RADAR TRACKING SYSTEM USING NEURAL NETWORKS

System Identification based on neural networks has become a very important field in research projects. An attempt has been made to use these neural networks based on a simple back propagation algorithm, with some modifications on input/output vectors, to track a moving object such as aircraft. Prediction was also made on the aircraft position, one step ahead in real time.

The capability of neural networks for approximating arbitrary input-output mappings give a simple way to identify unknown dynamic functions in order to predict the needed output one step ahead or more. In a tracking system, measured radar signals mostly have been mixed with additive white noise. In order to filter out or minimize this measured noise on-line and to predict the aircraft position one step ahead, a simple back propagation algorithm has been used.

where w(t) and v(t) are white noise processes[3]. f, G and h are known or unknown functions. Here, h is a known function as the radar measurement signal can be obtained. However, the enemy aircraft position signal changes are unknown functions. So, here we face a very critical problem i.e. how to train a neural network without using actual data as a guide to update the network weights. Therefore, some modifications have been made using some ideas from Kalman filter algorithm[7] and others [1],[2],[5],[6]. We have used the measurement data as the training target with some assumptions to predict the enemy aircraft position in real time within an acceptable accuracy.

A Multilayer Perceptrons (MLP) with only three layers, using back propagation algorithm as training scheme, is used to perform the tracking and predicting process.

Fig. 1: A single-hidden-layer MLP neural networks.

Below are the equations for back propagation algorithm used during training for a single vector pair [3].

i, j, k	denote the input layer, hidden layer and output layer units respectively.
h, o	denote the hidden layer and output layer respectively.
Net	denotes the total summation of Weights multiply input include bias.
W	denotes the network’s weight.
q	denotes the network’s bias.
I	denotes the activation output of Net for hidden layer.
O	denotes the activation output of Net for output layer.
f	denotes the activation function. In this paper, we used tanh(.) for hidden layer, linear function for output layer.
d	denotes the error term.
h	denotes learning factor.
a	denotes momentum factor.

1. Apply the input vector x = ( x₁, x₂, …,x_n) to the input units. Desired output vector is y = (y₁, y₂,…, y_m).

11. Calculate the total error for each output from the output layer units. If the error is less than the minimum value that has been set, stop the training process.

(2.9)

Function f is differentiable and can be used as an activation function for each output units from the hidden and output layers. Three typical functions are listed below:

The major problem when we train the neural network to become a "neural filter" that can minimize the measurement error, is the target value we want to use to train the networks. In [3], the actual values that are used to train the network are known. When the mean square error of the trained network reaches the threshold value that was set by the user, it becomes a neural filter and has the same function as Kalman filter.

In the test model of the radar system, there are no actual equations that can explain the enemy aircraft flight coordinates. So there will be no actual values that can be used to train the network. Only measurement values are available. In theory, for simulation purposes, the actual values and the measured values are available, so the network will be trained properly from the simulation data. This can be done only in off-line data. If the system is on-line, then training cannot be done without actual values or reference values.

To describe the above problem, here we present the simple equations in x-coordinate used to denote the aircraft position, if we know the dynamic process of the aircraft (for explanatory purposes).

Equation (3.3) shows the position measured at time t with T is the radar scan time. These measurement values contain measurement error v(t) with known variance and mean. Therefore, measured position is not accurate. The actual value is x(t). The objective of the designed neural filter is to minimize the error due to v(t) and the filter output values in order to approximate the actual values. Let’s say

is the output value from the neural filter when y(t) is used as an input value. Then error = x(t) -

. The desired target of the neural filter is x(t). So the minimization of the error x(t) -

is similar to the minimization of the v(t). The following equations can be verified.

Consequently, we can say that neural filter can be used to do the same function as Kalman filter.

Equation (3.1) shows how the aircraft positions change at time t+1. If T is small enough, equation (3.1) can be used to approximate the aircraft position after T seconds.

is the aircraft speed at t, the speed at t+1 changes because of u_x(t), where u_x(t) is the acceleration due to wind gusts or short-term irregularities in engine thrust. u_x(t) is zero-mean, stationary white noise process.

