Outlier-resistant variance-constrained $\mathit{H}_{\infty }$ state estimation for time-varying recurrent neural networks with randomly occurring deception attacks

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.

Appendices

Appendix A $H_{\infty }$ Performance analysis

Proof of Theorem 1:

Define

$$\begin{aligned} \bar{L}_{k}\,=\, & e_{k+1}^{T}Q_{k+1}e_{k+1}-e_{k}^{T}Q_{k}e_{k}. \end{aligned}$$

(A1)

To proceed, considering the EE dynamical system (6), we can get

$$\begin{aligned} \mathbb {E}\{\bar{L}_{k}\}\,=\, & \mathbb {E}\left\{ e_{k}^{T}A_{k}^{T}Q_{k+1}A_{k}e_{k} +f^{T}(e_{k}\right. \nonumber \\{} & {} \left. +\,\hat{x}_{k})B_{k}^{T}Q_{k+1}B_{k}f(e_{k}+\hat{x}_{k})+f^{T}(\hat{x}_{k})B_{k}^{T}\right. \nonumber \\{} & {} \left. \times \, Q_{k+1}B_{k}f(\hat{x}_{k})+v_{k}^{T}C_{k}^{T}Q_{k+1}C_{k}v_{k}\right. \nonumber \\{} & {} \left. +\,\bar{\sigma }_{k}^{T}K_{k}^{T}Q_{k+1}K_{k}\bar{\sigma }_{k}+2e_{k}^{T}A_{k}^{T}Q_{k+1} \right. \nonumber \\{} & {} \left. \times \, B_{k}f(e_{k}+\hat{x}_{k})-2e_{k}^{T}A_{k}^{T}Q_{k+1}K_{k}\bar{\sigma }_{k}-2f^{T}(e_{k}\right. \nonumber \\{} & {} \left. +\,\hat{x}_{k})B_{k}^{T}Q_{k+1}K_{k}\bar{\sigma }_{k} \right. \nonumber \\{} & {} \left. -\,2e_{k}^{T}A_{k}^{T}Q_{k+1}B_{k}f(\hat{x}_{k})-2f^{T}(e_{k}\right. \nonumber \\{} & {} \left. +\,\hat{x}_{k})B_{k}^{T}Q_{k+1}B_{k}f(\hat{x}_{k})+2\bar{\sigma }_{k}^{T}K_{k}^{T}\right. \nonumber \\{} & {} \left. \times \,Q_{k+1}B_{k}f(\hat{x}_{k})-e_{k}^{T}Q_{k}e_{k}\right\} , \end{aligned}$$

(A2)

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

$$\begin{aligned} \mathbb {E}\{\bar{L}_{k}\}\le & {} \mathbb {E}\left\{ 3e_{k}^{T}A_{k}^{T}Q_{k+1}A_{k}e_{k}\right. \nonumber \\{} & {} \left. +\,3f^{T}(e_{k}+\hat{x}_{k})B_{k}^{T}Q_{k+1}B_{k}f(e_{k}+\hat{x}_{k})\right. \nonumber \\{} & {} \left. +\,4f^{T}(\hat{x}_{k})B_{k}^{T}Q_{k+1}B_{k}f(\hat{x}_{k})\right. \nonumber \\{} & {} \left. +\,v_{k}^{T}C_{k}^{T}Q_{k+1}C_{k}v_{k}+4\bar{\sigma }_{k}^{T}K_{k}^{T}Q_{k+1}\right. \nonumber \\{} & {} \left. \times \, K_{k}\bar{\sigma }_{k}+2e_{k}^{T}A_{k}^{T}Q_{k+1}B_{k}f(e_{k}+\hat{x}_{k})-e_{k}^{T}Q_{k}e_{k}\right\} . \end{aligned}$$

(A3)

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

$$\begin{aligned} \mathbb {E}\{\bar{L}_{k}\}\le & {} \mathbb {E}\left\{ \begin{bmatrix} \Phi _{k}^{T} &{} v_{k}^{T} \\ \end{bmatrix} \tilde{\Psi }\begin{bmatrix} \Phi _{k} \\ v_{k} \\ \end{bmatrix} -\tilde{z}^{T}_{k}\tilde{z}_{k}+\gamma ^{2}v_{k}^{T}U_{\varphi }v_{k}\right\} , \end{aligned}$$

