An LMI Approach to Optimal Consensus Seeking in Multi-Agent Systems

In this paper an optimal control design strategy to guarantee consensus achievement in a multi-agent network is developed. Minimization of a global cost function for the entire network guarantees a stable consensus with an optimal control effort. In solving the optimization problem it is shown that the solution of the Riccati equation cannot guarantee the consensus achievement. Therefore, the linear matrix inequality (LMI) formulation is used to solve the corresponding optimization problem and simultaneously to address the consensus achievement constraint. Moreover, using the LMI formulation a controller specific structure based on the neighboring sets can be imposed as an additional LMI constraint. Therefore, the only information each controller needs is the one it receives from its associated neighbors in its neighboring set. The global cost function formulation provides more insight into the optimal performance of the entire network and would result in a "global" optimal (or suboptimal) solution. Simulation results are presented to illustrate the performance of the multi-agent team in achieving consensus.