## 3 Target State Estimation and Observer Trajectory Quality Metric  
### Target State Estimation  
Considering the advantage of using information filter in multiple sensor estimation and fusion problems, the estimation of target state is performed by using the EIF. The EIF is the information form of EKF. The EKF generates the target state estimation, \(\hat{x}_{t}(k\,|\,k)\), at time \(k\), together with a corresponding estimate covariance \(\mathbf{P}(k\,|\,k)\). The EIF is obtained by re-writing the target state estimate and covariance in terms of the information vector \(\hat{\mathbf{y}}\) and information matrix \(\mathbf{Y}\).  
\[\hat{\mathbf{y}}=\mathbf{P}^{-1}\hat{x}_{t},\ \mathbf{Y}=(\mathbf{P})^{-1} \tag{13}\]  
At time \(k\), each UAV \(A_{i}\) runs the following EIF algorithm.  
* Prediction phase \[\mathbf{Y}_{i,k|k-1}=\left(\mathbf{F}_{k}\mathbf{Y}_{i,k-1|k-1}^{-1}\mathbf{F}_{k}^{\text{T}}+ \mathbf{G}_{k}\mathbf{Q}_{k}\mathbf{G}_{k}^{\text{T}}\right)^{-1}\] (14) \[\hat{\mathbf{y}}_{i,k|k-1}=\mathbf{Y}_{i,k|k-1}\mathbf{F}_{k}\mathbf{Y}_{i,k-1|k-1}^{-1}\hat {\mathbf{y}}_{i,k-1|k-1}\] (15) \[\hat{x}_{i,k|k-1}=\mathbf{Y}_{i,k|k-1}^{-1}\hat{\mathbf{y}}_{i,k|k-1}\] (16)
* Measurement update \[\mathbf{Y}_{i,k|k}=\mathbf{Y}_{i,k|k-1}+\mathbf{I}_{i,k}\] (17) \[\hat{\mathbf{y}}_{i,k|k}=\hat{\mathbf{y}}_{i,k|k-1}+i_{i,k}\] (18) \[\hat{x}_{i,k|k}=\mathbf{Y}_{i,k|k}^{-1}\hat{\mathbf{y}}_{i,k|k}\] (19)  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Value** & **Unit** \\ \hline Transmission power \(P_{i}\) & 30 & dBm \\ Noise power \(N_{j}\) & \(-\)70 & dBm \\ SNR requirement \(\gamma\) & 10 & - \\ Antenna gain constant \(C_{ij}\) & 1 & - \\ Propagation loss factor \(\alpha\) & 3 & - \\ \hline \hline \end{tabular}
\end{table}
Table 1: The parameters of the realistic wireless communication model.  
where \(\hat{y}_{i}\) and \(Y_{i}\) represent the information state vector and information matrix of the UAV \(A_{i}\), respectively. At the time \(k\), \(A_{i}\) obtains measurement \(\mathbf{z}_{i,k}\), and performs the associated information state and matrix updates  
\[\mathbf{i}_{i,k}=\mathbf{H}_{i,k}^{T}\mathbf{R}_{i,k}^{-1}\left[\mathbf{z}_{i,k}-\mathbf{h}_{i}(k, \mathbf{x}_{i,k},\hat{\mathbf{x}}_{i,k,k|k-1})+\mathbf{H}_{i,k}\hat{\mathbf{x}}_{i,k,k|k-1}\right] \tag{20}\]  
\[\mathbf{I}_{i,k}=\mathbf{H}_{i,k}^{T}\mathbf{R}_{i,k}^{-1}\mathbf{H}_{i,k} \tag{21}\]  
Since its local estimation is not perfect, each UAV must send its estimate estimation to the base station to obtain the global optimal estimate, thus enhancing the tracking accuracy of the UAVs. The base station obtains the global optimal estimate by using following fusion algorithm:  
\[\hat{y}_{\text{B},k|k}=\hat{y}_{\text{B},k|k-1}+\sum_{i=1}^{N}\xi_{i,k}\cdot \mathbf{i}_{i,k} \tag{22}\]  
\[\mathbf{Y}_{\text{B},k|k}=\mathbf{Y}_{\text{B},k|k-1}+\sum_{i=1}^{N}\xi_{i,k}\cdot\mathbf{ I}_{i,k} \tag{23}\]  
where \(\hat{y}_{\text{B}}\) and \(\mathbf{Y}_{\text{B}}\) represent the information state vector and information matrix of the base station, respectively. In Equations (22) and (23), we use \(\xi_{i,k}\) to indicate whether UAV \(A_{i}\) has sent its own estimation to the base station successfully. \(\xi_{i,k}\) is a random variable taking on either 0 or 1 from the following Bernoulli distribution  
\[P(\xi_{i,k}=1)=\beta_{i,k};P(\xi_{i,k}=0)=1-\beta_{i,k} \tag{24}\]  
where \(\beta_{i,k}\) indicates the probability of the packet successfully getting through from \(A_{i}\) to the base station. \(\beta_{i,k}\) can be calculated by Equations (11) or (12).  
Hence, the global optimal estimate of the target state determined by the base station is:  
\[\hat{x}_{\text{t},k|k}=(\mathbf{Y}_{\text{B},k|k})^{-1}\cdot\hat{y}_{\text{B},k|k} \tag{25}\]