In networked control systems (NCSs), extensive data exchange between plants and controllers leads to an unnecessary usage of communication and computational resources. Aperiodic sample-and-hold methods such as event-triggered control (ETC) can reduce the number of transmissions, allowing more applications to operate within the same network. However, most existing event-triggering mechanisms enforce a Lyapunov function of the continuous-time closed-loop system to be (almost) always decreasing. We propose a relaxed triggering condition for periodic event-triggered control (PETC) based on bounding the Lyapunov function with an exponentially decaying reference function, which reduces the communications while guaranteeing the same decay rate as competing strategies. We provide sufficient global exponential and input-to-state stability conditions for linear time-invariant (LTI) systems under our event-based state feedback, giving explicit performance guarantees in the presence of additive disturbances. Finally, some simulation results illustrate the performance of the proposed control strategy.