Hybrid-automaton models for simulating systems with sliding motion: still a challenge

There is a gap between the modelling and the simulation of discontinuous dynamical systems (DDS) exhibiting sliding-mode behaviour. Several challenges arise. Mainly, the definition of unique solutions on the discontinuity surfaces and at their crossings. There are also numerical issues in the detection and location of the trajectory entering and going out of the discontinuity surface, as well as when it remains on the discontinuity surface (chattering phenomenon). The aim of this paper is establishing a semantics for DDS so that a deterministic dynamical behaviour can be defined for simulation purposes. Particularly, a class of DDS is modelled by using two different hybrid automata. One of them is called the DDS hybrid automaton, proposed by the author previously. The other is obtained by modifying the DDS hybrid automaton, and is inspired in computer-simulation-oriented friction models. The use of a computational model is an elegant way for specifying the multiple transitions in DDS. The modelling framework is specially useful for specifying transitions when multiple discontinuity surfaces are present. A system with discontinuous friction and stick-slip oscillations is used to validate the models. In the example, it is concluded that discontinuous systems can be appropriately simulated by using hybrid-automaton-based models without the problem of chattering. On the other hand, the system dynamical behaviour can change depending on the step size of the numerical integration method and on the hybrid-automaton representation used. The hybrid models are simulated by means of Simulink/Stateflow R © package.