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A New Adaptive Kalman Filter with Inaccurate Noise Statistics

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Abstract

In this paper, a new adaptive Kalman filter is proposed for a linear Gaussian state-space model with inaccurate noise statistics based on the variational Bayesian (VB) approach. Both the prior joint probability density function (PDF) of the one-step prediction and corresponding prediction error covariance matrix and the joint PDF of the mean vector and covariance matrix of measurement noise are selected as Normal-inverse-Wishart (NIW), from which a new NIW-based hierarchical Gaussian state-space model is constructed. The state vector, the one-step prediction and corresponding prediction error covariance matrix, and the mean vector and covariance matrix of measurement noise are jointly estimated based on the constructed hierarchical Gaussian state-space model using the VB approach. Simulation results show that the proposed filter has better estimation accuracy than existing state-of-the-art adaptive Kalman filters.

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Authors and Affiliations

  1. Harbin Institute of Technology, No. 2, Yikuang Street, Nangang District, Harbin, 150080, People’s Republic of China

    Dingjie Xu & Zhemin Wu

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

    Yulong Huang

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  1. Dingjie Xu
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  2. Zhemin Wu
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  3. Yulong Huang
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Correspondence to Zhemin Wu.

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This work was supported by the National Natural Science Foundation of China under Grant Nos. 61573117, 61374208 and 61673128, and the Ph.D. Student Research and Innovation Fund of the Fundamental Research Founds for the Central Universities under Grant No. HEUGIP201706.

Appendices

Appendices

1.1 A. Proofs of (29)–(33)

Substituting \({\varvec{\theta }}={\hat{{\varvec{x}}}}_{k|k-1}\), \({\varvec{\theta }}={\varvec{r}}\) and (28) in (26), \(\log q^{(i+1)}({\hat{{\varvec{x}}}}_{k|k-1})\) and \(\log q^{(i+1)}({\varvec{r}})\) are written as

$$\begin{aligned}&\log q^{(i+1)}({\hat{{\varvec{x}}}}_{k|k-1})=-\frac{1}{2}\mathrm {E}^{(i)} \left[ \left( {\varvec{x}}_{k}-{\hat{{\varvec{x}}}}_{k|k-1}\right) ^{\mathrm {T}} {\varvec{P}}_{k|k-1}^{-1}\left( {\varvec{x}}_{k}-{\hat{{\varvec{x}}}}_{k|k-1}\right) \right] \nonumber \\&\quad -\,\frac{1}{2}\alpha _{k}\left( {\hat{{\varvec{x}}}}_{k|k-1}-{\varvec{u}}_{k}\right) ^{\mathrm {T}} \mathrm {E}^{(i)}\left[ {\varvec{P}}_{k|k-1}^{-1}\right] \left( {\hat{{\varvec{x}}}}_{k|k-1}-{\varvec{u}}_{k}\right) +c_{{\hat{{\varvec{x}}}}_{k|k-1}}, \end{aligned}$$
(81)
$$\begin{aligned}&\log q^{(i+1)}({\varvec{r}})=-\frac{1}{2}\mathrm {E}^{(i)}\left[ \left( {\varvec{z}}_{k} -{\varvec{H}}_{k}{\varvec{x}}_{k}-{\varvec{r}}\right) ^{\mathrm {T}}{\varvec{R}}^{-1} \left( {\varvec{z}}_{k}-{\varvec{H}}_{k}{\varvec{x}}_{k}-{\varvec{r}}\right) \right] \nonumber \\&\quad -\,\frac{1}{2}\beta _{k}\left( {\varvec{r}}-{\varvec{\lambda }}_{k}\right) ^{\mathrm {T}} \mathrm {E}^{(i)}\left[ {\varvec{R}}^{-1}\right] \left( {\varvec{r}}-{\varvec{\lambda }}_{k}\right) +c_{{\varvec{r}}}. \end{aligned}$$
(82)

