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

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  • ReceivedJun 4, 2020
  • AcceptedOct 30, 2020
  • PublishedMay 18, 2021



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).



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 \begin{equation} f(x_{s})=f(x_{s|s})+F_{s}e_{s|s}+o(|e_{s|s}|), \tag{42}\end{equation} 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: \begin{equation} o(|e_{s|s}|)=D_{s}\aleph_{s}G_{s}e_{s|s}, \tag{43}\end{equation} 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 \begin{equation} (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}\end{equation} 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 \begin{equation} P_{s+1|s+1}\leq\Phi_{s+1|s+1}. \tag{51}\end{equation}

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: \begin{equation} \mathcal{V}_{s}(e_{s|s})=e_{s|s}^{\rm T}\Phi _{s|s}^{-1}e_{s|s}. \tag{53}\end{equation} 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 \begin{equation} {\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}\end{equation} 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 \begin{equation} \varrho\triangleq\frac{2j^{2}+2(1+\delta_{1})}{(1+\delta_{2})(1+\delta_{1})},\end{equation} and \begin{equation}\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}\end{equation}

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

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.


[1] Kalman R E. A New Approach to Linear Filtering and Prediction Problems. J Basic Eng, 1960, 82: 35-45 CrossRef Google Scholar

[2] Wan E A, Merwe R V D. The unscented Kalman filter for nonlinear estimation. In: Proceedings of the IEEE 2000 Adaptive Systems for Signal Processing, Communications, and Control Symposium, 2000. Google Scholar

[3] Wang X, He X, Bao Y. Parameter estimates of Heston stochastic volatility model with MLE and consistent EKF algorithm. Sci China Inf Sci, 2018, 61: 042202 CrossRef Google Scholar

[4] Li Q, Shen B, Wang Z. Event-triggered H state estimation for state-saturated complex networks subject to quantization effects and distributed delays. J Franklin Institute, 2018, 355: 2874-2891 CrossRef Google Scholar

[5] Appl 2017, 11: 2370--2382. Google Scholar

[6] Zhao S, Shmaliy Y S, Shi P. Fusion Kalman/UFIR Filter for State Estimation With Uncertain Parameters and Noise Statistics. IEEE Trans Ind Electron, 2017, 64: 3075-3083 CrossRef Google Scholar

[7] Caballero-águila R, Hermoso-Carazo A, Linares-Pérez J. New distributed fusion filtering algorithm based on covariances over sensor networks with random packet dropouts. Int J Syst Sci, 2017, 48: 1805-1817 CrossRef ADS Google Scholar

[8] S J Julier, J K Uhlmann. New extension of the Kalman filter to nonlinear systems. Signal processing, sensor fusion, and target recognition VI. International Society for Optics and Photonics, 1997, 3068: 182-193. doi: 10.1117/12.280797. Google Scholar

[9] Bai W, Xue W, Huang Y. On extended state based Kalman filter design for a class of nonlinear time-varying uncertain systems. Sci China Inf Sci, 2018, 61: 042201 CrossRef Google Scholar

[10] Mao W, Deng F, Wan A. Robust H 2/H global linearization filter design for nonlinear stochastic time-varying delay systems. Sci China Inf Sci, 2016, 59: 032204 CrossRef Google Scholar

[11] Liu Y J, Tong S. Barrier Lyapunov functions for Nussbaum gain adaptive control of full state constrained nonlinear systems. Automatica, 2017, 76: 143-152 CrossRef Google Scholar

[12] Hu J, Wang Z, Shen B. Quantised recursive filtering for a class of nonlinear systems with multiplicative noises and missing measurements. Int J Control, 2013, 86: 650-663 CrossRef ADS Google Scholar

[13] Rajasekaran P, Satyanarayana N, Srinath M. Optimum Linear Estimation of Stochastic Signals in the Presence of Multiplicative Noise. IEEE Trans Aerosp Electron Syst, 1971, AES-7: 462-468 CrossRef ADS Google Scholar

[14] Dong H, Wang Z, Ding S X. Event-Based $H_{\infty}$ Filter Design for a Class of Nonlinear Time-Varying Systems With Fading Channels and Multiplicative Noises. IEEE Trans Signal Process, 2015, 63: 3387-3395 CrossRef ADS Google Scholar

[15] Liu W. Optimal Estimation for Discrete-Time Linear Systems in the Presence of Multiplicative and Time-Correlated Additive Measurement Noises. IEEE Trans Signal Process, 2015, 63: 4583-4593 CrossRef ADS Google Scholar

[16] Zhang L, Zhu Y, Shi P. Resilient Asynchronous H Filtering for Markov Jump Neural Networks With Unideal Measurements and Multiplicative Noises.. IEEE Trans Cybern, 2015, 45: 2840-2852 CrossRef PubMed Google Scholar

