Intelligence, Surveillance, and Reconnaissance - Spawar

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INTELLIGENCE, SURVEILLANCE, AND RECONNAISSANCE
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where f i = is the output of the i th hidden node, x i is the
1 + exp(–x i)
dot product sum of the previous input layer’s output with the connecting
weights of the hidden layer, NN m is the m th output of the neural network,
w im is the m th output weight connected to the i th hidden node, w ki
is the k th input weight connected to the i th hidden node, and I k is the k th
input feeding the neural network.
Neural Extended Kalman Filter
The NEKF developed by Stubberud [1] is based on the Singhal and Wu
EKF neural-network trainer. The algorithm uses a Kalman filter to estimate
the states by using a linear system model and, at the same time,
using the Kalman filter to train a neural network to calculate the nonlinearities,
mismodeled dynamics, higher order modes, and other facets of a
system. Estimation of the system states is performed at once without the
necessity of modeling the nonlinearities a priori as in the case of the
extended Kalman filter. The inputs to the neural network are the updated
states of the filter. The inputs are passed through an input layer, a hidden
layer with nonlinear squashing functions, and an output layer. The outputs
of the neural network are nonlinear corrections to the linear predicted
state of the underlying Kalman filter.
NEKF IMM ALGORITHM
The NEKF IMM algorithm was developed under this ILIR project.
The IMM architecture allows for Kalman-filter models of different state
dimensions to be mixed together appropriately. In order to mix the
NEKF’s state vector and covariance matrix with the other two linear
Kalman-filter models, care had to be taken. To keep a positive definite
covariance matrix of the neural-network model, the mixing probability
of the interacting multiple-model architecture had to be applied to the
artificial-neural-network weights and covariance matrix. At first glance,
it was not obvious to weight probabilistically the weight vector and
weight covariance matrix of the NEKF by the mode probability. Once
the probability weighting was incorporated and had stabilized the neuralnetwork
covariance matrix, a different sigmoid squashing function had
to be implemented. By weighting the neural network with its mixing probability,
the weight vector tended toward zero, which caused the weights
to be untrainable. What caused the weights to be untrainable was using a
hyperbolic tangent function as a sigmoid-squashing function that varied
from [–1, 1]. A new sigmoid varying from [0, 1] allowed for a derivative
of 0.5 if the weights of the neural network were equal to 0.0. This allowed
the weights to be trainable if they were initialized to zero or tended
toward zero. Once this was incorporated, the neural network was able
to be trained inside an IMM architecture.
RESULTS
The scenario consisted of a single target moving in a circular motion for
500 seconds. The speed of the target was 50 meters per second, and the
angular rate of the turn was –10 degrees per second. The sampling rate of
the radar was one sample per second. There were 500 samples for each
Monte Carlo run. Figures 1 through 5 and Table 1 are the 100 Monte