(A4)

where

$$\begin{aligned} \Phi _{k}\,=\, & \begin{bmatrix} e_{k}^{T} &{} f^{T}(e_{k}+\hat{x}_{k})&{} 1 &{}\bar{\sigma }_{k}^{T}&{}\xi _{k}^{T}\\ \end{bmatrix}^{T},\nonumber \\ \tilde{\Psi }\,=\, & \begin{bmatrix} \tilde{\Psi }_{11} &{} A_{k}^{T}Q_{k+1}B_{k} &{} 0&{} 0&{} 0&{} 0\\ *&{} 3B_{k}^{T}Q_{k+1}B_{k} &{} 0 &{} 0 &{} 0&{} 0\\ *&{} *&{} \tilde{\Psi }_{33} &{} 0 &{} 0&{} 0\\ *&{} *&{} *&{} 4K_{k}^{T}Q_{k+1}K_{k}&{} 0&{} 0\\ *&{} *&{} *&{} *&{} 0&{} 0\\ *&{} *&{} *&{} *&{}*&{} \Psi _{66}\\ \end{bmatrix},\nonumber \\ \tilde{\Psi }_{11}\,=\, & 3A_{k}^{T}Q_{k+1}A_{k}+M_{k}^{T}M_{k}-Q_{k},\nonumber \\ \tilde{\Psi }_{33}\,=\, & 4f^{T}(\hat{x}_{k})B_{k}^{T}Q_{k+1}B_{k}f(\hat{x}_{k}) \end{aligned}$$

(A5)

with $\Psi _{66}$ defined below (12).

According to Lemma 1, the following form can be obtained

$$\begin{aligned} \mathbb {E}\{\bar{L}_{k}\}\le & {} \mathbb {E}\left\{ \begin{bmatrix} \Phi _{k}^{T} &{} v_{k}^{T} \\ \end{bmatrix} \tilde{\Psi }\begin{bmatrix} \Phi _{k} \\ v_{k} \\ \end{bmatrix} \right. \nonumber \\{} & {} \left. -\,\tilde{z}^{T}_{k}\tilde{z}_{k}+\gamma ^{2}v_{k}^{T}U_{\varphi }v_{k}-[e_{k}^{T}R_{1k}e_{k}+2e_{k}^{T}R_{2k}\right. \nonumber \\{} & {} \left. \times \, f(e_{k}+\hat{x}_{k})+2e_{k}^{T}R_{3k}^{T}-2f^{T}(e_{k}+\hat{x}_{k})f(\hat{x}_{k})\right. \nonumber \\{} & {} \left. +\,f^{T}(e_{k}+\hat{x}_{k})f(e_{k}+\hat{x}_{k})\right. \nonumber \\{} & {} \left. +\,f^{T}(\hat{x}_{k})f(\hat{x}_{k})]-\lambda \left\{ \bar{\sigma }_{k}^{T}\bar{\sigma }_{k}-\bar{g}[D_{k}e_{k}\right. \right. \nonumber \\{} & {} \left. \left. +\,\tilde{\beta }_{k}D_{k}(e_{k}+\hat{x}_{k})-\bar{\beta }D_{k}(e_{k}+\hat{x}_{k})\right. \right. \nonumber \\{} & {} \left. \left. +\,\tilde{\beta }_{k}\xi _{k}+\bar{\beta }\xi _{k}]^{T}\bar{\sigma }_{k}-\bar{\sigma }_{k}^{T}[D_{k}e_{k}\right. \right. \nonumber \\{} & {} \left. \left. +\,\tilde{\beta }_{k}D_{k}(e_{k}+\hat{x}_{k})-\bar{\beta }D_{k}(e_{k}+\hat{x}_{k})+\tilde{\beta }_{k}\xi _{k}\right. \right. \nonumber \\{} & {} \left. \left. +\,\bar{\beta }\xi _{k}]+\bar{g}[D_{k}e_{k}+\tilde{\beta }_{k}D_{k}(e_{k}+\hat{x}_{k})\right. \right. \nonumber \\{} & {} \left. \left. -\,\bar{\beta }D_{k}(e_{k}+\hat{x}_{k})+\tilde{\beta }_{k}\xi _{k}+\bar{\beta }\xi _{k}]^{T}\right. \right. \nonumber \\{} & {} \left. \left. \times \,[D_{k}e_{k}+\tilde{\beta }_{k}D_{k}(e_{k}+\hat{x}_{k})-\bar{\beta }D_{k}(e_{k}\right. \right. \nonumber \\{} & {} \left. \left. +\,\hat{x}_{k})+\tilde{\beta }_{k}\xi _{k}+\bar{\beta }\xi _{k}]\right\} \right\} \nonumber \\\,=\, & \mathbb {E}\left\{ \begin{bmatrix} \Phi _{k}^{T} &{} v_{k}^{T} \\ \end{bmatrix} \Psi \begin{bmatrix} \Phi _{k} \\ v_{k} \\ \end{bmatrix} -\tilde{z}^{T}_{k}\tilde{z}_{k}+\gamma ^{2}v_{k}^{T}U_{\varphi }v_{k}\right\} , \end{aligned}$$

