That dynamical systems can produce chaotic and erratic behavior is now an accepted feature of systems modeling and controller design. This behavior is not necessarily something to be avoided at all costs, but is an aspect which should be recognized and taken into account when dealing with a control system. Despite the intense interest in chaotic systems, there is no universally accepted mathematical definition of chaos. Common descriptions are in terms of positive topological entropy, Li-Yorke scrambled sets, sensitive dependence of initial conditions and positive Liapunov exponents. The precise relationships between these competing signatures is unclear and some notions are not even independent. Fuzzy dynamical systems occur in many areas of interest and it is important to understand the repertoire of possible behaviors of fuzzy systems. This review gives an overview of chaos in fuzzy dynamical systems, studying Ã¢â‚¬Â¢ Dynamical systems and definitions of chaos Ã¢â‚¬Â¢ Metric space of fuzzy sets (D", doo). Ã¢â‚¬Â¢ Chaotic mappings on D" and possible loss of information due to chaotic dynamics. Ã¢â‚¬Â¢ A general definition of fuzzification and level set, based on t- norms/conorms and their diagonal functions, and the effect of fuzzifying chaotic mappings. Ã¢â‚¬Â¢ A relationship between a very simple criterion, positive topological entropy and Li-Yorke chaos. Ã¢â‚¬Â¢ Examples of fuzzy chaos
In this chapter, we point out the symbolical-numerical duality of fuzzy logic and the rule-case duality of fuzzy rules in approximate reasoning. By the former, we can use a node of a neural network to represent a fuzzy proposition for the symbolic and the value passing the node (input or output) for the numeric. By the latter, we can construct a neural network structure for describing the relations of fuzzy rules and modifythe weights of the neural network to realize learning from cases (examples). The concept of the so-called approximate case-based reasoning' and its neural network implementation is set up on the above understanding. We first give the basic mechanism of approximate case-based reasoning and the neural network implementation, then extend it to more general and complex cases by several examples.
This chapter presents a new structure of intelligent control for robotic motion. This structure is analogous to the human cerebral control structure for intelligent control. Therefore, the system has a hierarchical structure as an integrated approach of Neuromorphic and Symbolic control, including an applied neural network for servo control, a knowledge based approximation, and a fuzzy set theory for a human interface. The neural network in the servo control level is numerical manipulation, while the knowledge based part is symbolic manipulation. In the neuromorphic control, the neural network compensates for the nonlinearities of the system and uncertainty in its environment. The knowledge base part develops control strategies symbolically for the servo level with a-priori knowledge. The fuzzy logic combined with the neural network is used between the servo control level and the knowledge based part to link numerals to symbols and express human skills through learning.
The accuracy of a robot arm is determined by its ability to move in a given particular task space to specific Cartesian positions that are not necessarily pretaught. As a consequence, the inverse kinematics is an important problem as it must be solved in real-time in order to position the end-effector at an appropriate Cartesian location. However, it is a difficult and challenging problem for it involves the determination whether or not at least one mathematical set of robot joint angle values exists that will produce a desired coordinate configuration. The mathematical solutions should be checked against the physical constraints associated with the manipulator. Many times, a solution many not be physically realizable in a constrained environment. The advent of artificial neural networks has made it possible to obtain general learning schemes which can be used to arrive at feasible solutions to inverse kinematics problem in a constrained environment independent of a robotic structure. In this paper, we present such a learning scheme using a dynamic neural processor (DNP). This neural model functionally mimics the subpopulation of biological neurons. For analytical simplicity, only two subpopulations of neurons, namely excitatory and inhibitory, are assumed to coexist. The DNP is a neural network structure consisting of two dynamic neural units coupled as excitatory and inhibitory neurons. It is demonstrated in this study that the DNP would avoid time consuming numerical calculations and provide, more or less, instant recall of the learned associations. The learning and adaptive nature of this neural approach is demonstrated for two- and three-linked robots.
