Continuous-Time Kalman Filter

Problem Statement

For UCLA’s MAE 271B final project, I implemented an online missile state estimation algorithm that estimates a missile’s position, velocity, and acceleration relative to a moving target. This is the first half of the rendezvous problem: the state estimation. The second half, the control, was not part of the project scope, so in the plots below you’ll see that the relative position between the missile and target does not go to zero (i.e. the missile does not hit the target). However, the filter performs well and is able to track the relative states between the missile and target accurately. Here are the position, velocity, acceleration, and sensor measurement plots for one run of the target tracking scenario.

The relative position between the missile and target over a ten second period. The missile and target diverge because there is no control effort applied to the missile to close the gap, but the missile's state estimator tracks the true state closely.

The relative velocity between the missile and target. The relative velocity is initialized by drawing from a zero-mean Gaussian with covariance 40,000 (ft/sec)^2.

The relative acceleration between the missile and target. The relative acceleration is initialized by drawing from a zero-mean Gaussian with covariance 10,000(ft/sec^2)^2

The true sensor measurement (Theta) and the unfiltered sensor measurment observed by the missile (Z). Theta is generated by Equation (3) from the paper. The angle blows up to infinity as the time-to-go approaches zero due to the structure of Equation (3). While Theta and Z look identical at the initial plot resolution, zooming into the low magnitude values of Z in the 10 - 2 second range, you'll see the noise in Z. Z is corrupted by fading and scintillation noise described in Equation (4) in the paper.

Monte Carlo Simulation

To further verify the performance of the filter, a Monte Carlo simulation with 10,000 realizations was performed. The Root Mean Square Error (RMSE) in position, velocity, and acceleration were compared with the a priori covariance values computed by the filter. The figures below show that the actual RMSE plots in the three states match the precomputed values, thus verifying the filter’s performance for the Gauss-Markov model.

The Full Report