Under some conditions in real world, precise parameters and/or initial values of dynamic systems are hard to be determined. Fuzzy Differential Equation (FDE) is a powerful tool to model dynamical systems with the uncertainty of impreciseness. This thesis presents the first numerical solution for Fuzzy Differential Equations with multiple fuzzy parameters and initial Values (FDEPIV) problems. Previous approaches for solving the FDEs only focused on FDEs with single fuzzy condition. In this thesis, we applied the proper fuzzy arithmetic on Runge-Kutta method for solving the FDEPIV problems with multiple fuzzy parameters and initial conditions. Furthermore, comparing with directly applying the extension principle in solving FDEPIV, the complexity of the proposed method is much lower, and parallelization of the proposed algorithm is feasible. Numerical examples of the FDEPIV problems are presented to demonstrate the effectiveness of the proposed method.

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Image segmentation is an important step in the image analysis process. Current image segmentation techniques, however, require that the user tune several parameters in order to obtain maximum segmentation accuracy, a computationally inefficient approach, especially when a large number of images must be processed sequentially in real time. Another major challenge, particularly with medical image analysis, is the discrepancy between objective measures for assessing and guiding the segmentation process, on the one hand, and the subjective perception of the end users (e.g., clinicians), on the other. Hence, the setting and adjustment of parameters for medical image segmentation should be performed in a manner that incorporates user feedback. Despite the substantial number of techniques proposed in recent years, accurate segmentation of digital images remains a challenging task for automated computer algorithms. Approaches based on machine learning hold particular promise in this regard because, in many applications, including medical image analysis, frequent user intervention can be assumed as a means of correcting the results, thereby generating valuable feedback for algorithmic learning. This thesis presents an investigation of the use of evolving fuzzy systems for designing a method that overcomes the problems associated with medical image segmentation. An evolving fuzzy system can be trained using a set of invariant features, along with their optimum parameters, which act as a target for the system. Evolving fuzzy systems are also capable of adjusting parameters based on online updates of their rule base. This thesis proposes three different approaches that employ an evolving fuzzy system for the continual adjustment of the parameters of any medical image segmentation technique. The first proposed approach is based on evolving fuzzy image segmentation (EFIS). EFIS can adjust the parameters of existing segmentation methods and switch between them or fuse their results. The evolving rules have been applied for breast ultrasound images, with EFIS being used to adjust the parameters of three segmentation methods: global thresholding, region growing, and statistical region merging. The results for ten independent experiments for each of the three methods show average increases in accuracy of 5\%, 12\% and 9\% respectively. A comparison of the EFIS results with those obtained using five other thresholding methods revealed improvements. On the other hand, EFIS has some weak points, such as some fixed parameters and an inefficient feature calculation process. The second approach proposed as a means of overcoming the problems with EFIS is a new version of EFIS, called self-configuring EFIS (SC-EFIS). SC-EFIS uses the available data to estimate all of the parameters that are fixed in EFIS and has a feature selection process that selects suitable features based on current data. SC-EFIS was evaluated using the same three methods as for EFIS. The results show that SC-EFIS is competitive with EFIS but provides a higher level of automation. In the third approach, SC-EFIS is used to dynamically adjust more than one parameter, for example, three parameters of the normalized cut (N-cut) segmentation technique. This method, called multi-parametric SC-EFIS (MSC-EFIS), was applied to magnetic resonance images (MRIs) of the bladder and to breast ultrasound images. The results show the ability of MSC-EFIS to adjust multiple parameters. For ten independent experiments for each of the bladder and the breast images, this approach produced average accuracies that are 8\% and 16\% higher respectively, compared with their default values. The experimental results indicate that the proposed algorithms show significant promise in enhancing image segmentation, especially for medical applications.

Cellular automata (CA) are discrete dynamical systems comprised of a lattice of finite-state cells. At each time step, each cell updates its state as a function of the previous state of itself and its neighbours. Fuzzy cellular automata (FCA) are a real-valued extension of Boolean cellular automata which "fuzzifies" Boolean logic in the transition function using real values between zero and one (inclusive). To date, FCA have only been studied in disjunctive normal form (DNF). In this thesis, we study FCA in conjunctive normal form (CNF). We classify FCA in CNF both analytically and empirically. We compare these classes to their DNF counterparts. We prove that certain FCA exhibit chaos in CNF, in contrast to the periodic behaviours of DNF FCA. We also briefly explore five different forms of fuzzy logic, and suggest further study. In support of this research, we introduce novel methods of simulating and visualizing FCA.

Under some conditions in real world, precise parameters and/or initial values of dynamic systems are hard to be determined. Fuzzy Differential Equation (FDE) is a powerful tool to model dynamical systems with the uncertainty of impreciseness. This thesis presents the first numerical solution for Fuzzy Differential Equations with multiple fuzzy parameters and initial Values (FDEPIV) problems. Previous approaches for solving the FDEs only focused on FDEs with single fuzzy condition. In this thesis, we applied the proper fuzzy arithmetic on Runge-Kutta method for solving the FDEPIV problems with multiple fuzzy parameters and initial conditions. Furthermore, comparing with directly applying the extension principle in solving FDEPIV, the complexity of the proposed method is much lower, and parallelization of the proposed algorithm is feasible. Numerical examples of the FDEPIV problems are presented to demonstrate the effectiveness of the proposed method.

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While modern engineering systems have become increasingly integrated and complex over the years, interest in the application of control techniques which specifically attempt to formulate and solve the control problem in its inherently uncertain environment has been moderate, at best. More specifically, although many control schemes targeted at Heating, Ventilating and Air-Conditioning (HVAC) systems have been reported in the literature, most seem to rely on conventional techniques which assume that a detailed, precise model of the HVAC plant exists, and that the control objectives of the controller are clearly defined. Experience with HVAC systems shows that these assumptions are not always justifiable, and that, in practice, these systems are usually characterized by a lack of detailed design data and a lack of a robust understanding of the processes involved. Motivated by the need to more efficiently control complex, uncertain systems, this thesis focuses on the development and evaluation of a new neuro-fuzzy model-based predictive control scheme, where certain variables used in the optimization remain in the fuzzy domain. The method requires no training data from the actual plant under consideration, since detailed knowledge of the plant is unavailable. Results of the application of the control scheme to the control of thermal comfort in a simulated zone and to the control of the supply air temperature of an air-handling unit in the laboratory are presented. It is concluded that precious resources (as measured by actuator activity, for example) need not be wasted when controlling these systems. In addition, it is also shown that a very precise (and sometimes not necessarily accurate) control value computed at each sample is unnecessary. Rather, by defining the system and its environment in the fuzzy domain, the fuzzy decision algorithms developed here may be employed to get an "acceptable" control performance.