Last modified: 2020-02-17

#### Abstract

The Kalman filter is the most well estimation algorithm. It is an optimal least-squares recursive estimation routine [1,2] . However, the Kalman filter is only optimal if the following assumptions are met: [1,2]

1. The system is linear

2. The noise on the measurement and the process are white and Gaussian

3. The measurements uncorrelated in time

The last assumption may seem as redundant with respect to the noise criterion, but, often, measurements in applications may be repeated from different sources which cause the filter to assume that it is receiving independent information and results in a significantly reduced uncertainty when it is not warranted.

In [3], the issue to remove measurements that were correlated in time as the input to the Kalman filter was presented. The paper looked at parallel Kalman filters designed to work together. This linear decorrelation algorithm removed the filtered data and produced a pseudo-measurement that removed the previous reported estimated. As seen in Figure 1, the Kalman filter is designed to remove the correlated information and then process the estimated new information into the local Kalman filter. As seen in Figure 1, the decorrelation algorithm creates a measurement from the second Kalman filter that is an estimate that has removed the influence from previously reported estimates. It would be as if the restarted Kalman filter restarted after it reported it last estimate. This would be the same as a tracklet in kinematic distributed sensor fusion [4]. If the dynamics of the all of the other Kalman filters were known precisely, the decorrelated pseudo-measurement is indistinguishable from the estimate based on only the new information.

The technique developed in [3], was expanded in [5] to handle the extended Kalman filter (EKF) and where the second Kalman filter is independently developed. As explained in [5], this would be important in target tracking [6].

Both of these techniques only dealt with systems that were designed to incorporate the nonlocal or offboard correlated data. They required the decorrelation be handled in the design of the filter. In both papers, this required the local filter system to be designed in the information version of the Kalman filter. With distributed tracking systems and interconnected systems that calibrate off of each other data, the core filters will often have a different design. Also, the decorrelation designs do not take into account system parameters being learned. This would be of concern of multiple systems calibrating off each other. As seen in Figure 2, a distributed tracking system could be used to monitor a number of targets. Three tracking systems report data on various targets. As seen in the figure, seven targets are present. One or more of the tracking systems tracks each target. However, each tracking system talks to the other two tracking systems. So Tracking System 1 can receive can receive its own data back from System 2 directly or from System 3 through System 2. If the technique of [5] was employed, all of the tracking system would have to be designed the same. In Figure 3, two inertial navigation systems (INSs) are using the location information from the other to help calibrate their parameters estimates. Again the Kalman estimators are corrupting themselves using the correlated information from the other INS along with its own data that the second INS has processed in its calibration. The techniques of [3,5] are not designed to handle parameter estimates being generated to calibrate the systems.

In this paper, the general decorrelation technique is developed that works in conjunction with existing Kalman filter designs and incorporate local parameter estimation. The first step is to reiterate the tracking design system of [5]. This will be followed be the generalized development of the decorrelation technique that can handle pre-designed tracking systems and parameter estimation calibration as discussed in Figure 2 and 3. In the final presentation and paper, the approach will be compared in a two-system tracking-problem that will be compared to the approach of [5].

Figure 1: The Kalman filter computes the difference of its own estimate from the measurement. If the difference greater than the defined measurement noise, then that is used to correct the state estimate.

Figure 2: Interconnected tracking systems look at the same targets and communicate their combined tracks to other systems and then receive that same information back.

Figure 3: Intercommunicating but decorrelated Kalman filters for parameter estimation