State Space Consistency and Differentiability Conditions for a Class of Causal Dynamical Input-Output Systems

TL;DR

This paper explores the properties of the natural state in causal input-output systems, focusing on conditions for system identification and differentiability, with implications for modeling physical systems.

Contribution

It provides sufficient conditions for identifying systems from natural states and examines differentiability properties, including a differential equation representation.

Findings

Counterexample showing system identification is not always possible from natural states

Sufficient conditions for system identification in time-invariant systems

Differentiability of natural states and trajectories established

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 report 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 is given. The results presented in this report 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.