Asynchronous $l_{2}$–$l_{\infty }$ Filtering for Discrete-Time Fuzzy Markov Jump Neural Networks with Unreliable Communication Links

Abstract

This paper investigates the problem of $l_{2}$–$l_{\infty }$ asynchronous filtering for a class of discrete-time fuzzy neural networks subject to Markov jump parameters and unreliable communication links. Due to the fact that neural networks possess the nonlinear dynamic characteristic, it is difficult to deal with such a nonlinear characteristic directly, so the Takagi–Sugeno fuzzy model is introduced to approximate the system. Directed against the unreliable communication links, the data packet loss depicted by a stochastic variable with Bernoulli distribution and the signal quantization phenomenon occurring in communication channels are taken into consideration simultaneously. The attention of this paper is mainly centered on devising an asynchronous $l_{2}$–$l_{\infty }$ filter for ensuring the $l_{2}$–$l_{\infty }$ performance of the studied system under asynchronous conditions. Some sufficient conditions for the existence of the asynchronous $l_{2}$–$l_{\infty }$ filter are presented. Finally, a numerical example is given to carry out the simulation experiment, which can verify the effectiveness of the obtained results.

Appendix

1.1 Proof of Theorem 1

Proof

Consider the following Lyapunov-Krasovskii function for system $\left( \breve{\Sigma }\right)$ as follows:

$$\begin{aligned} V\left( p\right) =\sum _{n=1}^{3}V_{n}\left( p\right) , \end{aligned}$$

(19)

where

$$\begin{aligned} V_{1}\left( p\right)= & {} \bar{x}^{T}\left( p\right) P_{hu}\bar{x}\left( p\right) ,\\ V_{2}\left( p\right)= & {} \sum _{s=p-d_{1}}^{p-1}\bar{x}^{T}\left( s\right) Q_{1}\bar{x}\left( s\right) +\sum _{s=p-d_{2}}^{p-1}\bar{x}^{T}\left( s\right) Q_{2}\bar{x}\left( s\right) +\sum _{s=p-d(p)}^{p-1}\bar{x}^{T}\left( s\right) Q_{3}\bar{x}\left( s\right) ,\\ V_{3}\left( p\right)= & {} \sum _{g=-d_{2}+1}^{-d_{1}}\sum _{l=p+g}^{p-1}\bar{x}^{T}\left( l\right) Q_{3}\bar{x}\left( l\right) . \end{aligned}$$

Calculating the value of $E\left\{ \Delta V\left( p\right) \right\} ,$ we have

$$\begin{aligned} E\left\{ \Delta V\left( p\right) \right\}= & {} E\left\{ V\left( p+1\right) \right\} -V\left( p\right) \nonumber \\= & {} E\left\{ \bar{x}^{T}\left( p+1\right) \tilde{P}_{hu}\bar{x}\left( p+1\right) -\bar{x}^{T} \left( p\right) P_{hu}\bar{x}\left( p\right) +\bar{x}^{T}\left( p\right) Q_{1}\bar{x}\left( p\right) \right. \nonumber \\&-\bar{x}^{T}\left( p-d_{1}\right) Q_{1}\bar{x}\left( p-d_{1}\right) +\bar{x}^{T}\left( p\right) Q_{2}\bar{x}\left( p\right) -\bar{x}^{T}\left( p-d_{2}\right) Q_{2}\bar{x}\left( p-d_{2}\right) \nonumber \\&+\bar{x}^{T}\left( p\right) Q_{3}\bar{x}\left( p\right) -\bar{x}^{T}\left( p-d\left( p\right) \right) Q_{3}\bar{x}\left( p-d\left( p\right) \right) \nonumber \\&\left. +\left( d_{2}-d_{1}\right) \bar{x}^{T}\left( p\right) Q_{3}\bar{x}\left( p\right) -\sum _{g=-d_{2}+1}^{-d_{1}}\bar{x}^{T}\left( p+g\right) Q_{3}\bar{x}\left( p+g\right) \right\} . \end{aligned}$$

(20)

Furthermore, according to Remark 3 in [45] and combining (13), (14). Then, one can find that

$$\begin{aligned} E\left\{ V\left( p+1\right) \right\} -V\left( p\right)\le & {} \breve{\Psi }^{T}\left( p\right) \Lambda _{hu}\breve{\Psi }\left( p\right) +\rho ^{2}\omega ^{T}\left( p\right) \omega \left( p\right) ,\nonumber \\\le & {} \rho ^{2}\omega ^{T}\left( p\right) \omega \left( p\right) , \end{aligned}$$

(21)

where

$$\begin{aligned} \breve{\Psi }\left( p\right) \triangleq&\left[ \begin{array}{cccccc} \overline{x}\left( p\right)&f\left( x\left( p\right) \right)&f\left( x\left( p-d\left( p\right) \right) \right)&\omega \left( p\right)&\overline{x}\left( p-d_{1}\right)&\overline{x}\left( p-d_{2}\right) \end{array}\right. \\&\left. \begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} \overline{x}\left( p-d_{1}-1\right)&\text { }\overline{x}\left( p-d_{1}-1\right)&\overline{x}\left( p-d_{2}-1\right) ,&\sum _{u=p-d_{2}+1}^{p-d_{1}+1}x\left( u\right)&\sum _{s=-d_{2}+1}^{-d_{1}+1}\sum _{u=p+s}^{p-d_{1}+1}x\left( u\right) \end{array}\right] . \end{aligned}$$

