SCIENCE CHINA Information Sciences, Volume 64 , Issue 7 : 172206(2021) https://doi.org/10.1007/s11432-020-3135-y

## Recursive filtering for nonlinear systems subject to measurement outliers

• AcceptedOct 30, 2020
• PublishedMay 18, 2021
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### Acknowledgment

This work was supported in part by National Natural Science Foundation of China (Grant Nos. 61873058, 61933007) and in part by Natural Science Foundation of Heilongjiang Province of China (Grant No. ZD2019F001).

### Supplement

Appendix

Proof of Theorem th:1

By means of (1) and (5), the OSPE is expressed as \begin{align} e_{s+1|s} =x_{s+1}-x_{s+1|s} =f(x_{s})+\sum_{i=1}^{j}\alpha_{i,s}A_{i,s}x_{s}-f(x_{s|s})+B_{s}w_{s}. \tag{41} \end{align} Expanding $f(x_{s})$ by Taylor formula around $x_{s|s}~$, one has $$f(x_{s})=f(x_{s|s})+F_{s}e_{s|s}+o(|e_{s|s}|), \tag{42}$$ where $F_{s}$ and $o(|e_{s|s}|)$ denote the Jacobian matrix and the high-order terms of Taylor series expression, respectively. Following the literature 1), the $o(|e_{s|s}|)$ is equivalent to the following equation: $$o(|e_{s|s}|)=D_{s}\aleph_{s}G_{s}e_{s|s}, \tag{43}$$ where $D_{s}$ is a scaling matrix, $G_{s}$ is a degree of freedom function, and $\aleph_{s}$ represents the linearization errors. Consequently, one has (19).

Next, taking (2), (6) and (11) into consideration, the FE $e_{s+1|s+1}$ is calculated as follows: \begin{align} e_{s+1|s+1} &= x_{s+1}-x_{s+1|s+1} \\ &= x_{s+1}-x_{s+1|s}-L_{s+1}\text{sat}_{\sigma_{s+1} }\left ( y_{s+1}-C_{s+1}x_{s+1|s} \right ) \\ &= x_{s+1}-x_{s+1|s}-L_{s+1}\Xi_{s+1}(C_{s+1}x_{s+1}+v_{s+1}-C_{s+1}x_{s+1|s})+L_{s+1}(\Xi_{s+1}-I) \sigma _{s+1} \\ &=\left (I-L_{s+1}\Xi_{s+1}C_{s+1} \right )e_{s+1|s}+L_{s+1}(\Xi_{s+1}-I)\sigma _{s+1}-L_{s+1}\Xi_{s+1}v_{s+1}. \tag{44} \end{align} Finally, the proof is complete.

Calafiore G. Reliable localization using set-valued nonlinear filters. IEEE Trans Syst Man Cybern, 2005, 35: 189–197.

