State-space fading memory

TL;DR

This paper introduces a state-space definition of fading memory (FM), linking it to stability concepts in nonlinear systems and demonstrating its relevance to memristors and system approximation.

Contribution

It formalizes FM in state-space, connecting it with incremental stability notions and extending existing theorems to this new framework.

Findings

Incremental input-to-state stability implies FM for time-invariant systems.

Boyd and Chua's approximation theorems apply to $ ext{delta}$ISS models.

Memristor models possess the FM property under mild conditions.

Abstract

The fading-memory (FM) property captures the progressive loss of influence of past inputs on a system's current output and has originally been formalized by Boyd and Chua in an operator-theoretic framework. Despite its importance for systems approximation, reservoir computing, and recurrent neural networks, its connection with state-space notions of nonlinear stability, especially incremental ones, remains understudied. This paper introduces a state-space definition of FM. In state-space, FM can be interpreted as an extension of incremental input-to-output stability ($\delta$IOS) that explicitly incorporates a memory kernel upper-bounding the decay of past input differences. It is also closely related to Boyd and Chua's FM definition, with the sole difference of requiring uniform, instead of general, continuity of the memory functional with respect to an input-fading norm. We demonstrate that incremental input-to-state stability ($\delta$ISS) implies FM semi-globally for time-invariant systems under an equibounded input assumption. Notably, Boyd and Chua's approximation theorems apply to $\delta$ISS state-space models. As a closing application, we show that, under mild assumptions, the state-space model of current-driven memristors possess the FM property.