(A6)

where $\Psi$ is defined in (12).

Summarizing both sides of (A6) from 0 to $N-1$ on k, we obtain

$$\begin{aligned} \sum _{k=0}^{N-1}\mathbb {E}\{\bar{L}_{k}\}\,=\, & \mathbb {E}\{e_{N}^{T}Q_{N}e_{N} -e_{0}^{T}Q_{0}e_{0}\}\nonumber \\\le & {} \mathbb {E}\left\{ \sum _{k=0}^{N-1}\begin{bmatrix} \Phi _{k}^{T} &{} v_{k}^{T} \\ \end{bmatrix} \Psi \begin{bmatrix} \Phi _{k} \\ v_{k} \\ \end{bmatrix} \right\} \\{} & {} -\,\mathbb {E}\left\{ \sum _{k=0}^{N-1}(\tilde{z}^{T}_{k}\tilde{z}_{k}-\gamma ^{2}v_{k}^{T}U_{\varphi }v_{k})\right\} . \end{aligned}$$

Furthermore, we derive the following form

$$\begin{aligned} J_{1}\le \mathbb {E}\left\{ \sum _{k=0}^{N-1}\begin{bmatrix} \Phi _{k}^{T} &{} v_{k}^{T} \\ \end{bmatrix} \Psi \begin{bmatrix} \Phi _{k} \\ v_{k} \\ \end{bmatrix} +\,e_{0}^{T}(Q_{0}-\gamma ^{2}U_{\phi })e_{0}\right\} -\mathbb {E}\{e_{N}^{T}Q_{N}e_{N}\}. \end{aligned}$$

(A7)

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:

$$\begin{aligned} X_{k+1}\,=\, & \mathbb {E}\left\{ e_{k+1}e_{k+1}^{T}\right\} \nonumber \\\,=\, & \mathbb {E}\left\{ A_{k}e_{k}e_{k}^{T}A_{k}^{T}+B_{k}f(e_{k})f^{T}(e_{k})B_{k}^{T}\right. \nonumber \\{} & {} \left. +\,C_{k}v_{k}v_{k}^{T}C_{k}^{T}+K_{k}\bar{\sigma }_{k}\bar{\sigma }_{k}^{T}K_{k}^{T}\right. \nonumber \\{} & {} \left. +\,B_{k}f(e_{k})e_{k}^{T}A_{k}^{T}+A_{k}e_{k}f^{T}(e_{k})B_{k}^{T}\right. \nonumber \\{} & {} \left. -\,K_{k}\bar{\sigma }_{k}e_{k}^{T}A_{k}^{T}-A_{k}e_{k}\bar{\sigma }_{k}^{T}K_{k}^{T}\right. \nonumber \\{} & {} \left. -\,K_{k}\bar{\sigma }_{k}f^{T}(e_{k})B_{k}^{T}-B_{k}f(e_{k})\bar{\sigma }_{k}^{T}K_{k}^{T}\right\} , \end{aligned}$$

