Decentralized estimation of Laplacian eigenvalues in multi-agent systems

Abstract

)>IJH=?J In this paper we present a decentralized algorithm to estimate the eigenvalues of the Laplacian matrix that encodes the network topology of a multi-agent system. We consider network topologies modeled by undirected graphs. The basic idea is to provide a local interaction rule among agents so that their state trajectory is a linear combination of sinusoids oscillating only at frequencies function of the eigenvalues of the Laplacian matrix. In this way, the problem of decentralized estimation of the eigenvalues is mapped into a standard signal processing problem in which the unknowns are the nite number of frequencies at which the signal oscillates.

Robust Consensus Analysis and Design Under Relative State Constraints or Uncertainties

“…Theorem 3: Suppose that Σ 1 = σ 1 I n and Σ 2 = σ 2 I n then the MAS (18) is robustly stabilized by the distributed stabilizing control law (19) and equivalently the robust consensus under relative-state constraints or uncertainties is achieved for the initial MAS (2) by the distributed controller (13) if there exist matrices X ∈ R n×n , Y ∈ R m×n and Z ∈ R m×m such that the following LMI problem is feasible with ǫ > 0,…”

“…The transformed edge dynamics (18) together with the robust stabilizing controller (19) can be rewritten in the following form of a network of Lur'e systems,…”

“…[19], [20], which propose different distributed methods to approximate the whole eigen-spectrum of the Laplacian matrix. These methods can be employed to estimate λ 2 and λ N before solving the LMI problems (21), (28).…”

Distributed Estimation of Graph Laplacian Eigenvalues by the Alternating Direction of Multipliers Method

“…The problem is then mapped into a signal processing one that can be efficiently and independently solved by each agent in applying the Fast Fourier Transform (FFT). The approach in Franceschelli et al (2013) involved twice communication burden compared to that in Sahai et al (2012). However, both methods inherit the limitations of the FFT-based algorithm.…”

“…In Aragues et al (2012), the algebraic connectivity was estimated by counting on the distributed computation of the powers of a deflated Laplacian matrix. In Sahai et al (2012) and Franceschelli et al (2013), Fast Fourier Transform (FFT)-based methods were suggested. The main idea in these works is to make the state of each agent oscillate only at frequencies corresponding to the eigenvalues of the Laplacian matrix associated with the network topology.…”