array}\right. \tag{10}\]  
where \(\mathbf{\Delta A}\) and \(\mathbf{\Delta B}\) are deviations in the original state and input matrices caused by parameter changes due to the gradual faults. It is necessary to state that the gradual faults caused deviation matrices have more adverse effect compared with the system model uncertainty under fault-free conditions. There is no \(\mathbf{\Delta C}\) in the above system description, as it is assumed that all of the sensors are operating normally. The system matrices and the fault module matrices can be expressed as:  
\[\left\{\begin{array}{l}\mathbf{A}=\left[\begin{array}{cccc}0&1&0&0\\ 0&a_{22}&a_{23}&0\\ 0&a_{32}&a_{33}&a_{34}\\ 0&0&a_{43}&a_{44}\end{array}\right]\\ \mathbf{B}=\left[\begin{array}{cccc}0&0&0&b_{41}\\ 0&0&0\\ 0&\Delta a_{22}&0&0\\ 0&0&0&0\\ 0&\Delta a_{32}&\Delta a_{33}&\Delta a_{34}\\ 0&0&0&0\end{array}\right]\\ \mathbf{\Delta B}=\left[\begin{array}{cccc}0&0&0&\Delta b_{41}\\ 0&0&0\\ 0&0&0\\ 0&\Delta a_{33}&\Delta a_{33}&\Delta a_{34}\\ 0&0&0&0\end{array}\right]\\ \mathbf{\Delta B}=\left[\begin{array}{cccc}0&0&0&\Delta b_{41}\\ 0&0&0\\ 0&0&0\\ 0&0&0\\ \end{array}\right]^{\rm T}\end{array}\right. \tag{11}\]  
where coefficients of the state and input matrices, \(\mathbf{A}\) and \(\mathbf{B}\), are defined in Equation (8); \(\Delta a_{22}=-\rm\Delta B_{eha}/(m_{eha}+m_{ema}+m_{d})\) represents damping fault induced factor; \(\Delta a_{32}=-4\rm\Delta e_{eha}A_{eha}/V_{eha}\) is the bulk modulus fault factor; \(\Delta a_{33}=-[4(\rm\Delta e_{eha})\cdot(\rm\Delta e_{ehal})]/V_{eha}\) represents the bulk modulus as well as the internal leakage fault factor; \(\Delta a_{34}=4\rm\Delta E_{eha}V_{P}/V_{eha}\) represents the bulk modulus fault factor; \(\Delta b_{41}=K_{m}/(D_{m}\Delta R_{e})\) represents the motor armature resistance fault factor. After making the following definitions \(A_{\rm F}=A+\mathbf{\Delta A}\) and \(B_{\rm F}=B+\mathbf{\Delta B}\), the model of NT-DRAS in Mode-1 can be described as a system with uncertainty in the following form:  
\[\left\{\begin{array}{l}\dot{\mathbf{x}}(t)=A_{\rm F}\mathbf{x}(t)+B_{\rm F}\mathbf{u}(t )+\mathbf{G}w(t)\\ \mathbf{y}(t)=\mathbf{C}\mathbf{x}(t)\end{array}\right. \tag{12}\]