Proof of Theorem the2

First of all, we need to solve the unknown terms of (21). According to Lemma lemma3, one obtains $$(F_{s}+D_{s}\aleph_{s}G_{s})P_{s|s}(F_{s}+D_{s}\aleph_{s}G_{s})^{\rm T}\leq F_{s}(P_{s|s}^{-1}-\gamma G_{s}^{\rm T}G_{s})^{-1}F_{s}^{\rm T}+\gamma ^{-1}D_{s}D_{s}^{\rm T}. \tag{45}$$ Applying (12) to the other uncertain term of the (21) yields \begin{align} \sum_{i=1}^{j}A_{i,s}{\rm E}\{x_{s}x_{s}^{\rm T}\}A_{i,s}^{\rm T} &\leq \sum_{i=1}^{j}A_{i,s}{\rm E}\{(1+\delta_{1})e_{s|s}e_{s|s}^{\rm T}+(1+\delta_{1}^{-1})x_{s|s}x_{s|s}^{\rm T}\}A_{i,s}^{\rm T} \\ &=\sum_{i=1}^{j}A_{i,s}[(1+\delta_{1})P_{s|s}+(1+\delta_{1}^{-1})x_{s|s}x_{s|s}^{\rm T}]A_{i,s}^{\rm T}, \tag{46} \end{align} which leads to \begin{align} P_{s+1|s} \leq &\, F_{s}(P_{s|s}^{-1}-\gamma G_{s}^{\rm T}G_{s})^{-1}F_{s}^{\rm T}+\gamma ^{-1}D_{s}D_{s}^{\rm T}+B_{s}Q_{s}B_{s}^{\rm T} \\ &+\sum_{i=1}^{j}A_{i,s}[(1+\delta_{1})P_{s|s}+(1+\delta_{1}^{-1})x_{s|s}x_{s|s}^{\rm T}]A_{i,s}^{\rm T}. \tag{47} \end{align} Next, we shall handle the uncertainty on the right side of (22) in a similar way. Again, it follows from (12) that \begin{align} {\rm E}\{\Upsilon_{s+1}+\Upsilon_{s+1}^{\rm T}\} \leq &\,\delta_{2}(I-L_{s+1}\Xi_{s+1}C_{s+1})P_{s+1|s}(I-L_{s+1}\Xi_{s+1}C_{s+1})^{\rm T} \\ &+\delta_{2}^{-1}L_{s+1}(\Xi_{s+1}-I){\rm E}\{\sigma _{s+1}\sigma _{s+1}^{\rm T}\}(\Xi_{s+1}-I)^{\rm T}L_{s+1}^{\rm T}. \tag{48} \end{align} According to (8), (9), and (16), one has \begin{align} {\rm E}\{\sigma_{s+1}\sigma _{s+1}^{\rm T}\}&\leq {\rm E}\left\{\sum_{i=1}^{m}\sigma_{i,s+1}^{2}I\right\} \\ &={\rm E}\left\{\bar{\sigma}_{s+1}\sum_{i=1}^{m}\frac{1}{\pi_{i}}I\right\} \\ &={\rm E}\left\{[\lambda\bar{\sigma}_{s}+(C_{s+1}e_{s+1|s}+v_{s+1})^{\rm T}R(C_{s+1}e_{s+1|s}+v_{s+1})]\sum_{i=1}^{m}\frac{1}{\pi_{i}}I\right\} \\ &={\rm E}\left\{[\lambda\bar{\sigma}_{s}+\text{Tr}\{R(C_{s+1}P_{s+1|s}C_{s+1}^{\rm T}+Z_{s+1})\}]\sum_{i=1}^{m}\frac{1}{\pi_{i}}I\right\} \\ &\leq\left[\lambda^{s+1}\overline{\sigma}_{0}+\sum_{i=0}^{s}\lambda^{s-i}\text{Tr}\{R(C_{i+1}\Phi_{i+1|i}C_{i+1}^{\rm T}+Z_{i+1})\}\right]\sum_{i=1}^{m}\frac{1}{\pi_{i}}I. \tag{49} \end{align} Substituting (48) and (49) into (22) yields \begin{align} P_{s+1|s+1} \leq &\left ( 1+\delta_{2}\right )\left ( I-L_{s+1}\Xi_{s+1}C_{s+1} \right )P_{s+1|s} \left ( I-L_{s+1}\Xi_{s+1}C_{s+1} \right )^{\rm T} \\ &+(1+\delta_{2}^{-1})\Lambda_{s+1}L_{s+1}(\Xi_{s+1}-I)(\Xi_{s+1}-I)^{\rm T}L_{s+1}^{\rm T}+L_{s+1}\Xi_{s+1}Z_{s+1}\Xi_{s+1}^{\rm T}L_{s+1}^{\rm T}. \tag{50} \end{align} Then, by utilizing Lemma le:4, it follows from (25) and (50) that $$P_{s+1|s+1}\leq\Phi_{s+1|s+1}. \tag{51}$$

Finally, making the partial derivative of the trace of (25) about $L_{s+1}$ equal to zero, one has \begin{align} \frac{\partial \text{Tr}\left \{\Phi _{s+1|s+1} \right \}}{\partial L_{s+1}} =&-2\left ( 1+\delta_{2}\right )\left ( I-L_{s+1}\Xi_{s+1}C_{s+1}\right )\Phi_{s+1|s}C_{s+1}^{\rm T}\Xi _{s+1}^{\rm T} \\ &+2(1+\delta_{2}^{-1})\Lambda_{s+1}L_{s+1}(\Xi_{s+1}-I)(\Xi_{s+1}-I)^{\rm T}+2L_{s+1}\Xi_{s+1}Z_{s+1}\Xi_{s+1}^{\rm T} \\ =&\,0. \tag{52} \end{align} In view of (52), the optimal $L_{s+1}$ is easily calculated in (27). The proof is now complete.

