# Hybrid System Modelling and Simulation with Dirac Deltas

## TL;DR

This paper explores the simulation of impulsive differential equations in hybrid systems, presenting symbolic and numerical approaches implemented in a Causal Block Diagram tool, addressing limitations of current simulators.

## Contribution

It provides a theoretically grounded implementation of symbolic and numerical impulse simulation methods within a Causal Block Diagram environment, comparing their effectiveness.

## Key findings

- Symbolic approach is more precise for certain systems.
- Numerical approach offers better performance in some scenarios.
- Guidelines for choosing the appropriate method are provided.

## Abstract

For a wide variety of problems, creating detailed continuous models of (continuous) physical systems is, at the very least, impractical. Hybrid models can abstract away short transient behaviour (thus introducing discontinuities) in order to simplify the study of such systems. For example, when modelling a bouncing ball, the bounce can be abstracted as a discontinuous change of the velocity, instead of resorting to the physics of the ball (de-)compression to keep the velocity signal continuous. Impulsive differential equations can be used to model and simulate hybrid systems such as the bouncing ball. In this approach, the force acted on the ball by the floor is abstracted as an infinitely large function in an infinitely small interval of time, that is, an impulse. Current simulators cannot handle such approximations well due to the limitations of machine precision. In this paper, we explore the simulation of impulsive differential equations, where impulses are first class citizens. We present two approaches for the simulation of impulses: symbolic and numerical. Our contribution is a theoretically founded description of the implementation of both approaches in a Causal Block Diagram modelling and simulation tool. Furthermore, we investigate the conditions for which one approach is better than the other.