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Application of Automatic Differentiation in the Dynamic Simulation and Parametric Sensitivity Analysis of Hybrid Discrete/Continuous Systems
by
John E. Tolsma
Massachusetts Institute of Technology, 77 Massachusetts Avenue,Cambridge MA 02139 USA
Coauthors: Paul I. Barton
Automatic differentiation (AD) has experienced widespread use in recent years and this trend is likely to continue and even expand in the future. However, while fast and accurate numerical derivatives are extremely important, they are only a piece of the puzzle when performing complicated numerical calculations using state-of-the-art algorithms. This is particularly true when performing dynamic simulation and parametric sensitivity analysis involving hybrid discrete/continuous ODEs and DAEs. Hybrid discrete/continuous systems are dynamical systems that exhibit both smooth and nonsmooth behavior. The discontinuous behavior, which occur at discrete points in time known as events, includes phenomenon such as nonsmooth forcing, switching of the vector field, and jumps in state. The problems associated with numerically integrating hybrid systems are well known and include integration inefficiencies and integration failures. The problems with performing parametric sensitivity analysis on hybrid systems is less known. Since the sensitivities often jump at events even if the states do not, not properly handling these events can result in calculation failures (if we are lucky) or completely erroneous results that may escape our attention. Proper simulation and sensitivity analysis of hybrid systems requires the user to provide a great deal of additional information other than the model and Jacobian matrix. Requiring the user to provide this information would be quite a burden particularly when the hybrid model contains existing legacy code. We present a set of components that apply to general Fortran-90 code that automatically generates this additional information (including extraction of the discontinuity functions and automatic generation of a discontinuity-locked model). Combined with AD, this new code can be used with novel numerical algorithms that perform hybrid simulation and parametric sensitivity analysis efficiently, robustly, and correctly. Several example problems are provided illustrating the importance of properly performing these calculations.
Date received: December 16, 1999
Copyright © 1999 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cads-15.