Grasping the Sampling Behaviour of Event-Triggered Control

Abstract

A fundamental challenge in networked control systems is reducing the amount of communications of each system in the network, so that bandwidth and energy are used efficiently. To address the challenge, the research community has shifted its focus to Event-Triggered Control (ETC), in which communication between the control system's different components takes place only when a state-dependent condition is satisfied. However, although ETC indeed often reduces communication, its communication times (or sampling times) are unknown beforehand and predictions thereof require intricate mathematical analysis on the system's (perturbed) dynamics. Nonetheless, predicting ETC's sampling is of paramount importance, as it enables:

• Self-Triggered Control (STC), which is a more economic implementation of ETC. In STC the controller, at each sampling time, decides the next sampling time, by employing 1-step predictions of ETC's sampling; given a state measurement it predicts ETC's next sampling time.

• Traffic scheduling, which is planning bandwidth allocation to each entity using the network and requires multi-step or infinite-step predictions of ETC's communication times. Without scheduling, many systems may access the network at the same time, resulting into network overflow and hindering the systems' stability.

• Formal assessment of an ETC-design's performance in terms of sampling and control, e.g. by computing associated long-term metrics such as the expected average intersampling time, which again requires multi-/infinite-step predictions of ETC's sampling.

This dissertation studies ETC's sampling behaviour and derives predictions thereof in all three aforementioned contexts.

First, we propose a novel STC scheme, termed region-based STC, for nonlinear systems with bounded disturbances and uncertainties. The system's state-space is partitioned into a finite number of regions, and to each region a uniform STC intersampling time is assigned. To decide the next sampling time, at each sampling time the controller simply checks to which region the measured state belongs. To derive the partition and corresponding intersampling times, we use approximations of so-called isochronous manifolds. To derive the approximations, we address theoretical issues of prior works and propose a computational algorithm, and, to account for disturbances/uncertainties, we employ differential inclusions.

Regarding traffic scheduling, our work follows the recently proposed abstraction-based approach. The sampling behaviour of a given ETC system is modeled by a finite-state system (the abstraction), offering an infinite-horizon prediction on ETC's sampling. In this work, we construct abstractions of (perturbed) nonlinear ETC systems. The system's state-space is partitioned into finitely many regions, representing the abstraction's states. For each region, a timing interval is determined, containing all intersampling times corresponding to states in the region. These intervals serve as the abstraction's output. Finally, the abstraction's transitions, given a starting region, indicate where the system's trajectories end up after an elapsed intersampling time. To determine the timing intervals and the transitions, we propose algorithms based on reachability analysis. Regarding state-space partitioning, we propose a partition similar to that of region-based STC, aiming at providing control over the timing intervals and improving their tightness.

Finally, on the formal-assessment front, we formally analyze the sampling behaviour of stochastic linear periodic ETC (PETC) systems by computing bounds on associated metrics. Specifically, we consider functions over sequences of state measurements and intersampling times that can be expressed as average, multiplicative or cumulative rewards, and introduce their expectations as metrics on PETC's sampling behaviour. We compute bounds on these expectations, by constructing appropriate Interval Markov Chains (IMCs) equipped with suitable reward structures, that abstract stochastic PETC's sampling behaviour, and employing value iteration over these IMCs.