## Online Dynamic Load Identification Method
### Online Force Identification Method  
The key point of the proposed method is to build a recursive algorithm for the dynamic load identification in the time domain. The system state is described as a series of time steps that are updated based on the measurement taken and the state unbiased estimation. We then identify the dynamic load using the weighted least square method to minimize covariance unbiased estimation. The state update, the load identification and the measurement update are achieved using the following three equations:  
\[\mathbf{x}_{k|k-1} =\mathbf{A}\mathbf{x}_{k-1|k-1}+\mathbf{B}\mathbf{\hat{u}}_{k-1} \tag{9}\] \[\mathbf{\hat{u}}_{k} =\mathbf{J}_{k}(\mathbf{y}_{k}-\mathbf{H}\mathbf{x}_{k|k-1})\] (10) \[\mathbf{x}_{k|k} =\mathbf{x}_{k|k-1}+\mathbf{K}_{k}(\mathbf{y}_{k}-\mathbf{H} \mathbf{x}_{k|k-1}-\mathbf{D}\mathbf{\hat{u}}_{k}) \tag{11}\]  
The matrices \(\mathbf{J}_{k}\) and \(\mathbf{K}_{k}\) are needed for unbiased estimates of the system state i.e., \(\mathbf{x}_{k|k}\) and the load estimates \(\mathbf{\hat{u}}_{k}\). The entries of \(\mathbf{J}_{k}\) are determined using the weighted least square method when identifying the load while those of \(\mathbf{K}_{k}\) are evaluated within the process of the state unbiased estimation of the minimum variance. Based on Equations (9) and (3), the error and covariance matrix of \(\mathbf{x}_{k|k-1}\) can be written as  
\[\widetilde{\mathbf{x}}_{k|k-1} =\mathbf{x}_{k}-\mathbf{x}_{k|k-1}=\mathbf{A}\widetilde{\mathbf{ x}}_{k-1|k-1}+\mathbf{B}\widetilde{\mathbf{u}}_{k-1}+\mathbf{w}_{k-1} \tag{12}\] \[\mathbf{P}_{k|k-1}^{x} =\left[ \begin{array}{cc}\mathbf{A}&\mathbf{B}\end{array} \right] \left[ \begin{array}{cc}\mathbf{P}_{k-1|k-1}^{x}&\mathbf{P}_{k-1}^{xu}\\ \mathbf{P}_{k-1}^{ux}&\mathbf{P}_{k-1}^{u}\end{array} \right]\left[ \begin{array} []{cc}\mathbf{A}^{\top}\\ \mathbf{B}^{\top}\end{array} \right]+\mathbf{G}_{k-1} \tag{13}\]  
The covariance matrix of response positions \(\mathbf{P}_{k-1|k-1}^{x}\) and \(\mathbf{P}_{k-1}^{ux}\) and of excitation positions \(\mathbf{P}_{k-1}^{u}\) and \(\mathbf{P}_{k-1}^{xu}\) are given by  
\[\mathbf{P}_{k-1|k-1}^{x} =\mathbf{E}(\widetilde{\mathbf{x}}_{k-1|k-1}\widetilde{\mathbf{ x}}_{k-1|k-1}^{\top})\] \[\mathbf{P}_{k-1}^{u} =\mathbf{E}(\widetilde{\mathbf{u}}_{k-1}\widetilde{\mathbf{u}}_{k -1}^{\top})\] \[\mathbf{P}_{k-1}^{ux} =(\mathbf{P}_{k-1}^{xu})^{\top}=\mathbf{E}(\widetilde{\mathbf{u}} _{k-1}\widetilde{\mathbf{u}}_{k-1|k-1}^{\top})\]  
We define the residual vector as  
\[\widetilde{\mathbf{y}}_{k}=\mathbf{y}_{k}-\mathbf{H}\mathbf{x}_{k|k-1} \tag{14}\]  
where the measurement \(\mathbf{y}_{k}\) including some error, is defined by  
\[\mathbf{y}_{k}=\mathbf{H}\mathbf{x}_{k}+\mathbf{D}\mathbf{u}_{k}+\mathbf{v}_{k} \tag{15}\]  
By substituting Equation (15) back into (14) we get  
\[\widetilde{\mathbf{y}}_{k}=\mathbf{D}\mathbf{u}_{k}+\mathbf{H}\widetilde{ \mathbf{x}}_{k|k-1}+\mathbf{v}_{k}=\mathbf{D}\mathbf{u}_{k}+\mathbf{e}_{k} \tag{16}\]  
Substituting Equation (14) into (10)  
\[\mathbf{\hat{u}}_{k}=\mathbf{J}_{k}\mathbf{D}\mathbf{u}_{k}+\mathbf{J}_{k} \mathbf{e}_{k} \tag{17}\]  
where \(\mathbf{x}_{k|k-1}\) is unbiased estimation, \(\mathbf{E}(\mathbf{v}_{k})=0\) and \(\mathbf{E}(\widetilde{\mathbf{y}}_{k})=\mathbf{D}\mathbf{E}(\mathbf{u}_{k})\). In addition, \(\mathbf{\hat{u}}_{k}\) is unbiased estimation then \(\mathbf{J}_{k}\mathbf{D}=\mathbf{I}\). We also define \(\widetilde{\mathbf{R}}_{k}\) as  
\[\widetilde{\mathbf{R}}_{k}\equiv\mathbf{E}(\mathbf{e}_{k}\mathbf{e}_{k}^{\top })=\mathbf{H}\mathbf{P}_{k|k-1}^{x}\mathbf{H}^{\top}+\mathbf{R}_{k}\qquad \text{with}\qquad\mathbf{R}_{k}=\mathbf{E}(\mathbf{v}_{k}\mathbf{v}_{k}^{\top}) \tag{18}\]  
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1: Given initial value \(\mathbf{X_{0|-1}}\),\(\mathbf{P^{x}}_{0|-1}\).