# Structural Modal Time Domain Identification Method Based on the Bayesian Uncertain Quantification
## 3 Bayesian Modal Identification Based on Modal State Space Prediction Models
### Uncertainty Analysis of Modal Parameters
The second derivative function of the function \(f\big{(}\overline{\theta}\big{)}\) with respect to \(l_{i}\) and \(\mathbf{\Phi}\) is as follows:  
\[\frac{\partial^{2}f\big{(}\overline{\theta}\big{)}}{\partial l_{i}\partial\bm {\Phi}}=2\mathbf{\Phi}^{\mathsf{T}}\Bigg{\{}\sum\limits_{k=1}^{N_{s}}\frac{ \partial^{\sim}_{l_{i},k}}{\partial l_{i}}\bigg{(}\mathbf{Q}_{i}-\mathbf{Q} _{i}^{\mathsf{T}}\bigg{)}\Bigg{]}\otimes\mathbf{I}_{n}\Bigg{\}}-2\sum\limits_{ k=1}^{N_{s}}\Bigg{[}\frac{\partial^{\sim}_{l_{i},k}}{\partial l_{i}}\mathbf{e}_{i} \otimes\mathbf{\hat{y}}_{k}^{\mathsf{T}}\Bigg{]} \tag{46}\]where \(\mathbf{e}_{i}\in R^{m}\) is a vector whose \(i\)th element is equal to one, and the remaining elements are equal to zero. The second derivative of the function \(f\left(\overline{\theta}\right)\) with respect to \(\mathbf{\Phi}\) is a matrix of \(mn\times mn\):  
\[\frac{\partial^{2}f\left(\overline{\theta}\right)}{\partial\mathbf{\Phi} \partial\mathbf{\Phi}}=2\left[\left(\sum_{k=1}^{N_{k}}\widetilde{\eta}_{k} ^{\top}\widetilde{\eta}_{k}^{\top}\right)\otimes\mathbf{I}_{n}\right] \tag{47}\]  
Equations (45)-(47) give semi-analytic expressions for the function \(f\left(\overline{\theta}\right)\) with respect to \(f_{i}\), \(\zeta_{i}\), and \(\mathbf{\Phi}\), which allows one to solve for the Hessian matrix \(\mathbf{H}_{f}\left(\overline{\theta}^{*}\right)\). Notice that in Equations (11) and (13), since the modal displacements and accelerations are nonlinear functions for \(\left\{f_{i},\zeta_{i}\right\}\), it is more difficult to give the analytical expressions of the functions. Therefore, it is necessary to solve for the first and second derivatives of the modal response \(\widetilde{\eta}_{i,k}\) with respect to \(f_{i}\) and \(\zeta_{i}\) by the finite difference method for the values of the function at the MPVs of \(\left\{f_{i},\zeta_{i}\right\}\). The Hessian matrix of \(f\left(\overline{\theta}\right)\) can be written as a chunking matrix as shown below:  
\[\mathbf{H}_{f}\left(\overline{\theta}^{*}\right)=\left[ \begin{matrix}\mathbf{H }_{\frac{\partial^{2}f\left(\overline{\theta}\right)}{\partial\zeta_{i} \partial\zeta_{j}}}\left(\overline{\theta}^{*}\right)&\mathbf{H}_{\frac{ \partial^{2}f\left(\overline{\theta}\right)}{\partial\zeta_{i}\partial\zeta} }\left(\overline{\theta}^{*}\right)\\ \left[\mathbf{H}_{\frac{\partial^{2}f\left(\overline{\theta}\right)}{ \partial\zeta_{i}\partial\zeta}}\left(\overline{\theta}^{*}\right)\right]^{ \top}&\mathbf{H}_{\frac{\partial^{2}f\left(\overline{\theta}\right)}{ \partial\zeta\partial\zeta}}\left(\overline{\theta}^{*}\right)\end{matrix} \right] \tag{48}\]  
Finally, the Hessian matrix \(\mathbf{H}_{L}(\theta^{*})\in R^{(2m+m+m+1)\times(2m+m+1)}\) can be written as a block diagonal matrix:  
\[\mathbf{H}_{L}(\theta^{*})=\left[ \begin{matrix}\frac{1}{2\left(\sigma_{e}^{2} \right)}\mathbf{H}_{f}\left(\overline{\theta}^{*}\right)&0\\ 0^{\text{T}}&\frac{nN_{t}}{2\left[\left(\sigma_{e}^{2}\right)^{*}\right]^{2}} \end{matrix} \right] \tag{49}\]  
where \(\mathbf{0}\in R^{2m+mn}\) is a zero vector. The posterior covariance matrix of parameters \(\theta\) can be solved from Equation (25). The first \(2m\) diagonal elements of \(\mathbf{C}^{*}\) are the posterior variances of \(\left\{f_{i},\zeta_{i}\right\}\), and the last diagonal element of \(\mathbf{C}^{*}\) is the posterior variance of \(\sigma_{e}^{2}\). For the signal noise, it mainly consists of random noise in the measurement hardware and the quantization error of the digitizer: when the sampling rate is large enough, the quantization error can be ignored; however, random noise is inevitable [30]. In frequency domain identification methods, the effect of random noise can be reduced by averaging multiple FFT results. In the method of this paper, noise reduction can be achieved by combining filtering techniques to avoid false modes caused by random noise.