Discontinuous Control of Multi-Variable Processes

Abstract

The research reported in this thesis is concerned with the control of physical systems which can be modelled by linear constant-coefficient vector differential equations of the form x = Ax + Gm where x and m are the n-dimensional state vector and the r-dimensional control vector, respectively. A is the n x n system matrix and G is the n x r control matrix. Examples of processes to which such equations apply are the control systems for small-angle-attitude motion of a satellite, the dynamical control of pilotless aircraft (see [5]), and the temperature-level control of processes. The behaviour of such systems depends upon the interaction of the matrices A and G, with the control domain [hand drawn symbol] which depends, in turn, upon the type of controllers used. This interaction is investigated in Chapter One to determine the large-scale behaviour of the system. In Chapter Two the problem of open-loop control is discussed, while in Chapter Three the time-optimal control of such systems in considered. The methods used to investigate these processes are geometric in nature. They are based upon the geometric properties of convex sets and hyperplanes in the n-dimensional state-space of the system. While the mathematical methods used to derive the results within may seem very abstract to the design engineer, it is hoped that the results themselves will provide useful insight into the behaviour of the systems as well as design criteria for the practising engineer.