(B8)

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

$$\begin{aligned} X_{k+1}\le & {} \mathbb {E}\left\{ 3A_{k}e_{k}e_{k}^{T}A_{k}^{T} +3B_{k}f(e_{k})f^{T}(e_{k})B_{k}^{T}\right. \\{} & {} \quad \left. +3K_{k}\bar{\sigma }_{k}\bar{\sigma }_{k}^{T}K_{k}^{T} +C_{k}v_{k}v_{k}^{T}C_{k}^{T}\right\} . \end{aligned}$$

It follows from Lemma 2 that

$$\begin{aligned} \mathbb {E}\{f(e_{k})f^{T}(e_{k})\}\le & {} \mathbb {E}\{\textrm{tr}(f(e_{k})f^{T}(e_{k}))\}I\\\,=\, & \mathbb {E}\{f^{T}(e_{k})f(e_{k})\}I \le Y\mathbb {E}\{e_{k}^{T}e_{k}\}I,\\ \mathbb {E}\{\bar{\sigma }_{k}\bar{\sigma }_{k}^{T}\}\le & {} \mathbb {E}\{\textrm{tr}(\bar{\sigma }_{k}^{T}\bar{\sigma }_{k})\}I, \end{aligned}$$

where Y is defined in (14).

According to (5), the following results can be obtained by calculation

$$\begin{aligned}{} & {} \bar{\sigma }_{k}^{T}\bar{\sigma }_{k}-\bar{g}[D_{k}e_{k}-\tilde{\beta }_{k}D_{k}(e_{k}\\{} & {} \quad +\,\hat{x}_{k})-\bar{\beta }D_{k}(e_{k}+\hat{x}_{k})+\tilde{\beta }_{k}\xi _{k} +\bar{\beta }\xi _{k}]^{T}\bar{\sigma }_{k}\\{} & {} \quad -\,\bar{\sigma }_{k}^{T}[D_{k}e_{k}-\tilde{\beta }_{k}D_{k}(e_{k}+\hat{x}_{k})\\{} & {} \quad -\,\bar{\beta }D_{k}(e_{k}+\hat{x}_{k})+\tilde{\beta }_{k}\xi _{k}+\bar{\beta }\xi _{k}]+\bar{g}[D_{k}e_{k}\\{} & {} \quad -\,\tilde{\beta }_{k}D_{k}(e_{k}+\hat{x}_{k})-\bar{\beta }D_{k}(e_{k}\\{} & {} \quad +\,\hat{x}_{k})+\tilde{\beta }_{k}\xi _{k}+\bar{\beta }\xi _{k}]^{T}[D_{k}e_{k}-\tilde{\beta }_{k}D_{k}(e_{k}\\{} & {} \qquad +\,\hat{x}_{k})-\bar{\beta }D_{k}(e_{k}+\hat{x}_{k})+\tilde{\beta }_{k}\xi _{k}+\bar{\beta }\xi _{k}]\le 0. \end{aligned}$$

Noticing $x^{T}y+y^{T}x<\varpi x^{T}x+\frac{1}{\varpi }y^{T}y$, we can derive that

