## 3 Results
### Multiplicative Fault Detection
#### 3.2.2 Indirect Estimation Procedure
\begin{bmatrix}\frac{R_{\alpha 2}}{R_{\text{el}}}\frac{(R_{ \alpha 2}B+K_{\alpha}K_{\alpha})^{2}}{((R_{1}+R_{\alpha})B+K_{\alpha}K_{\alpha})^{2}}\\ \frac{JK_{\alpha}K_{\alpha}(R_{1}+R_{\alpha})}{((R_{1}+R_{\alpha})B+K_{\alpha }K_{\alpha})^{2}}\\ \end{bmatrix} ;\mathbf{R}_{6}= \begin{bmatrix}\frac{db_{1}}{dR_{1}}\\ \frac{db_{1}}{dR_{1}}\\ \frac{db_{1}}{dR_{1}}\\ \frac{d_{1}}{dR_{1}}\\ \end{bmatrix} = \begin{bmatrix}\frac{R_{\alpha 2}}{R_{\text{el}}}\frac{R_{ \alpha 2}}{(R_{1}+R_{\alpha})B+K_{\alpha}K_{\alpha}}\\ \frac{(R_{1}+R_{\alpha})}{(R_{1}+R_{\alpha})B+K_{\alpha}K_{\alpha}}\\ \end{bmatrix} ; \tag{39}\]  
\[\mathbf{R}_{7}= \begin{bmatrix}\frac{db_{1}}{dR_{2}}\\ \frac{db_{1}}{dR_{2}}\\ \frac{da_{1}}{dR_{2}}\\ \end{bmatrix} = \begin{bmatrix}-\frac{R_{\alpha 2}}{R_{\text{el}}}\frac{(R_{ \text{T}}K_{\alpha}R_{1}}{(R_{1}+R_{\alpha})B+K_{\alpha}K_{\alpha})^{2}}\\ -\frac{R_{\alpha 2}}{R_{\text{el}}}\frac{JR_{\alpha}(R_{1}+R_{\alpha})}{((R_{1}+R_{ \alpha})B+K_{\alpha}K_{\alpha})^{2}}\\ -\frac{I(R_{1}+R_{\alpha})}{((R_{1}+R_{\alpha})B+K_{\alpha}K_{\alpha})^{2}} \\ \end{bmatrix} . \tag{40}\]  
After substituting the nominal values of the amplifier and motor model parameters (provided in Table 1), the Jacobian matrix with respect to the actual physical parameters was obtained as:  
\[\mathbf{R}= \begin{bmatrix}-6.5\cdot 10^{-3}&-1.197\cdot 10^{-4}&1.995\cdot 10^{-4}&2.39 \cdot 10^{3}&1.76\cdot 10^{-5}&0&-3.52\cdot 10^{3}\\ -1\cdot 10^{-4}&-1.9\cdot 10^{-6}&3.2\cdot 10^{-6}&0&5.1\cdot 10^{-3}&2.67\cdot 10^{4}&-4.83 \cdot 10^{2}\\ 2.1\cdot 10^{-4}&0&0&1.3\cdot 10^{3}&2.1\cdot 10^{-3}&1.26\cdot 10^{4}&-2.28\cdot 10^{2} \\ \end{bmatrix} . \tag{41}\]  
It has been concluded that modifying (35) into the form:  
\[\Delta\hat{\mathbf{\theta}}_{i}=(\mathbf{R}^{{}^{\prime}}\mathbf{R}_{i})^{-1}\mathbf{R}^{\prime}_ {i}\Delta\hat{\mathbf{\pi}}_{i}, \tag{42}\]  
leads to significantly more accurate parameter estimation, resulting in improved precision and reliability of the results compared to the original approach, where:  
\(\Delta\hat{\mathbf{\theta}}_{i}\) is the estimate of the change in the \(i\)-th physical parameter,\(\mathbf{R}_{i}\) is the \(i\)-th column of the Jacobian matrix (matrix of partial derivatives)  
\(\Delta\hat{\pi}_{i}=\hat{\pi}_{i}-\pi_{\text{n}}\) is the estimate of the change in model parameters due to the \(i\)-th fault,  
\(\pi_{\text{n}}\) is nominal model parameters,  
\(\hat{\pi}_{i}\) is estimated model parameters in the presence of a fault in the \(i\)-th physical parameter. Tables 11 and 12 present the estimates of the actual physical parameters of the amplifier and DC motor in the presence of individual multiplicative faults, using the described indirect method provided in (42). Thereby, the parameter estimates were calculated using:  
\[\hat{\theta}_{i}=\Delta\hat{\theta}_{i}+\Theta_{\text{n}}, \tag{43}\]  
where \(\Theta_{\text{n}}\) represent nominal values of the actual physical parameters of the system. Based on the values presented in Tables 11 and 12, it can be concluded that the indirect estimation of physical parameters \(R_{12}\), \(C_{8}\), and \(J\) is most accurate in the presence of multiplicative faults. This accuracy arises from the linear relationship with parameters \(b_{0}\), \(b_{1}\), and \(a_{1}\) of the continuous amplifier model. During the estimation of \(R_{1}\), \(R_{9}\), \(R_{\text{a}}\), and \(B\), due to the nonlinear relationship with the amplifier parameters and the large deviation of these parameters from their nominal values, the estimation results were less accurate. Nevertheless, the deviations obtained using the proposed solution (42) are significantly smaller than the completely unacceptable results produced by applying the approach available in the literature (35).