This paper studies the fuzzy modeling problem and the fault detection and diagnosis (FDD) algorithm for non-Gaussian stochastic distribution systems based on the nonlinear fuzzy filter design. Following spline function approximation for output probability density functions (PDFs), the T-S fuzzy model is built as a nonlinear identifier to describe the dynamic relationship between the control input and the weight vector. By combining the designed filter and the threshold value, the fault in T-S weight model can be detected and the stability of error system can also be guaranteed. Moreover, the novel adaptive fuzzy filter based on stochastic distribution function is designed to estimate the size of system fault. Finally, the simulation results can well verify the effectiveness of the proposed algorithm for the constant fault and the time-varying fault, respectively.

In order to improve the stability and the security of system, the fault detection and diagnosis (FDD) algorithm for the complex systems has been an important part in the field of control engineering. Many significant approaches have also been presented and applied to practical processes successfully (see [

In recent years, the well-known T-S fuzzy model was viewed as a popular and powerful modeling tool since it is a powerful solution that bridges the gap between linear and nonlinear control systems (see [

In this paper, we provide a novel FDD approach for non-Gaussian stochastic distribution systems. Based on B-spline approximation and T-S fuzzy modeling, the concerned FDD problem of the dynamic non-Gaussian systems can be transferred into a special nonlinear FDD problem for deterministic T-S weight dynamics. Instead of common nonlinear observer or filter in [

In this paper, for a square matrix

For a complex non-Gaussian stochastic process, we denote that

Similarly, with [

Due to the PDF constraint condition

It is obvious that the nonlinear term

In the following, we will find the dynamic relationship between the control input

To detect the fault existing in the output stochastic distribution, we construct the following fuzzy filter:

It can be seen that

In the fault detection phase, our objective is to find the gain

For the known parameters

Denote the Lyapunov function candidate as

By defining

It is noted that Theorem

By defining the threshold

Based on the results regarding fault detection, this part will estimate the size of the fault if the system has fault. We construct the following adaptive fuzzy filter:

Suppose that

Define the Lyapunov function as follows:

Furthermore, we can get that

By using Schur complement, the inequality (

So

Suppose that the output PDFs can be approximated using the following B-spline model:

In the simulation, it is supposed that

Furthermore, we choose the following Gaussian type functions as our member functions:

By defining

The responses of the residual signal are shown in Figure

The responses of residual signal.

Fault and its estimation value.

3D mesh plot of the PDFs.

Time-varying fault and its estimation value.

This paper presents a novel FDD algorithm for non-Gaussian stochastic distribution systems based on T-S fuzzy modeling and nonlinear fuzzy filter design. A series of LMIs based solution is presented such that the estimation error system is stable and the fault can be detected through a known threshold. Moreover, the adaptive filter based on the T-S fuzzy model is designed to estimate the size of system fault by optimizing the solutions for the concerned LMIs. Simulations are given to demonstrate the efficiency of the proposed approach.

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

This paper is supported by the National Natural Science Foundation of China under Grants (60904030, 61174046, and 61203195) and China Postdoctoral Science Foundation under Grants (201000470067, 201104541).