The first expectations in (81)–(82) are calculated as follows

$$\begin{aligned}&\mathrm {E}^{(i)}\left[ \left( {\varvec{x}}_{k}-{\hat{{\varvec{x}}}}_{k|k-1} \right) ^{\mathrm {T}}{\varvec{P}}_{k|k-1}^{-1}\left( {\varvec{x}}_{k}-{\hat{{\varvec{x}}}}_{k|k-1} \right) \right] \nonumber \\&\quad =\mathrm {tr}\left\{ \mathrm {E}^{(i)}\left[ \left( {\varvec{x}}_{k}-{\hat{{\varvec{x}}}}_{k|k-1} \right) \left( {\varvec{x}}_{k}-{\hat{{\varvec{x}}}}_{k|k-1}\right) ^{\mathrm {T}}\right] \mathrm {E}^{(i)}\left[ {\varvec{P}}_{k|k-1}^{-1}\right] \right\} \nonumber \\&\quad =\mathrm {tr}\left\{ \mathrm {E}^{(i)}\left[ \left( {\varvec{x}}_{k}-{\hat{{\varvec{x}}}}_{k|k}^{(i)} +{\hat{{\varvec{x}}}}_{k|k}^{(i)}-{\hat{{\varvec{x}}}}_{k|k-1}\right) \left( {\varvec{x}}_{k} -{\hat{{\varvec{x}}}}_{k|k}^{(i)}+\right. \right. \right. \left. \left. \left. {\hat{{\varvec{x}}}}_{k|k}^{(i)}-{\hat{{\varvec{x}}}}_{k|k-1}\right) ^{\mathrm {T}}\right] \times \right. \nonumber \\&\left. \mathrm {E}^{(i)}\left[ {\varvec{P}}_{k|k-1}^{-1}\right] \right\} \nonumber \\&\quad =\left( {\hat{{\varvec{x}}}}_{k|k}^{(i)}-{\hat{{\varvec{x}}}}_{k|k-1}\right) ^{\mathrm {T}} \mathrm {E}^{(i)}\left[ {\varvec{P}}_{k|k-1}^{-1}\right] \left( {\hat{{\varvec{x}}}}_{k|k}^{(i)} -{\hat{{\varvec{x}}}}_{k|k-1}\right) +c_{{\hat{{\varvec{x}}}}_{k|k-1}}, \end{aligned}$$
(83)
$$\begin{aligned}&\mathrm {E}^{(i)}\left[ \left( {\varvec{z}}_{k}-{\varvec{H}}_{k}{\varvec{x}}_{k} -{\varvec{r}}\right) ^{\mathrm {T}}{\varvec{R}}^{-1}\left( {\varvec{z}}_{k}-{\varvec{H}}_{k} {\varvec{x}}_{k}-{\varvec{r}}\right) \right] \nonumber \\&\quad =\mathrm {tr}\{\left. \mathrm {E}^{(i)}\left[ \left( {\varvec{z}}_{k} -{\varvec{H}}_{k}{\varvec{x}}_{k}-{\varvec{r}}\right) \left( {\varvec{z}}_{k}-{\varvec{H}}_{k}{\varvec{x}}_{k} -{\varvec{r}}\right) ^{\mathrm {T}}\right] \mathrm {E}^{(i)}\left[ {\varvec{R}}^{-1}\right] \right\} \nonumber \\&\quad =\mathrm {tr}\{\mathrm {E}^{(i)}[({\varvec{z}}_{k}-{\varvec{H}}_{k} {\hat{{\varvec{x}}}}_{k|k}^{(i)}-{\varvec{r}}-{\varvec{H}}_{k}({\varvec{x}}_{k} -{\hat{{\varvec{x}}}}_{k|k}^{(i)}))({\varvec{z}}_{k}-{\varvec{H}}_{k}{\hat{{\varvec{x}}}}_{k|k}^{(i)} -{\varvec{r}}- \nonumber \\&{\varvec{H}}_{k}({\varvec{x}}_{k}-{\hat{{\varvec{x}}}}_{k|k}^{(i)}))^{\mathrm {T}}] \mathrm {E}^{(i)}\left[ {\varvec{R}}^{-1}\right] \} \nonumber \\&\quad =\left( {\varvec{z}}_{k}-{\varvec{H}}_{k}{\hat{{\varvec{x}}}}_{k|k}^{(i)}-{\varvec{r}} \right) ^{\mathrm {T}}\mathrm {E}^{(i)}\left[ {\varvec{R}}^{-1}\right] \left( {\varvec{z}}_{k}-{\varvec{H}}_{k}{\hat{{\varvec{x}}}}_{k|k}^{(i)}-{\varvec{r}}\right) +c_{{\varvec{r}}}. \end{aligned}$$
(84)

