On <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.gif" display="inline" overflow="scroll"><mml:msub><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mi>∞</mml:mi></mml:mrow></mml:msub></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si5.gif" display="inline" overflow="scroll"><mml:msub><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:math> performance regions of multi-agent systems

This paper addresses the distributed H 2 and H ∞ control problems for multi-agent systems with linear or linearized dynamics. An undirected multigraph with loops is used to represent the communication topology of a multi-agent network. A distributed controller is designed, based on the relative states of neighboring agents and a subset of absolute states of the networked agents. The notions of H ∞ and H 2 performance regions are introduced and analyzed, respectively. A necessary and sufficient condition for the existence of a controller yielding an unbounded H ∞ performance region is derived. A multi-step procedure for suboptimal H ∞ controller synthesis is presented. It is also shown that the H ∞ performance limit of the network under the distributed controller is equal to the minimal H ∞ norm of a single agent achieved by using the state feedback controller. It is finally shown that, contrarily to the H ∞ case, the H 2 performance limit scales with the number of agents in the network.