Via operating iterative computations, it is referred from (21) that

$$\begin{aligned} E\left\{ V\left( p\right) \right\} \le E\left\{ V\left( 0\right) +\rho ^{2}\sum _{l=0}^{p-1}\omega ^{T}\left( l\right) \omega \left( l\right) \right\} , \end{aligned}$$

(22)

under zero initial conditions, it can be obtained that

$$\begin{aligned} E\left\{ V\left( p\right) \right\} \le E\left\{ V\left( 0\right) +\rho ^{2}\sum _{l=0}^{p-1}\omega ^{T}\left( l\right) \omega \left( l\right) \right\} \le \rho ^{2}\sum _{l=0}^{p-1}\omega ^{T}\left( l\right) \omega \left( l\right) \le \rho ^{2}\sum _{l=0}^{p}\omega ^{T}\left( l\right) \omega \left( l\right) . \end{aligned}$$

(23)

By using Schur complement to (16), it can see that

$$\begin{aligned} \hat{E}_{hu}^{T}\hat{E}_{hu}\le P_{hu}, \end{aligned}$$

(24)

then, from (22)-(24), it can be derived that the following inequality holds for any non-zero $\omega (p)\in l_{2}[0,\infty )$,

$$\begin{aligned} E\left\{ \bar{z}^{T}\left( p\right) \bar{z}\left( p\right) \right\}= & {} E\left\{ \bar{x}^{T}\left( p\right) \hat{E}_{hu}^{T}\hat{E}_{hu}\bar{x}\left( p\right) \right\} \\\le & {} E\left\{ \bar{x}^{T}\left( p\right) P_{hu}\bar{x}\left( p\right) \right\} =E\left\{ V_{1}\left( p\right) \right\} \\< & {} E\left\{ V\left( p\right) \right\} \\\le & {} \rho ^{2}\sum _{l=0}^{p}\omega ^{T}\left( l\right) \omega \left( l\right) . \end{aligned}$$

Clearly,the condition of Definition 1 can be guaranteed under the zero-initial conditions for any $\omega \left( p\right) \in \left( 0,\infty \right]$. This completes the proof.

1.2 Proof of Theorem 2

Proof

Pre- and post-multiplying (17) by

$$\begin{aligned} \text {diag}\left\{ I,I,I,I,I,I,I,I,I,I,I,\tilde{Z}_{hu}^{-1},\tilde{Z}_{hu}^{-1}\right\} \end{aligned}$$

and its transpose, then by using Schur complement, we have $\Lambda _{hu}<0.$ Obviously, it can be noted that from (17) that the condition (15) is ensured. This completes the proof.

1.3 Proof of Theorem 3

Proof

From $\left( J\tilde{P}_{hu}J^{T}-J\check{Z}_{hu}\right) \tilde{P}_{hu}^{-1}\left( J\tilde{P}_{hu}J^{T}-J\check{Z}_{hu}\right) ^{T}\ge 0,$ the condition holds as below for each $h\in \mathbf {R},u\in \mathbf {S}$,

$$\begin{aligned} -\check{Z}_{hu}\tilde{P}_{hu}^{-1}\check{Z}_{hu}^{T}\le J\tilde{P}_{hu}J^{T}-J\check{Z}_{hu}^{T}-\check{Z}_{hu}J^{T} \end{aligned}$$

(25)

where

$$\begin{aligned} \tilde{P}_{hu}\triangleq & {} \left[ \begin{array}{c@{\quad }c} \tilde{P}_{1hu} &{} \tilde{P}_{2hu}\\ *&{} \tilde{P}_{3hu} \end{array}\right] ,\text { }\tilde{P}_{1gv}\triangleq \sum _{g\in \mathbf {R}}\sum _{v\in \mathbf {S}}\pi _{hg}\chi _{uv}^{h}P_{1gv},\\ \tilde{P}_{2hu}\triangleq & {} \sum _{g\in \mathbf {R}}\sum _{v\in \mathbf {S}}\pi _{hg}\chi _{uv}^{h}P_{2gv},\text { }\tilde{P}_{3hu}\triangleq \sum _{g\in \mathbf {R}}\sum _{v\in \mathbf {S}}\pi _{hg}\chi _{uv}^{h}P_{3gv} \end{aligned}$$

Then let $\check{A}_{e,ju}=Z_{u}A_{e,ju}$, $\check{B}_{e,ju}=Z_{u}B_{e,ju}$, $\check{E}_{e,ju}=Z_{u}E_{e,ju}$, drawing support from Schur complement, Lemma 1, Lemma 2 as well as (25), from (18), it is clear that (17) is ensured. This completes the proof.