Consensus-based control for a network of diffusion PDEs with boundary local interaction

TL;DR

This paper develops decentralized consensus protocols for a network of agents modeled by boundary-controlled heat equations, enabling them to reach a common steady state despite boundary disturbances, with proven stability and effectiveness.

Contribution

It introduces novel local interaction rules for boundary-controlled PDE agents, including a sliding-mode protocol for disturbance rejection, advancing control strategies for PDE networks.

Findings

Protocols successfully achieve consensus in steady state

Sliding-mode control rejects boundary disturbances effectively

Lyapunov analysis confirms stability of the protocols

Abstract

In this paper the problem of driving the state of a network of identical agents, modeled by boundary-controlled heat equations, towards a common steady-state profile is addressed. Decentralized consensus protocols are proposed to address two distinct problems. The first problem is that of steering the states of all agents towards the same constant steady-state profile which corresponds to the spatial average of the agents initial condition. A linear local interaction rule addressing this requirement is given. The second problem deals with the case where the controlled boundaries of the agents dynamics are corrupted by additive persistent disturbances. To achieve synchronization between agents, while completely rejecting the effect of the boundary disturbances, a nonlinear sliding-mode based consensus protocol is proposed. Performance of the proposed local interaction rules are analyzed by applying a Lyapunov-based approach. Simulation results are presented to support the effectiveness of the proposed algorithms.