Elsevier

Information Sciences

Volume 181, Issue 7, 1 April 2011, Pages 1325-1347
Information Sciences

On the robustness of Type-1 and Interval Type-2 fuzzy logic systems in modeling

https://doi.org/10.1016/j.ins.2010.11.003Get rights and content

Abstract

Research on the robustness of fuzzy logic systems (FLSs), an imperative factor in the design process, is very limited in the literature. Specifically, when a system is subjected to small deviations of the sampling points (operating points), it is of great interest to find the maximum tolerance of the system, which we refer to as the system’s robustness. In this paper, we present a methodology for the robustness analysis of interval type-2 FLSs (IT2 FLSs) that also holds for T1 FLSs, hence, making it more general. A procedure for the design of robust IT2 FLSs with a guaranteed performance better than or equal to their T1 counterparts is then proposed. Several examples are performed to demonstrate the effectiveness of the proposed methodologies. It was concluded that both T1 and IT2 FLSs can be designed to achieve robust behavior in various applications, and preference one or the other, in general, is application-dependant. IT2 FLSs, having a more flexible structure than T1 FLSs, exhibited relatively small approximation errors in the several examples investigated. The methodologies presented in this paper lay the foundation for the design of FLSs with robust properties that will be very useful in many practical modeling and control applications.

Introduction

Fuzzy logic systems (FLSs) are well known for their ability to model linguistics and system uncertainties. Due to this ability, FLSs have been successfully used for identification, model-based control, input–output (I/O) mapping, and function approximation, to name a few, which has resulted their use in various modeling and control applications such as medicine, finance, communications, and operations research [28], [2], [29]. In recent years, there has been a growing interests in using FLSs in new applications [1], [10], [21].

More specifically, in the literature, type-2 FLSs (T2 FLSs) are particularly well-known for their capability in handling uncertainty. Specially, interval T2 FLSs (IT2 FLSs)1 have become very popular recently and have shown great promise to be used in various modeling and control applications with I/O system uncertainties [19]. Therefore, the analysis of IT2 FLSs systems has become a favorite topic for investigation. As a result, in recent years, researchers have been rigorously investigating the properties as well as the potentials of IT2 FLSs for numerous applications. Ying [25], [26] has shown that IT2 FLSs are universal approximators, hence proving the capability of such systems to be used on a larger scale of applications. In another work, the authors in [16], [15] presented the α-plane for T2 fuzzy sets (T2 FSs) which is useful for both theoretical and computational studies of these systems. Zhou et al. [32] proposed a new operator for linguistic terms in human decision-making modeled by T2 FSs. In a tutorial paper, Mendel [18] explains how to start solving problems involving IT2 FSs. IT2 FSs are also known as interval-valued fuzzy sets, please see [8], [5], [6], [9] for more information. Most recently, Mendel and Wu [17] have explained how IT2 FSs are used for perceptual computing and computing with words. The increasingly ongoing research on T2 and more specifically IT2 FLSs is facilitating the utilization of these FLSs in different applications.

Modeling is the main step in the identification of physical systems such as robotics, communications, medical systems, etc. More specifically, identification is vital in engineering systems estimation and control, both of which have found a wide range of applications in aerospace, automotive, micro-robotics, weather forecasting, etc. As stated earlier, FLSs have been largely used in modeling and identification of systems. They are known as one of the nonlinear identification tools due to their universal approximation properties. FLSs have also been used in control of nonlinear systems, especially in [7], [3], [22]. Therefore, to enable their utilization in a wider scale of modeling, identification as well as control applications, analysis of their robust behavior is of great interest.

In modeling or identification problems, usually some data (referred to as sampling points and often obtained by dedicated experiments) are used to capture the entire dynamics (or portions) of plants. To accurately determine the dynamics of the system considered, it is crucial to capture the behavior of systems through input/output (I/O) mappings. However, hardware/software limitations, unavoidable round off, and truncation of a system’s errors will ultimately result in deviation of the output from the optimum point for which the system was originally designed: the desired output. The output of a system can also change because of uncertainties, perhaps substantial ones, in sensory data. Moreover, in control, considerable deviations in the output change the closed-loop performance that, in severe cases, might even lead to an unstable system.

