1. Colorado Center for Astrodynamics Research The University of Colorado 1 STATISTICAL ORBIT DETERMINATION EKF and Observability ASEN 5070 LECTURE 23 10/21/09

2. Colorado Center for Astrodynamics Research The University of Colorado 2 Extended Kalman Filter (EKF) Computational Algorithm

3. Colorado Center for Astrodynamics Research The University of Colorado 3 Extended Kalman Filter (EKF) Computational Algorithm

4. Colorado Center for Astrodynamics Research The University of Colorado 4 The Prediction Residual

5. Colorado Center for Astrodynamics Research The University of Colorado 5 The Prediction Residual

6. Colorado Center for Astrodynamics Research The University of Colorado 6 The Prediction Residual This would be especially important in the case of the EKF

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13. Statistical Orbit Determination Copyright 2006 University of Colorado at Boulder 13 The mathematical conditions for parameter observability can be derived from the observability criterion for a linear dynamical system. The result is that the H matrix be full rank So that the information matrix is positive definite.

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16. Statistical Orbit Determination Copyright 2006 University of Colorado at Boulder 16 Rank of H matrix

17. Colorado Center for Astrodynamics Research The University of Colorado Effect on P as one parameter in the state becomes Less Observable Assume we have a system For simplicity assume we have two observations: Then,

18. Colorado Center for Astrodynamics Research The University of Colorado Effect on P as one parameter in the state becomes Less Observable *Note that A, B, and C are independent of. From the definition of P,

19. Colorado Center for Astrodynamics Research The University of Colorado Effect on P as one parameter in the state becomes Less Observable so and Note that and are independent of. In general, the uncertainty of other parameters in the state and the correlation coefficients of all parameters in the state are unaffected as one parameter becomes uniformly less observable.

20. Colorado Center for Astrodynamics Research The University of Colorado Observability Example

21. Colorado Center for Astrodynamics Research The University of Colorado Observability Example But, so, and Finally at as, will be rank 2 and the state vector is observable and

22. Colorado Center for Astrodynamics Research The University of Colorado Observability Example Same problem except h = α. That is, h is very small. If and and Then Now will contain a column of zeros and be rank = 1. Therefore, the state is unobservable.

23. Statistical Orbit Determination Copyright 2006 University of Colorado at Boulder 23 Observability Examples Start with a full-rank m x n H matrix. Simple case of estimating four parameters: n=4. Number of observations m is much larger than n. Make the H matrix nearly rank deficient and look at sigmas and correlation coefficients.

24. Statistical Orbit Determination Copyright 2006 University of Colorado at Boulder 24 Example 1 Multiply column two of H by . As  gets smaller, look at what happens to the sigmas and correlation coefficients. For small , the system is insensitive to parameter two in the state vector. In the following two plots, as  gets smaller: –The uncertainty in the second state parameter increases –The correlation cofficients remain unchanged.

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27. Statistical Orbit Determination Copyright 2006 University of Colorado at Boulder 27 Example 2 col 4 = (1-  )(col 2) + (  )(col 4) As  gets closer to 0, column 4 of H becomes nearly equal to column 2. For small , columns 2 and 4 are nearly linearly dependent. In the following two plots, as  gets smaller: –The uncertainty in the second and fourth state parameters increase. –The correlation cofficient  24 approaches 1.0.

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30. Statistical Orbit Determination Copyright 2006 University of Colorado at Boulder 30 Example 3 col 4 = (1-  )(col 2 + col 3) + (  )(col 4) As  gets closer to 0, column 4 of H becomes nearly equal to column 2 + column 3. For small , columns 2, 3, and 4 are nearly linearly dependent. In the following two plots, as  gets smaller: –The uncertainty in the second, third, and fourth state parameters increase. –The correlation cofficients  23,  24, and  34 approach 1.0.

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33. Statistical Orbit Determination Copyright 2006 University of Colorado at Boulder 33 Questions on Observability

34. a and b are known constants and we wish to solve for constants x 1 and x 2. For a least squares solution are these statements T or F? m>2 in all cases and i=1 … m. a)y i =asin(x 1 )+bsin(x 2 )+ Є i T or F _________ b)y i =asin(x 1 t i ) +bsin(x 2 t i )+Є i T or F _________ c)y i =x 1 sin(ωt i ) +x 2 cos(ωt i ) +Є i T or F _________ d)y i =x 1 /x 2 +x 2 x 1 + Є i T or F _________ e)y i =a+bsin(x 2 ) +Є i T or F _________ f)The same answers apply to a sequential filter solution T or F _________ 34