Adaptive Non-singleton Type-2 Fuzzy Logic Systems: A Way Forward for Handling Numerical Uncertainties in Real World Applications

Real world environments are characterized by high levels of linguistic and numerical uncertainties. A Fuzzy Logic System (FLS) is recognized as an adequate methodology to handle the uncertainties and imprecision available in real world environments and applications. Since the invention of fuzzy logic, it has been applied with great success to numerous real world applications such as washing machines, food processors, battery chargers, electrical vehicles, and several other domestic and industrial appliances. The first generation of FLSs were type-1 FLSs in which type-1 fuzzy sets were employed. Later, it was found that using type-2 FLSs can enable the handling of higher levels of uncertainties. Recent works have shown that interval type-2 FLSs can outperform type-1 FLSs in the applications which encompass high uncertainty levels. However, the majority of interval type-2 FLSs handle the linguistic and input numerical uncertainties using singleton interval type-2 FLSs that mix the numerical and linguistic uncertainties to be handled only by the linguistic labels type-2 fuzzy sets. This ignores the fact that if input numerical uncertainties were present, they should affect the incoming inputs to the FLS. Even in the papers that employed non-singleton type-2 FLSs, the input signals were assumed to have a predefined shape (mostly Gaussian or triangular) which might not reflect the real uncertainty distribution which can vary with the associated measurement. In this paper, we will present a new approach which is based on an adaptive non-singleton interval type-2 FLS where the numerical uncertainties will be modeled and handled by non-singleton type-2 fuzzy inputs and the linguistic uncertainties will be handled by interval type-2 fuzzy sets to represent the antecedents’ linguistic labels. The non-singleton type-2 fuzzy inputs are dynamic and they are automatically generated from data and they do not assume a specific shape about the distribution associated with the given sensor. We will present several real world experiments using a real world robot which will show how the proposed type-2 non-singleton type-2 FLS will produce a superior performance to its singleton type-1 and type-2 counterparts when encountering high levels of uncertainties.</jats:p

On transitioning from type-1 to interval type-2 fuzzy logic systems

Capturing the uncertainty arising from system noise has been a core feature of fuzzy logic systems (FLSs) for many years. This paper builds on previous work and explores the methodological transition of type-l (Tl) to interval type-2 fuzzy sets (IT2 FSs) for given "levels" of uncertainty. Specifically, we propose to transition from Tl to IT2 FLSs through varying the size of the Footprint Of Uncertainty (FOU) of their respective FSs while maintaining the original FS shape (e.g., triangular) and keeping the size of the FOU over the FS as constant as possible. The latter is important as it enables the systematic relating of FOU size to levels of uncertainty and vice versa, while the former enables an intuitive comparison between the Tl and T2 FSs. The effectiveness of the proposed method is demonstrated through a series of experiments using the well-known Mackey-Glass (MG) time series prediction problem. The results are compared with the results of the IT2 FS creation method introduced in [1] which follows a similar methodology as the proposed approach but does not maintain the membership function (MF) shape

Geometric Defuzzification Revisited

The file attached to this record is the author's final peer reviewed version. The Publisher's final version can be found by following the DOI link.In this paper the Geometric Defuzzification strategy for type-2 fuzzy sets is reappraised. For both discretised and geometric fuzzy sets the techniques for type-1, interval type-2, and generalised type-2 defuzzification are presented in turn. In the type-2 case the accuracy of Geometric Defuzzification is assessed through a series of test runs on interval type-2 fuzzy sets, using Exhaustive Defuzzification as the benchmark method. These experiments demonstrate the Geometric Defuzzifier to be wildly inaccurate. The test sets take many shapes; they are not confined to those type-2 sets with rotational symmetry that have previously been acknowledged by the technique’s developers to be problematic as regards accuracy. Type-2 Geometric Defuzzification is then examined theoretically. The defuzzification strategy is demonstrated to be built upon a fallacious application of the concept of centroid. This explains the markedly inaccurate experimental results. Thus the accuracy issues of type-2 Geometric Defuzzification are revealed to be inevitable, fundamental and significant

