Performance bounds for multi-vehicle networks with local integrators

In this work, we consider the problem of coordinating a collection of $n$th-order integrator systems. The coordination is achieved through the novel serial-consensus design, which can be seen as a method for achieving a stable closed-loop while only using local relative measurements. Earlier work has shown that second-order serial consensus can stabilize a collection of double integrators with scalable performance conditions, independent of the number of agents and topology. In this paper, we generalize these performance results to an arbitrary order $n\ge 1$. The derived performance bound depends on the condition number, measured in the vector-induced maximum matrix norm, of a general diagonalizing matrix. We provide an exact characterization of how a minimal condition number can be achieved. Third-order serial consensus is illustrated through a case study of PI-controlled vehicular formation, where the added integrators are used to mitigate the effect of unmeasured load disturbances. The theoretical results are illustrated through examples.