Event-Triggered Delay-Compensated Boundary Control of Reaction-Diffusion PDEs with Actuator Dynamics
Hongpeng Yuan, Ji Wang · 2024
We present an event-triggered delay-compensated boundary control scheme for a class of reaction-diffusion PDEs with actuator dynamics, where a time delay, whose length is arbitrary, exists between the PDE plant and the actuator. After treating the time delay as a transport PDE, the overall plant configuration becomes ODE-PDE-PDE. Combining PDE and ODE backstepping transformations, a three-step backstepping design is adopted to build the continuous-in-time control law. Then, a dynamic event-triggering mechanism is designed, based on the evaluation of the overall ODE-PDE-PDE system, to determine the updating times of the actuation signal. In the resulting event-based closed-loop system, a strictly positive lower bound of the minimal dwell time is found, which is independent of initial conditions. As a result, the absence of a Zeno behavior is guaranteed. Besides, exponential convergence to zero of all signals, including the $H^{1}$ norm of the transport PDE state, the $L^{2}$ norm of the reaction-diffusion PDE state, the ODE actuator state, the dynamic variable in the event-triggering mechanism, and the control input, is proved via Lyapunov analysis. The effectiveness of the proposed method is illustrated by numerical simulation.