In this paper we present a mathematical framework for our hybrid bond graph modeling methodology which combines continuous bond graph models with local discrete finite state automata [5]. We relate this mathematical model back to other hybrid dynamic systems modeling formalisms [1, 2]. Furthermore, we classify parameter and time scale abstractions and show how these result in different types of switching specifications based on a piori and a posteriori state vector values. Three types of operation of a hybrid dynamic systems are classified: mythical modes, pinnacles, and continuous modes, and facilitated by a recursive formula. Next, we augment the mathematical model into an implementation model that captures the idiosynchracies of physical system models and allows for a direct mapping of their model parts. The formal specifications are incorporated into a hybrid system simulation scheme that ensures the generation of correct system behavior.
Discontinuously changing state variables are systematically derived using the principle of conservation of state combined with explicitly defined interactions with the environment. Global specifications are derived dynamically based on systematic principles of invariance of state, divergence of time, and temporal evolution of states. This simplifies the modeling task and truly demonstrates the use of compositionality in defining system models. This is in contrast with the approach by Alur et al. which requires pre-defined global specifications of continuous system behavior in terms of differential equations [1]. Furthermore, global knowledge in specifying discrete behavior is required to ensure no mythical modes exist. Also, unlike the hybrid bond graph modeling paradigm, there is no support for systematic modeling based on physical principles (e.g., conservation of state). Future work will be directed toward applying this methodology in embedded (computer-based) control of physical systems.