Symbolic models for incrementally stable singularly perturbed hybrid affine systems

In this paper, we consider the problem of symbolic models design for the class of incrementally stable singularly perturbed hybrid affine systems. Contrarily to the existing results in the literature where only switching are taken into account, here we consider a more general class of hybrid systems including switches, impulsions and dynamics evolving in different timescales. Firstly, a discussion about incremental stability of the considered class of systems is given. Secondly, a new method for designing symbolic models for incrementally stable singularly perturbed hybrid affine systems is proposed. Inspired from singularly perturbed techniques based on decoupling the slow dynamics from the fast ones, the obtained symbolic abstraction is designed by discretizing only a part of the state space representing the slow dynamics. An epsilon-approximate bisimulation relation between the original singularly perturbed hybrid affine system and the symbolic model obtained by discretizing the slow dynamics is provided. Indeed, since the discrete abstraction is designed for a system of lower dimension, the number of its transitions is drastically reduced. Finally, an example is proposed in order to illustrate the efficiency of the proposed results.