Multi-Agent Consensus With Relative-State-Dependent Measurement Noises

Abstract

In this note, the distributed consensus corrupted by relativestate-dependent measurement noises is considered. Each agent can measure or receive its neighbors' state information with random noises, whose intensity is a vector function of agents' relative states. By investigating the structure of this interaction and the tools of stochastic differential equations, we develop several small consensus gain theorems to give sufficient conditions in terms of the control gain, the number of agents and the noise intensity function to ensure mean square (m. s.) and almost sure (a. s.) consensus and quantify the convergence rate and the steadystate error. Especially, for the case with homogeneous communication and control channels, a necessary and sufficient condition to ensure m. s. consensus on the control gain is given and it is shown that the control gain is independent of the specific network topology, but only depends on the number of nodes and the noise coefficient constant. For symmetric measurement models, the almost sure convergence rate is estimated by the Iterated Logarithm Law of Brownian motions.

Continuous‐time multi‐agent averaging with relative‐state‐dependent measurement noises: matrix intensity functions

“…Here, for a given link (j, i), the measurement noises of different state components are independent Brownian motions, which form an n-dimensional Brownian motion ξ ji (t) and coupled together by the matrix function f ji (·). Note that Assumptions 2.1 and 2.2 do not come down to a special case of Assumptions 2.1 and 2.2 of [19] and vice versa.…”

“…Here, we consider the case with n-dimensional state components, which means that the information state to be exchanged is a vector, not a scalar. If we construct the averaging algorithm for each state component and the communication channels of different state components are independent of each other, then the closedloop system degenerates to a special case considered in [19] (Theorem 4.2 of [19]). However, for the real communication environment, the communication channels of different state components may not be independent.…”

“…This is a reason why we study high-dimensional distributed averaging algorithms. Particularly, for the measurement model (2), the communication channels of different state components are coupled together and the noises of different state components are not the same one, which cannot be covered by Li et al [19].…”

“…For the case with analogue Gaussian fading measurements, the uncertainties of the measurement are also relative-state-dependent noises [6,18]. In [19], we considered the distributed averaging corrupted by relative-statedependent measurement noises. Each agent can measure or receive its neighbours' state information with relative-statedependent measurement random noises.…”

“…Especially, for the case with homogeneous communication and control channels, a necessary and sufficient condition to ensure mean square consensus on the control gain is given and it is shown that the control gain is independent of the specific network topology, but only depends on the number of nodes and the noise coefficient constant. For the measurement model of [19], the measurement noises of different state components are the same and the noise intensity is a vector function. This model cannot cover the case where the measurement noises for different state components are mutually different and coupled together.…”