Substituting (83)–(84) in (81)–(82) and using (33) yields

$$\begin{aligned}&\log q^{(i+1)}({\hat{{\varvec{x}}}}_{k|k-1})=-\frac{1}{2} \left( {\hat{{\varvec{x}}}}_{k|k}^{(i)}-{\hat{{\varvec{x}}}}_{k|k-1}\right) ^{\mathrm {T}} \left[ {\varvec{\bar{P}}}_{k|k-1}^{(i)}\right] ^{-1}\left( {\hat{{\varvec{x}}}}_{k|k}^{(i)} -{\hat{{\varvec{x}}}}_{k|k-1}\right) \nonumber \\&\quad -\,\frac{1}{2}\alpha _{k}\left( {\hat{{\varvec{x}}}}_{k|k-1}-{\varvec{u}}_{k}\right) ^{\mathrm {T}} \left[ {\varvec{\bar{P}}}_{k|k-1}^{(i)}\right] ^{-1}\left( {\hat{{\varvec{x}}}}_{k|k-1}-{\varvec{u}}_{k} \right) +c_{{\hat{{\varvec{x}}}}_{k|k-1}}, \end{aligned}$$
(85)
$$\begin{aligned}&\log q^{(i+1)}({\varvec{r}})=-\frac{1}{2}\left( {\varvec{z}}_{k} -{\varvec{H}}_{k}{\hat{{\varvec{x}}}}_{k|k}^{(i)}-{\varvec{r}}\right) ^{\mathrm {T}} \left[ {\varvec{\bar{R}}}_{k}^{(i)}\right] ^{-1}\left( {\varvec{z}}_{k}-{\varvec{H}}_{k} {\hat{{\varvec{x}}}}_{k|k}^{(i)}-{\varvec{r}}\right) \nonumber \\&\quad -\,\frac{1}{2}\beta _{k}\left( {\varvec{r}}-{\varvec{\lambda }}_{k}\right) ^{\mathrm {T}} \left[ {\varvec{\bar{R}}}_{k}^{(i)}\right] ^{-1}\left( {\varvec{r}}-{\varvec{\lambda }}_{k}\right) +c_{{\varvec{r}}}. \end{aligned}$$
(86)

Utilizing (85)–(86), we can obtain (29)–(32).

1.2 B. Proofs of (34)–(43)

Exploiting \({\varvec{\theta }}={\varvec{P}}_{k|k-1}\), \({\varvec{\theta }}={\varvec{R}}\) and (28) in (26), \(\log q^{(i+1)}({\varvec{P}}_{k|k-1})\) and \(\log q^{(i+1)}({\varvec{R}})\) are calculated as

