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This paper purposes the design of a fault detection filter for stochastic systems with mixed time-delays and parameter uncertainties. The main idea is to construct some new Lyapunov functional for the fault detection dynamics. A new robustly asymptotically stable criterion for the systems is derived through linear matrix inequality (LMI) by introducing a comprehensive different Lyapunov-Krasovskii functional. Then, the fault detection filter is designed in terms of linear matrix inequalities (LMIs) which can be easily checked in practice. At the same time, the error between the residual signal and the fault signal is made as small as possible. Finally, an example is given to illustrate the effectiveness and advantages of the proposed results.

Stochastic systems have strong practical relevance in mechanical systems, economics, systems with human operators, and other engineering areas. Meanwhile, the filtering problem has many applications in the areas of signal processing, signal estimation, pattern recognition, and many practical control systems. Therefore, the filter design problems and the stochastic systems have become important areas of research and received great attention over the past few years [

With the rising demand for higher safety and increasing demand for higher performance in the modern industries, the research on fault detection for dynamic systems has received more and more attention during the past two decades, like model-based schemes [

Besides, time-delay is one of the major sources of instability and poor performance of a practical control system. And many results on stochastic systems with time delays have been reported in the literature [

As far as we know, the delay-dependent criteria on fault detection filter design for delayed stochastic systems with parameter uncertainties have not been fully studied, which is still open. Motivated by the above discussion and in order to obtain less conservative results, we choose an appropriate new Lyapunov functional and establish a new integral inequality in the stochastic setting. What is more, because we have carefully considered the ranges for the time-varying delays, our criteria can be applicable to both fast and slow time-varying delays. The stability criteria obtained are in terms of linear matrix inequalities (LMIs) which can be checked efficiently via the LMI toolbox. Finally, a numerical example is also given to demonstrate the effectiveness and advantages of our theoretical results.

The simplest and most fundamental case considered is the problem of quadratic stabilization for the following system:

The nonlinear function

The stochastic variables

In a networked environment, it is quite common that the measurements

The map of the quantization process is

Consider the following full-order fault detection filter for system (

From (

In order to conduct the stability analysis for the above systems, it is necessary to make the following definitions and lemmas.

Consider the system (

Given a scalar

We further adopt a residual evaluation stage including an evaluation function

The parametric uncertainties

Let

Given constant matrices

For any constant matrix

In this section, we will investigate a sufficient condition on the performance analysis for the filter error system (

For nominal system of (

For simplicity, let

Taking the difference of the functional along the solution of the system, we obtain

It is obvious that

It can be deduced that there exists

Now, we are ready to prove the exponential stability of the system (

If

Next, we will establish the performance of the filtering error system (

Letting

This completes the proof.

Next, we are in a position to deal with the design of the filter output feedback controller for the system (

For given constants

If

The condition in (

This inequality implies that

Then, we introduce the following matrices:

Obviously

Let

Pre- and postmultiplying inequality (

When there are parameters uncertainties, we have the following corollary.

For given constants

When considering the system with uncertain parameters, we need to replace

Using Schur complement and lemma, we can derive (

In this section, a numerical example will be presented to illustrate the effectiveness of our results.

Consider the system (

Let the time-varying communication delays satisfy

For the parameters listed previously, by Corollary

To show the usefulness and effectiveness of the designed fault detection filter, let the external disturbance be

Residual signal without

Evolution of residual function

The fault detection filter design for stochastic systems with time-varying delays has been investigated in this research. Based on the Lyapunov functional method, sufficient conditions are obtained to ensure that the error systems are mean-square robustly asymptotically stable, and then the filters are designed in terms of LMIs. Numerical example has been given to illustrate the effectiveness of the proposed main results. The foregoing results have the potential to be useful for the study of stochastic systems.

This research was supported by the National Basic Research Program of China (2010 CB732501), the Fund of Sichuan Provincial Key Laboratory of Signal and Information Processing (SZJJ2009-002), and the Fund of Sichuan Provincial Key Laboratory of Signal and Information Processing (SGXZD0101-10-1).