## 5 Approaches of Optimal State Estimation Utilizing Filtering
### Multiple Quadrature Information Filters (MQIFs)  
Multiple Quadrature Information Filters (MQIFs) is a developed algorithm package of a recursion, non-linear Kalman filtering. The means and covariance of all conditional densities using the Gauss-Hermite quadrature rule in different degrees can be computed using this package [23; 123].  
In [124], the authors focused on the algorithm of Quadrature Kalman filter (QKF), which was recently developed. It is mentioned that with a Gaussian assumption and based on Gauss-Hermite quadrature rules, the QKF can tackle arbitrary non-linearities. The complexity reduction technique for the QKF based on the partitioning of the state-space, referred to as the Multiple QKF, was studied. The dimensionality reduction has been reduced with the aid of partitioning schemes.  
Figures 26 and 27 illustrate the performance validation of the non-linear filter considered, as well as the scenario and Geometric Dilution of Precision (GDOP) of the LEO satellite.  
Figure 26: Performance validation of EKF, 2 nd Order KF, UKF, CDKF, CKF with GHKFs algorithm. This study focused on a complexity reduction technique for the Bayesian filtering based on the partitioning of the state-space. It proved that the partitioning schemes can efficiently be used to overcome the curse of dimensionality in the GHKF. The simulation result shows that a nearly-optimal performance can be attained using GHKF with non-linear filtering problems. Authors’ own elaboration.  
Figure 27: The geometry of multi-sensors (LEO satellite) and GDOP of the sensors for achieving the most precise estimation of a moving target (UAV) proposed in the simulation scenario. Authors’ own elaboration.  
Scenario Simulation of RFI UAV Flying in the North Latitudes  
Numerous scenarios of geolocation and target tracking using considered filters based on TDOA, FDOA, TDOA/FDOA, and PDOA/FDOA measurements have been implemented. Multi LEO satellites were considered for geolocation and UAV tracking in these scenarios. Figure 28 shows a UAV embedded with RFI sources is flying under LEO satellite's visibility and sensitivity, which is subject to a problem statement of UAV target tracking by multiple satellites. Below, some simulation results are shown using high-degree non-linear filtering.  
In the simulation scenario, three (03) LEO satellites were considered as sensors to estimate UAV tracking flying above Canada. The initial point was selected from the reference station on the roof of ETS-Montreal. The simulation results are based on 100 Monte Carlo runs. The initial estimate \(x_{0}\) values with different uncertainty level values were introduced gradually. They are given as follows: \(x_{0}=[0\;m,0\;m/s,0\;m,0,0\text{deg}/s]^{T}\) and \(P_{0}\) being the initial covariance: \(P_{0}=\text{diag}(100\;m^{2},10\;m^{2}/s^{2},100\;m^{2},10\;m^{2}/s^{2},100\;m \;rad^{2}/s^{2})\). The metrics used to compare the performance of various filters was the average root mean square error (ARMSE).  
From Figures 29-33, it can be observed that the algorithm performance based on hybrid TDOA/FDOA measurements is the best compared to TDOA, FDOA, and TDOA/PDOA measurements.  
Figure 31: Three (03) Iridium satellites were employed to track a UAV based on a TDOA/FDOA measurement. Authors’ own elaboration.  
Figure 32: Target tracking and RMSE for verifying the algorithm performance. In the first simulation scenario, three (03) Iridium satellites were considered as sensors to estimate the trajectory and velocity of the UAV based on hybrid TDOA/PDOA measurements. Authors’ own elaboration.  
Figure 30: Three (03) Iridium satellites were employed to track a UAV based on an FDOA measurement. Authors’ own elaboration.