## 3 Proposed Adaptive KF-Based Tracking
### Analysis of the Adaptive Factor
Substituting (7) into (16), we have  
\[\hat{\mathbf{P}}_{k}^{-}\mathbf{H}^{\mathrm{T}}\bigg{\{}\mathbf{I}-\Big{(} \mathbf{H}\hat{\mathbf{P}}_{k}^{-}\mathbf{H}^{\mathrm{T}}+\mathbf{R}_{k} \Big{)}^{-1}\mathbf{C}_{k}\bigg{\}}=0 \tag{17}\]  
The sufficient condition for Equation (17) is as follows:  
\[\Big{(}\mathbf{H}\hat{\mathbf{P}}_{k}^{-}\mathbf{H}^{\mathrm{T}}+\mathbf{R}_{ k}\Big{)}^{-1}\bullet\mathbf{C}_{k}=\mathbf{I} \tag{18}\]  
Therefore,  
\[\mathbf{H}\hat{\mathbf{P}}_{k}^{-}\mathbf{H}^{\mathrm{T}}+\mathbf{R}_{k}= \mathbf{C}_{k} \tag{19}\]  
Substituting (14) into (19), it can be obtained that  
\[\mathbf{C}_{k}=\mathbf{H}\mathbf{\Phi}\hat{\mathbf{P}}_{k-1}\mathbf{\Phi}^{ \mathrm{T}}\mathbf{H}^{\mathrm{T}}+\mathbf{R}_{k}+\lambda_{k}\mathbf{H} \mathbf{Q}_{k-1}\mathbf{H}^{\mathrm{T}} \tag{20}\]  
Define  
\[\mathbf{A}_{k}\triangleq\mathbf{H}\mathbf{\Phi}\hat{\mathbf{P}}_{k-1}\mathbf{ \Phi}^{\mathrm{T}}\mathbf{H}^{\mathrm{T}}+\mathbf{R}_{k} \tag{21}\]  
\[\mathbf{B}_{k}\triangleq\mathbf{H}\mathbf{Q}_{k-1}\mathbf{H}^{\mathrm{T}} \tag{22}\]  
Then (20) can be rewritten as  
\[\mathbf{C}_{k}-\mathbf{A}_{k}=\lambda_{k}\mathbf{B}_{k} \tag{23}\]  
The trace of (23) is as follows:  
\[\text{tr}(\mathbf{C}_{k}-\mathbf{A}_{k})=\text{tr}(\lambda_{k}\mathbf{B}_{k}) \tag{24}\]  
where tr() represents the trace of the matrix. Then, \(\lambda_{k}\) is obtained:  
\[\lambda_{k}=\frac{\text{tr}(\mathbf{C}_{k}-\mathbf{A}_{k})}{\text{tr}( \mathbf{B}_{k})} \tag{25}\]  
The actual innovation covariance \(\mathbf{C}_{k}\) is difficult to obtain, which is typically estimated by \(\hat{\mathbf{C}}_{k}\) in (12). To decrease the computational burden, the recursive algorithm is applied in (12) to estimate the innovation covariance [28]:  
\[\hat{\mathbf{P}}_{k}^{-}\mathbf{H}^{\mathrm{T}}\bigg{\{}\mathbf{I}-\Big{(} \mathbf{H}\hat{\mathbf{P}}_{k}^{-}\mathbf{H}^{\mathrm{T}}+\mathbf{R}_{k} \Big{)}^{-1}\mathbf{C}_{k}\bigg{\}}=0 \tag{17}\]0  
where \(N\) is the window length of the sliding window, which should be correctly determined. Because if the window length is too small, it is difficult to obtain an unbiased estimate. On the contrary, if the window size is too large, the estimations cannot reflect the short-term statistics of innovation covariances. [PERSON] et al. [29] stated that a good window size is between 10 and 30. Then, the adaptive factor \(\lambda_{k}\) can be obtained as follows:  
\[\hat{\mathbf{P}}_{k}^{-}\mathbf{H}^{\mathrm{T}}\bigg{\{}\mathbf{I}-\Big{(} \mathbf{H}\hat{\mathbf{P}}_{k}^{-}\mathbf{H}^{\mathrm{T}}+\mathbf{R}_{k} \Big{)}^{-1}\mathbf{C}_{k}\bigg{\}}=0 \tag{17}\]1  
In dynamic and weak signal conditions, tracking systems often involve high levels of uncertainty in the system model. The larger adaptive factor can reduce its reliance on the system model and place more trust in the measurements, which can effectively remove the impact of model uncertainty. Accordingly, to keep the state estimation accuracy, the adaptive factor should not be smaller than 1. If the adaptive factor is larger than 1, indicating that the tracking system changes suddenly, the value of \(\mathbf{\hat{P}}_{k}^{-}\) will increase based on Equation (14). Meanwhile, the Kalman gain \(\mathbf{K}_{k}\) will increase according to Equation (7). Accordingly, based on Equation (8), more weight will be put on the measurement data, and then the accuracy of the state estimation will be improved [30]. A larger adaptive factor can respond more quickly to changes in the measurements. This can accelerate the convergence rate of the filter and provide more timely estimation results, adapting to the variations of the signal. Hence, the robustness and accuracy of the system can be improved. Then, the adaptive factor \(\lambda_{k}\) should be as follows:  
\[\hat{\mathbf{P}}_{k}^{-}\mathbf{H}^{\mathrm{T}}\bigg{\{}\mathbf{I}-\Big{(} \mathbf{H}\hat{\mathbf{P}}_{k}^{-}\mathbf{H}^{\mathrm{T}}+\mathbf{R}_{k} \Big{)}^{-1}\mathbf{C}_{k}\bigg{\}}=0 \tag{17}\]2