On the robustness of Type-1 and Interval Type-2 fuzzy logic systems in modeling
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]:where 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 defineThus (2.5) can be expressed asSuppose x∗ is deviated by a small Δx∗. The new input to the FLS would be x∗ + Δx∗. The deviated output, ΔY, is given by
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
Observe that
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:
Similarly, to compare the robustness performance of the two FLSs qualitatively, definewhere 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.
References (32)
- et al.
GP-COACH: genetic programming-based learning of compact and ACcurate fuzzy rule-based classification systems for high-dimensional problems
Information Sciences
(2010) Arithmetic operators in interval-valued fuzzy set theory
Information Sciences
(2007)- et al.
On the cardinalities of interval-valued fuzzy sets
Fuzzy Sets and Systems
(2007) A fuzzy asymmetric GARCH model applied to stock markets
Information Sciences
(2009)- et al.
Approximation and robustness of fuzzy finite automata
International Journal of Approximate Reasoning
(2008) - et al.
The development of a robust fuzzy inference mechanism
International Journal of Approximate Reasoning
(2005) On answering the question where do I start in order to solve a new problem involving interval type-2 fuzzy sets?
Information Sciences
(2009)- et al.
Hybrid learning for interval type-2 fuzzy logicnext term systems based on orthogonal least-squares and back-propagation method
Information Sciences
(2009) Is there a need for fuzzy logic?
Information Sciences
(2008)- et al.
Fuzzy logicnext term based algorithms for maximum covering location problems
Information Sciences
(2009)
On the stability of interval type-2 TSK fuzzy logic control systems
IEEE Transactions on Systems, Man, Cybernetics: Part B
Robustness of fuzzy reasoning and δ-equalities of fuzzy sets
IEEE Transactions on Fuzzy Systems
A survey on analysis and design of model-based fuzzy control systems
IEEE Transactions on Fuzzy Systems
A method of inference in approximate reasoning based on interval-valued fuzzy sets
Fuzzy Sets and Systems
Generation of interval-valued fuzzy and Atanassov’s institutionistic fuzzy connectives and from k-alpha operators: laws for conjunctions and disjunctions, amplitude
International Journal of Intelligent Systems
An approach to measure the robustness of fuzzy reasoning
International Journal of Intelligent Systems
Cited by (107)
Logic reasoning under data veracity concerns
2023, International Journal of Approximate ReasoningDesign and application of Nagar-Bardini structure-based interval type-2 fuzzy logic systems optimized with the combination of backpropagation algorithms and recursive least square algorithms
2023, Expert Systems with ApplicationsCitation Excerpt :As the T1 fuzzy sets can model uncertainties only to a limited extent, this transition to more complex forms of FL is necessary. Studies have illustrated that IT2 FLSs are better than T1 FLSs in coping with many fields (Castillo et al., 2019; Biglarbegian, Melek, &Mendel, 2011; Castillo & Melin, 2012; Tao et al., 2012) affected by uncertainties. Compared with the T1 FSs, the footprint of uncertainties (FOU) and third dimensional amplitude of IT2 FSs give them own more design freedom.
Smooth compositions are candidates for robust fuzzy systems
2022, Fuzzy Sets and SystemsA 3D Membership Function-Based Type-2 Fuzzy Brain Emotional Learning Predictor for Forecasting Taiwan Stock Price
2024, International Journal of Fuzzy SystemsTracking Control for Nonlinear Systems With Actuator Saturation via Interval Type-2 T-S Fuzzy Framework
2023, IEEE Transactions on CyberneticsIT2 fuzzy adaptive containment control for fractional-order heterogeneous multi-agent systems with input saturation
2023, Journal of Intelligent and Fuzzy Systems