Order-Adaptive Distributed Integral Control

We address a structural tradeoff in distributed dynamic coordination: when the target complexity is unknown, a low controller order saves states but may leave a persistent tracking error, whereas a high order improves tracking but may burden every agent with unnecessary dynamics. To remove this choice without resorting to computationally more involved nonlinear feedback or chattering-prone nonsmooth feedback, we develop an order-adaptive distributed integral controller (OADIC). Specifically, we start with proportional feedback and add integral states only when locally measurable relative errors show that the current order is inadequate. Meanwhile, we organize the candidate controllers in a nested form, thereby preserving the existing states and gains and maintaining continuous control inputs during order transitions. To provide a theoretical basis for this design, we first characterize the consistency of prescribed relative displacements on the augmented agent--target graph. Next, we construct gains that stabilize all admissible fixed-order subsystems and establish a uniform input-to-state stability bound for the variable-dimension closed loop. Furthermore, we prove that OADIC rejects every insufficient order after finitely many decision intervals and explicitly bound the rejection time of the critical order.