## 3 Proposed Adaptive KF-Based Tracking
### Analysis of the Adaptive Factor  
The innovation sequence can be used as a reliable indicator of the estimation performance, which is defined as the discrepancy between the actual measurement and the predicted result, as given below:  
\[\mathbf{d}_{k}=z_{k}-\mathbf{H}\hat{\mathbf{x}}_{k}^{-} \tag{10}\]  
Then, the innovation covariance is  
\[\mathbf{C}_{k}=\mathrm{E}[\mathbf{d}_{k}\mathbf{d}_{k}^{\mathsf{T}}] \tag{11}\]  
where \(\mathbf{C}_{k}\) is the actual innovation covariance. In practice, it is difficult to achieve the actual innovation covariance \(\mathbf{C}_{k}=\mathrm{E}[\mathbf{d}_{k}\mathbf{d}_{k}^{\mathsf{T}}]\), which can be estimated by the present and past sequences. The window function method is commonly applied for innovation covariance estimation, that is  
\[\mathbf{\hat{C}}_{k}=\frac{1}{N_{\mathrm{i}=k-N+1}}\sum_{i=k-N+1}^{k}\mathbf{ d}_{i}\mathbf{d}_{i}^{\mathsf{T}} \tag{12}\]  
where \(N\) represents the window length and \(\mathbf{\hat{C}}_{k}\) represents the estimated innovation covariance matrix. When there are no modeling errors, the innovation sequences follow the orthogonal principle [26], that is  
\[\mathrm{E}[\mathbf{d}_{k+j}\mathbf{d}_{k}^{\mathsf{T}}]=0,\;k=1,2\cdots,\;j= 1,2\cdots \tag{13}\]  
However, when modeling errors exist in the filtering system, the orthogonal principle cannot follow well. Therefore, the proposed adaptive tuning method should make the system obey the orthogonal principle. The orthogonal principle can make useful information of the innovation sequences completely extracted. Because the fixed \(\mathbf{Q}_{k}\) matrix of KF may be inconsistent with the variable practical situation, the adaptive factor is used to adjust the matrix \(\mathbf{Q}_{k}\). Accordingly, the predicted state error covariance matrix \(\mathbf{\hat{P}}_{k}^{-}\) is given as  
\[\mathbf{\hat{P}}_{k}^{-}=\mathbf{\Phi}\mathbf{\hat{P}}_{k-1}\mathbf{\Phi}^{ \mathsf{T}}+\lambda_{k}\mathbf{Q}_{k-1} \tag{14}\]  
where \(\lambda_{k}\) is the adaptive factor, which should be computed according to (13) so that the innovation sequences can follow the orthogonal principle. If the KF state estimations are quite accurate, the state estimate error \(\tilde{\textbf{x}}_{k}=\textbf{x}_{k}-\hat{\textbf{x}}_{k}\) could follow \(\mathsf{O}\bigg{[}\Big{|}\tilde{\textbf{x}}_{k}\Big{|}^{2}\Big{]}\ll\mathsf{O} \Big{[}\Big{|}\tilde{\textbf{x}}_{k}\Big{|}\Big{]}\). Then the following can be obtained [27]:  
\[ \begin{array}{c}\mathrm{E}[\textbf{d}_{k+j}\textbf{d}_{k}^{\mathrm{T}}] \approx\mathbf{H}\mathbf{\Phi}(\mathbf{I}-\mathbf{K}_{k+j-1}\mathbf{H}_{k+j-1 })\ \bullet\ \cdots\ \bullet\ \mathbf{\Phi}(\mathbf{I}-\mathbf{K}_{k+2}\mathbf{H}_{k+2})\ \bullet\\ \mathbf{\Phi}(\mathbf{I}-\mathbf{K}_{k+1}\mathbf{H}_{k+1})\mathbf{\Phi}( \tilde{\mathbf{p}}_{k}^{-}\mathbf{H}^{\mathrm{T}}-\mathbf{K}_{k}\mathbf{C}_{ k})\ \ j=1,\,2\cdots\end{array}  \tag{15}\]  
where \(\mathbf{I}\) denotes the identity matrix. Therefore, if the time-varying gain matrix \(\mathbf{K}_{k}\) is chosen appropriately for all \(j\) = 1, 2,... to make  
\[\hat{\mathbf{p}}_{k}^{-}\mathbf{H}^{\mathrm{T}}-\mathbf{K}_{k}\mathbf{C}_{k}=0 \tag{16}\]  
then the orthogonal principle \(\mathrm{E}[\textbf{d}_{k+j}\textbf{d}_{k}^{\mathrm{T}}]=0\) can be satisfied.