The Tethered Satellite System (TSS) consisting of the Shuttle, tether, and an Italian satellite, has complex and non-linear dynamics because of the spring-like behavior of the tether and its interaction with two six degrees-of-freedom end-bodies. The TSS has natural librational and longitudinal oscillations, whose characteristics change as the tether is deployed and retrieved. When the finite mass of the tether is properly modeled, the TSS also exhibits 'skip rope' motion, particularly if the motion is excited by the interaction between the geomagnetic field and electric current in the tether. A conventional control system based on the linearization of the system dynamics is employed in the onboard software. Its inputs are sensed length, length rate, and tension parameters and it is expected to maintain not only the length but also the in-plane librational angle. Due to the complexity of this problem and the recent success of fuzzy logic techniques in controlling systems with non-linear dynamics using imprecise measurements as input from sensors, a fuzzy logic based tether length controller has been developed and implemented into the TSS simulation to investigate the usefulness and robustness that can be achieved with fuzzy control. In this chapter, a concept which utilizes fuzzy membership functions for length, length rate, and voltage parameters, and the rulebase used in generating the control voltage signal is described. Initially, a simple proportional controller was created based on length error only. Later, the length rate error was included as input, and the rulebase was expanded. In order to have a fair comparison between the conventional and fuzzy controllers, tension feedback was not included as an input to the controllers since it is not an input to the conventional controller. Test results indicate that the fuzzy controller keeps the length error smaller throughout the mission, and the amplitude of the tension oscillations during the retrieval phase is also smaller. One surprising benefit, in addition to better length control, is smaller in-plane librational oscillations during the retrieval phase. Overall the results of the study indicate that fuzzy logic based control can provide the simplicity and robustness desired for complex tethered operations in space.
Uncertainty abounds in all aspects of computer vision. As a result, methods which explicitly model and manage that uncertainty have a better chance in producing meaningful results in such complex situations. Fuzzy set theory provides a framework to initially model the uncertainty in a vision problem, and many methods to exploit it in producing realistic results. One very important tool which has been used extensively in pattern recognition and computer vision is that of objective function based clustering. In this chapter, we will review classical and novel clustering methods as they apply to computer vision and show examples of their utility. In particular, we will focus on the use of clustering to detect and recognize regular boundaries of objects from images of edge magnitudes. The problem of fitting an unknown number of boundary curves to edge magnitude data is one of the major challenges of computer vision. We will show that fuzzy clustering can be readily adapted to the problem of curve detection, and that a new possibilistic clustering method introduced by the authors can produce significantly more accurate results than either crisp or fuzzy clustering in noisy situations.
A brief review of the applications of neural networks in various aspects of Image Processing (e.g., image segmentation, image restoration, texture segmentation, image coding etc.) along with their parallelism and fault tolerance characteristics will be provided first. Relevance of various fuzzy tools and methodologies for handling uncertainties and in providing soft decisions in image processing will be described. A discussion on making fusion of the merits of fuzzy logic and neural network technologies will then be made. Finally, a neuro fuzzy system for extracting objects from noisy images and its effectiveness will be described. The system consist of a multi-layer neural network with back-propagation of error and with feed back connections. In each layer there are M x N neurons (for an M x N image). Each neuron corresponds to a single pixel. Neurons in the same layer do not have any connection among themselves. Each neuron in a layer is connected to the corresponding neuron in the previous layer and to its neighbors over some window. Input is given as fuzzy set 'brightness'. The membership value for 'brightness' involves both global and local information of an image. Various measures of fuzziness of a set (like index of fuzziness, entropy, fuzzy correlation etc. ) are used to model the error in the output layer. The status of the neurons in the output layer is also considered to be a fuzzy set 'object pixels'.