When a system is subjected to small deviations around the sampling points (operating points), it is essential to find the maximum tolerance of the system with respect to those perturbations, referred to herein as the system’s robustness. Thus, in the context of modeling, robustness is a metric for measuring the impact of input deviations on the desired output. In the design of FLSs, usually some data (known as training data or nominal values), from a plant, a controller, or a function are selected and the FLS parameters are identified, accordingly. So far, a great deal of research has been dedicated with success to effectively design FLSs that can accurately capture the model/dynamics of systems [18], [14], [24]. However, it is also essential to identify (or model) a system as precisely as possible with maximum possible robustness to the given data points.

In this paper, robustness is defined as the maximum deviation of the output as a result of the deviation of the inputs [13]. In [13], a parameterized formulation of the fuzzy reasoning process was introduced. This parameterized formulation has a closed form, and it can be exploited to investigate the robustness characteristics of the fuzzy inference mechanism. Melek and Goldenberg [13] formulated the robustness problem by introducing several parameters into the fuzzy reasoning. By defining the bounds on inference parameters, they obtained maximum input deviation without reaching the maximum desired output. Although the authors have mathematically investigated the problem of robustness, certain assumptions have been made about the membership functions (MFs) and, hence, the approach cannot be applied in more general cases. Moreover, the approach presented in this paper lacks a systematic methodology for the design of robust T1 FLSs.

Robustness of FLSs with respect to fuzzy operators has been studied thoroughly. Nguyen et al. [20] introduced the robust properties of various fuzzy connectors. They also showed that min and max are the most robust operators. Ying [27] proposed the concepts of maximum and average perturbations of fuzzy sets that led to estimation of perturbation parameters of fuzzy reasoning. In another work, authors in [30] suggested that probabilistic or statistical approach can be used to evaluate the robustness of fuzzy reasoning methods for the case when analytical investigation of robustness is complex. Cai [4] investigated the robustness of various operators and inference rules in fuzzy reasoning and discussed how errors in premises affect conclusions. In another study, Li et al. [11], [12] introduced certain measures of robustness of fuzzy operators and discussed their relationships to perturbation. They showed that the robustness of fuzzy reasoning is directly dependent on the fuzzy connectives and implication operators. The measures defined in this work are useful in the context of robustness (or sensitivity) analysis; however, the focus of this work is only on the operators. Most recently, Zheng et al. [31] investigated the robustness of fuzzy system operators for small random perturbations. Their work is based on algebraic operators and proposes two methods for analyzing robustness for random deviations. Similar to the previous publications in the literature, the focus of this last paper is also on fuzzy operators as opposed to the design of robust FLSs.

In most of the existing FLSs in modeling and identification problems, robustness is not considered in the design process. Therefore, it is of great importance to determine the sensitivity of a FLS to its parameter variations. The question that will be answered in this paper is how robust a FLS is to the perturbations of the training data or operating points. Robustness analysis in this sense is more appealing for practical applications in which uncertainty, noise, disturbance, etc., not always considered in the design, are present.

Research on the robustness of FLSs is mainly limited to fuzzy operators and, hence more in-depth analyses into robust system design is required. Furthermore, we are concerned with a systematic methodology for robustness analysis as well as the design of robust systems that can be practical for modeling and control applications. More importantly, robustness of IT2 FLSs is an important concept that, to the best of our knowledge, has not been studied in the literature. Moreover, to our best knowledge, there is no metric for measuring the robustness of FLSs by the definition given in the manuscript (which will be completely explained in Section 3). Therefore, in this paper, the robustness of FLSs is investigated. The focus of this work is on IT2 FLSs, but since T1s are a special class of IT2s, our methodology is general and applies to those systems as well. A systematic methodology is then proposed for the analysis and design of robust IT2 FLSs. Several case studies are presented to describe the application of the proposed methodology in the design of robust FLSs. The contribution of this paper is (a) systematic robustness analysis of FLSs in the presence of parameter perturbations, and, (b) an algorithm for the design of robust FLSs that will enable engineers and researches to design more robust FLSs for modeling and control applications. Our results are general and can be used for both T1 and IT2 FLSs. The organization of this paper is as follows: Section 2 presents backgrounds on IT2 FLSs. Section 3 presents a definition and mathematical derivation of FLSs’ robustness. Section 4 provides the upper bounds of the maximum output deviation. Section 5 presents an algorithm for designing robust FLSs. Section 6 presents numerical examples. Finally, Section 8 provides the conclusions.

