(Ad/f,g/), (ad/fg/) and locally (ad/f,g/) invariant and controllability distributions

Over the past decade, it has been found that concepts from differential topology such as foliation and invariant distribution play a crucial role in the study of nonlinear systems. These concepts were first employed in studies of nonlinear controllability and observability. More recently, they have occurred in the study of decoupling and linearization via feedback. In connection with the widening area of application, requirements for a greater precision have arisen. The present paper represents an attempt to provide that precision. In this paper, an introduction is given of the basic concepts which are needed for an understanding of controllability, observability, and decoupling of nonlinear systems. Attention is given to mathematical preliminaries, aspects of invariance, nonlinear controllability and observability, disturbance decoupling, and controllability distributions.