Nie-Tan Method and its Improved Version: A Counterexample

Context: The bottleneck on interval type-2 fuzzy logic systems is the output processing when using Centroid Type-Reduction + Defuzzification (CTR+D method). Nie and Tan proposed an approximation to CTR+D (NT method). Recently, Mendel and Liu improved the NT method (INT method). Numerical examples (due to Mendel and Liu) exhibit the NT and INT methods as good approximations to CTR+D.Method: Normalization to the unit interval of membership function domains (examples and counterexample) and variables involved in the calculations for the three methods. Examples (due to Mendel and Liu) taken from the literature. Counterexample with piecewise linear membership functions. Comparison by means of error and percentage relative error.Results: NT vs. CTR+D: Our counterexample showed an error of 0.1014 and a percentage relative error of 30.53%. This is respectively 23 and 32 times higher than the worst case obtained in the examples. INT vs. CTR+D: Our counterexample showed an error of 0.0725 and a percentage relative error of 21.83%. This is respectively 363 and 546 times higher than the worst case obtained in the examples.Conclusions: NT and INT methods are not necessarily good approximations to the CTR+D method

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

Analysis and Applications of the Km Algorithm in Type-2 Fuzzy Logic Control and Decision Making

Ph.DDOCTOR OF PHILOSOPH

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

A framework for robust control of uncertainty in self-adaptive software connectors

Context and motivations. The desired behavior of a system in ubiquitous environments considers not only its correct functionality, but also the satisfaction of its non-functional properties, i.e., its quality of service. Given the heterogeneity and dynamism characterizing the ubiquitous environments and the need for continuous satisfaction of non-functional properties, self-adaptive solutions appear to be an appropriate approach to achieve interoperability. In this work, self-adaptation is adopted to enable software connectors to adapt the interaction protocols run by the connected components to let them communicate in a timely manner and with the required level of quality. However, this self-adaptation should be dependable, reliable and resilient to be adopted in dynamic, unpredictable environments with different sources of uncertainty. The majority of current approaches for the construction of self-adaptive software ignore the uncertainty underlying non-functional requirement verification and adaptation reasoning. Consequently, these approaches jeopardize system reliability and hinder the adoption of self-adaptive software in areas where dependability is of utmost importance. Objective. The main objective of this research is to properly handle the uncertainties in the non-functional requirement verification and the adaptation reasoning part of the self-adaptive feedback control loop of software connectors. This will enable a robust and runtime efficient adaptation in software connectors and make them reliable for usage in uncertain environments. Method. In the context of this thesis, a framework has been developed with the following functionalities: 1) Robust control of uncertainty in runtime requirement verification. The main activity in runtime verification is fine-tuning of the models that are adopted for runtime reasoning. The proposed stochastic approach is able to update the unknown parameters of the models at runtime even in the presence of incomplete and noisy observations. 2) Robust control of uncertainty in adaptation reasoning. A general methodology based on type-2 fuzzy logic has been introduced for the control of adaptation decision-making that adjusts the configuration of component connectors to the appropriate mode. The methodology enables a systematic development of fuzzy logic controllers that can derive the right mode for connectors even in the presence of measurement inaccuracy and adaptation policy conflicts. Results. The proposed model evolution mechanism is empirically evaluated, showing a significant precision of parameter estimation with an acceptable overhead at runtime. In addition, the fuzzy based controller, generated by the methodology, has been shown to be robust against uncertainties in the input data, efficient in terms of runtime overhead even in large-scale knowledge bases and stable in terms of control theory properties. We also demonstrate the applicability of the developed framework in a real-world domain. Thesis statement. We enable reliable and dependable self-adaptations of component connectors in unreliable environments with imperfect monitoring facilities and conflicting user opinions about adaptation policies by developing a framework which comprises: (a) mechanisms for robust model evolution, (b) a method for adaptation reasoning, and (c) tool support that allows an end-to-end application of the developed techniques in real-world domains

An Explainable Artificial Intelligence Approach Based on Deep Type-2 Fuzzy Logic System

Artificial intelligence (AI) systems have benefitted from the easy availability of computing power and the rapid increase in the quantity and quality of data which has led to the widespread adoption of AI techniques across a wide variety of fields. However, the use of complex (or Black box) AI systems such as Deep Neural Networks, support vector machines, etc., could lead to a lack of transparency. This lack of transparency is not specific to deep learning or complex AI algorithms; other interpretable AI algorithms such as kernel machines, logistic regressions, decision trees, or rules-based algorithms can also become difficult to interpret for high dimensional inputs. The lack of transparency or explainability reduces the effectiveness of AI models in regulated applications (such as medical, financial, etc.), where it is essential to explain the model operation and how it arrived at a given prediction. The need for explainability in AI has led to a new line of research that focuses on developing Explainable AI techniques. There are three main avenues of research that are being explored to achieve explainability; first, Deep Explanations, which involves the modification of existing Deep learning models to add explainability. The methods proposed to do Deep explanations generally provide details about all the input features that affect the output, generally in a visual format as there might be a large number of features. This type of explanation is useful for tasks such as image recognition, but in other tasks, it might be hard to distinguish the most important features. Second, Model induction, which involves methods that are model agnostic, but these methods might not be suitable for use in regulated applications. The third method is to use existing interpretable models such as decision trees, fuzzy logic, etc., but the problem with them is that they can also become opaque for high dimensional data. Hence, this thesis presents a novel AI system by combining the predictive power of Deep Learning with the interpretability of Interval Type-2 Fuzzy Logic Systems. The advantages of such a system are, first, the ability to be trained via labelled and unlabelled data (i.e., mixing supervised and unsupervised learning). Second, having embedded feature selection abilities (i.e., can be trained by hundreds and thousands of inputs with no need for feature selection) while delivering explainable models with small rules bases composed of short rules to maximize the model’s interpretability. The proposed model was developed with data from British Telecom (BT). It achieved comparable performance to the deep models such as Stacked Autoencoder (SAE) and Convolution Neural Networks (CNN). In categorical datasets, the model outperformed the SAE by 2%, performed within 2-3% of the CNN and outperformed Multi-Layer Perceptron (MLP) and IT2FLS by 4%. In the regression datasets, the model performed slightly worse than the SAE, MLP and CNN models, but it outperformed the IT2FLS with a 15% lower error. The proposed model achieved excellent interpretability in a survey where it was rated within 2% of the highly interpretable IT2FLS. It was also rated 20% and 17% better than Deep learning XAI tools LIME and SHAP, respectively. The proposed model shows a small loss in performance for significantly higher interpretability, making it a suitable replacement for the other AI models in applications with many features where interpretability is paramount

Fuzzy Controllers

Trying to meet the requirements in the field, present book treats different fuzzy control architectures both in terms of the theoretical design and in terms of comparative validation studies in various applications, numerically simulated or experimentally developed. Through the subject matter and through the inter and multidisciplinary content, this book is addressed mainly to the researchers, doctoral students and students interested in developing new applications of intelligent control, but also to the people who want to become familiar with the control concepts based on fuzzy techniques. Bibliographic resources used to perform the work includes books and articles of present interest in the field, published in prestigious journals and publishing houses, and websites dedicated to various applications of fuzzy control. Its structure and the presented studies include the book in the category of those who make a direct connection between theoretical developments and practical applications, thereby constituting a real support for the specialists in artificial intelligence, modelling and control fields