Abstract
This paper extends in two directions the results of prior work on generalized linear covariance analysis of both batch least-squares and sequential estimators. The first is an improved treatment of process noise in the batch, or epoch state, estimator with an epoch time that may be later than some or all of the measurements in the batch. The second is to account for process noise in specifying the gains in the epoch state estimator. We establish the conditions under which the latter estimator is equivalent to the Kalman filter.
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Notes
Here and subsequently we suppress the time arguments where there is no ambiguity so as to simplify the notation.
We use this notation in the sequel to denote a priori and a posteriori values of various quantities.
It should be kept in mind that if systematic errors cause e≠0, then P(t ∗) will describe the mean square error matrix, rather than the covariance.
We use the notation \(\sum _{i,j}(\cdot )\equiv \sum _{i}\sum _{j}(\cdot )\).
Remember that \(P_{w*}^{+}\) contains the correlations with the process noise in the a priori error at time t ∗.
In the same spirit, Φ s s (t i ,t ∗) and \( \tilde {H}_{si}\) in Ref. [1] can be replaced by \(\hat {\Phi }(t_{i},t_{*})\) and \(\hat {H}_{i,*}\), respectively.
References
Markley, F.L., Carpenter, J.R.: Generalized linear covariance analysis. J. Astronaut. Sci. 57 (1,2), 233–260 (2009)
Markley, F.L., Seidewitz, E., Nicholson, M.: A general model for attitude determination error analysis, NASA conference publication 3011: flight mechanics/estimation theory symposium, pp. 3–25. (1988)
Markley, F.L., Seidewitz, E., Deutschmann, J.: Attitude determination error analysis: general model and specific application. In: Proceedings of the CNES Space Dynamics Conference (Toulouse, France), pp. 251–266. (1989)
Gauss, K.F.: Theory of the motion of the heavenly bodies moving about the sun in conic sections. Dover, New York (1963)
Kalman, R.E.: A new approach to linear filtering and prediction problems. Trans. ASME J. Basic Eng. 82, 35–45 (1960)
Brown, R.G., Hwang, P.Y.: Introduction to random signals and applied kalman filtering, 3rd edn . Wiley, New York (1997)
Bierman, G.J.: Factorization methods for discrete sequential estimation. Academic Press, New York (1977)
Tapley, B.D., Schutz, B.E., Born, G.R.: Statistical orbit determination. Academic Press (2004)
McReynolds, S.: Covariance Projection Methods for Estimators, AIAA Guidance, Navigation and Control Conference, No. A9635623, AIAA Paper 963739 in AIAA Meeting Papers on Disk, American Institute of Aeronautics and Astronautics, Inc. (1996)
Maybeck, P.S.: Stochastic models, estimation, and control, vol. 1. Academic Press, New York (1979)
Carpenter, J.R., Markley, F.L., Alfriend, K.T., Wright, K.T., Arcido, J.: Sequential Probability Ratio Test for Collision Avoidance Maneuver Decisions Based on a Bank of Norm-Inequality-Constrained Epoch-State Filters, AAS/AIAA Astrodynamics Specialist Conference, No. AAS 11-437, Alyeska, AK (2011)
Zanetti, R., Majji, M., Bishop, R.H., Mortari, D.: Norm-constrained Kalman filtering. J. Guid. Control Dyn. 32 (5), 1458–1465 (2009)
Golub, G.H., Loan, C.F.V.: Matrix computations, 3rd edn. JHU Press, Baltimore (1996)
Montenbruck, O.: An epoch state filter for use with analytical orbit models of low earth satellites. Aerosp. Sci. Technol. 4, 277–287 (2000)
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Appendix: Consistency Condition for Sequential Implementation
Appendix: Consistency Condition for Sequential Implementation
We will first show that the consistency condition requires that either the estimator ignores process noise or that t i ≤t ∗ for i<k. We will use a very simple model with a one-component state vector and two scalar measurements with \(\hat {H}_{1,*}=\hat {H}_{2,*}=1\). The blocks
of \(\tilde {W}_{k}\) are scalars, so we have
The gains are given by Eq. (54d) as
The consistency condition, Eq. (54d) then gives
Canceling the common term of \((\hat {P}_{*|0}-\hat {Q}_{*,1}^{\rightarrow }){W}_{11}{W}_{22}\) from the two sides of the equation and rearranging gives
The quantity in square brackets is certainly not zero, because there is nothing to cancel the term R 1 in W 11. Thus we must have
This equation is satisfied if either the estimator ignores process noise entirely or if t 1≤t ∗. In the latter case it follows by by induction that we must have t i ≤t ∗ for all i<k.
The next task is to show that the consistency condition holds if \( \hat {Q}_{*,i}^{\rightarrow }= \hat {Q}_{*,i}\) for i≤k. We will show this in the general case. Eq. (54d) gives, with the subscripting notation used for the sequential implementation,
We use Eq. (54d) and the Sherman-Morrison-Woodbury matrix inversion lemma to write the upper left-hand corner of \(\tilde {W}_{k}^{-1}\) in the form
which is made up of the blocks
We also have
With these substitutions,
The first term on the right side of this equation is easily recognized to be K i|k−1. The sum over j in the second term can be written as
where Eq. (54d) (with equality) gives the last line. Putting all this together gives
recognizing Eq. (54d) in the last line. This is the desired relation.
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Markley, F.L., Carpenter, J.R. Linear Covariance Analysis and Epoch State Estimators. J of Astronaut Sci 59, 585–605 (2012). https://doi.org/10.1007/s40295-014-0006-0
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DOI: https://doi.org/10.1007/s40295-014-0006-0