Student: Paul Frihauf
Professors: Hiro Mukai and I. Norman Katz
Sponsor: MIT Lincoln Laboratory
Performed: September 2004 - May 2005
Problem Statement:
The states of a nonlinear system estimated using a sequence of linear approximations (e.g. extended Kalman Filter) are subject to the loss of observability due to its linearization even if there is no loss of observability for the original nonlinear system. Namely, at a given linearization point, the observability space based on the linearized model may collapse to null, leaving us with no means to estimate the state from the linearized model. We refer to this point as the Linearized Blind Spot (LBS). We conjecture that the particle filter will not be hindered by this collapse since it retains the original nonlinear model when calculating state estimates. A simple nonlinear system of a reentry vehicle with a vertical trajectory was used to test this conjecture.
Pictured is a graphical representaiton of the reentry vehicle as it descends vertically (blue arrow) and a sensor measures the reentry vehicle's range to the sensor, z(t). The sensor is fixed at a vertical height, H, and a horizontal distance from the trajectory, L.
The reentry vehicle's altitude is x1(t) and the LBS is represented by the purple square. Clearly, the LBS occurs when the reentry vehicle is perpendicular to the sensor.
Using only the range measurement, we estimate the reentry vehicle's altitude, velocity, and ballistic coefficient.
Divergence Issue:
An extended Kalman filter that requires linearizing the system to obtain state estimates may diverge as the observability space degenerates when the reentry vehicle nears the LBS. As the information from the measurements decreases, the filter's gains are amplified until they eventually cause instability withing the sensor. Below is an example of such a divergence. The red shading represents the region near the LBS.
Particle Filter Estimation:
The particle filter mitigates the effects of the degenerating observabiltiy space by retaining the system nonlinearities, but it is subject to its own drawbacks, mainly sample degeneracy and sample impoverishment. This study presents tuning results via diffusion covariance variation to overcome these drawbacks and prevent convergence to an incorrect estimate. Below is an example of a particle filter's estimates for a reentry vehicle's states. The red shading again represents the region near the LBS.
Monte Carlo simulations were performed to determine an appropriate tuning and to analyze the effects of the LBS region on filter performance by comparing the estimation error in this region to the steady state estimation error. The graphs below show that the LBS region does cause the estimation error to inflate, but the particle filter does not diverge and the estimates remain meaningful. The red shading is the region near the LBS and the green shading is the steady state region.
Summary and Future Work:
The study provides insights into why the particle filter can mitigate a degenerating observability space and details how the tuning process is related to the filter's effectiveness. This work may be extended to systems containing unobservable regions as opposed to just unobservable points. Other possible areas to pursue may be to automate the tuning process so that the particle filter can adapt to a changing estimation environment and to use a hybrid filter that combines the extended Kalman filter and the particle filter to take advantage of each filter's strengths.
This project was peformed using Matlab and the ReBEL Toolkit.