# Multi-UAVs Communication-Aware Cooperative Target Tracking
## 3 Information Fusion Based on Extended Information Filtering
### Extend Information Filter and Information Fusion Phase  
[PERSON]'s information matrix is a way of describing target state estimation and is widely used in the optimal estimation theory [3; 11]. It can be very convenient to do information fusion using fisher information matrix (FIM). For \(i\)-th UAV, the Fisher information matrix is defined as follows:  
\[Y_{i}[k]=P_{i}^{-1}[k] \tag{23}\]  
\[y_{i}[k]=Y_{i}[k]\hat{X}_{i}[k] \tag{24}\]  
where \(Y_{i}[k]\) represents the information matrix at time \(k\), \(y_{i}[k]\) the information state matrix, and \(P_{i}\) the matrix \(P_{k}\) in Equation (22). For the observed value \(z[k]\) at the discrete time \(k\), its gain for the information matrix and the information state matrix are:  
\[ \begin{array}{l}I_{i}[k]=H^{T}[k]R^{-1}[k]H[k]\\ i_{i}[k]=H^{T}[k]R^{-1}[k]z[k]\end{array} . \tag{25}\]  
Combined with Equations (23)-(25) and Equations (18)-(22), we get:  
\[ \begin{array}{l}Y_{i}^{s-}[k]=\left(P_{i}^{-}\right)^{-1}\\ \hat{y}_{i}^{s-}[k]=Y_{i}^{-}[k]\hat{X}_{i}[k-1]\end{array}  \tag{26}\]  
\[ \begin{array}{l}\Gamma_{i}[k]=H^{T}[k]R^{-1}[k]H[k]\\ \vec{i}^{s}i_{k}[k]=H^{T}[k]R^{-1}[k]z[k]\end{array}  \tag{27}\]  
\[ \begin{array}{l}Y_{i}^{s}[k]=Y_{i}^{-}[k]+I_{i}[k]\\ \vec{y}^{s}i_{k}[k]=\hat{y}_{i}^{-}[k]+i_{i}[k]\end{array} . \tag{28}\]The superscript \"\(s\)\" represents the information matrix before performed information fusion. It can be seen that the Fisher information matrix can easily superimpose the observation results into the information matrix through the information gain matrix, and this greatly reduces the computational complexity in the process of multi-UAVs information fusion. The posteriori state estimation can be obtained from Equation (29):  
\[\hat{x}_{i}[k]=Y^{-1}[k]\hat{y}[k]. \tag{29}\]  
Assuming that the observation of each UAV airborne sensors is not relevant, we can get the information fusion process shown in Figure 2.  
In the process of information fusion with communication noise, communication noise changes the gain matrix of information transmitted by other UAVs. For the \(i\)-th UAV, the information matrix \(Y_{i}\) and the information state matrix of the target state \(\hat{y}_{i}\) can be rewritten as:  
\[Y_{i}[k]=Y_{i}^{s}[k]+\sum_{j\
eq i}I_{ji}^{c}[k] \tag{30}\]  
\[\hat{y}_{i}[k]=\hat{y}_{i}^{s}[k]+\sum_{j\
eq i}\hat{t}_{ji}^{c}[k] \tag{31}\]  
where \(Y_{i}^{s}[k],\hat{y}_{i}[k]\) represent the information matrix and information state matrix from the \(i\)-th UAV, which can be calculated using Equation (28). \(I_{ji}^{c}[k]\), \(\hat{t}_{ji}^{s}[k]\) represent information gain matrix and information state gain matrix from the \(j\)-th UAVs to the \(i\)-th UAV. In the presence of communication noise, according to Equation (27), \(I_{ji}^{c}[k]\) and \(\hat{t}_{ji}^{c}[k]\) can be calculated using Equations (32) and (33). \[I_{ji,i}^{c}[k]=H_{j}^{T}[k](R_{j}+\Sigma_{ji}[k])^{-1}H_{j}[k] \tag{32}\]  
\[\hat{t}_{ji,i}^{c}[k]=H_{j}^{T}[k](R_{j}+\Sigma_{ji}[k])^{-1}z_{j}[k] \tag{33}\]  
Figure 2: Information fusion structure. where \(\Sigma_{ji}[k]\) represents communication noise covariance matrix from the \(j\)-th UAVs to the \(i\)-th UAV and can be calculated using Equations (11)-(14). From Equations (30)-(33), we get  
\[Y_{i}[k]=Y_{i}^{\rm s}[k]+\sum\limits_{j\
eq i}{H_{j}}^{T}[k](R_{j}+\Sigma_{ji} [k])^{-1}H_{j}[k] \tag{34}\]  
\[\hat{y}_{i}[k]=\hat{y}_{i}^{\rm s}[k]+\sum\limits_{j\
eq i}{H_{j}}^{T}[k](R_{j}+ \Sigma_{ji}[k])^{-1}z_{j}[k]. \tag{35}\]  
With Equations (30) and (32) we get:  
\[ \begin{array}{l}Y_{i}[k]=Y_{i}^{\rm s}[k]+\sum\limits_{j\
eq i}{H_{j}}^{T}[k ](R_{j}+\Sigma_{ji}[k])^{-1}H_{j}[k]\\ \qquad\qquad=\ Y_{i}^{\rm s-}[k]+I_{i}^{\rm s}+\sum\limits_{j\
eq i}\left\{I_{ j}^{\rm s}+H_{j}{}^{T}[k](\Sigma_{ji}[k])^{-1}H_{j}[k]\right\}\\ \qquad\qquad=\underbrace{Y_{i}^{\rm s-}[k]}_{local-estimate}+\underbrace{ \sum\limits_{i}I_{j}^{\rm s}}_{observation}+\underbrace{\sum\limits_{j\
eq i} \Gamma_{ji}^{\rm c}}_{communication-noise}\end{array}