This paper proposes an LMI-based algorithm for abstracting dynamical systems by timed automata, which enables automatic formal verification of linear systems.

The proposed abstraction is based on partitioning the state space of the system using positive invariant sets, generated by Lyapunov functions. This partitioning ensures that the vector field of the dynamical system is transversal to all facets of the cells, which induces some desirable properties of the abstraction.

The algorithm is based on identifying intersections of level sets of quadratic Lyapunov functions, and determining the minimum and maximum time that a trajectory of the system can stay in a set, defined as the set-difference of sub-level sets of Lyapunov functions.
The proposed algorithm applies for linear systems and can therefore be efficiently implemented using LMI-based tools.