# Automatic differentiation of hybrid models Illustrated by Diffedge Graphic Methodology. (Survey)

## TL;DR

This survey explores automatic differentiation techniques for hybrid models with delays, logical tests, and discontinuities, emphasizing high-level differentiation before code translation, and illustrates the approach with the Diffedge software for Simulink models.

## Contribution

It provides a comprehensive overview of differentiating hybrid models with complex components, highlighting high-level differentiation and the use of graphical methodology in Diffedge software.

## Key findings

- Differentiation at high-level objects simplifies code generation.
- The methodology handles models with delays, discontinuities, and logical tests.
- Diffedge software produces block diagrams for real-time embedded code.

## Abstract

We investigate the automatic differentiation of hybrid models, viz. models that may contain delays, logical tests and discontinuities or loops. We consider differentiation with respect to parameters, initial conditions or the time. We emphasize the case of a small number of derivations and iterated differentiations are mostly treated with a foccus on high order iterations of the same derivation. The models we consider may involve arithmetic operations, elementary functions, logical tests but also more elaborate components such as delays, integrators, equations and differential equations solvers. This survey has no pretention to exhaustivity but tries to fil a gap in the litterature where each kind of of component may be documented, but seldom their common use. The general approach is illustrated by computer algebra experiments, stressing the interest of performing differentiation, whenever possible, on high level objects, before any translation in Fortran or C code. We include ordinary differential systems with discontinuity, with a special interest for those comming from discontinuous Lagrangians. We conclude with an overview of the graphic methodology developped in the Diffedge software for Simulink hybrid models. Not all possibilities are covered, but the methodology can be adapted. The result of automatic differentiation is a new block diagram and so it can be easily translated to produce real time embedded programs. We welcome any comments or suggestions of references that we may have missed.