Stability of Open Multi-Agent Systems and Applications to Dynamic Consensus

TL;DR

A novel theoretical framework is provided to study the dynamical properties of open multiagent systems (OMASs), focusing on discrete-time evolutions of scalar agents, and proposes a suitable notion of stability and derives sufficient conditions for it.

Abstract

In this technical note we consider a class of multi-agent network systems that we refer to as Open Multi-Agent Systems (OMAS): in these multi-agent systems, an indefinite number of agents may join or leave the network at any time. Focusing on discrete-time evolutions of scalar agents, we provide a novel theoretical framework to study the dynamical properties of OMAS: specifically, we propose a suitable notion of stability and derive sufficient conditions to ensure stability in this sense. These sufficient conditions regard the arrival/departure of an agent as a disturbance: consistently, they require the effect of arrivals/departures to be bounded (in a precise sense) and the OMAS to be contractive in the absence of arrivals/departures. In order to provide an example of application for this theory, we re-formulate the well-known Proportional Dynamic Consensus for Open Multi-Agent Systems and we characterize the stability properties of the resulting Open Proportional Dynamic Consensus algorithm.

Figures (6)

• Figure 1: Time-varying number of agents.
• Figure 2: Evolution of the normalized open distance between average reference input and point of interest: $\frac{d(x^e_k, \bar{u}\mathbf{1})}{\sqrt{|V_k|}}$.
• Figure 3: Evolution of the normalized open distance between network state and point of interest: $\frac{d(x_k,x^e_k)}{\sqrt{|V_k|}}$.
• Figure 4: Evolution of the normalized open distance between network state and average reference input: $\frac{d(x_k,\bar{u}\mathbf{1}))}{\sqrt{|V_k|}}$.
• Figure 5: Evolution of the normalized open distance between network state and average reference input: $\frac{d(x_k,\bar{u}\mathbf{1}))}{\sqrt{|V_k|}}$ in the case of time-invariant number of agents $n=200$.

Theorems & Definitions (13)

• Definition 3.1: Trajectory of Points of Interest (TPI)
• Definition 3.2: Open Stability of a Trajectory of Points of Interest
• Definition 3.3: Contractive OMAS
• Definition 3.4: Open distance function
• Proposition 3.5: Properties of open distance functions
• proof
• Definition 3.6: Open sequence of bounded variation
• Definition 3.7: Trajectory of points of interest (TPI) of bounded variation
• Definition 3.8: Bounded join process
• Theorem 3.9: Stability of Open Multi-Agent Systems