Suboptimal distributed cooperative control for linear multi-agent system via Riccati design

In this paper, we propose a suboptimal distributed cooperative control scheme for the continuous-time linear multi-agent system (MAS) with a specified global quadratic cost functional over both undirected and directed graph scenarios. For undirected graphs, we first derive the cost functional for a given strictly linear feedback distributed protocol. It is shown that the cost functional is upper bounded by a quadratic form of the MAS's initial state, and the minimum upper bound can be derived by solving a parametric algebraic Riccati equation (PARE) depends solely on the algebraic connectivity of the graph and is independent of the largest eigenvalue compared with the existing work. Based on this, a suboptimal distributed design method is proposed, where the resulting cost functional is less than a specified positive scalar. Then, we extend the theoretical results and design method to the directed graph scenario by introducing the row and column Laplacian matrices associated with the directed graph. Finally, numerical examples are provided to verify the effectiveness of the obtained results.