# A Novel Real-Time Battery State Estimation Using Data-Driven Prognostics and Health Management
## 3 Battery SOC Model-Driven State-of-the-Art Method
### The Unscented Kalman Filter  
EKF can exhibit inaccurate behavior or even filter divergence, primarily due to linearization errors and the omission of higher-order terms in the Taylor series approximation. To overcome these limitations, the Sigma Point Kalman Filter (SPKF) was introduced. Unlike the EKF, SPKF does not require derivatives, as linearization is achieved through a set of sigma points [8]. SPKF provides a generalized approach to the Kalman filter for nonlinear system state estimation. Instead of relying on Taylor expansions for approximating covariance matrices, SPKF estimates these matrices through multiple function evaluations [20]. A widely used variant of SPKF is the Unscented Kalman Filter (UKF).  
Equations (6) and (7) are also applicable for a more generic case where neither the system or the noise can be assumed linear, nor the state a Gaussian distribution.  
\[x_{k} =f(x_{k-1})+w_{k} \tag{16}\] \[y_{k} =g(x_{k})+v_{k} \tag{17}\]  
The unscented transformation computes the possible values of a random variable as it undergoes a nonlinear transformation [9; 21].  
Consider propagating a random variable \(\mathbf{x}\), with dimension \(n\), through the nonlinear function given in (17). Let \(\mathbf{\hat{x}}\) and \(\mathbf{P_{x}}\) represent the mean and covariance of \(\mathbf{x}\), respectively. To compute the statistics of \(y_{k}\) in (17), a matrix \(\mathcal{X}\) comprising \(2n+1\)_sigma_ vectors (associated with corresponding weights \(W_{i}\)) is utilized, following the reasoning outlined below.  
Let \(\mathcal{X}_{k-1}\) represent a set of \(2n+1\) sigma points, where \(n\) denotes the dimension of the state space. The corresponding weights for these sigma points are defined as follows:  
\[\mathcal{X}_{k-1}=\{(\mathbf{x}_{k-1}^{j},\mathbf{W}^{j})|j=0\dots 2n\} \tag{18}\]  
The following sigma points are then selected to incorporate higher-order information into the chosen points:  
\[\mathcal{X}_{0}=\mathbf{\hat{x}}\] (19) \[-1<\ W^{0}\ <1\] (20) \[\mathbf{x}_{k-1}^{i} =\mathbf{\hat{x}}_{k-1}+\Psi_{i}\ \ \ i=1,\cdots,n\] (21) \[\mathbf{x}_{k-1}^{i+n} =\mathbf{\hat{x}}_{k-1}-\Psi_{i}\ \ \ i=1,\cdots,n\] (22) \[W^{j} =\frac{1-W_{0}}{2n}\ \ \ j=1,\cdots,2n\] (23) \[\sum_{j=0}^{2n}W^{j}=1\] (24) \[\Psi\(W^{0}\) determines the sigma points location: \(W^{0}\geqslant 0\) points move away from the origin, and \(W^{0}\leqslant 0\) points move closer to the origin.  
The nonlinear propagation of the sigma point is as follows:  
\[\hat{\mathbf{x}}_{k}^{i}=f(\mathbf{x}_{k-1}^{j}) \tag{26}\]  
The transformed sigma points are utilized to calculate the mean and covariance of the predicted value of \(\mathbf{x}_{k}\).  
\[\hat{\mathbf{x}}_{k}=\sum_{j=0}^{2n}W^{j}\hat{\mathbf{x}}_{k}^{j} \tag{27}\]  
\[\hat{\mathbf{P}}_{k}=\sum_{j=0}^{2n}W^{j}\{\hat{\mathbf{x}}_{k}^{j}-\bar{ \mathbf{x}}_{k}\}\{\hat{\mathbf{x}}_{k}^{j}-\bar{\mathbf{x}}_{k}\}^{T}+\mathbf{ Q}_{k-1} \tag{28}\]  
The sigma points are subsequently propagated through the nonlinear observation model:  
\[\hat{y}_{k-1}^{j}=g(x_{k-1}^{j}) \tag{29}\]