# Model-Free Identification of Nonlinear Restoring Force with Modified Observation Equation
## 2 Formulas of the Proposed Approach
Analogous to the traditional EKF, the priori state estimate \(\hat{\mathbf{Z}}_{k+1|k}\) can be given as:  
\[\hat{\mathbf{Z}}_{k+1|k}\ =\ \hat{\mathbf{Z}}_{k|k}+\int_{k\Delta t}^{(k+1) \Delta t}g\big{(}\hat{\mathbf{Z}}_{k|k},\ \hat{\mathbf{R}}_{k}^{non},\ k\Delta t\big{)} \mathrm{d}t \tag{12}\]  
Then, with the combination of Equation (5), the priori estimate error \(\varepsilon_{k+1|k}\) can be found as:  
\[\varepsilon_{k+1|k}\ =\ \mathbf{Z}_{k+1}-\hat{\mathbf{Z}}_{k+1|k}\ =\ \Big{(}\mathbf{I}+\Delta t\mathbf{U}_{k|k}\Big{)} \big{(}\mathbf{Z}_{k}-\hat{\mathbf{Z}}_{k|k}\big{)}+\Delta t\mathbf{W}_{k|k} \big{(}\mathbf{R}_{k}^{non}-\hat{\mathbf{R}}_{k}^{non}\big{)}+\Delta t\mathbf{ w}_{k} \tag{13}\]  
Focusing on the second term on the right-hand-side of Equation (13), the following transformation can be obtained:  
\[\Delta t\mathbf{W}_{k|k}\big{(}\mathbf{R}_{k}^{non}-\hat{\mathbf{R}}_{k}^{non }\big{)}\ =\ \Delta t\left[ \begin{array}{c}0\\ -\Big{[}\mathbf{h}(\mathbf{Z}_{k})-\mathbf{h}\Big{(}\hat{\mathbf{Z}}_{k|k} \Big{)}-\mathbf{v}_{k}\\ 0\end{array} \right] \tag{14}\]  
Similarly, \(\mathbf{h}(\mathbf{Z}_{k})\) can be linearized as follows:  
\[\mathbf{h}(\mathbf{Z}_{k})\ =\ \mathbf{h}\Big{(}\hat{\mathbf{Z}}_{k|k}\Big{)}+ \mathbf{H}_{k|k}\big{(}\mathbf{Z}_{k}-\hat{\mathbf{Z}}_{k|k}\big{)} \tag{15}\]  
where:  
\[\mathbf{H}_{k|k}\ =\ \ \frac{\partial\mathbf{h}(\mathbf{Z}_{k})}{\partial \mathbf{Z}_{k}}\Bigg{|}_{\mathbf{Z}_{k}=\hat{\mathbf{Z}}_{k|k}} \tag{16}\]  
Substituting Equations (14) and (15) into Equation (13) leads to:  
\[\varepsilon_{k+1|k}\ =\ \mathbf{A}_{1}\big{(}\mathbf{Z}_{k}-\hat{\mathbf{Z}}_{k|k} \big{)}+\mathbf{A}_{2}\mathbf{v}_{k}+\Delta t\mathbf{w}_{k}\ =\ \mathbf{A}_{1}\varepsilon_{k|k}+\mathbf{A}_{2}\mathbf{v}_{k}+\Delta t \mathbf{w}_{k} \tag{17}\]  
where:  
\[\mathbf{A}_{1}\ =\ \mathbf{I}+\Delta t\mathbf{U}_{k|k}+\Delta t\left[ \begin{array}{c}0\\ -\mathbf{H}_{k|k}\\ 0\end{array} \right]\! ;\ \mathbf{A}_{2}\ =\ \Delta t\left[ \begin{array}{c}0\\ -\mathbf{I}\\ 0\end{array} \right] \tag{18}\]  
The priori estimate error covariance matrix can be then determined as:  
\[\mathbf{P}_{k+1|k}\ =\ E\Big{(}\varepsilon_{k+1|k}\varepsilon_{k+1|k}^{ \mathrm{T}}\Big{)}\ =\ \mathbf{A}_{1}\mathbf{P}_{k|k}\mathbf{A}_{1}^{\mathrm{T}}+\mathbf{A}_{2}\bar{ \mathbf{R}}_{k}\mathbf{A}_{2}^{\mathrm{T}}+\Delta t^{2}\overline{Q}_{k} \tag{19}\]  
Based on Equation (11) and the principle of KF, the posteriori state estimate \(\hat{\mathbf{Z}}_{k+1|k+1}\) herein can be calculated as a linear combination of the priori state estimate \(\hat{\mathbf{Z}}_{k+1|k}\) and a weighted difference between the actual measurements and the corresponding predictions:  
\[\hat{\mathbf{Z}}_{k+1|k+1}\ =\ \hat{\mathbf{Z}}_{k+1|k}+\mathbf{G}_{k+1}\big{[} \mathbf{\Phi}\mathbf{y}_{k+1}-\mathbf{\Phi}\mathbf{h}\big{(}\hat{\mathbf{Z}}_{k +1|k}\big{)}\big{]} \tag{20}\]  
where \(\mathbf{G}_{k+1}\) is the gain matrix at time step \(t=(k+1)\times\Delta t\). Based on Equations (5), (15) and (20), the posteriori estimate error \(\varepsilon_{k+1|k+1}\) can be calculated as:  
\[\varepsilon_{k+1|k+1}\ =\ \mathbf{Z}_{k+1}-\hat{\mathbf{Z}}_{k+1|k+1}\ =\ \big{(} \mathbf{I}-\mathbf{G}_{k+1}\mathbf{\Phi}\mathbf{H}_{k+1|k}\big{)}\varepsilon_{k +1|k}-\mathbf{G}_{k+1}\mathbf{\Phi}\mathbf{v}_{k+1} \tag{21}\]where \(\mathbf{H}_{k+1|k}\) can be determined using Equation (16) when \(\mathbf{Z}_{k}~{}=~{}\hat{\mathbf{Z}}_{k+1|k}\). Accordingly, the posteriori estimate error covariance matrix can be obtained as:  
\[ \begin{split}\mathbf{P}_{k+1|k+1}&=E\Big{(}\epsilon_ {k+1|k+1}\epsilon_{k+1|k+1}^{\mathrm{T}}\Big{)}\\ &=\Big{(}\mathbf{I}-\mathbf{G}_{k+1}\mathbf{\Phi}\mathbf{H}_{k+1| k}\Big{)}\mathbf{P}_{k+1|k}\Big{(}\mathbf{I}-\mathbf{G}_{k+1}\mathbf{\Phi} \mathbf{H}_{k+1|k}\Big{)}^{\mathrm{T}}+\mathbf{G}_{k+1}\mathbf{\[PERSON]}\bar{ \mathbf{R}}_{k+1}\mathbf{\[PERSON]}^{\mathrm{T}}\mathbf{G}_{k+1}^{\mathrm{T}}\end{split}