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Design of Gaussian Approximate Filter and Smoother for Nonlinear Systems with Correlated Noises at One Epoch Apart

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Abstract

In this study, the authors investigate the filtering and smoothing problems of nonlinear systems with correlated noises at one epoch apart. A pseudomeasurement equation is firstly reconstructed with a corresponding pseudomeasurement noise, which is no longer correlated with the process noise. Based on the reconstructed measurement model, new Gaussian approximate (GA) filter and smoother are derived, from which Kalman filter and smoother can be obtained for linear systems. For nonlinear systems, different GA filters and smoothers can be developed through utilizing different numerical methods for computing Gaussian-weighted integrals involved in the proposed solution. Numerical examples concerning univariate nonstationary growth model, passive ranging problem, and target tracking show the efficiency of the proposed filtering and smoothing methods for nonlinear systems with correlated noises at one epoch apart.

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Author information

Authors and Affiliations

  1. College of Automation, Harbin Engineering University, No. 145, Nantong Street, Nangang District, Harbin, 150001, People’s Republic of China

    Yulong Huang, Yonggang Zhang, Ning Li & Zhen Shi

Authors
  1. Yulong Huang
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  2. Yonggang Zhang
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  3. Ning Li
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  4. Zhen Shi
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Corresponding author

Correspondence to Yonggang Zhang.

Additional information

This work was supported by the National Natural Science Foundation of China under Grant Nos. 61201409 and 61371173, China Postdoctoral Science Foundation Nos. 2013M530147 and 2014T70309, Heilongjiang Postdoctoral Funds LBH-Z13052 and LBH-TZ0505, and the Fundamental Research Funds for the Central Universities of Harbin Engineering University No. HEUCFQ20150407.

Appendix

Appendix

If the joint PDF of random vectors \(\varvec{a}_{i}\) and \(\varvec{b}_{j}\) conditioned on \(\varvec{Z}_{k}\) is Gaussian, i.e.,

$$\begin{aligned} p(\varvec{a}_{i},\varvec{b}_{j}|\varvec{Z}_{k}) =N\left( \left[ \begin{array}{c}\varvec{a}_{i}\\ \varvec{b}_{j} \end{array}\right] ; \left[ \begin{array}{c}{\hat{\varvec{a}}}_{i|k}\\ {\hat{\varvec{b}}}_{j|k} \end{array}\right] ,\left[ \begin{array}{c@{\quad }c} \varvec{P}_{i|k}^{aa}&{}\varvec{P}_{i,j|k}^{ab}\\ (\varvec{P}_{i,j|k}^{ab})^{{T}}&{}\varvec{P}_{j|k}^{bb} \end{array}\right] \right) \end{aligned}$$
(105)

then \(p(\varvec{a}_{i}|\varvec{b}_{j},\varvec{Z}_{k})\) can be computed as Gaussian PDF with mean vector \({\hat{\varvec{a}}}_{i|j,k}\) and corresponding covariance matrix \(\varvec{P}_{i|j,k}^{aa}\) as the unified form:

$$\begin{aligned} {\hat{\varvec{a}}}_{i|j,k}= & {} {\hat{\varvec{a}}}_{i|k} +\varvec{K}_{i}^{a}(\varvec{b}_{j}-{\hat{\varvec{b}}}_{j|k}), \end{aligned}$$
(106)
$$\begin{aligned} \varvec{P}_{i|j,k}^{aa}= & {} \varvec{P}_{i|k}^{aa}-\varvec{K}_{i}^{a}\varvec{P}_{j|k}^{bb}(\varvec{K}_{i}^{a})^{{T}},\end{aligned}$$
(107)
$$\begin{aligned} \varvec{K}_{i}^{a}= & {} \varvec{P}_{i,j|k}^{ab}(\varvec{P}_{j|k}^{bb})^{-1}. \end{aligned}$$
(108)

Proof

By the fact that if the joint PDF is Gaussian, then the marginal PDF is also Gaussian, and using (105), then the PDF \(p(\varvec{b}_{j}|\varvec{Z}_{k})\) can be marginalized as Gaussian, i.e.,

$$\begin{aligned} p(\varvec{b}_{j}|\varvec{Z}_{k})=N(\varvec{b}_{j};{\hat{\varvec{b}}}_{j|k},\varvec{P}_{j|k}^{bb}) \end{aligned}$$
(109)

Let

$$\begin{aligned} \varvec{\Sigma }= \left[ \begin{array}{c@{\quad }c} \varvec{P}_{i|k}^{aa}&{}\varvec{P}_{i,j|k}^{ab}\\ (\varvec{P}_{i,j|k}^{ab})^{{T}}&{}\varvec{P}_{j|k}^{bb} \end{array}\right] \end{aligned}$$
(110)

Rearranging (105), we can obtain

$$\begin{aligned} p(\varvec{a}_{i},\varvec{b}_{j}|\varvec{Z}_{k})=\frac{1}{\sqrt{\left| 2\pi \varvec{\Sigma }\right| }}{\mathrm {exp}}\left( -\frac{1}{2}\left[ {\tilde{\varvec{a}}}_{i|k}^{{T}}\;\tilde{\varvec{b}}_{j|k}^{{T}}\right] \varvec{\Sigma }^{-1}\left[ \begin{array}{c}{\tilde{\varvec{a}}}_{i|k}\\ {\tilde{\varvec{b}}}_{j|k} \end{array}\right] \right) . \end{aligned}$$
(111)

