Disturbance attenuation in discrete-time Kuramoto models

Abstract

This thesis reports the results of research into the stability of the all-to-all coupled discrete time Kuramoto model under constant, matched input disturbances. The discrete time Kuramoto model can be used as a dynamic, decentralized multi-agent orientation coordination system: once initialized, the agents will communicate their orientations to all other agents and calculate their own step-update based on the received data. A properly controlled and undisturbed Kuramoto model can direct agents to two final, stable sets of orientations: either all agents align to the same orientation or they form a balanced set of orientations with the characteristic that the centre of gravity of all orientations on the unit circle is at the origin.

These final states will not be reached when at least one of the agents is influenced by a disturbance. Not only will this agent be affected, but because of the networked system, the disturbance in one agent will influence other agents as well. Understanding the Kuramoto model enables the design of controllers that can attenuate the influence of disturbances. All controllers are designed with the assumption of constant matched input disturbances.
The first controller is an error feedback controller. For this strategy, the original Kuramoto model had to be modified. The resulting controller can attenuate the effects of matched input disturbances in a system of agents, but individual agents with matched input disturbances will not reach a steady state.

The second controller is based on predictor-error feedback and the Kuramoto characteristic that the average orientation in a Kuramoto model is constant. The controller is augmented with an algorithm that generates a one-step ahead prediction based on the known states and inputs. Since the disturbance is assumed to be constant, its effects can be calculated and attenuated in the next time step. This controller succeeds in directing the system to the same aligned set as the undisturbed system, although via a different trajectory. The controller also succeeds in directing the system to a balanced set, but for systems with N ≥ 4 agents that balanced set is different from the undisturbed set.

Since the second controller showed that a deviation from the undisturbed trajectory leads to a different balanced set, the third controller is designed with reference trajectories that do not use the actual states and inputs, but are generated fully autonomously. The difference between reference and actual state is processed by a proportional-integral algorithm to ensure zero steady state error. This controller however has the possibility of destabilizing the system, when not properly tuned.

All controllers have their merits: the first controller decouples the agents, thereby preventing that a disturbed agent affects others. Under constant disturbance, the second controller guarantees stability, but will let the agents follow different trajectories than the undisturbed system, leading to different balanced sets. The third controller can direct all agents to their undisturbed trajectory, but can negatively impact the stability properties of the Kuramoto controller when improperly tuned.