1. Prepared By: Kevin Meier Alok Desai
Using the Kalman Filter to Estimate the state of a Maneuvering Aircraft
ECEn -670 Stochastic Process
Prepared By:
Kevin Meier
Alok Desai
Instructor:
Dr. Brian Mazzeo
11/29/2011
ECEn -670 Stochastic Process

2. ECEn -670 Stochastic Process
Outlines
Kalman filter
Correlation Between the Process and Measurement Noise
Application of KF for estimating Bearing and Range
Simulation results
11/29/2011
ECEn -670 Stochastic Process

3. ECEn -670 Stochastic Process
Kalman Filter
Purpose: It is to use measurements observed over time, containing noise (random variations) and other inaccuracies, and produce values that tend to be closer to the true values of the measurements and their associated calculated values.
When system model and measurement model equations are linear, then to estimate the state vector recursively.
11/29/2011
ECEn -670 Stochastic Process

4. ECEn -670 Stochastic Process
Estimating States
System dynamic model:
Measurement model:
11/29/2011
ECEn -670 Stochastic Process

5. Kalman Filter Estimation
11/29/2011
ECEn -670 Stochastic Process

6. ECEn -670 Stochastic Process
Kalman Filter (Cont.)
State estimation:
Error covariance (a priori):
Kalman Gain:
Error covariance update (a posteriori):
State estimate update:
11/29/2011
ECEn -670 Stochastic Process

7. Correlation Between the Process and Measurement Noise
Correlation be given by
Prediction equation remain unchanged.
Measurement equation
11/29/2011
ECEn -670 Stochastic Process

8. Range and Bearing Estimation
Radars are used to track aircraft.
11/29/2011
ECEn -670 Stochastic Process

9. ECEn -670 Stochastic Process
Range = ct/2
11/29/2011
ECEn -670 Stochastic Process

10. How the Kalman filter applies to Radar
Radar is used to track the state of an aircraft
The state is the range, range rate, bearing and bearing rate
11/29/2011
ECEn -670 Stochastic Process

11. How to model the aircraft with no acceleration data
Model the acceleration as a uniform random variable using the singer model. Where the acceleration is correlated from sample to sample
11/29/2011
ECEn -670 Stochastic Process

12. How the Kalman filter applies to Radar
The radar uses sensors to measure the Range and Bearing angle. In this process there is sensor measurement noise
11/29/2011
ECEn -670 Stochastic Process

13. How the Kalman filter applies to Radar
The process and measurement noise are zero-mean white Gaussian random variables
11/29/2011
ECEn -670 Stochastic Process

14. ECEn -670 Stochastic Process
11/29/2011
ECEn -670 Stochastic Process

15. Error Covariance for Range
Error covariance (One prediction)
Error covariance (Multiple prediction)
11/29/2011
ECEn -670 Stochastic Process

16. Error Covariance of Bearing
Error covariance (One prediction)
Error covariance (Multiple prediction)
11/29/2011
ECEn -670 Stochastic Process

17. ECEn -670 Stochastic Process
Bearing Angle
Bearing Angle (One prediction)
Bearing Angle (Multiple prediction)
11/29/2011
ECEn -670 Stochastic Process

18. Vehicle Range Vehicle Range (One Prediction)
Vehicle Range (Multiple Prediction)
11/29/2011
ECEn -670 Stochastic Process

19. ECEn -670 Stochastic Process
Range Error
Range Error (One Prediction) c
Vehicle Range (Multiple Prediction)
11/29/2011
ECEn -670 Stochastic Process

20. Bearing Rate Bearing ( one prediction ) Bearing (multiple prediction )
11/29/2011
ECEn -670 Stochastic Process

21. Range Range (One prediction ) Range (Multiple prediction ) 11/29/2011
ECEn -670 Stochastic Process

22. Range Error and Range Rate with correlated noise
11/29/2011
ECEn -670 Stochastic Process

23. ECEn -670 Stochastic Process
Questions??
Thank you !
11/29/2011
ECEn -670 Stochastic Process