## 2 Equation of Motion and Its Solution Considering Structural Nonlinearity
 \tag{8}\]  
The new equation of motion in the mid-point between the time step of \(k\) and \(k+1\) for the structure subjected to the external force becomes  
\[\mathbf{M}\ddot{\mathbf{x}}_{k,\text{m}}+\mathbf{C}\dot{\mathbf{x}}_{k,\text{ m}}+\mathbf{R}_{k,\text{m}}(\mathbf{x})=\mathbf{LF}_{k,\text{m}} \tag{9}\]  
\[\mathbf{M}\ddot{\mathbf{x}}_{k,\text{m}}+\mathbf{C}\dot{\mathbf{x}}_{k,\text{ m}}+\mathbf{R}_{k,\text{m}}(\mathbf{x})=-\mathbf{M}\ddot{\mathbf{x}}_{g,k,\text{m}} \tag{10}\]  
Based on the assumption in Equation (5), the following equation can be obtained:  
\[\ddot{\mathbf{x}}_{\text{m}}=\frac{\ddot{\mathbf{x}}_{k+1}+\ddot{\mathbf{x}}_ {k}}{2}=\frac{1}{\Delta t}\big{(}\dot{\mathbf{x}}_{k+1}-\dot{\mathbf{x}}_{k} \big{)} \tag{11}\]  
Substitute Equation (11) into both sides of Equation (10) multiplied by the increment of the displacement \((\mathbf{x}_{k+1}-\mathbf{x}_{k})^{\text{T}}\), and a new equation will be obtained as follows. \[ \begin{array}{l}\frac{1}{2}\dot{\mathbf{x}}_{k+1}^{\text{T}}\mathbf{M}\dot{ \mathbf{x}}_{k+1}-\frac{1}{2}\dot{\mathbf{x}}_{k}^{\text{T}}\mathbf{M}\dot{ \mathbf{x}}_{k}+(\mathbf{x}_{k+1}-\mathbf{x}_{k})^{\text{T}}\mathbf{C}\big{(} \frac{\dot{\mathbf{x}}_{k+1}+\dot{\mathbf{x}}_{k}}{2}\big{)}^{\text{T}}\\ +(\mathbf{x}_{k+1}-\mathbf{x}_{k})^{\text{T}}\mathbf{R}_{\text{m}}(\mathbf{x}) =-(\mathbf{x}_{k+1}-\mathbf{x}_{k})^{\text{T}}\mathbf{M}\ddot{\mathbf{x}}_{g, \text{m}}\end{array}  \tag{12}\]  
Equation (12) makes the dynamic analysis relate to the energy of the structural system. Equation (12) illustrates the energy transfer and dissipation in the system. Based on Equation (12) it can be proved that with the assumptions shown in Equationswhere \(\mathbf{X}\) denotes the state vector including \(\mathbf{X}_{k}=\left[\mathbf{x}_{k}\quad\dot{\mathbf{x}}_{k}\right]\), \(\mathbf{u}\) is the input of the system, \(\mathbf{w}_{k-1}\) is the system process noise vector, and \(F\) is a nonlinear function of state vector \(\mathbf{X}\). It is noted that Equation (1) can also be expressed in state space as Equation (3). The discrete observed function can be written as  
\[\mathbf{y}_{k}=h(\mathbf{X}_{k},\mathbf{u}_{k},\mathbf{v}_{k}) \tag{14}\]  
where \(\mathbf{v}\) represents the observation noise assumed to follow a Gaussian white noise with zero mean and a covariance matrix \(\mathbf{R}\). The identification process can be initialized with the guess of \(\mathbf{X}_{0}=\mathrm{E}[\mathbf{X}]\) and \(\mathbf{P}_{0}=\mathrm{E}[(\mathbf{X}_{0}-\mathbf{\hat{X}}_{0})(\mathbf{X}_{ 0}-\mathbf{\hat{X}}_{0})^{\mathrm{T}}]\) based on unscented Kalman filter or other nonlinear estimators [22; 23; 24]. External excitations are always unknown for practical inverse problems. When the external load is unknown, the state equation of nonlinear hysteresis structural system cannot be solved with Equation (13). It is always difficult to directly identify the external force of nonlinear structure with UKF since it may be nonstationary and time variant. It is also impossible to build a relationship between \(\dot{\gamma}_{k}^{\prime}\) and \(\dot{\gamma}_{k-1}^{\prime}\) of external excitation but the excitation \(\mathbf{F}\) can be decomposed on a determined standard orthogonal basis. Since the orthogonal basis is known, the history of excitation will be re-constructed as if the coefficients of orthogonal basis can be identified. In this way, the external excitation identification is transformed to the identification of the coefficients of orthogonal decomposition. The excitation in Equation (1) can be decomposed as follows  
\[\mathbf{F}(t)\approx\sum_{m=1}^{N_{m}}w_{m}\mathbf{T}_{m}(t) \tag{15}\]  
in which \(w_{m}\) is the weighing coefficient of polynomial \(\mathbf{T}_{m}\). High order components in polynomials are ignored for the purpose of computational efficiency. Low frequency components in excitation would be a major contribution to the structural response for civil engineering structures. \(\mathbf{T}_{m}\) is the \(m\)th term orthogonal polynomial.