LettersFault detection for interval type-2 fuzzy systems with sensor nonlinearities☆
Introduction
It is well known that fuzzy logic control theory has been proposed as an effective approach to approximate the complex nonlinear systems for the control design objective [1], [2], [3], [4]. Over the past years, fuzzy logic control method has been widely used in many practical applications. Recently, the Takagi–Sugeno (T–S) [5] fuzzy systems have attached considerable attention because it is effective to analyze and synthesize nonlinear systems such as chemical processes, automotive systems, robotics systems and many manufacturing processes. The results on stability analysis, controller synthesis and filter design of fuzzy systems were reported in [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26]. To mention a few, the authors in [21] considered the reliable fuzzy control problem for uncertain suspension systems with actuator faults. Recently, the fault detection problem has been proposed in [27], [28], [29]. Based on a residual signal, the residual evaluation function is used to compare with a predefined threshold. An alarm of fault is presented such that the residual evaluation function has a value larger than the threshold. However, it should be mentioned that the above results are under the condition of type-1 T–S fuzzy sets and are available when the grades of membership are certain in the T–S fuzzy systems.
Therefore, it should be pointed out that the control problem of type-1 T–S fuzzy models cannot be addressed if the membership functions contain uncertainty information. Once the nonlinear plant is subject to parameter uncertainties, it will lead to the grades of membership uncertain in value. A basic IT2 fuzzy logic model was proposed in [30], which can be used to handle the nonlinear plants. It has been shown that the IT2 fuzzy logic systems have the superiority performance than the type-1 T–S fuzzy logic models in the aspect of handling parameter uncertainties. The IT2 fuzzy models have attached considerable attention and many control design results have been proposed in [31], [32]. Moreover, using the lower and upper membership functions, the authors in [33], [34] have dealt with the problem of the IT2 fuzzy systems subject to parameter uncertainties. However, the problem of sensor nonlinearities for IT2 fuzzy systems has not been studied and there are no results about fault detection for IT2 fuzzy systems. It is a valuable research direction to handle the problem of fault detection for IT2 fuzzy systems with sensor nonlinearities.
Motivated by the above discussion, this paper investigates the fault detection problem for the IT2 fuzzy systems subject to sensor nonlinearities. The output considered in this paper of IT2 fuzzy systems is a general sector-bounded nonlinearities. The IT2 fuzzy model and IT2 fuzzy fault detection filter do not require to share the same lower and upper membership functions. By using a general observer-based fault detection filter as a residual generator, the fault detection problem is described as a filter design problem. The fault detection filter is designed to guarantee the prescribed performance level. A decomposition approach is employed to handle the characteristic of sensor saturation. Using Lyapunov stability theory, a novel type of IT2 fault detection filter is designed to guarantee that the fault detection system is asymptotically stable with an performance. In the design procedure, the parameters of the IT2 filter can be solved by the standard software. A numerical example is provided to demonstrate the feasibility and effectiveness of the proposed method. The remaining of this paper is as follows. Section 2 introduces IT2 fuzzy systems, constructs the IT2 filter and presents the fault detection for IT2 fuzzy systems. Section 3 proposes stability conditions based on the Lyapunov stability theory for the IT2 fuzzy systems and Section 4 provides an illustrative example to show the effectiveness of the proposed results. Section 5 concludes this paper.
Notation: The superscripts “T” and “−1” stand for matrix transposition and inverse, respectively. Rn denotes the n-dimensional Euclidean space and the notation stands for a symmetric and positive definite (semi-definite). In symmetric block matrices or complex matrix expressions, we use an asterisk () to represent a term that is induced by symmetry and stands for a block-diagonal matrix. He(A) is defined as for simplicity. Matrices, if the dimensions are not explicitly stated, are assumed to be compatible for algebraic operations. The space of square-integrable vector functions over is denoted by , and for , its norm is denoted by .
Section snippets
IT2 T–S fuzzy model
Consider the following IT2 fuzzy model that represents a continuous-time nonlinear system with r rules:
Plant Rule is is :where is an IT2 fuzzy set of rule i corresponding to the function , ; ; p is a positive integer; is the system state vector, is the disturbance input and is the fault to be detected; is the measure output; Ai, Bi, B1
Main results
In this section, the stability condition with performance of the fault detection system (8) is first presented in the following theorem. Theorem 1 The membership functions satisfy and the fault detection system (8) is asymptotically stable with an performance level γ, if there exist matrices and with appropriate dimensions, such that the following linear matrix inequalities hold for :where
A numerical example
In this section, an example is used to illustrate the effectiveness of the proposed method. Consider a 2-rule IT2 fuzzy system in the form of (3) with (7), the matrices are listed below:Membership functions for Rules 1 and 2 are given as follows:
Conclusions
In this paper, the fault detection problem has been considered for a class of IT2 fuzzy systems with sensor nonlinearities. Firstly, the IT2 fuzzy systems and the fault detection filter have been constructed. By using a general observer-based fault detection filter as a residual generator, the fault detection problem has been described as a filter design problem. The fault detection filter has been designed to guarantee the prescribed performance level. In the design procedure, the
Yingnan Pan received the B.S. degree in mathematics from Bohai University, Jinzhou, China, in 2012. He is studying for the M.S. degree in applied mathematics in Bohai University, Jinzhou, China. His research interests include fuzzy control, robust control and their applications.
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2017, Journal of the Franklin InstituteCitation Excerpt :The filter design problem for IT2 T–S fuzzy systems with D stability constraints was investigated in [17]. The fault detection problem for IT2 T–S fuzzy systems subjected to sensor nonlinearities was studied in [18]. Notably, the influence of the time-delay was not considered in most existing studies IT2 T–S fuzzy systems.
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Yingnan Pan received the B.S. degree in mathematics from Bohai University, Jinzhou, China, in 2012. He is studying for the M.S. degree in applied mathematics in Bohai University, Jinzhou, China. His research interests include fuzzy control, robust control and their applications.
Hongyi Li received B.S. and M.S. degrees in mathematics from Bohai University, Jinzhou, China, in 2006 and 2009, respectively, and the Ph.D. degree in intelligent control from the University of Portsmouth, Portsmouth, UK, in 2012. He is currently a Professor of the College of Engineering, Bohai University. He was a Research Associate with the Department of Mechanical Engineering, University of Hong Kong and Hong Kong Polytechnic University, from June 2010 to September 2010 and from September 2012 to December 2012, respectively. His research interests include fuzzy control, robust control, and their applications. He is also an Associate Editor/Editorial Board member for several international journals, including Neurocomputing, Circuits, Systems, and Signal Processing, Shock and Vibration, etc.
Qi Zhou received the B.S. and M.S. degrees in mathematics from Bohai University, Jinzhou, China, in 2006 and 2009, and the Ph.D. degree in control theory from Nanjing University of Science and Technology, Nanjing, China, in 2013, respectively. She is presently a lecturer in the College of Information Science and Technology of Bohai University. Her research interest includes fuzzy logic control, stochastic control, and robust control.
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This work was partially supported by the National Natural Science Foundation of China (61203002, 61304003, 61333012, 61020106003), the Program for New Century Excellent Talents in University (NCET-13-0696), the NKTSP under Grant 2012BAF19G00, the Program for Liaoning Innovative Research Team in University (LT2013023) and the Program for Liaoning Excellent Talents in University (LR2013053).