Analogue-digital systems with modes of physical behaviour

Complex environments, processes and systems may exhibit several distinct modes of physical behaviour or operation. Thus, for 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 and, furthermore, not all physical modes may have a reliable model. 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/models can be determined, by constructing a 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. No plan survives contact with the enemy. After Helmuth von Moltke the Elder (1800-1891) There are known knowns; there are things we know that we know. There are known unknowns; that is to say, there are things that we now know we don’t know. But there are also unknown unknowns – there are things we do not know we don’t know. Donald Rumsfeld (2002)