$$\begin{aligned} \mathbb {E}\{\bar{\sigma }_{k}^{T}\bar{\sigma }_{k}\}\le & {} \frac{(1-2\varpi )\bar{g}+2\varpi (2\varpi -1)\bar{g}\bar{\beta }+(1+2\varpi \bar{\beta }+4\varpi ^{2}+4\varpi ^{2}\bar{g})\bar{g}\bar{\beta }+1+\bar{\beta }}{2\varpi -\varpi ^{2}(1+\bar{g}+3\bar{\beta }+3\bar{g}\bar{\beta })}\\{} & {} \times \,\textrm{tr}(D_{k}^{T}D_{k})e_{k}^{T}e_{k}+\frac{(3+4\bar{\beta }-2\varpi \bar{\beta })\bar{g}\bar{\beta }+\bar{\beta }+(4-2\varpi )\bar{g}\bar{\beta }(1-\bar{\beta })}{2\varpi -\varpi ^{2}(1+\bar{g}+3\bar{\beta }+3\bar{g}\bar{\beta })}\delta \\{} & {} +\,\frac{(3-2\varpi \bar{\beta })\bar{g}\bar{\beta }+\bar{\beta }+2(1+\varpi ^{2})\bar{g}\bar{\beta }^{2}+(2+2\varpi ^{2}-2\varpi )\bar{g}\bar{\beta }(1-\bar{\beta })}{2\varpi -\varpi ^{2}(1+\bar{g}+3\bar{\beta }+3\bar{g}\bar{\beta })}\\{} & {} \times \,\textrm{tr}(D_{k}^{T}D_{k})\hat{x}_{k}^{T}\hat{x}_{k}, \end{aligned}$$

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

$$\begin{aligned} \mathbb {E}\{\textrm{tr}(\bar{\sigma }_{k}^{T}\bar{\sigma }_{k})\}I \le (\iota _{1}\mathbb {E}\{e_{k}^{T}e_{k}\}+\iota _{2}\hat{x}_{k}^{T}\hat{x}_{k}+\iota _{3}\delta )I, \end{aligned}$$

where $\iota _{1}$, $\iota _{2}$ and $\iota _{3}$ are defined in (14). Furthermore, it can be obtained that

$$\begin{aligned} X_{k+1}\le & {} \mathbb {E}\left\{ 3A_{k}e_{k}e_{k}^{T}A_{k}^{T} +3YB_{k}e_{k}^{T}e_{k}B_{k}^{T}\right. \nonumber \\{} & {} \left. +\,3\iota _{1}K_{k}e_{k}^{T}e_{k}K_{k}^{T}+3\iota _{2}K_{k}\hat{x}_{k}^{T}\hat{x}_{k}K_{k}^{T}\right. \nonumber \\{} & {} \left. +\,3\iota _{3}\delta K_{k}K_{k}^{T}+C_{k}V_{k}C_{k}^{T}\right\} . \end{aligned}$$

(B9)

According to the feature of the trace, one has

$$\begin{aligned} \mathbb {E}\{e_{k}^{T}e_{k}\}\,=\, & \mathbb {E}\{\textrm{tr}(e_{k}e_{k}^{T})\}=\textrm{tr}(X_{k}),\nonumber \\ \mathbb {E}\{\hat{x}_{k}^{T}\hat{x}_{k}\}\,=\, & \mathbb {E}\{\textrm{tr}(\hat{x}_{k}\hat{x}_{k}^{T})\}. \end{aligned}$$

(B10)

Combining (B9) with (B10) results in

$$\begin{aligned} X_{k+1}\le & {} 3A_{k}X_{k}A_{k}^{T}+3\textrm{tr}(X_{k})B_{k}YB_{k}^{T}\\{} & {} +\,3\iota _{1}\textrm{tr}(X_{k})K_{k}K_{k}^{T}+3\iota _{2}\textrm{tr}(\hat{x}_{k}\hat{x}_{k}^{T})K_{k}K_{k}^{T}\\{} & {} +\,3\iota _{3}\delta K_{k}K_{k}^{T}+C_{k}V_{k}C_{k}^{T}\\\,=\, & \Theta (X_{k}). \end{aligned}$$

It is easy to get that $G_{0}\ge X_{0}$. Letting $G_{k}\ge X_{k}$, the following inequality can be derived as

$$\begin{aligned} \Theta (G_{k})\ge \Theta (X_{k})\ge X_{k+1}. \end{aligned}$$

(B11)

Then, from (13) and (B11), we obtain

$$\begin{aligned} G_{k+1}\ge & {} \Theta (G_{k})\ge \Theta (X_{k})\ge X_{k+1}. \end{aligned}$$

(B12)

The proof is complete. $\square$