# Adaptive Probabilistic Optimization Approach for Vibration-Based Structural Health Monitoring
## 2 Probabilistic Optimization Algorithm  
This section provides a detailed description of the algorithm for damage detection. The proposed algorithm integrates the use of the frequency response function (FRF) with optimization and a probabilistic model to identify the size of damage as well as the location of damage. The approach presented in this study uses the FRF to identify damage characteristics. The optimization problem is formulated to find the crack depth and crack location ratios by minimizing the difference between the measured FRF and the calculated FRF as follows:  
\[ \begin{array}{ll}\text{Find}:&Y=[\alpha,\beta]^{T}\\ \text{Minimize}:&Z(\alpha,\beta)=\left[H^{1},\ H^{2},H^{3},\ldots,H^{k-1},H^{ k}\right]^{T}\\ \text{Subject to}&H(\omega_{k})=\left|H_{m}(\omega_{k})-H_{c}(\omega_{k})\right| \\ &0\leq\alpha\leq 1\\ &0\leq\beta\leq 1\end{array}  \tag{1}\]  
In Equation (1), \(H_{m}(\omega_{k})\) is the measured FRF, \(H_{c}(\omega_{k})\) is the calculated FRF, \(\alpha\) is the crack size ratio, \(\beta\) is the crack location ratio, and \(H(\omega_{k})\) is the absolute difference between the measured and calculated FRFs over the frequency range, \(\omega_{k}\) for the kth segment. The objective function is minimized by using sequential quadratic programming (SQP), which is an iterative method suitable for constrained nonlinear optimization problems. An extensive amount of detail on SQP can be found in [27]. Prior to this optimization process, the measured FRF signals are de-noised by using the discrete wavelet transform with Daubechies mother wavelet, at level four wavelet decomposition. The frequency range and the number of samples in each segment are important factors that provide details about damage features and can also affect the confidence level in signal processing outcomes. Smaller segments can typically reveal more information but may be computationally expensive. This trade-off may need to be carefully evaluated when the data are being collected continuously, as is common in structural health monitoring. Thus, the segment size is defined as:  
\[N_{k+1}=N_{k}+\delta N \tag{2}\]  
In Equation (2), \(N_{k+1}\) is the next segment size, \(N_{k}\) is the current segment size, \(N\) is the minimum segment size, and \(\delta\) is the segment size factor. This factor controls the segment size by increasing or decreasing the number of samples that should be added or subtracted from the next segment depending on the magnitude of \(H(\omega_{k})\) that represents the level of agreement between the measured and calculated FRFs, i.e., the smaller \(H(\omega_{k})\) is, the larger the next segment will be. The segment size factor is defined as:  
\[\delta=\left\{ \begin{array}{ll}1&H^{k}<\delta_{lb}\\ 0&\delta_{lb}\leq H^{k}\leq\delta_{ub}\\ -1&H^{k}>\delta_{ub}\end{array} \right. \tag{3}\]  
The limits in Equation (3), \(\delta_{lb}\) and \(\delta_{ub}\), may vary from one application to another. The limits can be determined by the user to evaluate the appropriate limits for increasing and decreasing the segment size given the magnitude of error between measured and calculated FRFs. Since the segments vary in size, a weighting factor \(w_{i}\) is assigned to each segment. Therefore, large segments are associated with large weighting factors, i.e., the larger the segment, the higher the contribution it provides to the analysis of the damage identification process. The calculated FRF is a function of two design variables with upper bounds and lower bounds. By minimizing the objective function for each segment, the algorithm provides two vectors for each design variable. The size of each vector is the same as the number of segments created from the measured FRF. The algorithm was further developed to optimize the probability of crack location and crack size ratios while minimizing the uncertainty range to less than \(\eta_{\alpha}\) and \(\eta_{\beta}\), respectively,such that \(\eta_{\alpha}\) and \(\eta_{\beta}\) have a range of (0,1). For the crack location, the optimization problem is formulated as:  
\[