# 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  
Let \(V\) be the vector space composed of all \(2m+mn+1\) dimensional real vectors, denoted as \(\mathbf{R}^{2m+mn+1}\), which satisfies vector addition and multiplication closure. In this paper, the vector \(\mathbf{\theta}= \begin{bmatrix}f_{1}&\cdots&f_{m}&\zeta_{1}&\cdots&\zeta_{m}& \mathbf{\Phi}&\sigma_{\epsilon}^{2}\end{bmatrix} ^{\mathsf{T}}\in R^{2m+mn+1}\) is composed of the spectral parameters and mode parameters. In Equation (22), the following is defined:  
\[f\big{(}\overline{\mathbf{\theta}}\big{)}=\sum_{k=1}^{N_{\mathrm{s}}}\big{[} \hat{\mathbf{y}}_{k}-\mathbf{y}_{k}\big{(}\overline{\mathbf{\theta}}\big{)} \big{]}^{\mathsf{T}}\big{[}\hat{\mathbf{y}}_{k}-\mathbf{y}_{k}\big{(} \overline{\mathbf{\theta}}\big{)}\big{]} \tag{26}\]  
The first derivative of \(L(\mathbf{\theta})\) with respect to \(\sigma_{\epsilon}^{2}\) is as follows:  
\[\frac{\partial L(\mathbf{\theta})}{\partial\sigma_{\epsilon}^{2}}=\frac{nN_{ \mathrm{s}}}{2\sigma_{\epsilon}^{2}}-\frac{f\big{(}\overline{\mathbf{\theta} }\big{)}}{2\big{(}\sigma_{\epsilon}^{2}\big{)}^{2}} \tag{27}\]  
Based on the extremum theory of multivariate functions, when the first-order derivative of the likelihood function \(L(\mathbf{\theta})\) with respect to the variable \(\sigma_{\epsilon}^{2}\) is zero, an analytic expression for \(\sigma_{\epsilon}^{2}\) with respect to \(f\big{(}\overline{\mathbf{\theta}}\big{)}\) can be derived:  
\[\Big{(}\sigma_{\epsilon}^{2}\Big{)}^{*}=\frac{f\big{(}\overline{\mathbf{\theta }}\big{)}}{nN_{\mathrm{s}}} \tag{28}\]  
and, when \(f\big{(}\overline{\mathbf{\theta}}\big{)}\) takes the minimum value, \(\sigma_{\epsilon}^{2}\) also takes the minimum value. Let \(\ell\) denote \(f_{i}\), \(\zeta_{i}\), \(\gamma_{i}\), or \(\mathbf{\varphi}_{i}\), and the first derivative of the function \(L(\mathbf{\theta})\) with respect to \(\ell\) is as follows:  
\[\frac{\partial L(\mathbf{\theta})}{\partial\ell}=\frac{1}{2\sigma_{\epsilon}^ {2}}\frac{\partial f(\overline{\mathbf{\theta}})}{\partial\ell} \tag{29}\]  
Equation (29) indicates that \(\big{(}\partial f(\overline{\mathbf{\theta}})/\partial\ell\big{)}\big{|}_{ \ell=\ell^{*}}=0\) when \(\big{(}\partial L(\mathbf{\theta})/\partial\ell\big{)}\big{|}_{\ell=\ell^{*} }=0\). Therefore, the MPV of \(\overline{\mathbf{\theta}}\) needs to be solved for only by minimizing the function \(f\big{(}\overline{\mathbf{\theta}}\big{)}\). When minimizing \(f\big{(}\overline{\mathbf{\theta}}\big{)}\), the dynamic spectral and spatial parameters are coupled if the theoretical prediction model (Equation (15)) is directly substituted into Equation (26). When the number of modes to be identified and measurement channels are large, dozens of unknown parameters are generated, which poses a great challenge for solving the MPV of the parameters using an optimization algorithm. However, subsequent derivations demonstrate that vectorizing the spatial parameter matrix \(\overline{\mathbf{\Psi}}\) alters the mathematical structure of \(f\big{(}\overline{\mathbf{\theta}}\big{)}\), transforming it into a quadratic function for the unconstrained vector. This allows for the separation of the spatial and spectral parameters by solving for the extremum of the quadratic function. The specific derivation is shown below. For any matrix \(\mathbf{X}\in R^{x\times b}\), the vectorization operator \"\(vec\)\" stacks the elements \(x_{ij}\) of the matrix \(\mathbf{X}\) in columns. Applying the vectorization operator to the spatial parameter matrix \(\overline{\mathbf{\Psi}}\) gives the following:  
\[\mathbf{\Phi}=vec\big{(}\overline{\mathbf{\Psi}}\big{)}= \begin{cases}\psi_{1 }\\ \vdots\\ \psi_{m}\end{cases} \in R^{nm} \tag{30}\]  
According to the algorithm of the vectorization operator, \(vec\big{(}\mathbf{U}\mathbf{D}\mathbf{H}\big{)}=\big{(}\mathbf{H}^{\mathsf{T }}\otimes\mathbf{U}\big{)}vec\big{(}\mathbf{D}\big{)}\), where \(\mathbf{U}\in R^{d\times h}\), \(\mathbf{D}\in R^{h\times g}\), and \(\mathbf{H}\in R^{g\times u}\), and \(\otimes\) denotes the Kronecker product.