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This paper is concerned with the

It is well known that many phenomena in engineering have unavoidable uncertain factors that are modeled by the stochastic differential equation. And in recent years, the stochastic system has been widely studied. A great number of investigations on stochastic systems have been reported in the literature. For example, the adaptive back stepping controller has been addressed in [

By using T-S fuzzy model, nonlinear systems turn into linear input-output relations which could be handled easily by appropriate fuzzy sets. This method can be seen in the stirred tank reactor system in [

On the other hand, state estimation has been found in many practical applications and it has been extensively studied over decades. It aims at estimating the unavailable state variables or their combination for the given system [

Following above discussion, T-S fuzzy model could be used to divide the nonlinear stochastic systems into several subsystems. And during the past decade, many problems have been tackled. Reference [

As a consequence, this paper will focus on the robust fuzzy delay-dependent

Consider the time-delay T-S fuzzy stochastic system with time-varying parameter uncertainties as the following form:

It is worth to mention that there are two approaches for the filter design in fuzzy systems. The implementation of the filter could be chosen to depend on or not depend on the fuzzy rules when the fuzzy model is available or not. And it is obvious to see that the former filter related to the fuzzy rules is less conserve and more complex. So we assume that the fuzzy is known here, which means the fuzzy-rule-dependent filter is investigated in this paper as in (

Let

And the filtering error dynamic system can be written as

We intend to design sets of fuzzy filters in the form of (

Throughout the paper, we adopt the following definitions and lemmas, which help to complete the proof of the main results.

The system (

The robust stochastic mean-square stable system (

For the given matrices

First, we define the following variables for convenience:

The filtering error system (

Define the following Lyapunov-Krasovskii candidate for system (

When

By using the Newton-Leibnitz formula, the following equations can be got for any matrices

By the above formulas (

During the analysis, it can be seen that

Now we establish the

Then applying the Schur complement formula to (

Therefore, for all

The system we studied is a time-varying delay system containing the information of both the lower bound and the upper bound of time delay. By such a consideration, delay-dependent result is more reliable and approaches to reality that not all the delays begin with 0 moment.

It is worth mentioning that Theorem

Now we are in a position to present a sufficient condition for the solvability of robust

Consider the uncertain T-S fuzzy stochastic time-varying delay system (

When the LMIs (

Similar to [

Now using Lemma

The desired

In the proof of above Theorem, we adopt (

In this section, a numerical example is provided to show the effectiveness of the results obtained in the previous section.

Consider the T-S fuzzy stochastic system (

By using the Matlab LMI Control Toolbox, we have the robust

The simulation results of the state response of the plant and the filter are given in Figure

State responses of

Responses of the error signal

This paper considers the robust

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work is supported by the National Natural Science Foundation of China under Grants nos. 61203048, 61304047, and 61203047.

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