Observer-Based Event-Triggered Control of Infinite-Dimensional Linear Systems

Abstract

We investigate observer-based event-triggered stabilization of infinite-dimensional linear systems, such as those governed by partial differential equations. A Luenberger observer and a feedback gain are assumed to stabilize the closed loop in continuous time, and we study what happens when the control input is only updated at instants generated by a dynamic event-triggering mechanism. Unlike approaches that rely on state-dependent triggering rules, our condition only involves the observer state and the measured output, which makes it implementable when the full state is not measured. It compares the deviation of a monitored signal of the observer since the last update with a threshold made of two parts. The first one is a relative part, proportional to the current size of this signal, and the second one is a positive internal variable, which does not depend on the observer state. In particular, the case where the threshold reduces to the internal variable alone is covered. Under suitable assumptions, we establish the well-posedness of the closed-loop system and the absence of Zeno behavior for general choices of these two parts, and we prove its exponential stability under a small-gain condition on the relative part. We further analyse the existence of a dwell-time, namely a positive lower bound on the time elapsed between two consecutive triggering instants. When the sampled dynamics is bounded, as in finite-dimensional systems or systems with a finite-dimensional unstable part, we obtain a dwell-time independent of the initial data, together with exponential stability. For unbounded generators, we exhibit a system satisfying all our assumptions for which no triggering rule can achieve both a dwell-time independent of the initial data and bounded trajectories.