Traffic Characterization of Event-Triggered Control Systems: A Geometric-Algebraic Perspective

This paper characterizes the triggering behaviors of event-triggered control systems from a geometric-algebraic perspective. We first model the feasibility of inter-event time transition relations as a nonconvex quadratic constraint satisfaction problem and reformulate it as an equivalent linear cone problem, which provides a clearer geometric description of the feasible region, making subsequent analysis more reliable. Building on this formulation, we establish necessary and sufficient conditions that rigorously determine whether a given transition relation is feasible. Based on this condition, we propose an algorithm that computes the set of all feasible transition relations. Numerical simulations further demonstrate how the feasibility of specific transitions evolves with the control parameter \sigma, with visualizations of the feasible state space offering intuitive insight into parameter selection and system design.