## 2 EKF-Based Parameters Identification  
For a linear \(n\)-DOF dynamic system, the motion equation can be written as:  
\[\mathbf{M}\ddot{\mathbf{Y}}(t)+\mathbf{C}\dot{\mathbf{Y}}(t)+\mathbf{K}\mathbf{Y}(t)=\mathbf{F}(t) \tag{1}\]  
where \(\mathbf{M}\), \(\mathbf{C}\), and \(\mathbf{K}\) are the mass, damping, and stiffness matrices, respectively; vectors \(\mathbf{Y}(t)\), \(\dot{\mathbf{Y}}(t)\) and \(\ddot{\mathbf{Y}}(t)\) denote displacement, velocity, and acceleration response, respectively; \(\mathbf{F}(t)\) is the external force vector. Its state space equation can be expressed as:  
\[\dot{\mathbf{X}}(t)=\mathbf{A}\mathbf{X}(t)+\mathbf{B}\mathbf{F}(t) \tag{2}\]  
where \(\mathbf{X}(t)=\left\{ \begin{array}{c}\mathbf{Y}(t)\\ \dot{\mathbf{Y}}(t)\end{array} \right\}\), \(\mathbf{A}=\left[ \begin{array}{cc}\mathbf{0}_{n}&\mathbf{I}_{n}\\ -\mathbf{M}^{-1}\mathbf{K}&-\mathbf{M}^{-1}\mathbf{C}\end{array} \right]\), and \(\mathbf{B}=\left[ \begin{array}{cc}\mathbf{0}_{n}\\ \mathbf{M}^{-1}\end{array} \right]\), where \(\mathbf{0}_{n}\) and \(\mathbf{I}_{n}\) are an \(n\)-order zero matrix and an \(n\)-order unit matrix, respectively. The matrix exponential function of \(e^{-At}\) is introduced to solve Equation (2). Let \(\Phi(t,t_{0})=e^{\mathbf{A}(t-t_{0})}\), which is the state transition matrix. The system state at any time \(t\) can be obtained as:  
\[\mathbf{X}(t)=\Phi(t,t_{0})\mathbf{X}(t_{0})+\int_{t_{0}}^{t}\Phi(t,\tau)\mathbf{B}\mathbf{F}( \tau)d\tau \tag{3}\]  
Equation (3) is a continuous state equation. However, the continuous expression of external excitation time history function is often very difficult to be obtained, and discretization of the equation is therefore required. The discrete equation can be written as:  
\[\mathbf{X}(t_{k+1})=\Phi(t_{k+1},t_{k})\mathbf{X}(t_{k})+\int_{t_{k}}^{t_{k+1}}\Phi(t _{k+1},\tau)\mathbf{B}\mathbf{F}(\tau)d\tau \tag{4}\]The uniform sampling is usually assumed, and the external excitation can be regarded as linear within the small interval \([t_{k},t_{k+1}]\), and can be calculated by:  
\[\mathbf{F}(\tau)=F_{k}-\frac{F_{k}-F_{k+1}}{\Delta t}(\tau-t_{k}) \tag{5}\]  
where \(\Delta t=t_{k+1}-t_{k}\). Let \(s=\tau-t_{k}\). Equation (4) can be finally expressed as:  
\[\mathbf{X}_{k+1}=\Phi_{k}\mathbf{X}_{k}+\mathbf{\Gamma}_{k}F_{k}+\mathbf{C}_{k} \tag{6}\]  
\[\Phi_{k}=e^{\Delta t}\] \[\mathbf{\Gamma}_{k}=\big{(}e^{\Delta t\Delta t}-\mathbf{I}\big{)}\mathbf{A}^{-1}\mathbf{B}\] \[\mathbf{C}_{k}=\big{[}\Delta t-\big{(}e^{\Delta t}-\mathbf{I}\big{)}\mathbf{A }^{-1}\big{]}\mathbf{A}^{-1}\mathbf{B}(F_{k}-F_{k+1})/\Delta t\]  
where \(\mathbf{X}_{k}\) is the system state at time \(k\), and \(\mathbf{F}_{k}\) and \(\mathbf{F}_{k+1}\) are the sampled values at the time \(k\) and \(k+1\), respectively. Thus, the discrete-time state equation of linear dynamic system is then obtained. [PERSON] [15] proposed the traditional KF in 1960, which is an optimal linear recursive estimator designed for linear time-varying dynamic systems. By using the linearized models, EKF expands its identification ability to nonlinear systems. In order to identify the parameters of the deterministic dynamical system by the EKF, the structural parameters can be considered as the state variables. Thus, the corresponding discrete formulation of the parameters identification problem in the EKF can be expressed as:  
\[\mathbf{X}_{k+1}=\mathbf{X}_{k}+\int_{t_{k}}^{t_{k+1}}f(\mathbf{X},\mathbf{F})dt+\mathbf{w}_{k} \tag{7}\]  
\[\mathbf{Z}=\mathbf{h}(\mathbf{X})+\mathbf{v} \tag{8}\]  
where \(\mathbf{X}_{k+1}\) and \(\mathbf{Z}\) are the state vector and the observation vector, respectively; \(\mathbf{w}_{k}\) is the process noise represented by zero-mean white Gaussian noises with a covariance matrix \(\mathbf{Q}\), and \(\mathbf{v}\) is the observation noise represented by zero-mean white Gaussian noises with a covariance matrix \(\mathbf{R}\), and the function \(\mathbf{h}(\mathbf{X})\) is the theoretical solution of structural response at the measure point \(x\).