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On kalman filter for linear system with colored measurement noise

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Abstract

The Kalman filter for linear systems with colored measurement noises is revisited. Besides two well-known approaches, i.e., Bryson’s and Petovello’s, another measurement time difference-based approach is introduced. This approach is easy to be implemented and generalized to nonlinear system, and can provide filtering solutions directly. A unified view on these approaches is provided, and the equivalence between any two of the three is proved. In the case study part it is validated that, compared to the approach that neglects the time correlations, the approaches that take them into account not only avoid overly optimistically evaluating the estimate, but also improve the transient accuracy of the estimate.

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Acknowledgments

The author’s sincere thanks go to the associate editor and three anonymous reviewers for their thorough and critical comments which improved the ealier draft significantly; and also to Dr. Jiaxun Li of Tianjin Institute of Hydrographic Surveying and Charting, and Dr. Lubin Chang of the Department of Navigation Engineering, Naval University of Engineering, for checking the English. This work was supported in part by National Natural Science Foundation of China (No. 41374018).

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Authors and Affiliations

  1. Tianjin Institute of Hydrographic Surveying and Charting, Tianjin, China

    Guobin Chang

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  1. Guobin Chang
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Correspondence to Guobin Chang.

Appendix: proof of (48)

Appendix: proof of (48)

Substitute (27), (43), and (45) into (42) we have

$$\begin{aligned} \varvec{P}_{(3)}&= \varvec{\varPhi } \varvec{P}{\varvec{\varPhi }}^{T}+\varvec{Q}-\varvec{F}\varvec{K}_{(1)} \varvec{\varSigma }_{(1)} \varvec{K}_{(1)}^T \varvec{F}^{T}\nonumber \\&-\varvec{F}\varvec{K}_{(1)} {\varvec{\varSigma }}_{(1)} \varvec{J}^{T}-\varvec{J}{\varvec{\varSigma }}_{(1)} \varvec{K}_{(1)}^T \varvec{F}^{T}-\varvec{J}{\varvec{\varSigma }}_{(1)} \varvec{J}^{T}\nonumber \\ \end{aligned}$$
(60)

Substitute (10) into \(F\varvec{K}_{(1)} {\varvec{\varSigma }}_{(1)} \varvec{J}^{T}\) and \(\varvec{J}{\varvec{\varSigma }}_{(1)} {\varvec{K}}_{(1)}^T {\varvec{F}}^{T}\), (8) into \(J\varvec{\varSigma }_{(1)} \varvec{J}^{T}\), with (18) in mind, (60) can be rearranged as

$$\begin{aligned} \varvec{P}_{(3)}&= \varvec{\varPhi } \varvec{P}{\varvec{\varPhi }}^{T}+\varvec{Q}-\varvec{F}\varvec{K}_{(1)} {\varvec{\varSigma }}_{(1)} {\varvec{K}}_{(1)}^T \varvec{F}^{T}-\varvec{FPG}^{T}\varvec{J}^{T}\nonumber \\&-\varvec{JGPF}^{T}-\varvec{JGPG}^{T}\varvec{J}^{T} -\varvec{J}{\overline{\varvec{R}}}\varvec{J}^{T}\nonumber \\&= \varvec{\varPhi } \varvec{P}{\varvec{\varPhi }}^{T}+{\overline{\varvec{Q}}}-\varvec{F}\varvec{K}_{(1)} {\varvec{\varSigma }}_{(1)} \varvec{K}_{(1)}^T \varvec{F}^{T}-\varvec{FPG}^{T}\varvec{J}^{T}\nonumber \\&-\varvec{JGPF}^{T}-\varvec{JGPG}^{T}\varvec{J}^{T} \end{aligned}$$
(61)

With (15) in mind we notice that

$$\begin{aligned} \varvec{FPG}^{T}\varvec{J}^{T}+\varvec{JGPG}^{T}\varvec{J}^{T}= {\varvec{\varPhi }} \varvec{PG}^{T}\varvec{J}^{T} \end{aligned}$$
(62)

and its transpose

$$\begin{aligned} \varvec{JGPF}^{T}+\varvec{JGPG}^{T}\varvec{J}^{T}=\varvec{JGP}{\varvec{\varPhi }}^{T} \end{aligned}$$
(63)

So with (15) in mind, we have

$$\begin{aligned}&{\varvec{\varPhi }}\varvec{P}{\varvec{\varPhi }}^{T} -\varvec{FPG}^{T}\varvec{J}^{T}-\varvec{JGPF}^{T} -\varvec{JGPG}^{T}\varvec{J}^{T}\nonumber \\&\quad =\varvec{\varPhi }\varvec{P}{\varvec{\varPhi }}^{T}-\left( \varvec{FPG}^{T}\varvec{J}^{T} +\varvec{JGPG}^{T}\varvec{J}^{T}\right) \nonumber \\&\qquad - \left( \varvec{JGPF}^{T}+\varvec{JGPG}^{T}\varvec{J}^{T}\right) + \varvec{JGPG}^{T}\varvec{J}^{T}\nonumber \\&\quad =\varvec{\varPhi }\varvec{P}{\varvec{\varPhi }}^{T}-\varvec{\varPhi } \varvec{PG}^{T}\varvec{J}^{T}-\varvec{JGP}{\varvec{\varPhi }}^{T}+\varvec{JGPG}^{T}\varvec{J}^{T} \nonumber \\&\quad =(\varvec{\varPhi }-\varvec{JG})\varvec{P}({\varvec{\varPhi } -\varvec{JG}})^{T}=\varvec{FPF}^{T} \end{aligned}$$
(64)

Substitute (64) into (61), we have

$$\begin{aligned} \varvec{P}_{(3)} =\varvec{FPF}^{T}+{\overline{\varvec{Q}}}-\varvec{F}\varvec{K}_{(1)} {\varvec{\varSigma }}_{(1)}\varvec{K}_{(1)}^T \varvec{F}^{T}={\varvec{P}}_{(1)} \end{aligned}$$
(65)

So the equivalence between (20) and (42) are proved.

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Chang, G. On kalman filter for linear system with colored measurement noise. J Geod 88, 1163–1170 (2014). https://doi.org/10.1007/s00190-014-0751-7

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