In practice, we encounter numerous processes and systems described by parabolic partial differential equations (PDEs). Information diffusion over networks, spread of infectious diseases, multi-agent systems, heat flow, and Stefan problems are few such applications. During the past two decades, many significant results on the control of parabolic PDEs, including distributed and boundary control under both full-state and output feedback settings, have been obtained. Output feedback designs are available for both distributed and boundary sensing scenarios.
Usually, the controllers developed using continuous-time PDE models are implemented in a time-triggered fashion, in which control inputs are computed and updated periodically. However, it might not be necessary to execute the control task every period to guarantee the desired closed-loop performance. Therefore, the periodic execution of control tasks may often result in over-utilization of the limited communication resources and limited lifetime of battery recourses and actuators.
When the system is close to its desired setpoint, and no disturbances are acting on the system, there is no need to transmit sensor or controller data to the controller and the actuator, respectively. The feedback loop only needs to be closed when the system needs attention. Therefore, we must bring "feedback" into the system's sampling and communication process to identify the instances when the sensor and control updates should occur. This calls for the need to move from classical periodic control to aperiodic control, which has been proven to effectively trade-off the control performance and resource utilization.
Fig. 1. Comparison between periodic and aperiodic control. Arrows correspond to the time instances of control computation and update. (a) Periodic control. (b) Aperiodic control.
Event-triggered and self-triggered control are two such aperiodic control approaches. They consist of two elements: a feedback controller that computes the control input, and a triggering mechanism that determines when the control input has to be updated again. The difference between event-triggered control and self-triggered control is that the former is reactive, while the latter is proactive. In event-triggered control, a triggering condition based on current measurements is continuously monitored, and when violated, an event is triggered. In self-triggered control, the next update time is precomputed at a control update time based on predictions using previously received data and knowledge on the plant dynamics.
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