Type-2 fuzzy alpha-cuts

Type-2 fuzzy logic systems make use of type-2 fuzzy sets. To be able to deliver useful type-2 fuzzy logic applications we need to be able to perform meaningful operations on these sets. These operations should also be practically tractable. However, type-2 fuzzy sets suffer the shortcoming of being complex by definition. Indeed, the third dimension, which is the source of extra parameters, is in itself the origin of extra computational cost. The quest for a representation that allow practical systems to be implemented is the motivation for our work. In this paper we define the alpha-cut decomposition theorem for type- 2 fuzzy sets which is a new representation analogous to the alpha-cut representation of type-1 fuzzy sets and the extension principle. We show that this new decomposition theorem forms a methodology for extending mathematical concepts from crisp sets to type-2 fuzzy sets directly. In the process of developing this theory we also define a generalisation that allows us to extend operations from interval type-2 fuzzy sets or interval valued fuzzy sets to type-2 fuzzy sets. These results will allow for the more applications of type-2 fuzzy sets by expiating the parallelism that the research here affords

Type-2 Fuzzy Alpha-cuts

Systems that utilise type-2 fuzzy sets to handle uncertainty have not been implemented in real world applications unlike the astonishing number of applications involving standard fuzzy sets. The main reason behind this is the complex mathematical nature of type-2 fuzzy sets which is the source of two major problems. On one hand, it is difficult to mathematically manipulate type-2 fuzzy sets, and on the other, the computational cost of processing and performing operations using these sets is very high. Most of the current research carried out on type-2 fuzzy logic concentrates on finding mathematical means to overcome these obstacles. One way of accomplishing the first task is to develop a meaningful mathematical representation of type-2 fuzzy sets that allows functions and operations to be extended from well known mathematical forms to type-2 fuzzy sets. To this end, this thesis presents a novel alpha-cut representation theorem to be this meaningful mathematical representation. It is the decomposition of a type-2 fuzzy set in to a number of classical sets. The alpha-cut representation theorem is the main contribution of this thesis. This dissertation also presents a methodology to allow functions and operations to be extended directly from classical sets to type-2 fuzzy sets. A novel alpha-cut extension principle is presented in this thesis and used to define uncertainty measures and arithmetic operations for type-2 fuzzy sets. Throughout this investigation, a plethora of concepts and definitions have been developed for the first time in order to make the manipulation of type-2 fuzzy sets a simple and straight forward task. Worked examples are used to demonstrate the usefulness of these theorems and methods. Finally, the crisp alpha-cuts of this fundamental decomposition theorem are by definition independent of each other. This dissertation shows that operations on type-2 fuzzy sets using the alpha-cut extension principle can be processed in parallel. This feature is found to be extremely powerful, especially if performing computation on the massively parallel graphical processing units. This thesis explores this capability and shows through different experiments the achievement of significant reduction in processing time.The National Training Directorate, Republic of Suda

Type-2 fuzzy linear systems

Fuzzy Linear Systems (FLSs) are used in practical situations where some of the systems parameters or variables are uncertain. To date, investigations conducted on FLSs are restricted to those in which the uncertainty is assumed to be modeled by Type-1 Fuzzy Sets (T1FSs). However, there are many situations where considering the uncertainty as T1FSs may not be possible due to different interpretations of experts about the uncertainty. Moreover, solutions of FLSs are T1FSs which do not provide any information about a measure of the dispersion of uncertainty around the T1FSs. Therefore, in this research a model of uncertain linear equations system called a type-2 fuzzy linear system is presented to overcome the shortcomings. The uncertainty is represented by a special class of type-2 fuzzy sets – triangular perfect quasi type-2 fuzzy numbers. Additionally, conditions for the existence of a unique type–2 fuzzy solution to the linear system are derived. A definition of a type-2 fuzzy solution is also given. The applicability of the proposed model is illustrated using examples in the pulp and paper industry, and electrical engineering

Review of Type-1 and Type-2 Fuzzy Numbers

We review type-1 and type-2 fuzzy numbers in this chapter, and propose one way of perceiving the concept of fuzzy numbers by comparing with that of round numbers. There are some definitions of fuzzy numbers, but we particularly adopt the definition often used in fuzzy analysis. Thereby, we emphasize that fuzzy number theory can be reduced to an argument for interval analysis. Moreover, we explain type-2 fuzzy sets and list two specific type-2 fuzzy numbers, one is a (triangular) perfect quasi-type-2 fuzzy number and the other is a triangular shaped type-2 fuzzy number. Finally, we mention the importance and utility of using type-2 fuzzy numbers

An evolving feature weighting framework for radial basis function neural network models

