Output Feedback Periodic Event-Triggered and Self-Triggered Boundary Control of Coupled $2\times 2$ Linear Hyperbolic PDEs

Abstract

In this paper, we expand recently introduced observer-based periodic event-triggered control (PETC) and self-triggered control (STC) schemes for reaction-diffusion PDEs to boundary control of $2\times2$ coupled hyperbolic PDEs in canonical form and with anti-collocated measurement and actuation processes. The class of problem under study governs transport phenomena arising in water management systems, oil drilling, and traffic flow, to name a few. Relative to the state of the art in observer-based event-triggered control of hyperbolic PDEs, our contribution goes two steps further by proposing observer-based PETC and STC for the considered class of systems. These designs arise from a non-trivial redesign of an existing continuous-time event-triggered control (CETC) scheme. PETC and STC eliminate the need for constant monitoring of an event-triggering function as in CETC; PETC requires only periodic evaluations of the triggering function for event detection, whereas STC is a predictor-feedback that anticipates the next event time at the current event exploiting continuously accessible output measurements. The introduced resource-aware designs act as input holding mechanisms allowing for the update of the input signal only at events. Subject to the designed boundary output feedback PETC and STC control laws characterized by a set of event-trigger design parameters, the resulting closed-loop systems, which are inherently Zeno-free by design, achieve exponential convergence to zero in the spatial $L^2$ norm. We illustrate the feasibility of the approach by applying the control laws to the linearized Saint-Venant model, which describes the dynamics of shallow water waves in a canal and is used to design flow stabilizer feedback laws via gate actuation. The provided simulation results illustrate the proposed theory.