## 3 Results
### Multiplicative Fault Detection
#### 3.2.2 Indirect Estimation Procedure  
An alternative solution to the fault identification problem, which can also be applied in the case of discrete parameter estimation using the SWA, the actual physical parameters can be determined through one of the indirect estimation procedures [3]. This approach is fully applicable when the relationship between the estimated and physical parameters is linear. However, if the relationship is nonlinear (which is more common), the procedure remains applicable only when the change (i.e., the fault) in the physical parameter is relatively small compared to its nominal value. The starting point is the relationship between the physical parameters \(\theta\), whose number is \(L\), and the model parameters \(\pi\), whose number is \(N\), expressed through the relation:  
\[\pi=f(\theta). \tag{32}\]  
When \(L=N\), the actual physical parameters can be indirectly estimated according to:  
\[\Delta\hat{\theta}= \mathbf{R}^{-1}\Delta\hat{\pi}, \tag{33}\] \[\mathbf{R}=\left[\frac{\partial\pi}{\partial\theta}\right]_{\Delta \theta=0}=\left[ \begin{matrix}\frac{\partial\pi_{1}}{\partial\theta_{1}}& \cdots&\frac{\partial\pi_{1}}{\partial\theta_{1}}\\ \vdots&&\vdots\\ \frac{\partial\pi_{N}}{\partial\theta_{1}}&\cdots&\frac{\partial\pi_{N}}{ \partial\theta_{1}}\\ \end{matrix} \right]_{\Delta\theta=0}, \tag{34}\]  
Figure 21: Multiplicative fault isolation using a decision algorithm with the SIA. where:  
\(\Delta\hat{\mathbf{\pi}}=\hat{\mathbf{\pi}}-\mathbf{\pi}_{\text{n}}\) is estimation in model parameters variation,  
\(\mathbf{\pi}_{\text{n}}\) is the nominal model parameters,  
\(\hat{\mathbf{\pi}}\) is estimating model parameters,  
\(\Delta\hat{\mathbf{\theta}}\) is estimation of actual physical parameters. The partial derivatives in (34) are calculated using the nominal values of the parameters. If \(L<N\), a slightly modified form of (33) defined in [3] can be used:  
\[\Delta\hat{\mathbf{\theta}}=(\mathbf{R}^{{}^{\prime}}\mathbf{R})^{-1}\mathbf{R}^{{}^{\prime}} \Delta\hat{\mathbf{\pi}}. \tag{35}\]  
In the specific example, the vector of actual physical parameters is defined as:  
\[\mathbf{\theta}= \begin{bmatrix}R_{1}&R_{9}&R_{12}&C_{8}&R_{\text{a}}&J&B\end{bmatrix}  ^{\prime}, \tag{36}\]  
so, the columns of the Jacobian matrix (matrix of partial derivatives) are:  
\[\mathbf{R}_{1}= \begin{bmatrix}\frac{db_{1}}{dR_{1}}\\ \frac{db_{1}}{dR_{1}}\\ \frac{da_{1}}{dR_{1}}\\ \end{bmatrix} = \begin{bmatrix}-\frac{R_{\alpha 2}B}{R_{\text{el}}}\frac{R_{ \alpha 2}B+K_{\alpha}K_{\alpha}}{((R_{1}+R_{\alpha})B+K_{\alpha}K_{\alpha})^{2}}\\ \frac{R_{\alpha 2}}{R_{\text{el}}}\frac{R_{\alpha 2}B}{((R_{1}+R_{\alpha})B+K_{ \alpha}K_{\alpha})^{2}}\\ \frac{JK_{\alpha}K_{\alpha}(R_{1}+R_{\alpha})}{((R_{1}+R_{\alpha})B+K_{\alpha }K_{\alpha})^{2}}\\ \end{bmatrix} ;\mathbf{R}_{2}= \begin{bmatrix}\frac{db_{1}}{dR_{1}}\\ \frac{db_{1}}{dR_{1}}\\ \frac{db_{1}}{dR_{1}}\\ \frac{da_{1}}{dR_{1}}\\ \end{bmatrix} = \begin{bmatrix}-\frac{R_{\alpha 2}}{R_{\alpha}^{2}}\frac{R_{ \alpha 2}B+K_{\alpha}K_{\alpha}}{(R_{1}+R_{\alpha})B+K_{\alpha}K_{\alpha}}\\ \frac{db_{1}}{dR_{1}}\\ \end{bmatrix}  \tag{37}\]  
\[\mathbf{R}_{3}= \begin{bmatrix}\frac{db_{1}}{dR_{12}}\\ \frac{db_{1}}{dR_{12}}\\ \frac{db_{1}}{dR_{12}}\\ \end{bmatrix} = \begin{bmatrix}\frac{1}{R_{\text{el}}}\frac{R_{\text{el}}}{((R_ {1}+R_{\alpha})B+K_{\alpha}K_{\alpha})^{2}}\\ \frac{R_{\alpha 2}}{R_{\text{el}}}\frac{(R_{1}+R_{\alpha}K_{\alpha})}{((R_{1}+R_{ \alpha})B+K_{\alpha}K_{\alpha})^{2}}\\ \frac{JK_{\alpha}K_{\alpha}(R_{1}+R_{\alpha})}{((R_{1}+R_{\alpha})B+K_{\alpha }K_{\alpha})^{2}}\\ \end{bmatrix} ;\mathbf{R}_{4}= \begin{bmatrix}\frac{db_{1}}{dR_{5}}\\ \frac{db_{1}}{dR_{1}}\\ \frac{db_{1}}{dR_{1}}\\ \end{bmatrix} = \begin{bmatrix}\frac{R_{\alpha 2}}{R_{\text{el}}}\frac{R_{ \alpha 2}B+K_{\alpha}K_{\alpha}}{(R_{1}+R_{\alpha})B+K_{\alpha}K_{\alpha}}\\ \frac{d_{1}}{dR_{1}}\\ \end{bmatrix} ; \tag{38}\]  
\[\mathbf{R}_{5}= \begin{bmatrix}\frac{db_{1}}{dR_{2}}\\ \frac{db_{1}}{dR_{2}}\\ \frac{db_{1}}{dR_{1}}\\ \frac{da_{1}}{dR_{2}}\\ \end{bmatrix} =