A tool for analysis and synthesis of heterogeneous multi-agent systems under rank-deficient coupling

Abstract

The behavior of heterogeneous multi-agent systems is studied when the coupling matrices are possibly all different and/or singular, that is, its rank is less than the system dimension. Rank-deficient coupling allows exchange of limited state information, which is suitable for the study of multi-agent systems under output coupling. We present a coordinate change that transforms the heterogeneous multi-agent system into a singularly perturbed form. The slow dynamics is still a reduced-order multi-agent system consisting of a weighted average of the vector fields of all agents, and some sub-dynamics of agents. The weighted average is an emergent dynamics, which we call a blended dynamics. By analyzing or synthesizing the blended dynamics, one can predict or design the behavior of a heterogeneous multi-agent system when the coupling gain is sufficiently large. For this result, stability of the blended dynamics is required. Since stability of the individual agent is not asked, the stability of the blended dynamics is the outcome of trading off the stability among the agents. It can be seen that, under the stability of the blended dynamics, the initial conditions of the individual agents are forgotten as time goes on, and thus, the behavior of the synthesized multi-agent system is initialization-free and is suitable for plug-and-play operation. As a showcase, we apply the proposed tool to four application problems; distributed state estimation for linear systems, practical synchronization of heterogeneous Van der Pol oscillators, estimation of the number of nodes in a network, and a problem of distributed optimization.

Fully Distributed Resilient State Estimation Based on Distributed Median Solver

“…where Z and S are a given set of numbers {z i } i∈N and a given set of indicators {s i } i∈N respectively (with at least one index i such that s i = 1), and λ 2 (L) is the algebraic connectivity of the graph (i.e., the second smallest eigenvalue of the Laplacian matrix L that represents the graph). Remark 1: The insight behind the proposed distributed median solver (4) comes from the so-called 'blended dynamics' approach [15]. In this approach, the behavior of heterogeneous multi-agent systemṡ…”

“…The use of (centralized) median functions has been considered in [9] and [10] to present a fast polynomial-time algorithm for resilient state estimation. Taking advantage of the recent development of "blended dynamics" approach [15], the designed local observers approximately recover the median of the local state estimates in a distributed manner. It will be seen that the median is a resilient estimate of the state even under up to q corrupted sensor banks.…”

“…It will be seen that the median is a resilient estimate of the state even under up to q corrupted sensor banks. Thanks to one of the merits of blended dynamics approach [15], the proposed observer is different from [14] in that it allows the plug-and-play (initializationfree) operation; that is, the observer can perform resilient state estimation seamlessly even when some sensor banks or local observers intermittently join or leave the network as long as the proposed assumptions are maintained during the operation.…”

Utility of Edge-wise Funnel Coupling for Asymptotically Solving Distributed Consensus Optimization

“…where k > 0 is a design parameter, and N i is a subset of N whose elements are the indices of those agents which are connected to agent i within the network graph (the neighbors), and are hence able to share information with it. Remark 1: Insight into the proposed network (3) comes from the so-called 'blended dynamics' approach [13], [14]. In this approach, the behavior of heterogeneous multi-agent systemsẋ…”

“…In this paper, we present a network that finds the minimizer x * of f (•) asymptotically, with the assumption that f (•) is strictly convex even if each function f i (•) is not necessarily convex. This is obtained from the recent observation that, with a large coupling gain in the diffusive coupling, heterogeneous multi-agent systems behave like a single dynamical system whose vector field is simply the average of all agents' vector fields [13], [14]. By this observation, it is possible to trade stability among agents, and hence, to relax the assumptions on the individual cost functions.…”