# Damage Identification Method Using Additional Virtual Mass Based on Damage Sparsity
## 3 Improved OMP Damage Identification Method Based on Sparsity  
The traditional damage identification methods based on sparsity all have disadvantages. In Lasso regression model and ridge regression model with \(l_{1}\) norm and \(l_{2}\) norm as sparse constraints, respectively, the selection of the regularization coefficient \(\lambda\) directly affects the accuracy of the recognition results. The traditional methods for selecting \(\lambda\) based on the L-curve is more complicated, and there is no selection process for the damage substructure using the two traditional methods. The OMP method selects forward the column vector from sensitivity matrix based on the most significant correlation with the frequency residual. First, each selection step depends on the previous step selection result; therefore, the damage determined by this method is typically a local optimal result, and its integrity is insufficient. Second, because the OMP method needs to estimate the sparsity of the damage-factor variation to determine the iterative operation steps, the sparsity estimation accuracy directly confirms whether the damage recognition results are correct, which has certain logical defects. Moreover, the traditional OMP method only depends on the final pseudo-inverse calculation in determining the damage factors value, inducing a significant error. In this study, an improved OMP (IOMP) method was developed to overcome the shortcomings of traditional sparse damage identification methods. The damage identification process for this method is divided into three main steps. First, we determine the number of damaged substructures and consider the remain undamaged. Second, the damage factors corresponding to the undamaged substructures removed from the damage vector. Finally, the objective function (5) is used to determine the specific value of the damage factors. From Equation (8), it can be observed that the frequency residual had the following relationship with the sensitivity matrix and damage-factor variation. \[\Delta\mathbf{\omega}=\hat{\mathbf{\omega}}-\mathbf{\omega}(\mathbf{\mu}_{0})=\mathbf{R}\Delta\mathbf{ \mu}+\mathbf{e}_{taylor}+\mathbf{e}_{noise} \tag{15}\]  
It can be observed from Equation (7) that the sensitivity matrix is a full rank. The \(i\)th element, \(\Delta\mathbf{\mu}_{i}\), of the damage-factor variation \(\Delta\mathbf{\mu}\) is assumed to be zero, indicating that no damage occurred to the \(i\)th substructure. \(\Delta\mathbf{\mu}_{i}^{*}\) is the \((n-1)\times 1\) dimensional column vector after \(\Delta\mathbf{\mu}_{i}\) is removed from \(\Delta\mathbf{\mu}\). \(\mathbf{r}_{i}\) is the \(i\)th column of the sensitivity matrix \(\mathbf{R}\), \(\mathbf{R}_{i}^{*}\) is the \(k\times(n-1)\) dimension matrix of \(\mathbf{R}\) without the \(i\)th column, and \(\left(\mathbf{R}_{i}^{*}\right)^{+}\) is the pseudo inverse matrix of \(\mathbf{R}_{i}^{*}\). \[\left\{ \begin{array}{c}\Delta\mathbf{\mu}_{i}^{*}=\left(\mathbf{R}_{i}^{*}\right)^{ +}\Delta\mathbf{\omega}\\ \\ \left(\mathbf{R}_{i}^{*}\right)^{+}=\left(\left(\mathbf{R}_{i}^{*}\right)^{T}\mathbf{R}_{ i}^{*}\right)^{-1}\left(\mathbf{R}_{i}^{*}\right)^{T}\end{array} \right. \tag{16}\]  
When \(\Delta\mu_{i}\) is zero, the structural frequency difference is \(\Delta\mathbf{\omega}^{i}\), and the frequency residual vector is expressed as follows:  
\[ \begin{array}{c}\mathbf{\varepsilon}_{i}=\Delta\mathbf{\omega}-\Delta\mathbf{\omega}^{i }=\Delta\mathbf{\omega}-\mathbf{R}_{i}^{*}\left(\mathbf{R}_{i}^{*}\right)^{+}\Delta\mathbf{ \omega}\\ \\ \mathbf{\varepsilon}_{i}=\left(\mathbf{I}-\mathbf{R}_{i}^{*}\left(\mathbf{R}_{i}^{*}\right)^{ +}\right)\left(\mathbf{r}_{i}\Delta\mathbf{\mu}_{i}+\mathbf{e}\right)\end{array}  \tag{17}\]  
According to Equation (17), the residual vector \(\varepsilon_{i}\) is related to the ith element of damage-factor variation, \(\Delta\mu_{i}\).