Matrix-Scaled Consensus

TL;DR

This paper introduces a matrix-scaled consensus algorithm allowing agents to converge to distinct points in the state space, extending traditional scalar consensus by incorporating matrix weights and analyzing convergence for different agent dynamics.

Contribution

The paper generalizes scaled consensus algorithms by incorporating matrix weights, enabling convergence to a subspace rather than a line, with detailed analysis for single and double-integrator agents.

Findings

Agents converge to points in a subspace determined by their matrix weights.

The algorithm guarantees asymptotic convergence for single and double-integrator agents.

Simulation results validate the theoretical convergence analysis.

Abstract

This paper proposes matrix-scaled consensus algorithm, which generalizes the scaled consensus algorithm in \cite{Roy2015scaled}. In (scalar) scaled consensus algorithms, the agents' states do not converge to a common value, but to different points along a straight line in the state space, which depends on the scaling factors and the initial states of the agents. In the matrix-scaled consensus algorithm, a positive/negative definite matrix weight is assigned to each agent. Each agent updates its state based on the product of the sum of relative matrix scaled states and the sign of the matrix weight. Under the proposed algorithm, each agent asymptotically converges to a final point differing with a common consensus point by the inverse of its own scaling matrix. Thus, the final states of the agents are not restricted to a straight line but are extended to an open subspace of the state-space. Convergence analysis of matrix-scaled consensus for single and double-integrator agents are studied in detail. Simulation results are given to support the analysis.