Finite-step Approximately Bi-Similar Symbolic Model for Switched Systems

Obtaining an approximately bi-similar symbolic model for a continuous-state system is a crucial step in symbolic control. Symbolic control aims to design switching logic to achieve temporal logic specifications. Most existing approaches to approximate bi-simulation are developed over infinite time horizons. However, many control tasks are set within finite periods in practical scenarios. Hence, it makes sense to consider the behavior of trajectories within finite-time horizons. Therefore, this paper introduces two new notions: finite-time uniform incremental stability (FUI-stability) and finite-step approximate bi-simulation. Furthermore, a FUI-stability condition for switched system is derived using the multiple incremental Lyapunov-like functions technique. A symbolic model is constructed for switched system by the state space grid technique. Subsequently, a sufficient condition for finite-step approximate bi-simulation between the finite-time uniformly incrementally stable (FUI-stable) switched system and the constructed symbolic model is derived. Finally, the obtained results are illustrated by two numerical examples.