By: Amol G. Patil and Gautam A. Shah
This article explores the mathematical framework for developing consensus algorithms in multi-agent systems, using both fixed and switching communication graphs. Consensus refers to the agreement among agents achieved by sharing local information. Local interactions realize this global objective, a key issue in multi-agent control, also known as cooperative control. The consensus equation can be formulated in either continuous or discrete time domains. This article focuses on deriving the consensus equation in the discrete time domain using Perron-Frobenius theory. The discrete time consensus equation is dependent upon the underline structure of the communication graph. For achieving consensus, two types of communication graphs are considered: fixed communication graphs and switching communication graphs. Consensus values for switching communication graphs and fixed communication graphs are derived for random and fixed initial state information of agents. The convergence of the consensus algorithm depends upon the eigenstructure of the Frobenius matrix, and it is constructed for fixed and switch communication graphs. The eigenvalues of the Frobenius matrix lie within the unit circle, so the trajectory of state information of each agent is exponentially stable and converges to a common value known as the consensus value at steady state. The consensus value for fixed and switching graphs is the average of their initial state information, but the time required for convergence of the algorithm in the case of switching graphs is greater than that for fixed communication graphs. This theoretical finding is illustrated via simulations.
Keywords: Multiple agent system (MAS), consensus, graph Laplacian, Frobenius matrix and algebraic graph theory
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