## 4 Discussion
### Validation of Fault Detection Measurement Results
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**SWA** & **SIA** & **\(M_{\text{c}}\) = 0** &
\begin{tabular}{c} \(M_{\text{c}}\) = 0.48 mNm \\ **(\(M_{\text{c}}\) = 0.33 mNm)** \\ \end{tabular} \\ \hline
98.6\% & 98.59\% & 92.27\% & 91.92\% (92\%) \\ \hline \hline \end{tabular}
\end{table} 
Table 13: Percentage match of amplifier output before fault occurrence. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**SWA** & **SIA** & **\(M_{\text{c}}\) = 0** &
\begin{tabular}{c} \(M_{\text{c}}\) = 0.48 mNm \\ **(\(M_{\text{c}}\) = 0.33 mNm)** \\ \end{tabular} \\ \hline
10 \(\Omega\) & 95.71\% & 96.42\% & 70.78\% & 69\% (72.11\%) \\
56 \(\Omega\) & 81.65\% & 86.5\% & \(-30.61\%\) & 35.13\% (76.74\%) \\
100 \(\Omega\) & 74.44\% & 76.75\% & \(-150.2\%\) & \(-17.67\%\) (79\%) \\ \hline \hline \end{tabular}
\end{table} 
Table 14: Percentage match of system output during parametric fault in \(R_{1}\). \begin{table}
\begin{tabular}{c c c c} \hline \hline
**SWA** & **SIA** & **\(M_{\text{c}}\) = 0** &
\begin{tabular}{c} \(M_{\text{c}}\) = 0.48 mNm \\ **(\(M_{\text{c}}\) = 0.33 mNm)** \\ \end{tabular} \\ \hline
98.6\% & 98.59\% & 92.27\% & 91.92\% (92\%) \\ \hline \hline \end{tabular}
\end{table} 
Table 13: Percentage match of amplifier output before fault occurrence. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**SWA** & **SIA** & **\(M_{\text{c}}\) = 0** &
\begin{tabular}{c} \(M_{\text{c}}\) = 0.48 mNm \\ **(\(M_{\text{c}}\) = 0.33 mNm)** \\ \end{tabular} \\ \hline
10 \(\Omega\) & 95.71\% & 96.42\% & 70.78\% & 69\% (72.11\%) \\
56 \(\Omega\) & 81.65\% & 86.5\% & \(-30.61\%\) & 35.13\% (76.74\%) \\
100 \(\Omega\) & 74.44\% & 76.75\% & \(-150.2\%\) & \(-17.67\%\) (79\%) \\ \hline \hline \end{tabular}
\end{table} 
Table 15: Percentage match of system output during parametric fault in \(R_{9}\). Based on the results, it is clear that a multiplicative fault in resistor \(R_{1}\) = 100 \(\Omega\) (Table 14) significantly amplifies the effect of Coulomb friction (Figure 23). In this scenario, the model incorporating \(M_{\text{c}}\) = 0.33 \(\,\text{mNm}\) achieves the closest match to the measured output (79%), compared with the SIA-based model (76.8%) and the SWA-based model (74.4%). By contrast, for faults in \(C_{8}\), \(R_{9}\), or \(R_{12}\), including Coulomb friction makes virtually no difference: the percentage match to the measured output is effectively the same whether friction is modeled or not. A comparison of the fault-detection algorithms and the linearized models (with and without Coulomb friction) yields the following insights:  
* Although the SIA is computationally more complex and more sensitive to noise than the discrete-time approach, it proved almost as effective at detecting multiplicative faults. As it has been observed, it is important to ensure that the length of the identification dataset \(K\) is chosen to be sufficiently small yet greater than \(m\) (i.e., \(K>m\)). \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**B = 1.8\(\cdot\)10\({}^{-6}\) Nm/rad/s** & **SWA** & **SIA** & \(M_{\text{c}}\) = 0 &
\begin{tabular}{c} \(M_{\text{c}}\) = 0.48 mNm \\ (\(M_{\text{c}}\) = 0.33 mNm) \\ \end{tabular} \\ \hline
1.8\(\cdot\)10\({}^{-4}\) Nm/rad/s & 99.54\% & 99.21\% & 96.88\% & 96.84\% (96.86\%) \\
1.8\(\cdot\)10\({}^{-5}\) Nm/rad/s & 99.45\% & 99.1\% & 96.5\% & 96.2\% (96.35\%) \\ \hline \hline \end{tabular}
\end{table} 
Table 20: Percentage match of system output during parametric fault in \(B\). Figure 23: Measured and simulated outputs of the amplifier during the fault \(R_{1}\) = 100 \(\Omega\). Figure 22: Simulation models of the amplifier under a fault \(R_{1}\) = 100 \(\Omega\). * The SWA is computationally simpler and more robust to measurement noise, making it better suited for real-time applications. Its parameter estimates also showed strong agreement with the actual system response. * Obtained results confirm that, at the nominal resistance \(R_{1}=1\)\(\Omega\), Coulomb friction is negligible.