## 2 Background: Identification of Dynamic Systems by Unscented Kalman Filter  
### A Brief Review of Unscented Kalman Filter  
To overcome the aforementioned challenges in using EKF for estimation of nonlinear state-space models, [PERSON] and [PERSON] (2004) developed Unscented Kalman Filter (UKF) ([PERSON], 1982), also known as Sigma-Point Kalman Filter (SPKF). The unscented transformation of UKF is based on a statistical technique using sigma points (SPs), which provides derivative-free state estimation for a nonlinear state-space model. UKF is known to achieve the second-order accuracy with the same computational complexity as that of EKF. For details of UKF, let us consider a dynamic state-space model governed by a set of discrete-time nonlinear equations ([PERSON], 1982)  
\[ \begin{array}{c}\mathbf{x}_{k+1}=f(\mathbf{x}_{k})+\mathbf{w}_{k}\\ \mathbf{z}_{k+1}=h(\mathbf{x}_{k+1},\ \mathbf{u}_{k+1})+\mathbf{v}_{k+1}\end{array}  \tag{1}\]  
where \(k\) is the time index; \(\mathbf{x}_{k}\in\mathbb{R}^{n_{\text{x}}\times 1}\) is the state vector representing displacement, velocity, or system parameters such as elasticity and damping coefficient at time \(t_{k}\); \(f(\cdot)\) and \(h(\cdot)\) respectively denote nonlinear system-process model and measurement-output model; \(\mathbf{w}_{k}\in\mathbb{R}^{n_{\text{x}}\times 1}\) represents the artificial process noise modeled by a Gaussian random vector with zero means and covariance matrix \(\mathbf{Q}_{k}\in\mathbb{R}^{n_{\text{x}}\times n_{\text{x}}}\), i.e., \(\mathbf{w}_{k}\sim N(\mathbf{0},\mathbf{Q}_{k})\); \(z_{k}\in\mathbb{R}^{n_{\text{x}}\times 1}\) is the vector of the measured quantities such as acceleration; \(\mathbf{u}_{k}\) is the vector of inputs; and \(\mathbf{v}_{k}\in\mathbb{R}^{n_{\text{x}}\times 1}\) denotes artificial measurement noise modelled as \(\mathbf{v}_{k}\sim N(\mathbf{0},\mathbf{R}_{k})\). The noises \(\mathbf{w}_{k}\) and \(\mathbf{v}_{k}\) are assumed to be uncorrelated with each other. Figure 1 illustrates the general algorithm of UKF ([PERSON], 1982). In the figure, \(\mathbf{\hat{x}}_{k\text{jk}}\) and \(\mathbf{P}_{k\text{jk}}^{\text{xx}}\) respectively denote the mean (column) vector and covariance matrix of the states at the \(k\)-th time step (\(k=0,1,\ldots\)) before the updates are performed. At each time step, sigma points (SPs) \(\chi_{k\text{jk}}^{[i]}\), \(i=0,\ldots,2n_{\text{x}}\), are first generated based on \(\mathbf{\hat{x}}_{k\text{jk}}\) and \(\mathbf{P}_{k\text{jk}}^{\text{xx}}\) as  
\[ \begin{array}{c}\mathbf{\chi}_{k\text{jk}}^{[i]}=\mathbf{\hat{x}}_{k\text{ jk}}\\ \mathbf{\chi}_{k\text{jk}}^{[i]}=\mathbf{\hat{x}}_{k\text{jk}}+\left(\sqrt{(n _{\text{X}}+\kappa)\mathbf{P}_{k\text{jk}}^{\text{xx}}}\right)^{i},\ i=1, \ldots,n_{\text{X}}\\ \mathbf{\chi}_{k\text{jk}}^{[i]}=\mathbf{\hat{x}}_{k\text{jk}}-\left(\sqrt{(n _{\text{X}}+\kappa)\mathbf{P}_{k\text{jk}}^{\text{xx}}}\right)^{-i\text{nx}}, i=n_{\text{X}}+1,\ldots,2n_{\text{X}}\end{array}  \tag{2}\]  
where the square-root symbol denotes the \"matrix square root\", which can be computed by Cholesky factorization ([PERSON], 1982), and \((\ )^{i}\) indicates the \(i\)-th column vector of the matrix. The tuning parameter \(\kappa\) is often used to consider high order moments in the approximation and assumed to be zero in this paper. Next, using the system-process model \(f(\cdot)\), \(\mathbf{\hat{x}}_{k\text{jk}}\) and \(\mathbf{P}_{k\text{jk}}^{\text{xx}}\) are updated to \(\mathbf{\hat{x}}_{k+1\text{jk}}\) and \(\mathbf{P}_{k+1\text{jk}}^{\text{xx}}\), respectively, via the sigma points in Equation (2).