A Mixed-Boolean Hybrid Mathematical Model with Discontinuous States

Abstract

Hybrid mathematical models are often represented as continuous functions with discontinuous inputs, or they are visualised as state machines or petri-nets comprising continuous models linked by discontinuous mappings. The analysis and simulation of hybrid (or nonsmooth dynamical) models is plagued with difficulty, necessitating careful consideration of energy losses and state reinitialisation on commutation. The author proposes an alternative model, where states are discontinuous. The engineer familiar with techniques such as signal flow graphs or bond graphs can clearly visualise discontinuities as breaks (or joins) in power flow between parts of the model. A mixed-Boolean state equation can be derived which reflects the physics of switching behaviour. This has two advantages: first, by considering the physics incrementally about the discontinuity it can be simulated without the need for state reinitialisation algorithms, and second, it can be analysed for structural control properties to show how they change with commutation.