Introduction

Abstract

A causal input–output system may be described by a function space for inputs, a function space for outputs, and a causal operator mapping the input space into the output space. A particular representation of the state of such a system at any instant has been defined as an operator from the space of possible future inputs to that of future outputs. This representation is called the natural state. The purpose of this book is to investigate additional properties of the natural state in two areas. The first area has to do with the possibility of determining the input–output system from its natural state set. A counterexample where this is not possible is given. Sufficient conditions for identifying the system from its natural state set are given. The results in this area are mostly for time-invariant systems. There are also some preliminary observations on reachability. The second area deals with differentiability properties involving the natural state inherited from the input–output system, including differentiability of the natural state and natural state trajectories. A differential equation representation for the trajectories is given. The results presented in this book may be considered as aids in modeling physical systems because system identification from state set holds in many models and is only tacitly assumed; also, differentiability is a useful property for many systems, for example, those used by engineers.