## 3 Adaptive Ensemble Square Root Kalman Filter  
### Ensemble Kalman Filter  
Regarding the nonlinear systems and in order to track their state trajectories, the extended Kalman filter and the nonlinear filtering method which combines the techniques of unscented transformation and numerical difference without computing Jacobian matrix are widely used. However, when the system's nonlinearity is strong and the noise is non-Gaussian, the performance of these algorithms will be degraded significantly, resulting in unreliable state estimation. Therefore, ensemble Kalman filter, a sub-optimal filtering method, has attracted more and more attention. Based on Kalman filtering, ensemble prediction is introduced into ensemble Kalman filtering. Sequential Monte Carlo simulation method is employed to create the initial set of samples representing the state statistics. The state transition function is used to calculate the new set of states at the current time for each sample in the set, then the samples of the new set of states are calculated. Mean and covariance are used to get the optimal value of the current time estimate [18]. In the traditional Kalman filter, the calculation formulas of state prediction error covariance and analysis error covariance are presented as  
\[ \begin{array}{l}P^{f}=\overline{\left(X^{f}-X^{t}\right)\left(X^{f}-X^{t} \right)^{T}}\\ P^{a}=\overline{\left(X^{a}-X^{t}\right)\left(X^{a}-X^{t}\right)^{T}}\end{array}  \tag{12}\]where \(X\) denotes the state vector, \(P\) represents the state covariance, \(T\) is the matrix transposition. The superscript \(f\) indicates the prediction state, the superscript \(a\) is the analysis state, and the superscript \(t\) is the real state. In the EnKF, the posterior distribution of the state function is approximated by the ensemble elements, and the degree of approximation depends on the number of elements in the ensemble. Assuming that the number of set elements is \(N\), \(X\!E^{f}\) represents the set storing all predicted states, and \(X\!E^{a}\) represents the set storing all analyzed states. When \(N\) tends to infinity, the real value of the state can be replaced by the mean value of the set of state estimators. It follows from [19] that the prediction error covariance and the analysis error covariance can be calculated by  
\[ \begin{array}{l}P^{f}\approx\frac{1}{N-1}\left(X\!E^{f}-\overline{X\!E^{f}} \right)\!\left(X\!E^{f}-\overline{X\!E^{f}}\right)^{T}\\ P^{a}\approx\frac{1}{N-1}\left(X\!E^{a}-\overline{X\!E^{a}}\right)\!\left(X\! E^{a}-\overline{X\!E^{a}}\right)^{T}\end{array}  \tag{13}\]  
The EnKF uses the finite element in the ensemble for the estimation of the state error covariance, thus avoiding the prediction of the covariance matrix. In the process of EnKF, firstly, according to the known state priori information, an initial state set of \(N\) elements is obtained by SMC method, then a prediction set \(X\!E^{f}\) is obtained by predicting each element in the initial state set through Kalman filter, in which the elements are expressed by \(X\!^{f_{i}}\). Secondly, the observation set of state by SMC method as well, and the prediction state set is modified by covariance of \(X\!E^{f}\) to get the analysis set \(X\!E^{a}\) of state. The elements are expressed by \(X^{a_{i}}\), and the average value of the state analysis set \(X\!E^{a}\) is the optimal state estimation value. The analysis set of states is used for prediction [20; 21]. The iteration formula of ensemble Kalman filter algorithm is given as follows  
(1) State prediction:  
\[X\!^{f_{i}}_{k+1}=f(X\!^{a_{i}}_{k},U_{k})+W\!^{i}_{k} \tag{14}\]  
\[\overline{X}\!^{f}_{k+1}=\frac{1}{N}\!\sum_{i=1}^{N}X\!^{f_{i}}_{k+1} \tag{15}\]  
\[E\!^{f}_{X\! respectively, \(P^{f}_{XZ,k+1}\) is the covariance of predictive state error and observation state error at \(k+1\) time, \(P^{f}_{ZZ,k+1}\) is the covariance of observation state error at \(k+1\) time.