Firstly, according to (110), we can reformulate \(\varvec{\Sigma }\) as follows:

$$\begin{aligned} \varvec{\Sigma }=\left[ \begin{array}{c@{\quad }c@{\quad }c}\varvec{I}_{L}&{}\varvec{K}_{i}^{a}\\ \varvec{0}_{s\times {L}}&{}\varvec{I}_{s}\end{array}\right] \left[ \begin{array}{c@{\quad }c@{\quad }c}\varvec{P}_{i|j,k}^{aa}&{}\varvec{0}_{L\times {s}}\\ \varvec{0}_{s\times {L}}&{}\varvec{P}_{j|k}^{bb}\end{array}\right] \left[ \begin{array}{c@{\quad }c@{\quad }c}\varvec{I}_{L}&{}\varvec{0}_{L\times {s}}\\ (\varvec{K}_{i}^{a})^{{T}}&{}\varvec{I}_{s}\end{array}\right] \end{aligned}$$
(112)

and

$$\begin{aligned} \left| \varvec{\Sigma }\right| =\left| \varvec{P}_{i|j,k}^{aa}\right| \left| \varvec{P}_{j|k}^{bb}\right| , \end{aligned}$$
(113)

where L and s are the dimensions of vectors \(\varvec{a}_{i}\) and \(\varvec{b}_{j}\), respectively, and \(\varvec{K}_{i}^{a}\) and \(\varvec{P}_{i|j,k}^{aa}\) are given by (107)–(108). Exploiting (112), \(\varvec{\Sigma }^{-1}\) can be computed as

$$\begin{aligned} \varvec{\Sigma }^{-1}= \left[ \begin{array}{c@{\quad }c@{\quad }c}\varvec{I}_{L}&{}\varvec{0}_{L\times {s}}\\ -(\varvec{K}_{i}^{a})^{{T}}&{}\varvec{I}_{s}\end{array}\right] \left[ \begin{array}{c@{\quad }c@{\quad }c}(\varvec{P}_{i|j,k}^{aa})^{-1}&{}\varvec{0}_{L\times {s}}\\ \varvec{0}_{s\times {L}}&{}(\varvec{P}_{j|k}^{bb})^{-1}\end{array}\right] \left[ \begin{array}{c@{\quad }c@{\quad }c}\varvec{I}_{L}&{}-\varvec{K}_{i}^{a}\\ \varvec{0}_{s\times {L}}&{}\varvec{I}_{s}\end{array}\right] \end{aligned}$$
(114)

Substituting (113, 114) into (111) and using (109), we have

$$\begin{aligned}&p(\varvec{a}_{i},\varvec{b}_{j}|\varvec{Z}_{k})=\frac{1}{\sqrt{\left| 2\pi \varvec{P}_{i|j,k}^{aa}\right| \left| 2\pi \varvec{P}_{j|k}^{bb}\right| }}\times \nonumber \\&\mathrm {exp}\left\{ -\frac{1}{2}({\tilde{\varvec{a}}}_{i|k} -\varvec{K}_{i}^{a}{\tilde{\varvec{b}}}_{j|k})^{{T}}(\varvec{P}_{i|j,k}^{aa})^{-1} ({\tilde{\varvec{a}}}_{i|k}-\varvec{K}_{i}^{a}{\tilde{\varvec{b}}}_{j|k}) -\frac{1}{2}({\tilde{\varvec{b}}}_{j|k})^{{T}}(\varvec{P}_{j|k}^{bb})^{-1}{\tilde{\varvec{b}}}_{j|k}\right\} \nonumber \\&\qquad \quad =N(\varvec{a}_{i};{\hat{\varvec{a}}}_{i|j,k},\varvec{P}_{i|j,k}^{aa}) p(\varvec{b}_{j}|\varvec{Z}_{k}) \end{aligned}$$
(115)

Employing the Bayesian rule and using (115) yields

$$\begin{aligned} p(\varvec{a}_{i}|\varvec{b}_{j},\varvec{Z}_{k})=\frac{p(\varvec{a}_{i}, \varvec{b}_{j}|\varvec{Z}_{k})}{p(\varvec{b}_{j}|\varvec{Z}_{k})} =N(\varvec{a}_{i};{\hat{\varvec{a}}}_{i|j,k},\varvec{P}_{i|j,k}^{aa}) \end{aligned}$$
(116)

where \({\hat{\varvec{a}}}_{i|j,k}\) and \(\varvec{P}_{i|j,k}^{aa}\) are given by (106, 107). \(\square \)

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Huang, Y., Zhang, Y., Li, N. et al. Design of Gaussian Approximate Filter and Smoother for Nonlinear Systems with Correlated Noises at One Epoch Apart. Circuits Syst Signal Process 35, 3981–4008 (2016). https://doi.org/10.1007/s00034-016-0256-0

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