Physical systems are by nature continuous. Discontinuities introduced in system models are in reality nonlinear behaviors that are linearized to: (i) prevent stiffness when performing numerical simulation, (ii) reduce computational complexity in system configurations and behavior analysis, and (iii) apply qualitative reasoning methods to gain a better understanding of overall system behavior. Our hybrid modeling framework combines bond graphs, controlled junctions, and finite state automata to model and analyze discontinuities in physical system behavior[9, 10]. In some situations, the hybrid models seem to violate the basic physical principles of conservation of energy and conservation of momentum. A primary contribution of this paper is the development of a systematic approach to analyze the correctness of the mode transition algorithm (MMA), and also verify the consistency of physical system models.
This approach is based on characterizing discontinuous transitions as those that involve (i) no change in the independence of buffers in the system, (ii) two or more buffers within the system become dependent, and (iii) explicitly modeled Dirac source and source-buffer dependencies occur, where energy exchange takes place with the environment. It is shown that these effects are correctly handled by the mythical mode algorithm.
Complex hybrid models may contain multiple controlled junctions, and, in general, there is no guarantee that a switching process, once initiated, will ever terminate. We have discussed that these situations violate the principle of divergence of time and, therefore, indicate modeling inconsistencies. A multiple energy phase space analysis technique is developed to analyze the divergence of time in individual model modes, and validated against the principle of invariance of state. Therefore, this research extends previous work by Alur, Henzinger, and Nicollin[6] who show divergence of time only for models that have a constant rate of change of variable values.
Iwasaki et al. introduce the concept of hypertime to represent the instantaneous switching time stamp as an infinitely short interval of time[7]. Switches are then said to occur in hypertime, where there is an infinitely small interval of hypertime between each. Effectively this introduces small time constants of parasitic effects in the simulation engine that were abstracted away from the model. This renders it possible to simulate physically inconsistent models, only valid by merit of execution semantics and the legitimacy of the results is questionable since energy exchange takes place during switching. Moreover, this does not eliminate the problem of numerical stiffness and disregards the alternatives of tightening switching conditions or modifying landmarks to ensure consistency.
In summary, we have investigated the nature, effects, and consistency of discontinuities in physical system models, and developed algorithms and systematic evaluation methods for validating our algorithm and model. Future work will focus on more general analysis of hybrid models in terms of reachability as well as the issue of time complexity by partitioning the model in instantaneously independent parts.