## A Modeling and Simulation Methodology
for Hybrid Dynamic Physical Systems

Pieter J. Mosterman

Institute for Robotics and System Dynamics

DLR Oberpfaffenhofen
Gautam Biswas

Department of Computer Science

Vanderbilt University

### Abstract

This paper develops a mathematical framework for specification of
hybrid physical system models to develop a robust simulation
methodology for hybrid system analyses.
Our framework for modeling hybrid dynamic physical
systems encompasses *time scale* and *parameter* abstractions,
and results in behavior patterns that undergo discrete
transitions between modes of continuous behavior evolution.
Using this framework we develop
formal execution semantics for characterizing hybrid behaviors in terms
of three distinct modes of system operation: (i) *continuous*,
(ii) *pinnacles*, and (iii) *mythical*. Continuous modes
represent normal physical system behavior, where the system variables
evolve continuously in time. Pinnacles, an artifact of time scale
abstraction, define behaviors at a point in real time. Mythical
modes, an artifact of parameter abstractions, are defined by sequences of
instantaneous local switching transitions in the hybrid model, and have
no real existence on the time line. A key aspect of the work is the
link established between the switching transitions and the
*a priori* and *a posteriori* state vector
values, which leads to the definition of recursive mode switching functions
that govern the interactions between the continuous and discrete components
of the system models. The mathematical specifications are developed into an
implementation model that allows for a direct mapping of system
components onto model fragments, and facilitates simulation of
system behavior. Discrete switching in the simulation model is
implemented as instantaneous transition functions, and continuous
behavior generation is based on differential equation models.
We demonstrate the effectiveness of this approach on a number of physical
system models.
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