Guaranteed cost consensus for second-order multi-agent systems with fixed topologies

The consensus regulation performance for second-order multi-agent systems was analyzed in the literature, but there were few literatures to consider the control energy consumption. In the current paper, guaranteed cost consensus problems for second-order multi-agent systems with fixed topologies are investigated to find a tradeoff between the consensus regulation performance and the control energy consumption. Firstly, based on states errors among neighboring agents and control inputs of all the agents, a cost function is constructed. Secondly, by the state space decomposition approach and the Lyapunov method, a sufficient condition for the guaranteed cost consensus is given and an upper bound of the cost function is determined. It should be pointed out that the condition is related to the second smallest and the maximum eigenvalues of the Laplacian matrix of the interaction topology. Thirdly, an approach is presented to obtain the consensus function when second-order multi-agent systems achieve guaranteed cost consensus. Finally, the theoretical results are validated by simulations.