# An Interactive Transient Model Correction Active Sonar Target Tracking Method
## 2 Description of the TMC Tracking Method
### Transient Correction  
When the system state equation deviates from the actual situation, the system uncertainty increases, the estimation error of the KF filter becomes larger, and the reliability of the one-step prediction estimation \(\hat{\mathbf{X}}(k|k-1)\) decreases. At this time, KF adjusts the estimation by changing the filtering gain \(\mathbf{K}(k)\) to ensure that the filtered estimation \(\hat{\mathbf{X}}(k|k)\) is as close as possible to the actual state \(\mathbf{X}(k)\). When the motion state of the underwater target changes, it causes an increase in the system process noise described in Section 3.1.1. As a result, the covariance matrix \(\mathbf{P}(k|k-1)\) of the one-step advance prediction error becomes larger, and subsequently, the filtering gain \(\mathbf{K}(k)\) also increases. Considering an extreme case where \(\mathbf{P}(k|k-1)\) approaches infinity, we take the limit of Equation (22) as follows:  
\[\lim_{\mathbf{P}^{-}\rightarrow\infty}\mathbf{K}(k)=\lim_{\mathbf{P}^{-} \rightarrow\infty}\frac{\mathbf{P}(k|k-1)\mathbf{H}^{T}}{\mathbf{H}\mathbf{P} (k|k-1)\mathbf{H}^{T}+\mathbf{R}(k)}=\mathbf{H}^{-1},\ \big{\{}\mathbf{P}^{-}\ \text{Rep} \mathbf{P}(k|k-1)\big{\}}. \tag{28}\]  
By combining Formulas (25), (26) and (28), we can rewrite Formula (20) as \(\hat{\mathbf{X}}(k|k)\ =\ \mathbf{Z}(k)\mathbf{H}^{-1}\). This indicates that when the system error reaches its limit, the KF system adjusts the reliability of the one-step prediction estimate to the lowest level, while the weight of the measurement variables reaches its maximum. Therefore, the residual covariance \(\mathbf{S}(k)\) of the variable gain KF filter can be used as a signal indicator, and the residuals of the constant gain filter can be used as a reference to correct the KF filter gain matrix and reset the state model. This process can be iterated. The comparison judgment threshold of the residual covariance \(\mathbf{S}(k)\) in Figure 1 can be obtained through error analysis of the target measurement history data of the sonar system. Normally, after the design of the sonar is completed, its measurement error is known. Therefore, the comparison threshold for \(\mathbf{S}(k)\) can be determined beforehand. The constant gain filter selects the \(\alpha\)he filtering method. The \(\alpha\)il filter is a constant residual filter that has good convergence properties and can rapidly track maneuvering targets over a wide range. The \(\alpha\)-\(\beta\) filtering equation is as follows:  
\[ \begin{cases}\hat{\mathbf{X}}_{\alpha\beta}(k)=(1-\alpha)\big{[}\hat{\mathbf{ X}}_{\alpha\beta}(k-1)+\mathbb{T}\hat{\mathbf{\sigma}}_{\alpha\beta}(k-1) \big{]}+\alpha Z(k),\\ \hat{\mathbf{\sigma}}_{\alpha\beta}(k)=(1-\beta)\hat{\mathbf{\sigma}}_{ \alpha\beta}(k-1)+\beta\big{[}Z(k)-\hat{\mathbf{X}}_{\alpha\beta}(k-1)\big{]} /T.\end{cases}  \tag{29}\]  
In the equation, \(\hat{\mathbf{X}}_{\alpha\beta}(k)\) and \(\hat{\mathbf{X}}_{\alpha\beta}(k-1)\) represent the current and previous moment's target position filtered estimation values, respectively. \(\alpha\) and \(\beta\) are the filtering coefficients, while \(\hat{\mathbf{\sigma}}_{\alpha\beta}(k)\) and \(\hat{\mathbf{\sigma}}_{\alpha\beta}(k-1)\) represent the current and previous moment's target velocity filtered estimation values, respectively. \(\mathbf{Z}(k)\) is the current moment's target position measurement value, and T is the measurement update period. The \(\alpha\)s filter gain is a constant \(\mathbf{K_{\alpha\beta}}=\left[\alpha\ \beta/\mathrm{T}\right]^{T}\), and the residual \(\mathbf{d_{\alpha\beta}}(k)\) is defined as:  
\[\mathbf{d_{\alpha\beta}}(k)=\mathbf{Z}(k)-\left[\hat{\mathbf{X}}_{\alpha\beta} (k-1)+\mathrm{T}\hat{\eta}_{\alpha\beta}(k-1)\right] \tag{30}\]  
When the residual covariance \(\mathbf{S}(k)\) of the KF exceeds the comparison judgment threshold, we select the \(N\) residual values before time \(k\) to perform mean processing on \(\mathbf{d}(k-1)\ \sim\mathbf{d}(k-N)\).