Design and Implementation of Object-Oriented ... - Automatic Control

Chapter 2. Modeling Techniques
tion. Numerical methods for solving ODEs are mostly based on more sophisticated
discrete approximations. Further details on discretization of
continuous time models can be found in [Åström and Wittenmark, 1990].
Hybrid systems combine the continuous behavior specified by differential
equations with discontinuous changes specified by discrete event
switching logic or finite difference equations at sample times. For successful
modeling of hybrid phenomena, the semantics of the hybrid modeling
language elements have to be well understood. In certain difficult cases
even an understanding of the implementation of the computational solution
procedure are necessary. At least three modeling situations lead to
hybrid phenomena in systems modeling:
• Combination of continuous physical systems with discrete time, computer
implemented control logic.
• Discrete approximations to physical phenomena which exhibit steep
changes within short time- or length scales, e. g., the time scale abstractions
mentioned in Section 1.3 like colliding objects in mechanics.
• The modeled device changes its characteristic behavior in the interesting
range of operation so much that different states or even a
different number of states are needed for an adequate description
of the behavior. An example could be an evaporator that is flooded
with liquid at startup.
Modeling on a macroscopic scale often gives rise to empirical relations
that are piecewise functions. Typical examples are friction, pressure drop
or heat transfer correlations. As long as the relations do not change the
causality of the calculation, these are fairly easy to handle in simulation.
Hybrid phenomena are still the topic of intensive research activity, even
fundamental issues such as the existence and uniqueness of executions
of hybrid automata are not understood in all details. Good introductions
to ongoing research in hybrid systems is found in [Johansson et al., 1999]
and [Zhang et al., 2000].
From the point of view of numerical mathematics, the following classification
can be made:
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1. ODE or index 1 DAE where a discrete event or discontinuous function
does not change the index or the size of the state vector.
(a) Only the right hand side of the ODE/DAE changes discontinuously,
all states are C 0 continuous. The values of the states
before and after the event are the same. Computational causality
of algebraic variables in an index 1 DAE might change also.