Hybrid automata: an insight into the discrete abstraction of discontinuous systems

We develop a novel computational–dynamical framework for the modelling of a class of discontinuous dynamical systems (DDSs). In particular, what is referred to as the DDS hybrid automaton with inputs and outputs is proposed. This is a general hybrid automaton that provides a suitable mathematical model for DDSs with discontinuous state derivatives and sliding motions. The chief characteristic of this model is that, following the computational divide-and-conquer principle, a system with multiple discontinuous elements can be represented by the composition of several DDS hybrid automata. Although discontinuous, non-smooth or switched dynamical systems have been well-investigated within different frameworks, it is still a challenge to give satisfactory solutions for specifying the transitions between the different modes of operation of these systems. We propose a new way of solving this problem, which is especially effective for systems with multiple switching elements. An example is used to illustrate these ideas. Several simulations are presented. The simulations results are obtained with Stateflow® and Modelica®.