CONSTITUTIVE HYBRID PROCESSES P.J.L. Cuijpers Technical University Eindhoven Eindhoven, Netherlands P.J.L.Cuijpers@tue.nl J.F. Broenink University of Twente Enschede, Netherlands J.F.Broenink@el.utwente.nl P.J. Mosterman The MathWorks, Inc. Natick, MA Pieter.Mosterman@mathworks.com Keywords: Process Algebra, Constitutive Equations, Hybrid Systems Abstract This paper studies constitutive relations of physical processes with time-scale abstractions by giving a process algebraic foundation for hybrid bond graph models. For a number of standard physical elements, it is shown how discontinuous constitutive relations that describe the behavior of hybrid bond graph models can be derived from existing continuous constitutive relations. These discontinuous and continuous descriptions are then combined using the hybrid process algebra HyPA, which may be used to reason symbolically about both the continuous behavior, and the mode switching that results from the time-scale abstraction. 1 Introduction A physical system is often modeled by aggregating constitutive relations on pertinent phenomena. At a macroscopic level, the corresponding behavior can be considered continuous, and the constitutive relations can be stated as diŽerential equations possibly extended by algebraic constraints. When part of the continuous behavior occurs very fast it may be beneficial to describe this behavior as discontinuous. In this case, the constitutive relations may be extended with explicit re-initialization constraints (e.g. for a bouncing ball that reverses its velocity after a collision, v+ = -v-). In this paper, the constitutive relations of a set of fundamental phenomena in physical modeling are described using the hybrid process algebra HyPA [Cuijpers and Reniers 2004]. This algebra allows for the description of combinations of continuous and discontinuous behavior as one, hybrid, process. The algebra builds on bond graphs [Paynter 1961], a formalism that unifies diŽerent domains in physics, e.g., electronics, hydraulics, and mechanics. In particular, bond graphs that are extended with elements to describe discontinuous behavior [Breedveld 1996, Mosterman et al.1998, Stromberg et al. 1993] are used. This paper, can therefore also be considered an attempt to give a process algebraic semantics to such hybrid bond graphs. This paper assumes familiarity with the bond graph formalism (see, e.g., [Karnopp et al. 1990]). In Section 2, a subset of the hybrid process algebra HyPA [Cuijpers and Reniers 2004] is discussed briefly, with a focus on the constructs for describing physical behavior. In Section 3, hybrid constitutive relations of a number of bond graph elements are 1 .....