# An Interactive Transient Model Correction Active Sonar Target Tracking Method
## 2 Description of the TMC Tracking Method
### System Model
#### 2.1.1 State Equation  
When a slowly varying target like a submarine is moving underwater, most of the time, the speed changes are weak. Therefore, it can be assumed that most of the time, it is in a uniform motion state, and the acceleration can be regarded as a random disturbance with Gaussian white noise characteristics, usually with a zero mean and a variance of \(q\). Therefore, the continuous-time state equation of the target can be established as:  
\[\dot{\mathbf{X}}(t)=\mathbf{A}\mathbf{X}(t)+\mathbf{F}w(t). \tag{3}\]  
In the formula, \(\mathbf{X}(t)\) represents the system state vector, and \(\mathbf{X}(t)=\begin{bmatrix}x(t)&\dot{x}(t)&y(t)&\dot{y}(t)\end{bmatrix}^{T}\). \(x(t)\) and \(y(t)\) represent the target's displacement on the X- and Y-axis, respectively, while \(\dot{x}(t)\) and \(\dot{y}(t)\) represent the target's velocity on the X- and Y-axis, respectively. \(\dot{\mathbf{X}}(t)=\begin{bmatrix}\dot{x}(t)&\ddot{x}(t)&\ddot{y}(t)\end{bmatrix} ^{T},\ddot{x}(t)\) and \(\ddot{y}(t)\) represent the accelerations expressed in terms of random noise \(w(t)\), with mean and variance given by:  
\[\mathbf{E}[w(t)]=0, \tag{4}\]  
\[\mathbf{E}\Big{[}w(t)w^{T}(\tau)\Big{]}=q\delta(t-\tau) \tag{5}\]  
\(\mathbf{A}\) is the state transition matrix.  
\[\mathbf{A}=\begin{bmatrix}0&1&0&0\\ 0&0&0&0\\ 0&0&0&1\\ 0&0&0&0\end{bmatrix} \tag{6}\]  
\(\mathbf{F}\) is the process noise input matrix: \(\mathbf{F}=\begin{bmatrix}0&1&0&1\end{bmatrix}^{T}\).  
The discrete-time state equation for a constant system is:  
\[\mathbf{X}(k+1)=\mathbf{\Phi}\mathbf{X}(k)+\mathbf{\Gamma}w(k) \tag{7}\]  
In the formula, \(\mathbf{X}(k)\) represents the discrete state vector, and \(\mathbf{X}(k)=\begin{bmatrix}r_{x}&v_{x}&r_{y}&v_{y}\end{bmatrix}^{T}\), \(r_{x}\) and \(r_{y}\) represent the target's displacement on the X- and Y-axis, respectively, while \(v_{x}\) and \(v_{y}\) represent the target's velocity on the X- and Y-axis, respectively. The state transition matrix \(\mathbf{\Phi}\) is defined as:  
\[\mathbf{\Phi}=e^{\mathbf{A}\mathbf{T}}=\begin{bmatrix}1&T&0&0\\ 0&1&0&0\\ 0&0&1&\text{T}\\ 0&0&0&1\end{bmatrix} \tag{8}\]  
Figure 2: Schematic diagram of target movement situation.  
The process noise input matrix \(\mathbf{\Gamma}\) is:  
\[\mathbf{\Gamma}=\int_{0}^{\mathrm{T}}e^{\mathbf{A}(\mathrm{T}-\tau)}\mathbf{F}d \tau=\int_{0}^{\mathrm{T}}\begin{bmatrix}1&\mathrm{T}-\tau&0&0\\ 0&1&0&0\\ 0&0&1&\mathrm{T}-\tau\\ 0&0&0&1\end{bmatrix}\begin{bmatrix}0\\ 1\\ 0\\ 1\end{bmatrix}d\tau=\int_{0}^{\mathrm{T}}\begin{bmatrix}T-\tau\\ 1\\ T-\tau\\ 1\end{bmatrix}d\tau=\begin{bmatrix}\mathrm{T}^{2}/2\\ T\\ \mathrm{T}^{2}/2\\ T\end{bmatrix} \tag{9}\]  
The process noise matrix \(\mathbf{W}(k)\) is defined as:  
\[\mathbf{W}(k)=\int_{0}^{\mathrm{T}}e^{\mathbf{A}(\mathrm{T}-\tau)}\mathbf{F}w (k\mathrm{T}+\tau)d\tau=\int_{0}^{\mathrm{T}}\begin{bmatrix}T-\tau\\ 1\\ T-\tau\\ 1\end{bmatrix}w(k\mathrm{T}+\tau)d\tau \tag{10}\]  
The covariance matrix \(\mathbf{Q}(k)\) of the process noise \(w(k)\) is defined as:  
\[\mathbf{Q}(k)=\mathbf{E}\Big{[}\mathbf{W}(k)\mathbf{W}^{\mathrm{T}}(k)\Big{]} =\int_{0}^{\mathrm{T}}\begin{bmatrix}T-\tau\\ 1\\ T-\tau\\ 1\end{bmatrix}[\mathrm{T}-\tau\ 1\ \mathrm{T}-\tau\ 1]qd\tau \tag{11}\] \[=\begin{bmatrix}\mathrm{T}^{3}/3\mathrm{T}^{2}/2\ \mathrm{T}^{3}/3 \mathrm{T}^{2}/2\\ \mathrm{T}^{2}/2\ \mathrm{T}\ \mathrm{T}^{2}/2\ \mathrm{T}\\ \mathrm{T}^{3}/3\mathrm{T}^{2}/2\ \mathrm{T}^{3}/3\ \mathrm{T}^{2}/2\\ \mathrm{T}^{2}/2\ \mathrm{T}\ \mathrm{T}^{2}/2\ \mathrm{T}\end{bmatrix}q\]