## 5 Simulation Analysis
### Analysis of Gradual Fault Effects in Time-Domain  
The effects of gradual faults of different severity levels and different command inputs are presented and discussed in this section. The system response to a square-wave input and a sine-wave are studied, and the results, displacement response and the tracking error, are presented (see Figure 9). The minor fault upper bounds matrices are set as stated in Table 6. The system response to the square-wave input and the response to the compound sine-wave input indicate that, under the minor fault conditions, the tracking performance decreases slightly and the tracking errors increase compared with the system response under normal operating conditions.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Fault Degree** & **System Status** & \multicolumn{3}{c}{**System Matrices**} & **Control Matrices** \\ \hline Degree-1 & Normal system & \(A=\begin{bmatrix}0&1&0&0&0\\ 0&-31.408&2.0704\times 10^{-6}&0&0\\ 0&-3.20\times 10^{10}&-5.9864\times 10^{2}&5.8776\times 10^{13}\\ 0&0&-9.9472\times 10^{-5}&-1.0\times 10^{2}\end{bmatrix}\) & \(B=\begin{bmatrix}0\\ 0\\ 0\\ 456.63\end{bmatrix}\) \\ \hline Degree-2 & Minor failure & \(|\Delta A|_{\text{bound}}^{1}=\begin{bmatrix}0&0&0&0\\ 0&28.169&0&0\\ 0&0.11\times 10^{10}&1.7959\times 10^{2}&1.7633\times 10^{13}\\ 0&0&0&0\end{bmatrix}\) & \(|\Delta B|_{\text{bound}}^{1}=\begin{bmatrix}0\\ 0\\ 0\\ 11.6233\end{bmatrix}\) \\ \hline Degree-3 & Moderate failure & \(|\Delta A|_{\text{bound}}^{2}=\begin{bmatrix}0&0&0&0\\ 0&70.423&0&0\\ 0&0.22\times 10^{10}&3.5978\times 10^{2}&3.5266\times 10^{13}\\ 0&0&0&0\end{bmatrix}\) & \(|\Delta B|_{\text{bound}}^{2}=\begin{bmatrix}0\\ 0\\ 0\\ 23.2466\end{bmatrix}\) \\ \hline \hline \end{tabular}
\end{table}
Table 6: NT-DRAS system parameters with different fault upper bound scenarios.  
The system response is also analyzed under the condition of a moderate fault (see Figure 10), where the upper bound matrices are set as shown in Table 6. The same square-wave input and the compound sine-wave are used to analyze the system response. The main difference between the system responses under different fault levels is that under moderate fault conditions, tracking performance decreases faster while tracking errors increase to a higher degree compared with those under normal conditions.  
To calculate the quantized value under integrated fault conditions, the start and end times are set as \(t_{1}=0\) and \(t_{2}=20\), all evaluation results are shown in Table 7. It can be seen that with the fault degree increasing, the quantized value \(e_{\text{perf}}(t)\) also shows an increasing trend. Under minor fault conditions, the value increases 102% compared with normal state by calculation, while under moderate fault conditions it increases 134%. Since gradual faults have apparent impacts on the system performance and it is difficult to detect the specific fault mode, the NT-DRAS cannot work as effectively for a system under the fault  
Figure 10: Tracking responses under moderate fault conditions: **(a1,a2)** Displacement response to square-wave input; **(b1,b2)** Displacement response to compound sine-wave input.  
Figure 9: Tracking responses under minor fault conditions: **(a1,a2)** Displacement response to square-wave input; **(b1,b2)** Displacement response to compound sine-wave input.  
conditions as under the normal conditions. Hence, it can be concluded that an appropriate FTC strategy should be developed and implemented.