Proof of Theorem the3

Define the quadratic function like the following: $$\mathcal{V}_{s}(e_{s|s})=e_{s|s}^{\rm T}\Phi _{s|s}^{-1}e_{s|s}. \tag{53}$$ Combining (19), (44) and $x_{s}~=~x_{s|s}~+~e_{s|s}$, one has \begin{align} e_{s+1|s+1}= &\,\mathfrak{H}_{s+1}\mathfrak{M}_{s}e_{s|s}+\sum_{i=1}^{j}\alpha _{i,s}\mathfrak{H}_{s+1}A_{i,s}(x_{s|s}+e_{s|s})+\mathfrak{H}_{s+1}B_{s}w_{s} \\ &+L_{s+1}(\Xi_{s+1}-I)\sigma _{s+1}-L_{s+1}\Xi _{s+1}v_{s+1}, \tag{54} \end{align} where $\mathfrak{H}_{s+1}\triangleq~I-L_{s+1}\Xi_{s+1}C_{s+1}$ and $\mathfrak{M}_{s}\triangleq~F_{s}+D_{s}\aleph_{s}G_{s}$.

Next, calculating $\mathcal{V}_{s+1}(e_{s+1|s+1})$ and taking the mathematical expectation, one has \begin{align} &{\rm E}\left \{ \mathcal{V}_{s+1}(e_{s+1|s+1})|e_{s|s} \right \} \\ &= {\rm E}\Bigg\{\sigma_{s+1}^{\rm T}\left ( \Xi _{s+1}-I \right )^{\rm T}L_{s+1}^{\rm T}\Phi _{s+1|s+1}^{-1}\mathfrak{H}_{s+1}\mathfrak{M}_{s}e_{s|s}+\sigma _{s+1}^{\rm T}\left ( \Xi _{s+1}-I \right )^{\rm T}L_{s+1}^{\rm T}\Phi _{s+1|s+1}^{-1}L_{s+1}\left ( \Xi_{s+1}-I \right )\sigma _{s+1} \\ & +\sum_{i=1}^{j}\sum_{l=1}^{j}\left ( x_{s|s}+e_{s|s} \right )^{\rm T}A_{i,s}^{\rm T}\mathfrak{H}_{s+1}^{\rm T}\Phi _{s+1|s+1}^{-1}\mathfrak{H}_{s+1}A_{l,s}\left ( x_{s|s}+e_{s|s} \right )+w_{s}^{\rm T}B_{s}^{\rm T}\mathfrak{H}_{s+1}^{\rm T}\Phi _{s+1|s+1}^{-1}\mathfrak{H}_{s+1}B_{s}w_{s} \\ & +e_{s|s}^{\rm T}\mathfrak{M}_{s}^{\rm T}\mathfrak{H}_{s+1}^{\rm T}\Phi _{s+1|s+1}^{-1}\mathfrak{H}_{s+1}\mathfrak{M}_{s}e_{s|s}+e_{s|s}^{\rm T}\mathfrak{M}_{s}^{\rm T}\mathfrak{H}_{s+1}^{\rm T}\Phi _{s+1|s+1}^{-1}L_{s+1}\left ( \Xi _{s+1}-I \right )\sigma _{s+1} \\ & +v_{s+1}^{\rm T}\Xi _{s+1}^{\rm T}L_{s+1}^{\rm T}\Phi _{s+1|s+1}^{-1}L_{s+1}\Xi _{s+1}v_{s+1}\Bigg\}. \tag{55} \end{align} To deal with the cross terms in (55), one has \begin{align} &{\rm E}\big\{e_{s|s}^{\rm T}\mathfrak{M}_{s}^{\rm T}\mathfrak{H}_{s+1}^{\rm T}\Phi _{s+1|s+1}^{-1}L_{s+1}\left ( \Xi _{s+1}-I \right )\sigma _{s+1}+\sigma _{s+1}^{\rm T}\left ( \Xi _{s+1}-I \right )^{\rm T}L_{s+1}^{\rm T}\Phi _{s+1|s+1}^{-1}\mathfrak{H}_{s+1}\mathfrak{M}_{s}e_{s|s} \big\} \\ &\leq {\rm E}\big\{e_{s|s}^{\rm T}\mathfrak{M}_{s}^{\rm T}\mathfrak{H}_{s+1}^{\rm T}\Phi _{s+1|s+1}^{-1}\mathfrak{H}_{s+1}\mathfrak{M}_{s}e_{s|s}+\sigma _{s+1}^{\rm T}\left ( \Xi _{s+1}-I \right )^{\rm T}L_{s+1}^{\rm T}\Phi _{s+1|s+1}^{-1}L_{s+1}\left ( \Xi _{s+1}-I \right )\sigma _{s+1}\big\}. \tag{56} \end{align} Similarly, the third item in (55) satisfies \begin{align} &{\rm E}\left\{\sum_{i=1}^{j}\sum_{l=1}^{j}\left ( x_{s|s}+e_{s|s} \right )^{\rm T}A_{i,s}^{\rm T}\mathfrak{H}_{s+1}^{\rm T}\Phi _{s+1|s+1}^{-1}\mathfrak{H}_{s+1}A_{l,s}\left ( x_{s|s}+e_{s|s} \right )\right\} \\ &\leq {\rm E}\left\{2\sum_{i=1}^{j}\sum_{l=1}^{j}e_{s|s}^{\rm T}A_{i,s}^{\rm T}\mathfrak{H}_{s+1}^{\rm T}\Phi _{s+1|s+1}^{-1}\mathfrak{H}_{s+1}A_{l,s}e_{s|s}+2\sum_{i=1}^{j}\sum_{l=1}^{j}x_{s|s}^{\rm T}A_{i,s}^{\rm T}\mathfrak{H}_{s+1}^{\rm T}\Phi _{s+1|s+1}^{-1}\mathfrak{H}_{s+1}A_{l,s}x_{s|s}\right\} \\ &\leq {\rm E}\left\{2j\sum_{i=1}^{j}e_{s|s}^{\rm T}A_{i,s}^{\rm T}\mathfrak{H}_{s+1}^{\rm