Geometric Fuzzy Logic Systems

There has recently been a significant increase in academic interest in the field oftype-2 fuzzy sets and systems. Type-2 fuzzy systems offer the ability to model and reason with uncertain concepts. When faced with uncertainties type-2 fuzzy systems should, theoretically, give an increase in performance over type-l fuzzy systems. However, the computational complexity of generalised type-2 fuzzy systems is significantly higher than type-l systems. A direct consequence of this is that, prior to this thesis, generalised type-2 fuzzy logic has not yet been applied in a time critical domain, such as control. Control applications are the main application area of type-l fuzzy systems with the literature reporting many successes in this area. Clearly the computational complexity oftype-2 fuzzy logic is holding the field back. This restriction on the development oftype-2 fuzzy systems is tackled in this research. This thesis presents the novel approach ofdefining fuzzy sets as geometric objects - geometric fuzzy sets. The logical operations for geometric fuzzy sets are defined as geometric manipulations of these sets. This novel geometric approach is applied to type-I, type-2 interval and generalised type-2 fuzzy sets and systems. The major contribution of this research is the reduction in the computational complexity oftype-2 fuzzy logic that results from the application of the geometric approach. This reduction in computational complexity is so substantial that generalised type-2 fuzzy logic has, for the first time, been successfully applied to a control problem - mobile robot navigation. A detailed comparison between the performance of the generalised type-2 fuzzy controller and the performance of the type-l and type-2 interval controllers is given. The results indicate that the generalised type-2 fuzzy logic controller outperforms the other robot controllers. This outcome suggests that generalised type-2 fuzzy systems can offer an improved performance over type-l and type-2 interval systems

Novel fuzzy techniques for modelling human decision making

Standard (type-1) fuzzy sets were introduced to resemble human reasoning in its use of approximate information and uncertainty to generate decisions. Since knowledge can be expressed in a more natural by using fuzzy sets, many decision problems can be greatly simplified. However, standard type-1 fuzzy sets have limitations when it comes to modelling human decision making. In many applications involving the modelling of human decision making (expert systems) the more traditional membership functions do not provide a wide enough choice for the system developer. They are therefore missing an opportunity to produce simpler or better systems. The use of complex non-convex membership functions in the context of human decision making systems were investigated. It was demonstrated that non-convex membership functions are plausible, reasonable membership functions in the sense originally intended by Zadeh. All humans, including ‘experts’, exhibit variation in their decision making. To date, it has been an implicit assumption that expert systems, including fuzzy expert systems, should not exhibit such variation. Type-2 fuzzy sets feature membership functions that are themselves fuzzy sets. While type-2 fuzzy sets capture uncertainty by introducing a range of membership values associated with each value of the base variable, but they do not capture the notion of variability. To overcome this limitation of type-2 fuzzy sets, Garibaldi previously proposed the term ‘non-deterministic fuzzy reasoning’ in which variability is introduced into the membership functions of a fuzzy system through the use of random alterations to the parameters. In this thesis, this notion is extended and formalised through the introduction of a notion termed a non-stationary fuzzy set. The concept of random perturbations that can be used for generating these non-stationary fuzzy sets is proposed. The footprint of variation (FOV) is introduced to describe the area covering the range from the minimum to the maximum fuzzy sets which comprise the non-stationary fuzzy sets (this is similar to the footprint of uncertainty of type-2 sets). Basic operators, i.e. union, intersection and complement, for non-stationary fuzzy sets are also proposed. Proofs of properties of non-stationary fuzzy sets to satisfy the set theoretic laws are also given in this thesis. It can be observed that, firstly, a non-stationary fuzzy set is a collection of type-1 fuzzy sets in which there is an explicit, defined, relationship between the fuzzy sets. Specifically, each of the instantiations (individual type-1 sets) is derived by a perturbation of (making a small change to) a single underlying membership function. Secondly, a non-stationary fuzzy set does not have secondary membership functions, and secondary membership grades. Hence, there is no ‘direct’ equivalent to the embedded type-2 sets of a type-2 fuzzy sets. Lastly, the non-stationary inference process is quite different from type-2 inference, in that non-stationary inference is just a repeated type-1 inference. Several case studies have been carried out in this research. Experiments have been carried out to investigate the use of non-stationary fuzzy sets, and the relationship between non-stationary and interval type-2 fuzzy sets. The results from these experiments are compared with results produced by type-2 fuzzy systems. As an aside, experiments were carried out to investigate the effect of the number of tunable parameters on performance in type-1 and type-2 fuzzy systems. It was demonstrated that more tunable parameters can improve the performance of a non-singleton type-1 fuzzy system to be as good as or better than the equivalent type-2 fuzzy system. Taken as a whole, the techniques presented in this thesis represent a valuable addition to the tools available to a model designer for constructing fuzzy models of human decision making