## 4 Discussion  
### Validation of Fault Detection Measurement Results  
To evaluate the effectiveness of different methods for detecting multiplicative faults in the motor and amplifier, the outputs of the amplifier model (3), obtained by substituting system parameters into Equations (4) and (5) for three different values of Coulomb friction [45] (\(M_{\text{c}}=0\), \(M_{\text{c}}=0.48\) mNm and \(M_{\text{c}}=0.33\) mNm), with the outputs of model (6), which incorporates parameters estimated using the SIA, and model (8), based on the SWA results, have been compared. These outputs were compared with the measured output voltage \(U_{\text{m}}\), and the percentage agreement was calculated using MATLAB's System Identification Toolbox, as shown in Tables 13-20. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline
**Parameter** & \(R_{\text{a}}=7.41\) \(\Omega\) & \multicolumn{3}{c}{\(J=1.43\cdot 10^{-6}\) kgm\({}^{2}\)} & \multicolumn{3}{c}{\(B=1.8\cdot 10^{-6}\) Nm/rad/s} \\ \hline Fault & 0.741 \(\Omega\) & 74.1 \(\Omega\) & 1.43\(\cdot 10^{-4}\) kgm\({}^{2}\) & 1.43\(\cdot 10^{-8}\) kgm\({}^{2}\) & 1.8\(\cdot 10^{-4}\) Nm/rad/s & 1.8\(\cdot 10^{-5}\) Nm/rad/s \\ \hline Estimated fault & 0.62 \(\Omega\) & 64.4 \(\Omega\) & 1.43\(\cdot 10^{-4}\) kgm\({}^{2}\) & 1.45\(\cdot 10^{-8}\) kgm\({}^{2}\) & 0.57\(\cdot 10^{-4}\) Nm/rad/s & 1.53\(\cdot 10^{-5}\) Nm/rad/s \\ \hline \hline \end{tabular}
\end{table} 
Table 12: Estimation of motor physical parameters under multiplicative faults. \begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline
**Parameter** & \(R_{\text{1}}=1\) \(\Omega\) & \multicolumn{3}{c}{\(R_{\text{9}}=10\) k\(\Omega\)} & \multicolumn{3}{c}{\(R_{\text{12}}=10\) k\(\Omega\)} & \multicolumn{3}{c}{\(C_{\text{8}}=47\) pF} \\ \hline Fault & 10 \(\Omega\) & 56 \(\Omega\) & 100 \(\Omega\) & 33 k\(\Omega\) & 56 k\(\Omega\) & 100 k\(\Omega\) & 2 k\(\Omega\) & 3.3 k\(\Omega\) & 5.6 k\(\Omega\) & 1 \(\mu\)F & 10 \(\mu\)F & 47 \(\mu\)F \\ \hline Estimated fault & 9.8 \(\Omega\) & 49 \(\Omega\) & 79 \(\Omega\) & 17 k\(\Omega\) & 18.2 k\(\Omega\) & 19 k\(\Omega\) & 2 k\(\Omega\) & 3.3 k\(\Omega\) & 5.6 k\(\Omega\) & 0.97 \(\mu\)F & 9.7 \(\mu\)F & 46 \(\mu\)F \\ \hline \hline \end{tabular}
\end{table} 
Table 11: Estimation of amplifier physical parameters under multiplicative faults. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**\(R_{9}\) = 10 k\(\Omega\)** & **SWA** & **SIA** & **\(M_{\text{c}}\) = 0** &
\begin{tabular}{c} \(M_{\text{c}}\) = 0.48 mNm \\ **(\(M_{\text{c}}\) = 0.33 mNm)** \\ \end{tabular} \\ \hline
33 k\(\Omega\) & 97.7\% & 97.14\% & 90.54\% & 89.01\% (89.52\%) \\
56 k\(\Omega\) & 97.28\% & 96.72\% & 91.07\% & 89.26\% (89.87\%) \\
100 k\(\Omega\) & 95.89\% & 98.59\% & 92\% & 91.4\% (91.6\%) \\
5.6 k\(\Omega\) & 98.77\% & 97.96\% & 92.48\% & 91.52\% (91.89) \\ \hline \hline \end{tabular}
\end{table} 
Table 16: Percentage match of system output during parametric fault in \(R_{12}\). \begin{table}
\begin{tabular}{c c c c} \hline \hline
**\(R_{8}\) = 47 pF** & **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 \(\mu\)F & 97.85\% & 90.6\% & 75.41\% & 75.4\% (75.43\%) \\
10 \(\mu\)F & 95.07\% & 97.57\% & 88.53\% & 89.04\% (88.98\%) \\
47 \(\mu\)F & 87.96\% & 96.72\% & 96.3\% & 96.69\% (96.7\%9 \\ \hline \hline \end{tabular}
\end{table} 
Table 17: Percentage match of system output during parametric fault in \(C_{8}\). \begin{table}
\begin{tabular}{c c c c} \hline \hline
**\(R_{8}\) = 47 pF** & **SWA** & **SIA** & **\(M_{\text{c}}\) = 0** &
\begin{tabular}{c} \(M_{\text{c}}\) = 0.48 mNm \\ **(\(M_{\text{c}}\) = 0.33 mNm)** \\ \end{tabular} \\ \hline
0.741 \(\Omega\) & 99.68\% & 89.53\% & 95.43\% & 95.01\% (95.25\%) \\
74.1 \(\Omega\) & 95.84\% & 99.31\% & 99.75\% & 98.53\% (98.93\%) \\ \hline \hline \end{tabular}
\end{table} 
Table 18: Percentage match of system output during parametric fault in \(R_{8}\).