Consensus conditions of continuous-time multi-agent systems with time-delays and measurement noises

TL;DR

This paper establishes necessary and sufficient conditions for stochastic consensus in continuous-time multi-agent systems with time-delays and measurement noises, using differential resolvent functions, martingale convergence, and Lyapunov functionals.

Contribution

It provides new theoretical criteria for achieving stochastic weak and strong consensus under various noise and delay conditions in multi-agent systems.

Findings

Necessary and sufficient conditions for stochastic weak consensus.

Stochastic strong consensus can be achieved under certain conditions.

Control gain can be tuned to ensure consensus despite delays and noises.

Abstract

This work is concerned with stochastic consensus conditions of multi-agent systems with both time-delays and measurement noises. For the case of additive noises, we develop some necessary conditions and sufficient conditions for stochastic weak consensus by estimating the differential resolvent function for delay equations. By the martingale convergence theorem, we obtain necessary conditions and sufficient conditions for stochastic strong consensus. For the case of multiplicative noises, we consider two kinds of time-delays, appeared in the measurement term and the noise term, respectively. We first show that stochastic weak consensus with the exponential convergence rate implies stochastic strong consensus. Then by constructing degenerate Lyapunov functional, we find the sufficient consensus conditions and show that stochastic consensus can be achieved by carefully choosing the control gain according to the noise intensities and the time-delay in the measurement term.