$$\begin{aligned}&\log q^{(i+1)}({\varvec{P}}_{k|k-1})=-\frac{1}{2}(\omega _{k}+n+3)\log \left| {\varvec{P}}_{k|k-1}\right| \nonumber \\&\quad -\,\frac{1}{2}\mathrm {E}^{(i)}\left[ \left( {\varvec{x}}_{k}-{\hat{{\varvec{x}}}}_{k|k-1} \right) ^{\mathrm {T}}{\varvec{P}}_{k|k-1}^{-1}\left( {\varvec{x}}_{k}-{\hat{{\varvec{x}}}}_{k|k-1} \right) \right] \nonumber \\&\quad -\,\frac{1}{2}\alpha _{k}\mathrm {E}^{(i)}\left[ \left( {\hat{{\varvec{x}}}}_{k|k-1} -{\varvec{u}}_{k}\right) ^{\mathrm {T}}{\varvec{P}}_{k|k-1}^{-1}({\hat{{\varvec{x}}}}_{k|k-1} -{\varvec{u}}_{k})\right] \nonumber \\&\quad -\,\frac{1}{2}\mathrm {tr}\left\{ {\varvec{\varSigma }}_{k}{\varvec{P}}_{k|k-1}^{-1}\right\} +c_{{\varvec{P}}_{k|k-1}}, \end{aligned}$$
(87)
$$\begin{aligned}&\log q^{(i+1)}({\varvec{R}})=-\frac{1}{2}(\nu _{k}+m+3)\log \left| {\varvec{R}}\right| \nonumber \\&\quad -\,\frac{1}{2}\mathrm {E}^{(i)}\left[ \left( {\varvec{z}}_{k}-{\varvec{H}}_{k}{\varvec{x}}_{k} -{\varvec{r}}\right) ^{\mathrm {T}}{\varvec{R}}^{-1}\left( {\varvec{z}}_{k}-{\varvec{H}}_{k}{\varvec{x}}_{k} -{\varvec{r}}\right) \right] \nonumber \\&\quad -\,\frac{1}{2}\beta _{k}\mathrm {E}^{(i)}\left[ \left( {\varvec{r}}-{\varvec{\lambda }}_{k} \right) ^{\mathrm {T}}{\varvec{R}}^{-1}\left( {\varvec{r}}-{\varvec{\lambda }}_{k}\right) \right] -\frac{1}{2}\mathrm {tr}\left\{ {\varvec{\varDelta }}_{k}{\varvec{R}}^{-1}\right\} +c_{{\varvec{R}}}. \end{aligned}$$
(88)

Substituting (40)–(43) in (87)–(88) gives

$$\begin{aligned} \log q^{(i+1)}({\varvec{P}}_{k|k-1})= & {} -\frac{1}{2}(\omega _{k}+n+3)\log \left| {\varvec{P}}_{k|k-1}\right| \nonumber \\&\quad -\,\frac{1}{2}\mathrm {tr}\left\{ \left( {\varvec{A}}_{k}^{(i+1)}+{\varvec{B}}_{k}^{(i+1)} +{\varvec{\varSigma }}_{k}\right) {\varvec{P}}_{k|k-1}^{-1}\right\} +c_{{\varvec{P}}_{k|k-1}},\qquad \end{aligned}$$
(89)
$$\begin{aligned} \log q^{(i+1)}({\varvec{R}})= & {} -\frac{1}{2}(\nu _{k}+m+3)\log \left| {\varvec{R}}\right| \nonumber \\&\quad -\,\frac{1}{2}\mathrm {tr}\left\{ \left( {\varvec{C}}_{k}^{(i+1)} +{\varvec{D}}_{k}^{(i+1)}+{\varvec{\varDelta }}_{k}\right) {\varvec{R}}^{-1}\right\} +c_{{\varvec{R}}}. \end{aligned}$$
(90)

According to (89)–(90), we can obtain (34)–(39).

1.3 C. Proofs of (44)–(47)

Using \({\varvec{\theta }}={\varvec{x}}_{k}\) and (28) in (26), \(\log q^{(i+1)}({\varvec{x}}_{k})\) can be formulated as

