Nonlinear Control Systems By Matthew

Control systems are prevelant in nature and in man-made systems. Natural regulation occurs in biological and chemical processes, and may serve to maintain the various constituents at their appropriate levels, for example. In the early days of the industrial revolution, governors were devised to regulate the speed of steam engines, while in modern times, computerised control systems have become commonplace in industrial plants, robot manipulators, aircraft and spacecraft, etc. Indeed, the highly maneuverable X-29 aircraft using forward swept wings is possible only because of its control systems, and moreover, control theory has been crucial in NASA’s Apollo and Space Shuttle programmes. Control systems such as in these examples use in an essential way the idea of feedback, the central theme of this chapter. Control theory is the branch of engineering/science concerned with the design and analysis of control systems. Linear control theory treats systems for which an underlying linear model is assumed, and is a relatively mature subject, complete with firm theoretical foundations and a wide range of powerful and applicable design methodologies; see e.g., Anderson & Moore (1990), Kailath (1980). In contrast, nonlinear control theory deals with systems for which linear models are not adequate, and is relatively immature, especially in relation to applications. In fact, linear systems techniques are frequently employed in spite of the presence of nonlinearities. Nonetheless, nonlinear control theory is exciting and vitally important, and is the subject of a huge and varied range of research worldwide. The aim of this chapter is to convey to readers of Complex Systems something of the flavour of the subject, the techniques, the computational issues, and some of the applications. To place this chapter in perspective, in relation to the other chapters in this book, it is worthwhile citing Brockett’s remark that control theory is a prescriptive science, whereas physics, biology, etc, are descriptive sciences, see Brockett (1976b). Computer science shares some of the prescriptive qualities of control theory, in the sense that some objective is prescribed, and means are sought to fulfill it. It is this design aspect that is most important here. Indeed, control systems are designed to influence the behaviour of the system being controlled in order to achieve a desired level of performance. Brockett categorised control theory briefly: (i) To express models in input-output form, thereby identifying those variables which can be manipulated and those which can be observed. (ii) To develop methods for regulating the response of systems by modifying the dynamical nature of the system—e.g. stabilisation. (iii) To optimise the performance of the system relative to some performance index. In addition, feedback design endevours to compensate for disturbances and uncertainty. This chapter attempts to highlight the fundamental role played by feedback in control theory. Additional themes are stability, robustness, optimisation, information and com-