[17] Li Y, Karimi H R, Zhong M. Fault Detection for Linear Discrete Time-Varying Systems With Multiplicative Noise: The Finite-Horizon Case. IEEE Trans Circuits Syst I, 2018, 65: 3492-3505 CrossRef Google Scholar

[18] Appl 2016, 10: 1161--1169. Google Scholar

[19] Yang F, Dong H, Wang Z. A new approach to non-fragile state estimation for continuous neural networks with time-delays. Neurocomputing, 2016, 197: 205-211 CrossRef Google Scholar

[20] Gao H, Han F, Jiang B. Recursive filtering for time-varying systems under duty cycle scheduling based on collaborative prediction. J Franklin Institute, 2020, 357: 13189-13204 CrossRef Google Scholar

[21] Wang C, Han F, Zhang Y. An SAE-based resampling SVM ensemble learning paradigm for pipeline leakage detection. Neurocomputing, 2020, 403: 237-246 CrossRef Google Scholar

[22] Li J, Dong H, Wang Z. Partial-Neurons-Based Passivity-Guaranteed State Estimation for Neural Networks With Randomly Occurring Time Delays. IEEE Trans Neural Netw Learning Syst, 2020, 31: 3747-3753 CrossRef Google Scholar

[23] Gao Y, Luo W, Liu J. Integral sliding mode control design for nonlinear stochastic systems under imperfect quantization. Sci China Inf Sci, 2017, 60: 120206 CrossRef Google Scholar

[24] Li X, Han F, Hou N. Set-membership filtering for piecewise linear systems with censored measurements under Round-Robin protocol. Int J Syst Sci, 2020, 51: 1578-1588 CrossRef ADS Google Scholar

[25] Peng H, Lu R, Xu Y. Dissipative non-fragile state estimation for Markovian complex networks with coupling transmission delays. Neurocomputing, 2018, 275: 1576-1584 CrossRef Google Scholar

[26] Rakkiyappan R, Sivaranjani K. Sampled-data synchronization and state estimation for nonlinear singularly perturbed complex networks with time-delays. NOnlinear Dyn, 2016, 84: 1623-1636 CrossRef Google Scholar

[27] Zhou D, Qin L, He X. Distributed sensor fault diagnosis for a formation system with unknown constant time delays. Sci China Inf Sci, 2018, 61: 112205 CrossRef Google Scholar

[28] Meinhold R J, Singpurwalla N D. Robustification of Kalman Filter Models. J Am Statistical Association, 1989, 84: 479-486 CrossRef Google Scholar

[29] Huang Y, Zhang Y, Xu B. A New Outlier-Robust Student's t Based Gaussian Approximate Filter for Cooperative Localization. IEEE/ASME Trans Mechatron, 2017, 22: 2380-2386 CrossRef Google Scholar

[30] De Palma D, Indiveri G. Output outlier robust state estimation. Int J Adapt Control Signal Process, 2017, 31: 581-607 CrossRef Google Scholar

[31] Alessandri A, Zaccarian L. Results on stubborn Luenberger observers for linear time-invariant plants. In: Proceedings of European Control Conference (ECC), Linz, 2015. 2920--2925. Google Scholar

[32] Alessandri A, Zaccarian L. Stubborn state observers for linear time-invariant systems. Automatica, 2018, 88: 1-9 CrossRef Google Scholar

[33] Gao H, Dong H, Wang Z. An Event-Triggering Approach to Recursive Filtering for Complex Networks With State Saturations and Random Coupling Strengths.. IEEE Trans Neural Netw Learning Syst, 2020, 31: 4279-4289 CrossRef PubMed Google Scholar

[34] Li W, Sun J, Jia Y. Variance-constrained state estimation for nonlinear complex networks with uncertain coupling strength. Digital Signal Processing, 2017, 67: 107-115 CrossRef Google Scholar

[35] Wang Y, Xie L, de Souza C E. Robust control of a class of uncertain nonlinear systems. Syst Control Lett, 1992, 19: 139-149 CrossRef Google Scholar

[36] Wang L, Wang Z, Huang T. An Event-Triggered Approach to State Estimation for a Class of Complex Networks With Mixed Time Delays and Nonlinearities.. IEEE Trans Cybern, 2016, 46: 2497-2508 CrossRef PubMed Google Scholar

[37] Reif K, Gunther S, Yaz E. Stochastic stability of the discrete-time extended Kalman filter. IEEE Trans Automat Contr, 1999, 44: 714-728 CrossRef Google Scholar

[38] S. Liu. Schur Complement. In: Encyclopedia of Statistical Sciences. Hoboken: John Wiley & Sons,Inc., 2008. Google Scholar


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