Interval type-2 fuzzy logic controllers (IT2 FLCs) have recently been attracting a lot of research attention. Many reported results have shown that IT2 FLCs are better able to handle uncertainties than their type-1 (T1) counterparts. A challenging question is the following: What are the fundamental differences between IT2 and T1 FLCs? Once the fundamental differences are clear, we can better understand the advantages of IT2 FLCs and, hence, make better use of them. This paper explains two fundamental differences between IT2 and T1 FLCs: 1) Adaptiveness, meaning that the embedded T1 fuzzy sets used to compute the bounds of the type-reduced interval change as input changes; and 2) Novelty, meaning that the upper and lower membership functions of the same IT2 fuzzy set may be used simultaneously in computing each bound of the type-reduced interval. T1 FLCs do not have these properties; thus, a T1 FLC cannot implement the complex control surface of an IT2 FLC given the same rulebase. We also present several methods to visualize and analyze the effects of these two fundamental differences, including the control surface, the P-map, the equivalent generalized T1 fuzzy sets, and the equivalent PI gains. Finally, we examine five alternative type reducers for IT2 FLCs and explain why they do not capture the fundamentals of IT2 FLCs.

On the Fundamental Differences Between Interval Type-2 and Type-1 Fuzzy Logic Controllers

https://isiarticles.com/bundles/Article/pre/pdf/46311.pdf

A review on interval type-2 fuzzy logic applications in intelligent control

The purpose of this tutorial paper is to make general type-2 fuzzy logic systems (GT2 FLSs) more accessible to fuzzy logic researchers and practitioners, and to expedite their research, designs, and use. To accomplish this, the paper 1) explains four different mathematical representations for general type-2 fuzzy sets (GT2 FSs); 2) demonstrates that for the optimal design of a GT2 FLS, one should use the vertical-slice representation of its GT2 FSs because it is the only one of the four mathematical representations that is parsimonious; 3) shows how to obtain set theoretic and other operations for GT2 FSs using type-1 (T1) FS mathematics (α- cuts play a central role); 4) reviews Mamdani and TSK interval type-2 (IT2) FLSs so that their mathematical operations can be easily used in a GT2 FLS; 5) provides all of the formulas that describe both Mamdani and TSK GT2 FLSs; 6) explains why center-of sets type-reduction should be favored for a GT2 FLS over centroid type-reduction; 7) provides three simplified GT2 FLSs (two are for Mamdani GT2 FLSs and one is for a TSK GT2 FLS), all of which bypass type reduction and are generalizations from their IT2 FLS counterparts to GT2 FLSs; 8) explains why gradient-based optimization should not be used to optimally design a GT2 FLS; 9) explains how derivative-free optimization algorithms can be used to optimally design a GT2 FLS; and 10) provides a three-step approach for optimally designing FLSs in a progressive manner, from T1 to IT2 to GT2, each of which uses a quantum particle swarm optimization algorithm, by virtue of which the performance for the IT2 FLS cannot be worse than that of the T1 FLS, and the performance for the GT2 FLS cannot be worse than that of the IT2 FLS.

General Type-2 Fuzzy Logic Systems Made Simple: A Tutorial

Interval type-2 fuzzy logic systems (IT2 FLSs) have demonstrated better abilities to handle uncertainties than their type-1 (T1) counterparts in many applications; however, the high computational cost of the iterative Karnik-Mendel (KM) algorithms in type-reduction means that it is more expensive to deploy IT2 FLSs, which may hinder them from certain cost-sensitive real-world applications. This paper provides a comprehensive overview and comparison of three categories of methods to reduce their computational cost. The first category consists of five enhancements to the KM algorithms, which are the most popular type-reduction algorithms to date. The second category consists of 11 alternative type-reducers, which have closed-form representations and, hence, are more convenient for analysis. The third category consists of a simplified structure for IT2 FLSs, which can be combined with any algorithms in the first or second category for further computational cost reduction. Experiments demonstrate that almost all methods in these three categories are faster than the KM algorithms. This overview and comparison will help researchers and practitioners on IT2 FLSs choose the most suitable structure and type-reduction algorithms, from a computational cost perspective. A recommendation is given in the conclusion.

