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. This global objective is realized through local interactions, a key issue in multi-agent control, often referred to 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. Discrete time consensus equation is depending upon underline structure of communication graph. For achieving consensus two types of communication graphs are considered Fixed communication graph and Switching communication graph. Consensus values for switching communication graph and fixed communication graph is derived for random and fixed initial state information of agents. The convergence of consensus algorithm is depend upon eigen structure of Frobenius matrix and it is constructed for fixed and switch communication graph. The eigen values of Frobenius matrix lie within the unit circle so trajectory of state information of each agent exponentially stable and convergence to common value known as consensus value at steady state. The consensus value for fixed and switching graph is average of their initial state information but time required for convergence of algorithm in case of switching graph is greater than fixed communication graph. This theoretical finding is illustrated via simulations.
Keywords: Multiple agent system (MAS), Consensus, Graph Laplacian, Frobenius Matrix and algebraic graph theory
Citation:
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