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


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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