Delay-Adaptive Compensator for 3-D Space Formation of Multi-Agent Systems with Leaders Actuation

TL;DR

This paper develops a delay-adaptive control method for 3-D multi-agent systems with leader actuation delays, using PDE backstepping and Lyapunov techniques to ensure stability and formation control.

Contribution

It introduces a novel delay-adaptive PDE controller for 3-D multi-agent systems with unknown actuator delays, extending previous 1-D PDE delay control results to 2-D PDEs.

Findings

The controller effectively compensates for unknown delays in simulations.

The method guarantees local boundedness and state regulation in $H^2$ norm.

Numerical results demonstrate improved formation stability with delay compensation.

Abstract

This paper focuses on the control of collective dynamics in large-scale multi-agent systems (MAS) operating in a 3-D space, with a specific emphasis on compensating for the influence of an unknown delay affecting the actuated leaders. The communication graph of the agents is defined on a mesh-grid 2-D cylindrical surface. We model the agents' collective dynamics by a complex- and a real-valued reaction-advection-diffusion 2-D partial differential equations (PDEs) whose states represent the 3-D position coordinates of the agents. The leader agents on the boundary suffer unknown actuator delay due to the cumulative computation and information transmission time. We design a delay-adaptive controller for the 2-D PDE by using PDE backstepping combined with a Lyapunov functional method, where the latter is employed to design an update law that generates real-time estimates of the unknown delay. Capitalizing on our recent result on the control of 1-D parabolic PDEs with unknown input delay, we use Fourier series expansion to bridge the control of 1-D PDEs to that of 2-D PDEs. To design the update law for the 2-D system, a new target system is defined to establish the closed-loop local boundedness of the system trajectories in $H^2$ norm and the regulation of the states to zero assuming a measurement of the spatially distributed plant's state. We illustrate the performance of delay-adaptive controller by numerical simulations.