Proceedings of the 2003 Winter Simulation Conference S. Chick, P. J. Sánchez, D. Ferrin, and D. J. Morrice, eds. MODE TRANSITION BEHAVIOR IN HYBRID DYNAMIC SYSTEMS Pieter J. Mosterman The MathWorks, Inc. 3 Apple Hill Dr. Natick, MA 01760, U.S.A. ABSTRACT Physical system modeling benefits from the use of implicit equations because it is often an intuitive way to describe physical constraints and behaviors. To achieve efficient models, model abstraction may lead to idealized compo- nent behavior that switches between modes of operation (e.g., an electrical diode may be on or off) based on in- equalities (e.g., voltage > 0). In an explicit representation, the combination of these local mode switches leads to a combinatorial explosion of the number of global modes. It is shown how an implicit formulation can be used to formulate these mode switches, thereby circumventing the combinatorial problem. This leads to the use of differen- tial and algebraic equations (DAEs) for each of the modes. In case these DAEs are of high index, jumps in general- ized state variables may occur. In combination with the inequalities that define mode switching, this leads to rich and complex mode transition behavior. An overview of this mode switching behavior and an ontology is presented. 1 INTRODUCTION Modern engineered systems have reached a complexity that requires systematic design methodologies and model based approaches to ensure correct and competitive realization. In addition, the use of digital controllers has become critical. Embedded software, however, has proven to be difficult to manage since small errors may lead to catastrophic fail- ures. Furthermore, the interdependencies in software that implements the control algorithms are difficult to oversee, which only exacerbates with the increasing size of em- bedded software. Similarly, the interdependencies between controllers scattered about the control system are difficult to distill. Their effects as well as the subtle interaction be- tween controllers and the physical environment are difficult to analyze. Modeling can be the mortar to combine the controller software and hardware of the controlled system, the plant, but different modeling paradigms are used for the different domains. To model the plant, differential and algebraic equations (DAEs) are the method of choice. The controller, on the other hand, is typically modeled by a discrete time or discrete event formalism. In early design stages, contin- uous models may be preferred because of the analysis and synthesis benefits, but when moving to a software imple- mentation, at one point a discretized version has to be de- rived. The combined controller/plant analysis then requires mixed continuous/discrete formalisms, or so-called hybrid dynamic systems (Benedetto and Sangiovanni-Vincentelli 2001, Lynch and Krogh 2000, Vaandrager and van Schup- pen 1999). In addition, hi-fidelity plant models often include highly nonlinear behaviors that complicate analyses. In a more ef- ficient model, these nonlinearities may be linearized around one or more operating points. Switching between these linearized models then requires a discrete mode switching control structure combined with the continuous models in each of the modes, leading to a hybrid dynamic system as well. Another ground for using discrete switching effects in plant models is to model perceived physical discontinu- ities such as valves, overflows, and collisions. In many cases, it is more convenient to model such phenomena as discontinuities although it may require quite some addi- tional conceptual investment compared to a more detailed continuous model (Breedveld 1996). There is a distinct difference between hybrid dynamic systems that arise from combining controller models with plant models and those that emerge because of including discontinuities in the plant models of physical systems. Controller models are by design of an explicit nature. Plant models, on the other hand, often contain implicit constraints that should be satisfied, without explicitly stating how these are used to generate behavior. For example, Newton’s collision law says that the difference of velocities of two colliding bodies after a collision, v, equals their difference before, v − , given some coefficient of restitution, ǫ , v =−ǫv − (1)