A model of systems with modes and mode transitions

We propose a method of classifying the operation of a system into finitely many modes. Each mode has its own objectives for the system's behaviour and its own mathematical models and algorithms designed to accomplish its objectives. A central problem is deciding when to transition from one mode to some other mode, a decision that may be contested and involve partial or inconsistent information or evidence. We model formally the concept of modes for a system and derive a family of data types for analysing mode transitions. The data types are simplicial complexes, both abstract and realised in euclidean space $\mathbb{R}^{n}$. In the data type, a mode is represented by a simplex. Each state of a system can be evaluated relative to different modes by mapping it into one or more simplices. This calibration measures the extent to which distinct modes are appropriate for the state and can decide on a transition. We explain this methodology based on modes, introduce the mathematical ideas about simplicial objects we need and use them to build a theoretical framework for modes and mode transitions. To illustrate the general model in some detail, we work though a case study of an autonomous racing car.