# Robust Structural Damage Detection Using Analysis of the CMSE Residual's Sensitivity to Damage
## 6 Conclusions  
This paper presented a CMSE-based damage identification scheme. An analysis of the defined residual function's sensitivity to damage was performed to measure how the actual damage level vector satisfied each CMSE equation. A sensitivity index was formulated to discriminate and eliminate the redundant equations that do not contribute to damage detection but complicate the system. The numerical and experiment robustness of the proposed damage identification scheme against measurement noise was investigated. Two main aspects were emphasized by the results.  
Figure 23: Damage identification results of damage case II, with an iterative Tikhonov solution: (**a**) CMSE method; (**b**) RCMSE method.  
Figure 24: Damage identification results of damage case III, with an iterative Tikhonov solution: (**a**) CMSE method; (**b**) RCMSE method.  
First, the damage identification performance of the classical CMSE method was clearly improved by eliminating redundant equations from the original CMSE system, confirming the validity of the sensitivity analysis process. The improvement of damage identification might inherently benefit from an improvement of the originally ill-conditioned problem by a reasonable reduction of the dimension of the CMSE system.  
Second, for the proposed CMSE-based damage identification scheme, the detectability of damage depended highly upon the modal parameter changes that the damage caused. For different locations of damage even with the same damage level, the resulting modal parameter change in different modes varied, and the employed modes were also different. This is the reason for the discrepancy of the damage localization performance for different damage locations. Besides, the damage identification probability gradually decreased as the damage level decreased. The phenomenon is due to the fact that smaller damage always results in a smaller change in structural modal parameters and leads to lower damage detectability.  
Overall, this study provides a contribution towards a clear and simple guideline for eliminating redundant equations in order to enhance the robustness of the CMSE system, which is solved by the ITR method to obtain spare solutions. Since a broad range of regression techniques [37; 38] that can solve the CMSE system are now available, it is extremely necessary to compare the performance of the ITR method and these state-of-art techniques.  
Funding acquisition, S.W. and Y.L.; Investigation, M.X., J.G. and Y.L.; Methodology, M.X. and S.W.; Software, M.X., J.G. and S.W.; Supervision, S.W.; Validation, S.W. and Y.L.; Experiment, S.W.; Writing--Original draft, M.X.; Writing--Review and Editing, M.X., Y.L. and S.W. All authors have read and agreed to the published version of the manuscript.  
This work was supported by the National Science Fund for Distinguished Young Scholars (51625902), the National Key Research and Development Program of China (2019 YFC0312404), the Major Scientific and Technological Innovation Project of Shandong Province (2019 JZZY010820), the National Natural Science Foundation of China (51809134), the Natural Science Foundation of Shandong Province (ZR2017 MEE007), and the Taishan Scholars Program of Shandong Province (TS201511016).  
The authors declare no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.