# Robust Structural Damage Detection Using Analysis of the CMSE Residual's Sensitivity to Damage
## 3 Robust Damage Identification Scheme  
From the standpoint of solving a linear system of equations, optimizing the assemblage of the system by eliminating redundant equations serves as an effective tool to improve identification of damage. However, this approach was rarely investigated in previous research. Therefore, this section approaches discrimination and elimination of redundant equations, thereby enhancing the robustness of the CMSE system against noise. If the actual and predicted damage level vectors of the measured damaged structure are denoted as \(\mathbf{\alpha}_{0}\) and \(\mathbf{\hat{\alpha}}\), respectively, then \(\mathbf{\hat{\alpha}}=\mathbf{\alpha}_{0}+\Delta\mathbf{\alpha}\) is due to measurement errors, in which \(\Delta\mathbf{\alpha}\) is a small perturbation in \(\mathbf{\hat{\alpha}}\). In the context of the least-squares problem, \(\mathbf{\hat{\alpha}}\) instead of \(\mathbf{\alpha}_{0}\) is the optimal solution that can minimize the Euclidean norm \(\|\mathbf{C}^{*}\mathbf{\alpha}-\mathbf{b}^{*}\|_{2}\). One would also expect that \(\mathbf{\alpha}_{0}\) could nearly minimize \(\|\mathbf{C}^{*}\mathbf{\alpha}-\mathbf{b}^{*}\|_{2}\) so that \(\mathbf{\alpha}_{0}\) can be obtained by solving \(\mathbf{C}^{*}\mathbf{\alpha}=\mathbf{b}^{*}\) via a specific tool, such as the ITR method. In view of this, narrowing the difference between \(\|\mathbf{C}^{*}\mathbf{\hat{\alpha}}-\mathbf{b}^{*}\|_{2}\) and \(\|\mathbf{C}^{*}\mathbf{\alpha}_{0}-\mathbf{b}^{*}\|_{2}\) may raise the possibility of obtaining \(\mathbf{\alpha}_{0}\). Hence, the following CMSE residual function is defined:  
\[\mathbf{R}(\mathbf{\alpha})=\mathbf{C}^{*}\mathbf{\alpha}-\mathbf{b}^{*} \tag{22}\]  
where \(\mathbf{R}(\mathbf{\alpha})\) collects \(N_{q}\) residual sub-functions \(\mathbf{R}_{m}(\mathbf{\alpha})=\mathbf{C}_{m}^{*}\mathbf{\alpha}-b_{m}^{*}\). Sensitivity analyses are widely used in engineering to evaluate the effect of changes of one variable on another variable. Herein, a sensitivity analysis of \(\mathbf{R}(\mathbf{\alpha})\) with respect to \(\mathbf{\alpha}\) is performed to measure the difference between \(\|\mathbf{R}(\mathbf{\hat{\alpha}})\|_{2}\) and \(\|\mathbf{R}(\mathbf{\alpha}_{0})\|_{2}\) to some degree. If \(\mathbf{R}(\mathbf{\alpha})\) has a large sensitivity at \(\mathbf{\hat{\alpha}}\), a small perturbation \(\Delta\mathbf{\alpha}\) tends to cause a big change of \(\|\mathbf{R}(\mathbf{\alpha})\|_{2}\), or a great difference between \(\|\mathbf{R}(\mathbf{\hat{\alpha}})\|_{2}\) and \(\|\mathbf{R}(\mathbf{\alpha}_{0})\|_{2}\). In other words, \(\mathbf{\alpha}_{0}\) will not satisfy the currently assembled CMSE system. Hence, a linear system that corresponds to a small residual sensitivity to damage, i.e., \(\|\mathbf{R}^{{}^{\prime}}(\mathbf{\hat{\alpha}})\|_{2}\) is expected. To further illustrate this point, one considers two special residual sub-functions \(Y_{1}=R_{1}(\alpha)\) and \(Y_{2}=R_{2}(\alpha)\) in a CMSE linear system. It should be noted that all symbols are presented by a scalar, indicating the measured damaged structure only has one assumed damage location. The curves of the Euclidean norm of these two residual functions, i.e., \(l_{1}:\;y_{1}=\|R_{1}(\alpha)\|_{2}\) and \(l_{2}:\;y_{2}=\|R_{2}(\alpha)\|_{2}\), are plotted against \(\alpha\), respectively, as shown in Figure 1. Because \(\hat{\alpha}\) is the least-squares solution of the CMSE linear system, one assumes that both \(y_{1}\) and \(y_{2}\) have their near-minimum value \(R_{min}\) when \(\alpha=\hat{\alpha}\). However, their right derivations, i.e., \(\|R_{1}^{\prime}(\hat{\alpha})\|_{2}\) and \(\|R_{2}^{\prime}(\hat{\alpha})\|_{2}\), are different, more specifically, \(\|R_{2}^{\prime}(\hat{\alpha})\|_{2}>\|R_{1}^{\prime}(\hat{\alpha})\|_{2}\). It can be observed by comparing curves \(l_{1}\) and \(l_{2}\) that, at the actual solution \(\alpha_{0}\), \(\|R_{1}(\alpha_{0})\|_{2}\) is closer to the minimum residual \(R_{min}\) than \(\|R_{2}(\alpha_{0})\|_{2}\).