Fuzzy Logic for Non-smooth Dynamical Systems

Fuzzy Logic for Non-smooth Dynamical Systems

Kamyar Mehran (Newcastle University, UK), Bashar Zahawi (Newcastle University, UK) and Damian Giaouris (Newcastle University, UK)
Copyright: © 2011 |Pages: 22
DOI: 10.4018/978-1-61692-797-4.ch008
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Dynamical system theory has proved to be a powerful tool in the analysis and comprehension of a diverse range of problems. Over the past decade, a significant proportion of these systems have been found to contain terms that are non-smooth functions of their arguments. These problems arise in a number of practical systems ranging from electrical circuits to biological systems and even financial markets. It has also been demonstrated that Fuzzy engineering can be effectively employed to identify or even predict an array of uncertainties and chaotic phenomena caused by discontinuities typical of this class of system. This chapter presents a review of the most recent developments concerned with the confluence of these two fields through real-life examples and current advances in research.
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1 Introduction

Bridging the two seemingly unrelated concepts, fuzzy logic and nonlinear piecewise-smooth dynamical systems theory is chiefly motivated by the concept of soft computing (SC), initiated by Lotfi A. Zadeh, the founder of fuzzy set theory. The principal components of SC, as defined in his initiative for soft computing1, are fuzzy logic (FL), neural network theory (NN) and probabilistic reasoning (PR), with the latter subsuming parts of belief networks, genetic algorithms, chaos theory and learning theory. SC is essentially distant from traditional, immutable (hard) computing and is much more aligned to the main ideas of Kansei Engineering in that the imprecision, uncertainty and partial truth reflecting the working of the human mind are incorporated into the computing process to form a new paradigm to tackle highly complex, nonlinear systems. The aim of this chapter is to combine FL from general Soft Computing theory and piece-wise smooth dynamical systems from the general theory of nonlinear dynamical systems. The main target being to examine their relationship and interaction and to demonstrate that the blending of the two concepts can be effectively used to analyse and control the chaotic and other nonlinear behaviours typical of such systems.

Over the past few decades, there has been a substantial level of interest in fuzzy systems technology and dynamical systems theory shown by almost all hard and soft-science research communities such as theoretical and experimental physicists, applied mathematicians, meteorologists, climatologists, physiologists, psychologists and engineers. More specifically, fuzzy system technology has emerged as an effective methodology to solve many problems ranging from control engineering, robotics, and automation to system identification, medical image/signal processing and Kansei engineering. Meanwhile, dynamical systems theory has proved to be a powerful tool to analyze and understand the behaviour of a diverse range of real life problems. The vast majority of these problems can only be modelled with dynamical systems whose behaviour is characterized by instantaneous changes and discontinuities. These practically ubiquitous dynamical systems are usually referred to as piece-wise smooth or non-smooth dynamical systems. Examples include mechanical systems with friction, robotic systems, electric and electronic systems employing electronic switching devices, unmanned vehicle systems, biological and biomedical systems, climate modelling and even financial forecasting. The study of these systems is of a great importance, primarily because they have captivating dynamics with significant practical applications and show rich nonlinear phenomena such as quasi-periodicity and chaos.

Unfortunately, many of the mathematical tools developed for smooth dynamical systems have to-date proven inadequate when dealing with the discontinuities present in non-smooth systems. Furthermore, certain specific approaches are not well-established and are still in their early stages of development. Not surprisingly, fuzzy system technology, in the context of highly complex nonlinear systems, can be critically intuitive and useful. This belief arises from the fact that fuzzy logic resembles human reasoning in its use of approximate information so it can embody the uncertainty which is the essential part of the mainly event-driven, non-smooth system and its chaotic dynamics. That’s why it is possible to believe that FL can lead to a general theory of uncertainty2.

The chapter does not intend – in fact, is not able – to provide a thorough explanation of the intrinsic relationship between fuzzy logic and dynamical system theory, but attempts to give some heuristic research results and insightful ideas, shedding some light on the subject and attracting more attention to the topic.

The chapter is organised in the following way: it commences with an outline of the fundamental concepts of nonlinear dynamical systems theory, and specifically non-smooth dynamical systems, with examples showing the richness and uniqueness of their nonlinear behaviours. The inherent difficulties in modelling these dynamical systems will be highlighted. This is followed by an examination of how the Takagi-Sugeno fuzzy modelling concept could be extended to overcome these problems. The fuzzy approach is further extended and applied to the stability analysis (in the Lyapunov framework) to predict the onset of structural instability or so-called bifurcations in the evolution of the dynamical system. The chapter ends with an evaluation of the proposed approach in nonlinear system modelling and analysis in general and further discussion about potential forthcoming research.

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