T}\Phi _{s+1|s+1}^{-1}\mathfrak{H}_{s+1}A_{i,s}e_{s|s}+2j\sum_{i=1}^{j}x_{s|s}^{\rm T}A_{i,s}^{\rm T}\mathfrak{H}_{s+1}^{\rm T}\Phi _{s+1|s+1}^{-1}\mathfrak{H}_{s+1}A_{i,s}x_{s|s}\right\}. \tag{57} \end{align} Applying Lemma lemma3 to (24) and (25), one obtains \begin{align} &\Phi_{s+1|s}\geq\mathfrak{M}_{s}\Phi_{s|s}\mathfrak{M}_{s}^{\rm T}, \tag{58} \\ &\Phi_{s+1|s}\geq(1+\delta_{1})\sum\limits_{i=1}^{j}A_{i,s}\Phi_{s|s}A_{i,s}^{\rm T}, \tag{59} \\ &\Phi_{s+1|s+1}\geq(1+\delta_{2})\mathfrak{H}_{s+1}\Phi_{s+1|s}\mathfrak{H}_{s+1}^{\rm T}, \tag{60} \end{align} which follows from Lemma lemma7 that \begin{align} &\mathfrak{M}_{s}^{\rm T}\Phi_{s+1|s}^{-1}\mathfrak{M}_{s}\leq\Phi_{s|s}^{-1}, \tag{61} \\ &\sum\limits_{i=1}^{j}A_{i,s}^{\rm T}\Phi_{s+1|s}^{-1}A_{i,s}\leq\frac{j}{1+\delta_{1}}\Phi_{s|s}^{-1}, \tag{62} \\ &\mathfrak{H}_{s+1}^{\rm T}\Phi_{s+1|s+1}^{-1}\mathfrak{H}_{s+1}\leq\frac{1}{1+\delta_{2}}\Phi_{s+1|s}^{-1}. \tag{63} \end{align} Based on Lemma lemma5, one has \begin{align} {\rm E}\{v_{s+1}^{\rm T}L_{s+1}^{\rm T}\Phi _{s+1|s+1}^{-1}L_{s+1}v_{s+1}\} &=\text{Tr}\big\{{\rm E}\{L_{s+1}^{\rm T}\Phi _{s+1|s+1}^{-1}L_{s+1}v_{s+1}v_{s+1}^{\rm T}\}\big\} \\ &=\text{Tr}\big\{{\rm E}\{L_{s+1}^{\rm T}\Phi _{s+1|s+1}^{-1}L_{s+1}Z_{s+1}\}\big\} \\ &=\text{Tr}\big\{L_{s+1}^{\rm T}\Phi _{s+1|s+1}^{-1}L_{s+1}Z_{s+1}\big\}. \tag{64} \end{align} Similarly, one obtains $${\rm E}\{w_{s}^{\rm T}B_{s}^{\rm T}\Phi _{s+1|s}^{-1}B_{s}w_{s}\}=\text{Tr}\big\{B_{s}^{\rm T}\Phi _{s+1|s}^{-1}B_{s}Q_{s}\big\}, \tag{65}$$ and \begin{align} {\rm E}\{ \sigma _{s+1}^{\rm T}L_{s+1}^{\rm T}\Phi _{s+1|s+1}^{-1}L_{s+1}\sigma _{s+1}\} &=\text{Tr}\{L_{s+1}^{\rm T}\Phi _{s+1|s+1}^{-1}L_{s+1}{\rm E}\{\sigma _{s+1}\sigma _{s+1}^{\rm T}\}\} \\ &\leq\text{Tr}\{L_{s+1}^{\rm T}\Phi _{s+1|s+1}^{-1}L_{s+1}\Lambda_{s+1}\}. \tag{66} \end{align} Substituting (56)–(66) into (55) yields \begin{align} &{\rm E}\left \{ \mathcal{V}_{s+1}(e_{s+1|s+1})|e_{s|s} \right \} \\ &\leq \frac{2}{1+\delta_{2}}e_{s|s}^{\rm T}\mathfrak{M}_{s}^{\rm T}\Phi _{s+1|s}^{-1}\mathfrak{M}_{s}e_{s|s}+{\rm E}\Bigg\{\frac{1}{1+\delta_{2}}w_{s}^{\rm T}B_{s}^{\rm T}\Phi _{s+1|s}^{-1}B_{s}w_{s}+v_{s+1}^{\rm T}L_{s+1}^{\rm T}\Phi _{s+1|s+1}^{-1}L_{s+1}v_{s+1} \\ & + \frac{2j}{1 + \delta_{2}}\sum_{i=1}^{j}e_{s|s}^{\rm T}A_{i,s}^{\rm T}\Phi _{s+1|s}^{-1}A_{i,s}e_{s|s} + \frac{2j}{1 + \delta_{2}}\sum_{i=1}^{j}x_{s|s}^{\rm T}A_{i,s}^{\rm T}\Phi _{s+1|s}^{-1}A_{i,s}x_{s|s} + 2\sigma _{s+1}^{\rm T}L_{s+1}^{\rm T}\Phi _{s+1|s+1}^{-1}L_{s+1}\sigma _{s+1}\Bigg\} \\ &\leq \varrho e_{s|s}^{\rm T}\Phi _{s|s}^{-1}e_{s|s}+\kappa, \tag{67} \end{align} where $$\varrho\triangleq\frac{2j^{2}+2(1+\delta_{1})}{(1+\delta_{2})(1+\delta_{1})},$$ and \begin{aligned} \kappa\triangleq&\,\frac{1}{1+\delta_{2}}\text{Tr}\big\{B_{s}^{\rm T}\Phi _{s+1|s}^{-1}B_{s}Q_{s}\big\}+\text{Tr}\big\{L_{s+1}^{\rm T}\Phi _{s+1|s+1}^{-1}L_{s+1}Z_{s+1}\big\} \\ &+\frac{2j}{1 + \delta_{2}}\sum_{i=1}^{j}x_{s|s}^{\rm T}A_{i,s}^{\rm T}\Phi _{s+1|s}^{-1}A_{i,s}x_{s|s} +2\text{Tr}\{L_{s+1}^{\rm T}\Phi _{s+1|s+1}^{-1}L_{s+1}\Lambda_{s+1}\}. \end{aligned} \tag{68}

