An Operational Semantics for Hybrid Systems Involving Behavioral Abstraction Simon Bliudze École Polytechnique Fédérale de Lausanne INJ Building, Station 14 1015 Lausanne, Switzerland Sébastien Furic LMS Imagine S.A. 7 place des Minimes 42300 Roanne, France Abstract We discuss the challenges of building a simulation framework for hybrid systems, in particular the well- known Zeno effect and correct composition of models idealised by abstracting irrelevant behavioural details (e.g. the bounce dynamics of a bouncing ball or the process of fuse melting in an electrical circuit). We ar- gue that the cornerstone of addressing these challenges is the definition of a semantic framework with an ap- propriate underlying model of time. Using two simple examples, we illustrate the prop- erties of such a model and explain why existing models are not sufficient. Finally, we propose a new Zeno-free semantic model that allows mixing discrete and con- tinuous behaviour in a rigorous way and provides for the compositional behavioural abstraction. Although it is based on non-standard analysis, we explain how our semantic model can be used to de- velop hybrid system simulators. Keywords: Hybrid Modeling Languages; Non- Standard Analysis; Models of Signals; Behavioral Ab- straction; Operational Semantics 1 Introduction A large number of modelling, verification and sim- ulation frameworks for hybrid systems have been designed in the past years. Although, a complete overview is beyond the scope of our paper, we ob- serve that they broadly fall in two categories: those that put special emphasis on a rigorous model defi- nition, such as, for instance, the Ptolemy project [6] (based on [14]), the Zélus synchronous language [3] (based on the semantics in [1]) and SpaceEx [7]; and those that have chosen a more pragmatic, informal ap- proach, such as the Modelica language [8] and the as- sociated tools, and the Scicos block-diagram modeller and simulator [4]. All the associated tools share the same basic model of execution alternating between continuous phases and sequences of ‘run-to-completion’ discrete actions [3] as formalised by the notion of hybrid automata [11]. None of these approaches attempts to include the operational semantics of differential equations in their core semantic model: execution of the continu- ous phases is delegated to numerical solvers, which are used to advance physical time and compute the values of physical signals. Except for Zélus, none of the above semantic mod- els are Zeno-free, which means that, as explained in the next sections, they do not reflect the fact that time diverges and rely on analysing the solver output to de- tect and advance past the Zeno points [13] (cf. Sec- tion 2). This poses a fundamental problem, since the solver behaviour at this point is usually unspecified. Furthermore, none of the above proposals al- lows compositional behavioural abstraction: idealised models do not account for the physical nature of phenomena, in particular the fact that original, high- fidelity signals are continuous. However, this property is assumed by most users and validated by most real- life systems. Hence, it must be a fundamental property of signals and should be reflected in their idealisations. In our view, to achieve maximum robustness of a simulation framework, it is crucial to define the se- mantic model before designing either the language or the simulator. Thus, the design of a hybrid simulation framework should involve the following steps. First of all, one must define a semantic model that properly accounts for the expected elementary prop- erties of systems to be simulated. This includes dy- namic behaviour properties, but also “higher level” ones, such as modularity. The second step consists in designing a simulator DOI 10.3384/ECP14096693 Proceedings of the 10 th International ModelicaConference March 10-12, 2014, Lund, Sweden 693