Abstract
This paper investigates the outlier-resistant variance-constrained \(H_{\infty }\) state estimation problem for a class of discrete time-varying recurrent neural networks with randomly occurring deception attacks. The randomly occurring deception attacks are modeled by a series of random variables satisfying the Bernoulli distribution with known probability. In addition, the saturation function is introduced to reduce the negative impact from the measurement outliers onto the estimation performance. The objective of this paper is to propose an outlier-resistant finite-horizon state estimation scheme without utilizing the augmentation method such that, in the presence of measurement outliers and randomly occurring deception attacks, some sufficient criteria are obtained ensuring both the desired \(H_{\infty }\) performance index and the error variance boundedness. Finally, a numerical example is used to illustrate the feasibility of the presented outlier-resistant variance-constrained \(H_{\infty }\) state estimation algorithm.
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Acknowledgements
This work was supported in part by the National Natural Science Foundation of China under Grant 12171124 and 72001059, the Natural Science Foundation of Heilongjiang Province of China under Grant ZD2022F003, the Heilongjiang Provincial Key Laboratory of Complex Intelligent System and Integration of China under Grant HPKL-CICS-202203, the Postdoctoral Science Foundation of Heilongjiang Province of China under Grant LBH-Z22199, the Fundamental Research Funds in Heilongjiang Provincial Universities of China under Grant 135509121, the Educational Research Project of the Qiqihar University of China under Grant YB201904, and the Alexander von Humboldt Foundation of Germany.
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Appendices
Appendix A \(H_{\infty }\) Performance analysis
Proof of Theorem 1:
Define
To proceed, considering the EE dynamical system (6), we can get
where \(\bar{\sigma }_{k}=\sigma [D_{k}e_{k}-\tilde{\beta }_{k}D_{k}(\hat{x}_{k}+e_{k})-\bar{\beta }D_{k}(\hat{x}_{k}+e_{k})+\tilde{\beta }_{k}\xi _{k}+\bar{\beta }\xi _{k}].\)
Using the fundamental inequality \(2x^{T}Py\le x^{T}Px+y^{T}Py\) \((P>0)\), we can get the following results
Adding the zero term \(\tilde{z}^{T}_{k}\tilde{z}_{k}-\gamma ^{2}v_{k}^{T}U_{\varphi }v_{k}-\tilde{z}^{T}_{k}\tilde{z}_{k}+\gamma ^{2}v_{k}^{T}U_{\varphi }v_{k}\) to \(\mathbb {E}\big \{\bar{L}_{k}\big \}\) yields
where
with \(\Psi _{66}\) defined below (12).
According to Lemma 1, the following form can be obtained
where \(\Psi\) is defined in (12).
Summarizing both sides of (A6) from 0 to \(N-1\) on k, we obtain
Furthermore, we derive the following form
Noting \(\Psi <0\), \(Q_{N}>0\) and \(Q_{0}\le \gamma ^{2}U_{\phi }\), it follows that \(J_{1}<0\). \(\square\)
Appendix B Boundedness analysis of error variance
Proof of Theorem 2
According to (7), we can calculate the EE covariance matrix \(X_{k}\) as follows:
where \(\bar{\sigma }_{k}=\sigma [D_{k}e_{k}-\tilde{\beta }_{k}D_{k}(\hat{x}_{k}+e_{k})-\bar{\beta }D_{k}(\hat{x}_{k}+e_{k})+\tilde{\beta }_{k}\xi _{k}+\bar{\beta }\xi _{k}].\)
Using the inequality \(xy^{T}+yx^{T}\le xx^{T}+yy^{T}\), it can be obtained
It follows from Lemma 2 that
where Y is defined in (14).
According to (5), the following results can be obtained by calculation
Noticing \(x^{T}y+y^{T}x<\varpi x^{T}x+\frac{1}{\varpi }y^{T}y\), we can derive that
where \(0<\varpi <\frac{2}{1+\bar{g}+3\bar{\beta }+3\bar{g}\bar{\beta }}\).
Based on the above derivation results, we can get
where \(\iota _{1}\), \(\iota _{2}\) and \(\iota _{3}\) are defined in (14). Furthermore, it can be obtained that
According to the feature of the trace, one has
Combining (B9) with (B10) results in
It is easy to get that \(G_{0}\ge X_{0}\). Letting \(G_{k}\ge X_{k}\), the following inequality can be derived as
Then, from (13) and (B11), we obtain
The proof is complete. \(\square\)
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Gao, Y., Hu, J., Yu, H. et al. Outlier-resistant variance-constrained \(\mathit{H}_{\infty }\) state estimation for time-varying recurrent neural networks with randomly occurring deception attacks. Neural Comput & Applic 35, 13261–13273 (2023). https://doi.org/10.1007/s00521-023-08419-x
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DOI: https://doi.org/10.1007/s00521-023-08419-x