## Modeling Systems With Variable Algebraic Constraints for Explicit Integration Methods

Pieter J. Mosterman, Peter Neumann, and Carsten Preusche

Institute for Robotics and Mechatronics

DLR Oberpfaffenhofen

### Abstract

Efficient models are not necessarily the most detailed ones.
The purpose of a model is to solve a problem and it needs
to be just detailed enough to achieve this. In many cases
this may require simplifying models by removing nonlinear
continuous behaviors
by means of a piecewise linearization. As a consequence,
the model operates in a number of different continuous modes where
different equations describe system behavior. When models
are simplified even further and fast continuous
mode transition behaviors are removed,
the dynamic coupling between state variables of interest
may reduce to algebraic constraints, causing a reduction
of degrees of freedom of the system when mode changes occur.
To generate behaviors for such variable structure systems
requires algebraic manipulations to derive the reduced
order system. These algebraic manipulations may include differentiation
of equations that is inefficient when performed during
behavior generation. The alternative of pre-compilation is
restricted to systems with few modes to avoid enumeration
problems because of the combinatorial explosion. This paper
presents a method to handle variable structure systems with
varying algebraic constraints (i.e., run-time index changes)
by means of explicit integration methods complemented by
a projection in the impulse space that is consistent with
the instantaneous dynamics of the vector field. The method
is demonstrated by modeling and simulation of an AC induction motor.
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