A Closed-Loop Design for Scalable High-Order Consensus

This paper studies the problem of coordinating a group of
$n^{\text{th}}$
-order integrator systems. As in the well-studied conventional consensus problem, we consider linear and distributed control with only local and relative measurements. We propose a closed-loop dynamic that we call serial consensus and prove it achieves
$n^{\text{th}}$
-order consensus regardless of model order and underlying network graph. This alleviates an important scalability limitation in conventional consensus dynamics of order
$n\geq 2$
, whereby they may lose stability if the underlying network grows. The distributed control law which achieves the desired closed loop dynamics is shown to be localized and obey the limitation to relative state measurements. Furthermore, through use of the small-gain theorem, the serial consensus system is shown to be robust to both model and feedback uncertainties. We illustrate the theoretical results through examples.