The fundamentals of hybrid systems modelers

Hybrid systems modelers have become the corner stone of embedded system development, with Simulink a de facto standard and Modelica a new player. Such tools still raise a number of issues that, we believe, require more fundamental understanding. In this paper we propose using non standard analysis as a semantic domain for hybrid systems — non standard analysis is an extension of classical analysis in which infinitesimals (the ε and η in the celebrated generic sentence ∀ε∃η… in college maths) can be manipulated as first class citizens. This allows us to provide a denotational semantics and a constructive semantics for hybrid systems, thus establishing simulation engines on a firm mathematical basis. In passing, we cleanly separate the job of the numerical analyst (solving differential equations) from that of the computer scientist (generating execution schemes).1