Event-triggered and self-triggered control for linear systems based on reachable sets

We propose novel aperiodic control schemes for additively perturbed discrete-time linear systems based on the evaluation of set-membership conditions related to disturbance reachable sets. The goal is to reduce the rate of communication between the sensor and the actuator, while guaranteeing that a certain set in the state space is asymptotically stabilized. In particular, we prescribe this set to be the minimal robust positively invariant set under a given feedback law updated at every time, multiplied by a factor that acts as a tuning parameter. This way, we achieve a trade-off between the communication rate and the worst-case asymptotic bound on the system state in the closed-loop system. We employ a novel stability concept that captures how much the system dynamics are explicitly dependent on past system states. This allows us to quantitatively compare the stability properties guaranteed by an all-time updated (static) feedback controller with those guaranteed by a (dynamic) aperiodic controller. We use the proposed framework to design both event-triggered and self-triggered controllers under the assumption of state feedback or output feedback.