$$\begin{aligned} \log q^{(i+1)}({\varvec{x}}_{k})= & {} -\frac{1}{2}\mathrm {tr}\left\{ \mathrm {E}^{(i+1)} \left[ \left( {\varvec{z}}_{k}-{\varvec{H}}_{k}{\varvec{x}}_{k}-{\varvec{r}}\right) \times \right. \right. \left. \left. \left( {\varvec{z}}_{k}-{\varvec{H}}_{k}{\varvec{x}}_{k} -{\varvec{r}}\right) ^{\mathrm {T}}\right] \times \mathrm {E}^{(i+1)}\left[ {\varvec{R}}^{-1}\right] \right\} \nonumber \\&\quad -\,\frac{1}{2}\mathrm {tr}\left\{ \mathrm {E}^{(i+1)}\left[ \left( {\varvec{x}}_{k} -{\hat{{\varvec{x}}}}_{k|k-1}\right) \left( {\varvec{x}}_{k}-{\hat{{\varvec{x}}}}_{k|k-1} \right) ^{\mathrm {T}}\right] \right. \left. \mathrm {E}^{(i+1)} \left[ {\varvec{P}}_{k|k-1}^{-1}\right] \right\} +c_{{\varvec{x}}_{k}}, \nonumber \\ \end{aligned}$$
(91)

where the first and third expectations are calculated as

$$\begin{aligned}&\mathrm {E}^{(i+1)}\left[ \left( {\varvec{z}}_{k}-{\varvec{H}}_{k}{\varvec{x}}_{k} -{\varvec{r}}\right) \left( {\varvec{z}}_{k}-{\varvec{H}}_{k}{\varvec{x}}_{k} -{\varvec{r}}\right) ^{\mathrm {T}}\right] \nonumber \\&\quad =\,\mathrm {E}^{(i+1)} \left[ \left( {\varvec{z}}_{k}-{\varvec{H}}_{k}{\varvec{x}}_{k}-\right. \left. {\hat{{\varvec{\lambda }}}}_{k}^{(i+1)}+{\hat{{\varvec{\lambda }}}}_{k}^{(i+1)} -{\varvec{r}}\right) \left( {\varvec{z}}_{k}-{\varvec{H}}_{k}{\varvec{x}}_{k} -{\hat{{\varvec{\lambda }}}}_{k}^{(i+1)}+{\hat{{\varvec{\lambda }}}}_{k}^{(i+1)} -{\varvec{r}}\right) ^{\mathrm {T}}\right] \nonumber \\&\quad =\left( {\varvec{z}}_{k}-\right. \left. {\hat{{\varvec{\lambda }}}}_{k}^{(i+1)} -{\varvec{H}}_{k}{\varvec{x}}_{k}\right) \left( {\varvec{z}}_{k}-{\hat{{\varvec{\lambda }}}}_{k}^{(i+1)} -{\varvec{H}}_{k}{\varvec{x}}_{k}\right) ^{\mathrm {T}}+{\hat{{\varvec{\varOmega }}}}_{k}^{(i+1)}, \end{aligned}$$
(92)
$$\begin{aligned}&\mathrm {E}^{(i+1)}\left[ \left( {\varvec{x}}_{k}-{\hat{{\varvec{x}}}}_{k|k-1}\right) \left( {\varvec{x}}_{k}-{\hat{{\varvec{x}}}}_{k|k-1}\right) ^{\mathrm {T}}\right] \nonumber \\&\quad =\mathrm {E}^{(i+1)}\left[ \left( {\varvec{x}}_{k}-\right. \right. \left. \left. {\hat{{\varvec{u}}}}_{k}^{(i+1)}+{\hat{{\varvec{u}}}}_{k}^{(i+1)}-{\hat{{\varvec{x}}}}_{k|k-1}\right) \left( {\varvec{x}}_{k}-{\hat{{\varvec{u}}}}_{k}^{(i+1)}+{\hat{{\varvec{u}}}}_{k}^{(i+1)}-\right. \right. \left. \left. {\hat{{\varvec{x}}}}_{k|k-1}\right) ^{\mathrm {T}}\right] \nonumber \\&\quad =\left( {\varvec{x}}_{k}-{\hat{{\varvec{u}}}}_{k}^{(i+1)}\right) \left( {\varvec{x}}_{k} -{\hat{{\varvec{u}}}}_{k}^{(i+1)}\right) ^{\mathrm {T}}+{\hat{{\varvec{U}}}}_{k}^{(i+1)}. \end{aligned}$$
(93)