Approaches for Reducing the Computational Cost of Interval Type-2 Fuzzy Logic Systems: Overview and Comparisons

Type-reduction algorithms are very important for type-2 fuzzy sets and systems. The earliest one, and also the most popular one, is the Karnik-Mendel Algorithm, which is iterative and computationally intensive. In the last a few years researchers have proposed several other more efficient type-reduction algorithms. In this paper we also propose a new algorithm which improves over the latest results. Experiments show that it is the most efficient one to use in practice. Particularly, when the number of elements in type-reduction is smaller than 100, which is true in most practical type-reduction computations, our proposed algorithm can save over 50% computational cost over the Karnik-Mendel Algorithms. We also give the Matlab implementation of our most efficient algorithm in the Appendix. It includes preprocessing steps to eliminate numerical problems, and also improved testing criteria to prevent possible infinite loops. This program will be very helpful in promoting the popularity of type-2 fuzzy sets and systems.

Comparison and practical implementation of type-reduction algorithms for type-2 fuzzy sets and systems

This paper introduces an alternative type-reduction method for interval type-2 (IT2) fuzzy logic systems (FLSs), with either continuous or discrete secondary membership function. Unlike the Karnik-Mendel type reducer which is based on the wavy-slice representation of a type-2 fuzzy set, the proposed type reduction algorithm is developed using the vertical-slice representation. One advantage of the approach is the output of the type reducer can be expressed in closed form, thereby providing a tool for the theoretical analysis of IT2 FLSs. The computational complexity of the proposed method is also lower than the uncertainty bounds method and the enhanced Karnik-Mendel method. To assess the feasibility of the proposed type-reducer, it is used to calculate the output of an IT2 fuzzy logic controller (FLCs). Results from a simulated coupled tank experiment demonstrated that IT2 FLCs that employ the proposed type reduction algorithm share similar robustness properties as FLCs based on the Karnik-Mendel type reducer.

Towards an efficient type-reduction method for interval type-2 fuzzy logic systems

This paper presents the analytical structure of a class of interval type-2 (IT2) fuzzy proportional derivative (PD) and proportional integral (PI) controllers that have symmetrical rule base and symmetrical consequent sets. Two assumptions are made: 1) The Zadeh AND operator is employed as the t-norm operator; 2) type-reduction is performed by the Karnik-Mendel (KM) type-reduction method. The main contributions are the methodology that identifies the input conditions, where the KM algorithm uses a new switch point to compute the bounds of the type-reduced set, the closed-form expressions that relate the inputs and output of an IT2 fuzzy controller, and insights into the potential performance improvement because of the inclusion of the footprint of uncertainty (FOU). Compared with its T1 counterpart, two additional FOU parameters generate 31 extra local regions, each providing a unique relationship between the inputs and output signals. The generation of a relatively large number of local regions at the cost of two extra design parameters indicates that an IT2 fuzzy controller may be able to provide better performance. Furthermore, by comparing the analytical structure with the corresponding T1 counterpart, the potential advantages to use the IT2 over the T1 fuzzy controller are studied. Four interesting characteristics are identified, and they provide insights into why the IT2 fuzzy controller may better balance the conflicting aims of fast rise time and small overshoot.

Analytical Structure and Characteristics of Symmetric Karnik–Mendel Type-Reduced Interval Type-2 Fuzzy PI and PD Controllers

Stable adaptive fuzzy PD plus PI controller for nonlinear uncertain systems

This paper introduces the analytical structure of a class of interval type-2 (IT2) fuzzy PD and PI controllers that have symmetrical rule base. Two assumptions are made: (1) Zadeh AND operator is employed as the t-norm operator; (2) the centroid type-reduction method is used. The main contribution of this paper is the analytical relationship between the inputs and output of this IT2 fuzzy controller. Comparing with its T1 counterpart, the addition of two FOU parameters generated 31 extra local regions that each provides a unique relationship between the inputs and output signals. The relatively large number of local regions that results, at the cost of two extra design parameters, means that an IT2 fuzzy controller has the potential to provide better performance. Another outcome is the analysis of the centroid type-reduction algorithm yielded closed form expressions for each partition. This result may provide a platform for further theoretical study.

Maowen Nie

Papers

Towards an efficient type-reduction method for interval type-2 fuzzy logic systems

Comparison and practical implementation of type-reduction algorithms for type-2 fuzzy sets and systems

Analytical Structure and Characteristics of Symmetric Karnik–Mendel Type-Reduced Interval Type-2 Fuzzy PI and PD Controllers

Stable adaptive fuzzy PD plus PI controller for nonlinear uncertain systems

Derivation of the analytical structure of symmetrical IT2 fuzzy PD and PI controllers