This example shows how to use a Kalman filter to estimate an aircraft’s position and velocity from noisy radar measurements.
The example model has three main functions. It generates aircraft position, velocity, and acceleration in polar (range-bearing) coordinates; it adds measurement noise to simulate inaccurate readings by the sensor; and it uses a Kalman filter to estimate position and velocity from the noisy measurements.
Run the model. At the end of the simulation, a figure displays the following information:
– The actual trajectory compared to the estimated trajectory
– The estimated residual for range
– The actual, measured, and estimated positions in X (North-South) and Y (East-West)
Kalman Filter Block
Estimation of the aircraft’s position and velocity is performed by the ‘Radar Kalman Filter’ subsystem. This subsystem samples the noisy measurements, converts them to rectangular coordinates, and sends them as input to the DSP System Toolbox™ Kalman Filter block.
The Kalman Filter block produces two outputs in this application. The first is an estimate of the actual position. This output is converted back to polar coordinates so it can be compared with the measurement to produce a residual, the difference between the estimate and the measurement. The Kalman Filter block smoothes the measured position data to produce its estimate of the actual position.
The second output from the Kalman Filter block is the estimate of the state of the aircraft. In this case, the state is comprised of four numbers that represent position and velocity in the X and Y coordinates.
Experiment: Initial Velocity Mismatch
The Kalman Filter block works best when it has an accurate estimate of the aircraft’s position and velocity, but given time it can compensate for a bad initial estimate. To see this, change the entry for the Initial condition for estimated state parameter in the Kalman Filter. The correct value of the initial velocity in the Y direction is 400. Try changing the estimate to 100 and run the model again.
Observe that the range residual is much greater and the ‘E-W Position’ estimate is inaccurate at first. Gradually, the residual becomes smaller and the position becomes more accurate as more measurements are gathered.
Experiment: Increasing the Measurement Noise
In the present model, the noise added to the range estimate is rather small compared to the ultimate range. The maximum magnitude of the noise is 300 ft, compared to a maximum range of 40,000 ft. Try increasing the magnitude of the range noise to an larger value, for example, 5 times this amount or 1500 ft. by changing the first component of the Gain parameter in the ‘Meas. Noise Intensity’ Gain block.
Observe that the blue lines representing the estimated positions have moved farther from the red lines representing the actual positions, and the curves have become much more ‘bumpy’ and ‘jagged’. We can partially compensate for the inaccuracy by giving the Kalman Filter block a better estimate of the measurement noise. Try setting the Measurement noise covariance parameter of the Kalman Filter block to 1500 and run the model again.
Observe that when the measurement noise estimate is better, the E-W and N-S position estimate curves become smoother. The N-S position curve now consistently underestimates the position. Given how noisy the measurements are compared to the value of the N-S coordinate, this is expected behavior.