# Impact Load Sparse Recognition Method Based on Mc Penalty Function
## 3 Impact Load Sparse Identification Model  
### Impact Load Sparse Identification Model of MC Penalty Function  
The Tikhonov regularization method usually makes the ill-posed problem well-posed by adding constraints [29; 30]. Its core idea is to find a weighted factor \(\lambda\) called regularization parameter between the residual norm \(||\mathbf{H}\mathbf{f}-\mathbf{y}||\) and the norm \(||\mathbf{f}||_{1}\), and then correct the singular value to improve the stability of the solution [31; 32]. The objective function of Tikhonov regularization is:  
\[\underset{f}{\text{minimize}}||\mathbf{f}||_{2};\text{ subject to }\mathbf{y}=\mathbf{H}\mathbf{f} \tag{20}\]  
In the formula, \(\mathbf{f}\) represents the impact load vector to be identified, \(\mathbf{Y}\) represents the vector of sparse recognition result, and \(\mathbf{H}\) represents the function transfer matrix. Considering that the response data inevitably contains noise, Equation (20) can be written:  
\[\underset{f}{\text{minimize}}||\mathbf{f}||_{2};\text{ subject to }||\mathbf{y}-\mathbf{H}\mathbf{f}||_{1}\leq\mathbf{\delta} \tag{21}\]  
where \(\mathbf{\delta}\) is the noise level. Equation (21) can be written the unconstrained equivalent form:  
\[\underset{f}{\text{minimize}}||\mathbf{H}\mathbf{f}-\mathbf{y}||_{2}^{2}+\lambda||\mathbf{f} ||_{1} \tag{22}\]  
where \(||\mathbf{f}||_{1}\) represents the L1 norm penalty term. The residual error \(||\mathbf{H}\mathbf{f}-\mathbf{y}||_{2}^{2}\) and sparse regularization operator \(||\mathbf{f}||_{1}\) trade-off is established. The L1 norm term \(||\mathbf{f}||_{1}\) in Equation (22) has prompted a smaller numerical zero role in unknown impact load, thus inducing get sparse solution of impact load. However, in practice, to solve when the dimension is larger, or when the condition is complicated, in addition to the fixed threshold function, L1 threshold function with the same function [33] has a certain distance, causing the L1 penalty function to lead to an undervalued signal amplitude. The situation has some weak shock, which may be ignored,and it is easy to cause misjudgment of early fault detection. This indirectly causes the loss of equipment and the increase of cost, especially due to the rotor part of the gas turbine. Once it has failed, the consequences are unimaginable. The selection of penalty function is a key part of the sparse representation of impact load identification. The biggest feature of the MC penalty function sparse decomposition model is that, based on Tikhonov regularization, MC penalty function is used to reconstruct the regularized sparse identification model, so that the regularization method can effectively avoid generating local minimum value, and can more easily obtain sub-global optimal solutions. The approximate operator of the MC penalty function:  
\[\Theta(\mathbf{y},\lambda,\mathbf{a}):=\operatorname*{argmin}_{\mathbf{x}}\left\{ \frac{1}{2}(\mathbf{y}-\mathbf{f})^{2}+\lambda\mathbf{\phi}(\mathbf{f}\colon\mathbf{a})\right\} \tag{23}\]  
The MC penalty function is used to replace the solution; a represents the proximal operator parameters of MC penalty function. The MC penalty function makes up for the deficiency of L1 norm, and the proximal operator of MC is consistent with the same function [34], so that the signal amplitude will not be underestimated. The penalty function MC is selected as the penalty term of the objective function to regularize the objective function.