Distributed Time- and Event-Triggered Observers for Linear Systems: Non-Pathological Sampling and Inter-Event Dynamics

TL;DR

This paper introduces a distributed observer framework for linear systems that combines time-triggered sampling with event-triggered communication, establishing conditions for convergence and avoiding Zeno behavior.

Contribution

It provides necessary and sufficient conditions for sampling periods ensuring convergence, and designs an event-triggering mechanism with analytical inter-event behavior characterization.

Findings

Sampling period critically affects observer convergence.

Event-triggering mechanism guarantees error convergence without Zeno behavior.

Analytical relationships among sampling, triggering, and inter-event times are established.

Abstract

For an autonomous linear time-invariant (LTI) system, a distributed observer with time-triggered periodic observations and event-triggered communication is proposed to estimate the state of the system. It is shown that the sampling period is critical for the existence of desirable observers. A necessary and sufficient condition is established to give all feasible sampling periods that lead to convergent error dynamics and characterize delicate relationships among sampling periods, topologies, and system matrices. An event-triggering mechanism based on locally sampled data is designed to regulate the communication among agents, and the convergence of the estimation errors under the mechanism holds for a class of positive and convergent triggering functions, which include the commonly used exponential function as a special case. The mixed time- and event-triggered architecture naturally excludes the existence of Zeno behavior as the system updates at discrete instants. When the triggering function is bounded by exponential functions, analytical characterization of the relationship among sampling, event triggering, and inter-event behaviour is established. Finally, several examples are provided to illustrate the effectiveness and merits of the theoretical results.