# Robust Structural Damage Detection Using Analysis of the CMSE Residual's Sensitivity to Damage
## 1 Introduction  
Structural damage identification is a fundamental element of structural health monitoring (SHM) that has become a vital tool in maintaining the safety and integrity of structures [1; 2; 3; 4; 5; 6; 7]. Research on vibration-based damage identification has been rapidly expanding over recent decades. The basic idea behind this technology is that modal parameters (notably frequencies, mode shapes, and modal damping) are functions of the physical properties of the structure. Therefore, changes in the physical properties (such as stiffness reduction caused by damage) will cause detectable changes in the modal properties. Conversely, these changes can be used to reflect damage. Vibration-based structural damage identification can be formulated as a linear inverse problem [8; 9; 10; 11; 12], which requires the determination of the unknown input (i.e., structural damage) to a linear system from the known output (i.e., extracted modal parameters from the vibration measurements of the structure). The discretization of the linear inverse problem [12; 13] typically gives rise to the linear system of equations with a very ill-conditioned matrix \(\mathbf{C}^{*}\)[14],  
\[\mathbf{C}^{*}\boldsymbol{\alpha}=\boldsymbol{b}^{*},\;\mathbf{C}^{*}\in \mathbb{R}^{N_{b}\times N_{d}},\;\boldsymbol{\alpha}\in\mathbb{R}^{N_{d}},\; \boldsymbol{b}^{*}\in\mathbb{R}^{N_{b}}, \tag{1}\]  
where the coefficient matrix \(\mathbf{C}^{*}\) and the right-hand side vector \(\boldsymbol{b}^{*}\) are both assembled by the identified modal parameters and will be given in Equation (16); \(\boldsymbol{\alpha}\) is the unknown damage level vector to besolved, whose definition will be given in Equation (13); and the superscript \"*\" indicates a damaged version of physical or modal parameters. In other words, these parameters are dependent on \(\mathbf{\alpha}\); and \(N_{q}\) and \(N_{d}\) are the number of linear equations in the system and the potentially damaged elements, respectively. Usually, \(N_{q}\geq N_{d}\) is required to ensure that the linear system of equations has a unique solution. In general, the computation of a meaningful approximate solution of the linear system requires that the system be replaced by a nearby system that is less sensitive to perturbations. This replacement is referred to as regularization. Tikhonov regularization is one of the most popular regularization techniques that can damp out the small measurement errors and single out a relatively accurate solution [15]. So far, it has been widely applied to damage detection [16; 17; 18] and other related research, such as model updating [19; 20] and load identification [21]. [PERSON] [22] gave a general form of Tikhonov regularization as follows:  
\[\mathbf{\left(C^{*T}C^{*}+\xi^{2}L^{T}L\right)}\mathbf{\alpha}=C^{*T}b^{*} \tag{2}\]  
where \(\xi^{2}\) and \(L\) are the regularization parameter and the regularization operator, respectively. Then, the solution \(\mathbf{\alpha}\) of Equation (2) satisfies the minimization problem  
\[\min_{\mathbf{\alpha}\in\mathbb{N}^{N_{d}}}\{\|\mathbf{\left(C^{*}\mathbf{\alpha}-b^{*} \right)}\|^{2}+\xi^{2}\|\mathbf{L}\mathbf{\alpha}\|^{2}\} \tag{3}\]  
Here and below, \(\|\cdot\|\) denotes the Euclidean norm. The determination of a suitable value of \(\xi^{2}\) and \(\mathbf{L}\) is an important task. In many previously developed methods, \(\xi^{2}\) is selected using an L-curve criterion [12; 15] or the generalized cross-validation method [20; 23], but both methods are time-consuming computational tasks. Besides, the regularization parameter \(\mathbf{L}\) is usually selected as an identity operator and thus leads to over-smoothing of the solution, which goes against the sparse feature of isolated damage at the early stage of structural deterioration [18]. [PERSON] et al. [24] proposed the iterative Tikhonov regularization (ITR) method specialized for the identification of the isolated damage to structures.