# Model-Free Identification of Nonlinear Restoring Force with Modified Observation Equation
## 2 Formulas of the Proposed Approach  
The general equation of motion of a discrete \(n\)-DOFs nonlinear structure under external excitations can be expressed as:  
\[\mathbf{M}\ddot{\mathbf{x}}\Big{(}t\!+\!\mathbf{C}\dot{\mathbf{x}}(t)\!+\! \mathbf{K}\mathbf{x}(t)\!+\!\mathbf{\eta}^{u}\mathbf{R}^{non}(t)=\!\boldsymbol {\varphi}\mathbf{f}(t) \tag{1}\]  
in which \(\mathbf{M}\) is the mass matrix and assumed to be known; \(\mathbf{C}\) and \(\mathbf{K}\) are the damping matrix and stiffness matrix, respectively; \(\ddot{\mathbf{x}}(t)\), \(\dot{\mathbf{x}}(t)\) and \(\mathbf{x}(t)\) are acceleration, velocity and displacement responses, respectively; \(\mathbf{R}^{non}(t)\) is the vector of NRF; \(\mathbf{f}(t)\) is the external excitation and assumed to be available in this study; \(\mathbf{\eta}^{u}\) and \(\boldsymbol{\varphi}\) are the influence matrix associated with \(\mathbf{R}^{non}(t)\) and \(\mathbf{f}(t)\), respectively. Here, the state vector of the structure is defined as \(\mathbf{Z}(t)\ =\ \left[\mathbf{x}(t)^{\mathrm{T}}\,\ \dot{\mathbf{x}}(t)^{ \mathrm{T}},\mathbf{\theta}^{\mathrm{T}}\right]^{\mathrm{T}}\), where \(\mathbf{\theta}\) is the unknown time-invariant structural parameters (i.e., stiffness and damping coefficients). Then, the following state equation can be found:  
\[\dot{\mathbf{Z}}(t)\ =\ \left[ \begin{array}{c}\dot{\mathbf{x}}(t)\\ -\mathbf{M}^{-1}\Big{[}\mathbf{C}\dot{\mathbf{x}}(t)+\mathbf{K}\mathbf{x}(t) +\eta^{\mathrm{u}}\mathbf{R}^{non}(t)-\varphi\mathbf{f}(t)\ \Big{]}\\ 0\end{array} \right]=\mathbf{g}[\mathbf{Z}(t),\mathbf{R}^{non}(t).t]+\mathbf{w}(t) \tag{2}\]  
where \(\mathbf{w}(t)\) is model noise vector with zero mean and a covariance matrix \(\overline{Q}(t)\). It should be noted, besides the unknown structural parameters, the NRF \(\mathbf{R}^{non}_{k}\) is assumed to be unavailable as well. Since the NRF is treated as 'unknown fictitious inputs' herein, it can be found the prior knowledge or intuitive assumptions of any nonlinear model class of NRF is not required. Let \(\hat{\mathbf{Z}}_{ikk}\) and \(\hat{\mathbf{R}}^{non}_{k}\) be the estimations of \(\mathbf{Z}_{k}\) and \(\mathbf{R}^{non}_{k}\), respectively. Using Taylor expansion, Equation (2) can be linearized at time instant \(t=k\times\Delta t\) with the sampling interval of \(\Delta t\):  
\[\mathbf{g}\Big{(}\mathbf{Z}_{k},\mathbf{R}^{non}_{k},k\Delta t\Big{)}\ \approx\ \mathbf{g}\Big{(}\hat{\mathbf{Z}}_{ikk},\hat{\mathbf{R}}^{non}_{k},k\Delta t \Big{)}\ +\ \mathbf{U}_{kkk}\Big{(}\mathbf{Z}_{k}-\hat{\mathbf{Z}}_{ikk}\Big{)}+\mathbf{W} _{kkk}\Big{(}\mathbf{R}^{non}_{k}-\hat{\mathbf{R}}^{non}_{k}\Big{)} \tag{3}\]  
where the subscript \(k\) denotes the \(k\)-th time step:  
\[\mathbf{U}_{ikk}\ =\ \frac{\partial\mathbf{g}\big{(}\mathbf{Z}_{k},\mathbf{R}^{ non}_{k},k\Delta t\big{)}}{\partial\mathbf{Z}_{k}}\bigg{|}\ \mathbf{Z}_{k}\ =\ \mathbf{\hat{Z}}_{ikk}\ \ \ ;\ \mathbf{W}_{ikk}\ =\ \frac{\partial \mathbf{g}\big{(}\mathbf{Z}_{k},\mathbf{R}^  
where **I** is an identity matrix; \(\mathbf{D}\big{(}\mathbf{D}^{\mathrm{T}}\mathbf{D}\big{)}^{-1}\mathbf{D}^{\mathrm{T}}\) is known as the projection matrix. As a limit, the error shown in Equation (10) should tend to be zero, leading to:  
\[\mathbf{\Phi}\mathbf{y}_{\mathrm{k}}\ =\ \mathbf{\Phi}\mathbf{h}(\mathbf{Z}_{k})+ \mathbf{\Phi}\mathbf{v}_{\mathrm{k}} \tag{11}\]  
in which \(\mathbf{\Phi}\ =\ \mathbf{I}-\mathbf{D}\big{(}\mathbf{D}^{\mathrm{T}}\mathbf{D} \big{)}^{-1}\mathbf{D}^{\mathrm{T}}\). As compared with Equation (6), it can be found that multiple unknown issues have been transformed into a single unknown problem. Then, the EKF technique can be used to deal with such single regression problem.