Employing (31) and (92)–(93) in (91) yields

$$\begin{aligned}&\log q^{(i+1)}({\varvec{x}}_{k})=-\frac{1}{2}\left( {\varvec{z}}_{k} -{\hat{{\varvec{\lambda }}}}_{k}^{(i+1)}-{\varvec{H}}_{k}{\varvec{x}}_{k}\right) ^{\mathrm {T}} \left[ {\varvec{\bar{R}}}_{k}^{(i)}\right] ^{-1}\left( {\varvec{z}}_{k} -{\hat{{\varvec{\lambda }}}}_{k}^{(i+1)}-{\varvec{H}}_{k}{\varvec{x}}_{k}\right) \nonumber \\&\quad -\,\frac{1}{2}\left( {\varvec{x}}_{k}-{\hat{{\varvec{u}}}}_{k}^{(i+1)}\right) ^{\mathrm {T}} \times \left[ {\varvec{\bar{P}}}_{k|k-1}^{(i)}\right] ^{-1}\left( {\varvec{x}}_{k} -{\hat{{\varvec{u}}}}_{k}^{(i+1)}\right) +c_{{\varvec{x}}_{k}}, \end{aligned}$$
(94)

According to (94), we can obtain (44)–(47), where (45)–(47) is given by the measurement update of the Kalman filter.

1.4 D. Proofs of (55)–(58)

Exploiting (29)–(30) and (44), the auxiliary parameters \({\varvec{A}}_{k}^{(i+1)}\), \({\varvec{B}}_{k}^{(i+1)}\), \({\varvec{C}}_{k}^{(i+1)}\) and \({\varvec{D}}_{k}^{(i+1)}\) in (40)–(43) can be, respectively, calculated as

