# Structural Modal Time Domain Identification Method Based on the Bayesian Uncertain Quantification
## 3 Bayesian Modal Identification Based on Modal State Space Prediction Models
### Decoupling Method for Modal Identification
Thus, \(\overline{\mathbf{\Psi}}\widetilde{\mathbf{\eta}}_{k}=vec\big{(}\overline{ \mathbf{\Psi}}\widetilde{\mathbf{\eta}}_{k}\big{)}=vec\big{(}\mathbf{I}_{ \mathbf{n}}\widetilde{\mathbf{\Psi}}\widetilde{\mathbf{\eta}}_{k}\big{)}\), and Equation (15) can be rewritten as follows:  
\[\mathbf{y}_{k}\big{(}\overline{\mathbf{\theta}}\big{)}=\widetilde{\mathbf{\Psi }}\widetilde{\mathbf{\eta}}_{k}=\bigg{(}\widetilde{\mathbf{\eta}}_{k}^{ \mathsf{T}}\otimes\mathbf{I}_{\mathbf{n}}\bigg{)}\mathbf{\Phi} \tag{31}\]Substituting Equation (31) into Equation (26) and expanding it yields the following functional expression:  
\[f\big{(}\overline{\theta}\big{)}=\mathbf{\Phi}^{\mathrm{T}}\mathbf{G}\mathbf{\Phi}-2\mathbf{ \Phi}^{\mathrm{T}}\mathbf{R}+\sum_{k=1}^{N_{k}}\Big{(}\tilde{\mathbf{\eta}}_{k}^{ \mathrm{T}}\tilde{\mathbf{\eta}}_{k}\Big{)} \tag{32}\]  
where  
\[\mathbf{G}=\left[\left(\sum_{k=1}^{N_{k}}\widetilde{\mathbf{\eta}}_{k}^{\sim} \widetilde{\mathbf{\eta}}_{k}^{\sim\mathrm{T}}\right)\otimes\mathbf{I}_{n}\right] \in R^{nm\times nm},\mathbf{R}=\sum_{k=1}^{N_{k}}\left(\widetilde{\mathbf{\eta}}_{k }\otimes\tilde{\mathbf{\eta}}_{k}\right)\in R^{nm\times 1} \tag{33}\]  
For the algorithm of Kronecker product, please refer to Appendix A. In the real vector space, for a symmetric positive definite matrix \(\overline{\mathbf{\Lambda}}\in R^{n_{r}\times n_{r}}\) and nonzero vectors \(\overline{\mathbf{\kappa}}\in R^{n_{r}}\), \(\overline{\mathbf{\mathfrak{b}}}\in R^{n_{r}}\), and \(c\in R\), a quadratic function \(F(\overline{\mathbf{\kappa}})=\overline{\mathbf{\kappa}}^{\mathrm{T}}\overline{\mathbf{ \Lambda}}\overline{\mathbf{\kappa}}+2\overline{\mathbf{\mathfrak{b}}}^{\mathrm{T}} \overline{\mathbf{\kappa}}+c\) can be constructed [30]. From Equation (32), it can be seen that \(f(\overline{\theta})\) is a quadratic function with respect to the mode vector \(\mathbf{\Phi}\). Since \(\mathbf{\Phi}\) is an unconstrained vector, based on the law of derivation of a function with respect to vectors [31], an analytical expression for \(\mathbf{\Phi}\) can be derived when the gradient of \(f(\overline{\theta})\) with respect to the vector \(\mathbf{\Phi}\) is zero. The specific derivation is as follows:  
\[\frac{\partial f(\overline{\theta})}{\partial\mathbf{\Phi}}=2\Big{(}\mathbf{\Phi}^{ \mathrm{T}}\mathbf{G}-\mathbf{R}^{\mathrm{T}}\Big{)} \tag{34}\]  
According to \(\left(\partial f\big{(}\overline{\theta}\big{)}/\partial\mathbf{\Phi}\right)|_{ \mathbf{\Phi}=\mathbf{\Phi}^{*}}=0\), the analytic solution of the vector \(\mathbf{\Phi}^{*}\) can be given by the following:  
\[\mathbf{\Phi}^{*}(f_{i},\zeta_{i})=\mathbf{G}^{-1}\mathbf{R} \tag{35}\]  
The function \(f(\overline{\theta})\) has a minimum at \(\mathbf{\Phi}^{*}=\mathbf{G}^{-1}\mathbf{R}\). In Equation (35), since the matrices \(\mathbf{G}\) and \(\mathbf{R}\) are implicit functions for \(f_{i}\) and \(\zeta_{i}\), \(\mathbf{\Phi}^{*}\) is also a function for \(f_{i}\) and \(\zeta_{i}\). By evaluating the function \(J(\theta)\) at \(\mathbf{\Phi}^{*}\), the optimization objective function \(f(f_{i},\zeta_{i})\) only for \(f_{i}\) and \(\zeta_{i}\) can be given by the following:  
\[f(f_{i},\zeta_{i})=-[\mathbf{R}(f_{i},\zeta_{i})]^{\mathrm{T}}[\mathbf{G}(f_{ i},\zeta_{i})]^{-1}\mathbf{R}((f_{i},\zeta_{i}))+\sum_{k=1}^{N_{k}}\Big{(} \tilde{\mathbf{\gamma}}_{k}^{\mathrm{T}}\tilde{\mathbf{\gamma}}_{k}\Big{)} \tag{36}\]  
Equation (36) realizes the decoupling of the dynamic spectral parameters \(\{f_{i},\zeta_{i}\}\) and the spatial parameters \(\{\mathbf{\varphi}_{i},\gamma_{i}\}\). In Equation (32), a total of \(2m+nm\) unknown parameters are involved. By contrast, Equation (36) solely incorporates \(2m\) unknown parameters, substantially reducing the number of variables to be identified. This transformation significantly improves the computational efficiency of the modal parameter identification.