Nonlinear decision boundaries separating similar regions in unlabeled data sets are encountered in many real applications but are extremely difficult to generate by statistical decision models. Supervised multilayered neural networks are capable of generating some nonlinear decision boundaries but require extensive training procedures leading to computational burden as well. Unsupervised self-organizing neural nets are capable of forming crisp clusters from unlabeled data sets. A recent trend has been to integrate the concept of fuzzy sets with adaptive learning inherent to neural nets. Such integration yields not only better clustering of similar data groups, it may also provide a method for generating nonlinear boundaries among clusters in close proximity. The weaknesses and strengths of such integrated self-organizing neuro-fuzzy models for adaptive pattern recognition are described. Successful classification of standard data sets and generation of nonlinear decision boundaries among neighboring clusters from computer generated data are demonstrated with a recently developed integrated adaptive fuzzy clustering (IAFC) model.
Processing gray-scale realizations of images that are ide- ally binary (such as gray-scale realizations of printed characters) is problematic due to the fact that gray-scale processing should be con- sistent with the binary nature of the ideal image. Essentially, any final decision (such as the recognition of a specific character at a spe- cific location) should reflect the content of the ideal image, which is generally unknown. Too often, a gray-scale realization of an ideal binary image is processed using methods appropriate for gray-scale re- alizations of ideal gray-scale images. These should not be expected to lead, ipso facto, to decision procedures appropriate to binary images. Fuzzy morphological algorithms do not assume probabilistic knowledge of the degradation process; however, they mirror the processing that one would have performed were the ideal binary image known. Thus, they lead to decision procedures consistent with those that would have been taken following processing of the ideal binary image. The theory of fuzzy mathematical morphology has been fully char- acterized in terms of a set of axioms for subset fuzzzfication. Within the general framework there exists an infinite number of realizations of fuzzy mathematical morphology and the determination of which the- ory is relevant is application dependent; nevertheless, there does exist a general paradigm for the construction of fuzzy algorithms from cor- responding binary algorithms. In this chapter 'we discuss this general paradigm for algorithm construction, and in particular, we discuss morphological algorithms for smoothing, geometric filtering, edge de- tection, object detection, and peak detection. The efficacy of these algorithms are underlined by the various simulations presented herein.
The segmentation and representation of complex features in higher dimensional data sets is of paramount importance for machine recognition and human perception of image informa- tion. A multiresolution and multiÃ¢â‚¬â€aspect data representation paradigm, a morphological skeleton, is used in this paper to provide a hierarchical framework for efficient representa- tion and visualization of data for machine recognition and human perception of data features. The utilization of fuzzy operators establishes a basis within this framework for organizing packets, or fuzzy sets, of approximate information by minimum coverings or maximal sub- tense. 3Ã¢â‚¬â€D and 2Ã¢â‚¬â€D image data are used to demonstrate applications of these techniques on higher dimensional data sets. Grayscale mathematical morphology provides an established basis and an algebra for fuzzy operators due to its representation of fuzzy maps over a set support. Specifically, the applica- tion of morphological operators in scale and orientation paradigms with tractable support shapes provides an ordered basis for topological analysis and user perception of data. To eliminate precision loss grayscale morphology utilizes only set operations requiring only computer addition and comparison.
The tools of mathematical morphology originally developed for image analysis can be generalized for structural analysis of the general class of multidimensional fuzzy data sets. In this chapter we discuss morphological tools useful for clustering analysis of fuzzy sets. Morphological techniques can perform cluster segmentations that are stable with respect to relative scale changes of the axes of the multidimensional space. We briefly explain a few simple approaches to convert data sets from continuous spaces to discrete spaces where the morphological algorithms can be applied. Morphological filters can be designed to eliminate noise which can cause large clusters to split into smaller clusters. Connectivity preserving filters can play an important role in removing noise while preserving important features of the data set which may be vulnerable to filtering processes. Morphological tools can segment complex data sets such as those consisting of shell type of clusters. We have reviewed several morphological algorithms useful for clustering of binary and fuzzy data sets.