$$\begin{aligned} {\varvec{A}}_{k}^{(i+1)}= & {} \mathrm {E}^{(i)}\left[ \left( {\varvec{x}}_{k} -{\hat{{\varvec{x}}}}_{k|k}^{(i)}+{\hat{{\varvec{x}}}}_{k|k}^{(i)}-{\hat{{\varvec{u}}}}_{k}^{(i+1)} +{\hat{{\varvec{u}}}}_{k}^{(i+1)}-\right. \right. \left. \left. {\hat{{\varvec{x}}}}_{k|k-1}\right) \left( {\varvec{x}}_{k}-{\hat{{\varvec{x}}}}_{k|k}^{(i)}\right. \right. \nonumber \\&\quad +\,\left. \left. {\hat{{\varvec{x}}}}_{k|k}^{(i)}-{\hat{{\varvec{u}}}}_{k}^{(i+1)} +{\hat{{\varvec{u}}}}_{k}^{(i+1)}-{\hat{{\varvec{x}}}}_{k|k-1}\right) ^{\mathrm {T}}\right] \nonumber \\= & {} {\varvec{P}}_{k|k}^{(i)}+\left( {\hat{{\varvec{x}}}}_{k|k}^{(i)}-{\hat{{\varvec{u}}}}_{k}^{(i+1)} \right) \left( {\hat{{\varvec{x}}}}_{k|k}^{(i)}-{\hat{{\varvec{u}}}}_{k}^{(i+1)}\right) ^{\mathrm {T}} +{\hat{{\varvec{U}}}}_{k}^{(i+1)}, \end{aligned}$$
(95)
$$\begin{aligned} {\varvec{B}}_{k}^{(i+1)}= & {} \alpha _{k}\mathrm {E}^{(i+1)} \left[ \left( {\hat{{\varvec{x}}}}_{k|k-1}-{\hat{{\varvec{u}}}}_{k}^{(i+1)}+{\hat{{\varvec{u}}}}_{k}^{(i+1)} -{\varvec{u}}_{k}\right) \right. \nonumber \\&\quad \times \left. \left( {\hat{{\varvec{x}}}}_{k|k-1}-{\hat{{\varvec{u}}}}_{k}^{(i+1)}+{\hat{{\varvec{u}}}}_{k}^{(i+1)} -{\varvec{u}}_{k}\right) ^{\mathrm {T}}\right] \nonumber \\= & {} \alpha _{k}{\hat{{\varvec{U}}}}_{k}^{(i+1)}+\alpha _{k}\left( {\hat{{\varvec{u}}}}_{k}^{(i+1)} -{\varvec{u}}_{k}\right) \left( {\hat{{\varvec{u}}}}_{k}^{(i+1)}-{\varvec{u}}_{k}\right) ^{\mathrm {T}}, \end{aligned}$$
(96)
$$\begin{aligned} {\varvec{C}}_{k}^{(i+1)}= & {} \mathrm {E}^{(i)}\left[ \left( {\varvec{z}}_{k}-{\varvec{H}}_{k} {\hat{{\varvec{x}}}}_{k|k}^{(i)}-{\hat{{\varvec{\lambda }}}}_{k}^{(i+1)}-{\varvec{H}}_{k}({\varvec{x}}_{k} -{\hat{{\varvec{x}}}}_{k|k}^{(i)})+{\hat{{\varvec{\lambda }}}}_{k}^{(i+1)}-{\varvec{r}}\right) \right. \nonumber \\&\quad \times \left( {\varvec{z}}_{k}-{\varvec{H}}_{k}{\hat{{\varvec{x}}}}_{k|k}^{(i)}-{\hat{{\varvec{\lambda }}}}_{k}^{(i+1)} -{\varvec{H}}_{k}({\varvec{x}}_{k}-{\hat{{\varvec{x}}}}_{k|k}^{(i)})\right. \left. \left. +{\hat{{\varvec{\lambda }}}}_{k}^{(i+1)}-{\varvec{r}}\right) ^{\mathrm {T}}\right] \nonumber \\= & {} {\varvec{H}}_{k}{\varvec{P}}_{k|k}^{(i)}{\varvec{H}}_{k}^{\mathrm {T}}+{\hat{{\varvec{\varOmega }}}}_{k}^{(i+1)} +\left( {\varvec{z}}_{k}-{\varvec{H}}_{k}{\hat{{\varvec{x}}}}_{k|k}^{(i)}-{\hat{{\varvec{\lambda }}}}_{k}^{(i+1)} \right) \left( {\varvec{z}}_{k}-{\varvec{H}}_{k}{\hat{{\varvec{x}}}}_{k|k}^{(i)}- {\hat{{\varvec{\lambda }}}}_{k}^{(i+1)}\right) ^{\mathrm {T}}, \end{aligned}$$
(97)
$$\begin{aligned} {\varvec{D}}_{k}^{(i+1)}= & {} \beta _{k}\mathrm {E}^{(i+1)}\left[ \left( {\varvec{r}} -{\hat{{\varvec{\lambda }}}}_{k}^{(i+1)}+{\hat{{\varvec{\lambda }}}}_{k}^{(i+1)} -{\varvec{\lambda }}_{k}\right) \right. \left. \left( {\varvec{r}}-{\hat{{\varvec{\lambda }}}}_{k}^{(i+1)} +{\hat{{\varvec{\lambda }}}}_{k}^{(i+1)}-{\varvec{\lambda }}_{k}\right) ^{\mathrm {T}}\right] \nonumber \\= & {} \beta _{k}{\hat{{\varvec{\varOmega }}}}_{k}^{(i+1)}+\beta _{k} \left( {\hat{{\varvec{\lambda }}}}_{k}^{(i+1)}-{\varvec{\lambda }}_{k}\right) \left( {\hat{{\varvec{\lambda }}}}_{k}^{(i+1)}-{\varvec{\lambda }}_{k}\right) ^{\mathrm {T}}. \end{aligned}$$
(98)

According to (95)–(98), we can obtain (55)–(58).

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Xu, D., Wu, Z. & Huang, Y. A New Adaptive Kalman Filter with Inaccurate Noise Statistics. Circuits Syst Signal Process 38, 4380–4404 (2019). https://doi.org/10.1007/s00034-019-01053-w

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