Linear Temporal Logic for Hybrid Dynamical Systems: Characterizations and Sufficient Conditions

TL;DR

This paper develops a framework of operators, semantics, and conditions to verify temporal logic specifications in hybrid dynamical systems, enabling reasoning about complex system behaviors without relying on explicit solutions.

Contribution

It introduces solution-independent conditions and characterizations for temporal logic in hybrid systems, enhancing verification methods for continuous and discrete dynamics.

Findings

Characterizations in terms of invariance and attractivity

Solution-independent sufficient conditions for formula satisfaction

Decomposition approach for complex temporal logic formulas

Abstract

This paper introduces operators, semantics, characterizations, and solution-independent conditions to guarantee temporal logic specifications for hybrid dynamical systems. Hybrid dynamical systems are given in terms of differential inclusions -- capturing the continuous dynamics -- and difference inclusions -- capturing the discrete dynamics or events -- with constraints. State trajectories (or solutions) to such systems are parameterized by a hybrid notion of time. For such broad class of solutions, the operators and semantics needed to reason about temporal logic are introduced. Characterizations of temporal logic formulas in terms of dynamical properties of hybrid systems are presented -- in particular, forward invariance and finite time attractivity. These characterizations are exploited to formulate sufficient conditions assuring the satisfaction of temporal logic formulas -- when possible, these conditions do not involve solution information. Combining the results for formulas with a single operator, ways to certify more complex formulas are pointed out, in particular, via a decomposition using a finite state automaton. Academic examples illustrate the results throughout the paper.