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This paper is concerned with fault detection problem for a class of network control systems (NCSs) with multiple communication delays and stochastic missing measurements. The missing measurement phenomenon occurs in a random way and the occurrence probability for each measurement output is governed by an individual random variable. Besides, the multiple communication delay phenomenon reflects that networked control systems have different communication delays when the signals are transferred via different channels. We aim to design a fault detection filter so that the overall fault detection dynamics is exponentially stable in the mean square. By constructing proper Lyapunov-Krasovskii functional, we acquire sufficient conditions to guarantee the stability of the fault detection filter for the discrete systems, and the filter parameters are also derived by solving linear matrix inequality. Finally, an illustrative example is provided to show the usefulness and effectiveness of the proposed design method.

Over the past few decades, the fault detection problem has been attracting extensive research attention from scholars [

Considering the fault detection problem in a class of networked control systems, some new problems have merged out [

Summarizing the above discussion, in this paper, we are motivated to study the fault detection problem for a class of network control systems with multiple communication delays and stochastic missing measurements. A fault detection filter is constructed through the establishment of the existing model; then the addressed fault detection problem is converted into an auxiliary

In this paper, we consider the fault detection problem for a class of network control systems with multiple communication delays and stochastic missing measurements; then a NCS model can be represented by the following dynamic model:

Because missing measurements in system occur in a stochastic way,

Then, we can easily describe a NCS model with multiple communication delays and stochastic missing measurements as

The key step of fault detection schemes is the construction of a dynamic system called a fault detection observer or filter, in which the residual signal is generated in order to decide whether a fault has occurred or not. In this paper, according to the above formula, we build a fault detection filter whose model can be described as follows:

Then, we can get the overall fault detection dynamics governed by the following system:

There is a big probability of the existence of errors between theoretical and practical systems due to unexpected factors in NCSs. In order to overcome this phenomenon, it is natural to assume system uncertainties. In this paper, we assume that the uncertainties occur on not only regular item of the system but also time-delay item. Therefore, this is a more general description for the NCSs.

The main purpose of this paper is to design a fault detection filter such that the overall fault detection dynamics is exponentially stable in the mean square and, at the same time, the error between the residual signal and the fault signal is made as small as possible. Until now, the fault detection problem to be addressed in this paper can be described by the following two steps.

For system (

We set up a fault detection measure to judge whether a fault occurs. In this paper, we adopt two variables: an evaluation function

First of all, let us introduce the following lemmas which will be used in deriving our main results.

Given constant matrices

Let

For convenience of presentation, we first discuss the nominal system without parameter uncertainties

Consider the nominal system model (

Choose a Lyapunov functional for system (

Then, along the trajectory of augmented system (

In the following, we first prove the exponential stability of the fault detection dynamics system (

Now, we are in a position to deal with the

Letting

According to the analysis results established, we will deal with the fault detection filter design problem.

Consider the nominal system model (

First, let us rewrite the parameters in Theorem

Pre- and postmultiplying inequalities (

Now, according to previous Theorems

Consider the uncertain fault detection system (

According to result (

From Theorem

The main results in Theorems

In this section, we present an illustrative example to demonstrate the effectiveness of the proposed algorithm. Consider the following networked system with multiple communication delays and stochastic missing measurements:

By applying Theorem

To further illustrate the effectiveness of the designed fault detection filter, we give a fault signal; for

First, we assume our initial conditions as

Residual signal

Evolution of residual evaluation function

Next, we consider that the disturbance is given by

Residual signal

Evolution of residual evaluation function

Selecting a threshold as

In this paper, we have addressed the fault detection problem for a class of network control systems comprising multiple communication delays and stochastic missing measurements. Our purpose is to build up a fault detection filter through an existing model of NCSs such that the overall fault detection dynamics is exponentially stable while preserving a guaranteed performance; at the same time, the error between the residual signal and the fault signal is made as small as possible. At the end, an illustrative simulation example has been given to demonstrate the effectiveness of the fault detection techniques presented in this paper.

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

This work has been supported by the National Natural Science Foundation of China (Grant no. 61104109), the Natural Science Foundation of Jiangsu Province of China (Grant no. BK2011703), the Support of Science and Technology and Independent Innovation Foundation of Jiangsu Province of China (Grant no. BE2012178), the NUST Outstanding Scholar Supporting Program, and the Doctoral Fund of the Ministry of Education of China (Grant no. 20113219110027).