Via Granular Computing (GrC), one can create effective computational frameworks for obtaining information from data, motivated by the human perception of combining similar objects. Combining knowledge gained via GrC with a Fuzzy inference engine (Neural-Fuzzy) enable us to develop a transparent system. While weighting variables based on their importance during the iterative data granulation process has been proposed before (W-GrC), there is no work in the literature to demonstrate effectiveness and impact on Type-2 Fuzzy Logic systems (T2-FLS). The main contribution of this paper is to extend W-GrC, for the first time, to both Type-1 and Type-2 models known as Radial Basis Function Neural Network (RBFNN) and General Type-2 Radial Basis Function Neural Network (GT2-RBFNN). The proposed framework is validated using popular datasets: Iris, Wine, Breast Cancer, Heart and Cardiotocography. Results show that with the appropriate selection of feature weight parameter, the new computational framework achieves better classification accuracy outcomes. In addition, we also introduce in this research work an investigation on the modelling structure's interpretability (via Nauck's index) where it is shown that a good balance of interpretability and accuracy can be maintained

Online Coordinated Charging of Plug-In Electric Vehicles in Smart Grid to Minimize Cost of Generating Energy and Improve Voltage Profile

This Ph.D. research highlights the negative impacts of random vehicle charging on power grid and proposes four practical PEV coordinated charging strategies that reduce network and generation costs by integrating renewable energy resources and real-time pricing while considering utility constraints and consumer concerns

Fuzzy Logic Based Software Product Quality Model for Execution Tracing

This report presents the research carried out in the area of software product quality modelling. Its main endeavour is to consider software product quality with regard to maintainability. Supporting this aim, execution tracing quality, which is a neglected property of the software product quality at present in the quality frameworks under investigation, needs to be described by a model that offers possibilities to link to the overall software product quality frameworks. The report includes concise description of the research objectives: (1) the thorough investigation of software product quality frameworks from the point of view of the quality property analysability with regard to execution tracing; (2) moreover, extension possibilities of software product quality frameworks, and (3) a pilot quality model developed for execution tracing quality, which is capable to capture subjective uncertainty associated with the software quality measurement. The report closes with concluding remarks: (1) the present software quality frameworks do not exhibit any property to describe execution tracing quality, (2) execution tracing has a significant impact on the analysability of software systems that increases with the complexity, and (3) the uncertainty associated with execution tracing quality can adequately be expressed by type-1 fuzzy logic. The section potential future work outlines directions into which the research could be continued. Findings of the research were summarized in two research reports, which were also incorporated in the thesis, and submitted for publication: 1. Tamas Galli, Francisco Chiclana, Jenny Carter, Helge Janicke, “Towards Introducing Execution Tracing to Software Product Quality Frameworks,” Acta Polytechnica Hungarica, vol. 11, no. 3, pp. 5-24, 2014. doi: 10.12700/APH.11.03.2014.03.1 2. Tamas Galli, Francisco Chiclana, Jenny Carter, Helge Janicke “Modelling Execution Tracing Quality by Means of Type-1 Fuzzy Logic,” Acta Polytechnica Hungarica, vol. 10, no. 8, pp. 49-67, 2013. doi: 10.12700/APH.10.08.2013.8.

Circumventing the fuzzy type reduction for autonomous vehicle controller

Fuzzy type-2 controllers can easily deal with systems nonlinearity and utilise humans’ expertise to solve many complex control problems; they are also very good at processing uncertainty, which exists in many robotic systems, such as autonomous vehicles. However, their computational cost is high, especially at the type reduction stage. In this research, it is aimed to reduce the computation cost of the type reduction stage, thus to facilitate faster performance speed and increase the number of actions able to be operated in one microprocessor. Proposed here are adaptive integration principles with a binary successive search technique to locate the straight or semi-straight segments of a fuzzy set, thus to use them in achieving faster weighted average computation. This computation is very important because it runs frequently in many type reductions. A variable adaptation rate is suggested during the type reduction iterations to reduce the computation cost further. The influence of the proposed approaches on the fuzzy type-2 controller’s error has been mathematically analysed and then experimentally measured using a wall-following behaviour, which is the most important action for many autonomous vehicles. The resultant execution time-gain of the proposed technique has reached to 200%. This evaluated with respect to the execution time of the original, unmodified, type reduction procedure. This study develops a new accelerated version of the enhanced Karnik-Mendel type reducer by using better initialisations and better indexing scheme. The resulting performance time-gain reached 170%, with respect to the original version. A further cut in the type reduction time is achieved by proposing a One-Go type reduction procedure. This technique can reduce multiple sets altogether in one pass, thus eliminating much of the redundant calculations needed to carry out the reduction individually. All the proposed type reduction enhancements were evaluated in terms of their execution time-gain and performance error using every possible fuzzy firing level combination. Tests were then performed using a real autonomous vehicle, navigates in a relatively complex arena field with acute, right, obtuse, and reflex angled corners, to assure evaluating wide variety of operation conditions. A simplified state hold technique using Schmitt-trigger principles and dynamic sense pattern control was suggested and implemented to assure small rule base size and to obtain more accurate evaluation of the type reduction stages