Analogue-digital systems with modes of physical behaviour

TL;DR

This paper introduces a general framework for analogue-digital systems with multiple physical modes, using topological data types to model transitions and handle diverse, possibly unmodelled, physical behaviors.

Contribution

It proposes a novel, unified definition of analogue-digital systems with modes and a topological approach to model mode transitions.

Findings

A topological data type based on simplicial complexes models mode transitions.

The framework accommodates diverse models, including unmodelled physical modes.

Illustrations include applications to driverless racing cars.

Abstract

Complex environments, processes and systems may exhibit several distinct modes of physical behaviour or operation. Thus, for example, in their design, a set of mathematical models may be needed, each model having its own domain of application and representing a particular mode of behaviour or operation of physical reality. The models may be of disparate kinds { discrete or continuous in data, time and space. Furthermore, some physical modes may not have a reliable model. Physical measurements determine modes of operation. We explore the question: What is a mode of behaviour? How do we specify algorithms and software that monitor or govern a complex physical situation with many modes? How do we specify a portfolio of modes, and the computational problem of transitioning from using one mode to another mode as physical modes change? We propose a general definition of an analogue-digital system with modes. We show how any diverse set of modes { with or without models { can be bound together, and how the transitions between modes can be determined, by constructing a topological data type based upon a simplicial complex. We illustrate the ideas of physical modes and our theory by reflecting on simple examples, including driverless racing cars.