Global Dynamics Achievable by Feedback Controls: Some Preliminary Results

Abstract

Any control system that uses a feedback control law leads to globally defined control dynamics. To understand control systems, it is therefore important to understand the global dynamics possible through the use of feedback controls. This is a point of view different, in many respects, from that of established aproaches such as that of controllability theory. We are no longer interested in a, single trajectory, or trajectories that originate or terminate in a given point, but in the behaviour of ensembles of trajectories.In this paper, we formulate the basic definitions and prove some theorems that constitute a preliminary treatment of this very difficult problem. After defining our topic and describing the main classes of dynamics, we come to a discussion of 'closeness' and of structural stabilty, adapted to the control problem. Having separated the 'transient' from the 'asymptotic' part of the dynamics, we then turn to a discussion of the elementary kinds of asymptotic behaviour, namely isolated equilibrium points and limit cycles.What makes this treatment possible is the input from dynamical systems theory. It provides a point of view and motivates many of the results.