Furthermore, by means of the properties of the conditional expectations, one has $${\rm E}\left \{ \mathcal{V}_{s+1}(e_{s+1|s+1})\right \}\leq \varrho \mathcal{V}_{s}(e_{s|s})+\kappa. \tag{69}$$

As can be seen from (8), (9), (61), and (63) that $\Phi_{s+1|s}^{-1}$, $\Phi_{s+1|s+1}^{-1}$, $L_{s+1}$, and $\sigma~_{s+1}$ are all bounded. At this point, according to Lemma lemma6 and condition (40), the MSEB of FE is confirmed which completes the proof.

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• Figure 1

(Color online) Actual states $x_{s}$ and their estimates. (a) Actual state $x_{s}^{(1)}$ and its estimate $\hat{x}_{s}^{(1)}$; (b) actual state $x_{s}^{(2)}$ and its estimate $\hat{x}_{s}^{(2)}$.

• Figure 2

(Color online) Dynamic saturation levels of (a) $\sigma_{1}$ and (b) $\sigma_{2}$.

• Figure 3

(Color online) (a) $\text{lg}(\text{FECM})$ and (b) $\text{lg}(\text{MSEB})$ of FE and their UB.

• Figure 4

(Color online) RMSEs of (a) $x_{s}^{(1)}$ and (b) $x_{s}^{(2)}$.

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