Section snippets

Background

The purpose of this paper is to present a systematic methodology to investigate the robustness of FLSs (both T1 and IT2). The robustness analysis will exploit a general structure of IT2 TSK FLSs, and since T1 TSK FLSs are a special case of IT2 TSK FLSs, the presented results can be readily used for T1s as well.

The general structure of an IT2 TSK fuzzy logic model is as follows [14]:Ifx1isF1iandx2isF2iandandxpisFpi,Thenyi=a0i+a1ix1++apixp,where i=1,,M,Fji represents the IT2 FS of

Robustness of FLSs

In this section, we formulate the robustness of fuzzy logic systems and derive the governing equations. The final expressions derived herein are used for the analysis and design of robust FLSs in the subsequent sections.

To begin, we definew̲i(x)f̲i(x)i=1Mf̲i(x),w¯i(x)f¯i(x)i=1Mf¯i(x).Thus (2.5) can be expressed asY(x)=mi=1Mw̲i(x)yi(x)+ni=1Mw¯i(x)yi(x).Suppose x∗ is deviated by a small Δx∗. The new input to the FLS would be x + Δx∗. The deviated output, ΔY, is given byΔY=Y(x+Δx)

Upper bound of the output deviation

In this section, an upper bound of ∣ΔY∣ is found. This result will be of interest to designers for estimating the maximum deviations expected for a given problem. To do so, it is only needed to find the upper bounds of ΔA and ΔB. Here, the derivation for the upper bound of ∣ΔA∣ is presented (similar analysis can be performed to obtain ΔB). ∣ΔA∣ can be written as|ΔA|=|m|i=1Mw̲i(x+Δx)yi(x+Δx)-i=1Mw̲i(x)yi(x)|m|i=1Mw̲i(x+Δx)yi(x+Δx)+|m|i=1Mw̲i(x)yi(x).

Observe thati=1Mw̲i(x+Δx)yi

Algorithm to design robust FLSs

This section presents an algorithm to design robust TSK FLSs. This algorithm provides a methodology that can be used for general TSK FLSs.

First, an optimum T1 TSK FLS is designed using available software tools such as ANFIS/Matlab. We exploit the model-structure of the developed T1 for the design of an IT2 TSK.

Next, define the error vector as the difference between the sampling (training) output vector, Ysampling, and the T1 output vector, YT1, as eT1  Ysampling  YT1. The total error for the T1

Examples

In this section, several examples are presented to demonstrate how the methodology presented in Section 5 is used for the analysis and design of the robustness of FLSs.

We define the error performance index (EPI) (in percentage) as:%EPI|eT1|-|eIT2||eT1|×100.

Similarly, to compare the robustness performance of the two FLSs qualitatively, define%RPIimax|Δxi|IT2-max|Δxi|T1max|Δxi|T1×100,where RPIi is the robustness performance improvement and i corresponds to the ith input.

In the following

Summary of the numerical analyses

In this section, we summarize the conclusions and lessons learned from conducting extensive numerical examples and case studies.

  • IT2 clearly reveals a better tool for modeling systems and plants, as it was shown in numerous cases to outperform T1 considerably.

  • Both T1 and IT2 can be used as robust FLSs and an ‘absolute’ preference of one to another depends on the application. However, IT2 showed to have a great potential to be used in the design of robust systems compared to T1, as several case

Conclusion

This paper presented a rigorous mathematical analysis of the robustness of T1 and IT2 FLSs. Robustness of FLSs was formulated for the case when the sampling points are subjected to small deviations. An efficient algorithm was introduced that can be used for the design of robust FLSs. Several examples verified the effectiveness of the proposed methodologies. It was shown that both T1 and IT2 FLSs reveal robust behaviors, and preference one or the other, in general, is application-dependant.

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