In radar system, position calculation at t+1 must be fast and accurate. The problem of the original algorithm i.e. prediction of the true position coordinates converges slowly to the desired value. This is because the target position always change. Therefore, we modified the target of the neural network to the position gaps at t+1 from the true value. The justification of the modification is that the position gaps are more static than the true values. Also the scan time of the radar, T has to be small enough for the desired accuracy. The error becomes Dx(t) -

. The minimization of the error is then reduces the minimization of the v(t) - v(t-1) = D v(t),

y(t) - y (t-1) = x(t) - x(t-1) + v(t) - v(t-1)
x(t) - x (t-1) - { y(t) - y(t-1)} = - {v(t) - v(t-1)}

After the original equations have changed to the proposed equation, the error due to the white noise becomes D v(t). This solves a portion of the tracking problem. However, there are no actual values for the tracking problem.

The probability of the enemy aircraft flying from one position to another position is large and cannot be modelled easily by equations in 3 dimension. So that is no way to train the network with the actual values and to predict the next coordinates of the aircraft. Therefore, it is inevitable to use the measured values to train the neural network. Here, we find that it is the most practical approach. The measured values become the target values of the network output. If we want the prediction to be accurate, the measured values must be in the acceptable error percentage. If the measurement is not accurate enough, the prediction is not accurate either. The second reason is that the flight of the aircraft is not static. The prediction of the position is impossible if real time training and predicting cannot be done. If the Cartesian coordinates of the neural network’s target are measured values, filtering can still be done with certain assumption. Let’s say, there are 3 measured values y(t-1), y(t) and y(t+1) then,

y(t) - y(t-1) = x(t) - x(t-1) - [ v(t) - v(t-1)]
y(t+1) - y(t) = x(t+1) - x(t) - [ v(t+1) - v(t)]

Since the actual values of x(t+1), x(t) and x(t-1) are not available, comparison cannot be made and the weights of the network cannot be updated. If,

i.e. the difference in positions between time t+1 and t is approximately the same as the position difference between time t and t-1. This means the speed of the aircraft does not change abruptly. The target of the network is [y(t+1) - y(t)].

where v₁(t+1) is the noise term due to the measurement error. Input data are [y(t) - y(t-1)] and the output of the network is the prediction value of [y(t+1) - y(t)] , then

the reduction of the error D y(t+1)-is similar to the reduction of the v₁(t+1). However in this case, the v₁(t+1) may become 4 times larger than the original measurement error due to the arithmetic operation after the modification. Thus, more data are needed to be input to the neural network to have an accurate prediction.

In the designed neural network based radar tracking system, input vector is in y(t)=[ D x(t) D y(t) D z(t) D x(t-1) D y(t-1) D z(t-1) … D x(t-m) D y(t-m) D z(t-m) ]^T, and the desired output is in y(t+1)=[ D x(t+1) D y(t+1) D z(t+1)]^T, where D x, D y, and D z denote the measured position gap in x, y and z coordinates of the aircraft, t denotes the discrete time and m denotes how many time lag variables are included in calculation. Fig.2 shows the neural network based radar tracking system that has been designed.

Fig. 2: The Neural Network Based Radar tracking System Structure.

Normally, radar measured data are in polar coordinates, so conversion from polar coordinates to Cartesian coordinates has to be made first. As we take the position gaps as the input to the neural network, the value of the input vector has to be in a certain range to make sure that the overflow does not happen in the calculation of updating the weights. Therefore, normalization of the input vector based on radar scan range is used.

Among the various runs of the simulations, one typical set of output is shown in fig. 3, 4, 5, 6. Fig.3 shows the measurement and prediction position of the aircraft in (x, y, z) coordinates vs. time. We can see that the predicted values and the measured values are very closed. Fig. 4, 5 and 6 show the prediction error of the aircraft. Comparison between predicted values to the measured values is made. In these figures, we can see that the errors are small.

Fig. 3: Aircraft measurement and prediction positions in (x, y, z) vs. time

Fig. 4: x axis error.

Fig. 5: y axis error

Fig. 6: z axis error.

Note: measurement error = 0.1% of radar scan range, 20km, aircraft speed = 200m/s, 1cycle = 1 second. All figures in section V use the normalized values based on the maximum scan range. Therefore, actual value = graph value ´ Scan range for x and y coordinates, z actual value = graph value ´ 1000 because z maximum value set to 1000m in the simulation program.

The idea of using the neural network to do filtering concept and prediction of one step ahead is to test whether the neural network can be used to replace the function of Kalman filter. After some modifications have been made to the input/output vectors of the neural network, the problem of the radar tracking system using only measured data is solved. The simulation results after the modifications show the tracking error percentages are in the acceptable range.

Radar Tracking System Using Neural Networks