Control of large 1D networks of double integrator agents: role of heterogeneity and asymmetry on stability margin

TL;DR

This paper analyzes how heterogeneity and asymmetry in control gains affect the stability margin of large 1D networks of double integrator agents, revealing that asymmetry can significantly improve stability scaling.

Contribution

It demonstrates that asymmetry in velocity feedback gains enhances stability margins in large heterogeneous agent networks, unlike symmetry which causes decay to zero.

Findings

Symmetric control leads to stability margin decay as O(1/N^2).

Small asymmetry in velocity feedback improves decay to O(1/N).

Proper asymmetry can maintain stability for arbitrarily large N.

Abstract

We consider the distributed control of a network of heterogeneous agents with double integrator dynamics to maintain a rigid formation in 1D Euclidean space. The control signal at a vehicle is allowed to use relative position and velocity with its two nearest neighbors. Most of the work on this problem, though extensive, has been limited to homogeneous networks, in which agents have identical masses and control gains, and symmetric control, in which information from front and back neighbors are weighted equally. We examine the effect of heterogeneity and asymmetry on the closed loop stability margin, which is measured by the real part of the least stable pole of the closed-loop system. By using a PDE (partial differential equation) approximation in the limit of large number of vehicles, we show that heterogeneity has little effect while asymmetry has a significant effect on the stability margin. When control is symmetric, the stability margin decays to 0 as $O(1/N^2)$, where $N$ is the number of agents, even when the agents are heterogeneous in their masses and control gains. In contrast, we show that arbitrarily small amount of asymmetry in the velocity feedback gains can improve the decay of the stability margin to $O(1/N)$. Poor design of such asymmetry makes the closed loop unstable for sufficiently large $N$. With equal amount of asymmetry in both position and velocity feedback gains, the closed loop is stable for arbitrary $N$ and the stability margin scaling trend can also be improved to $O(1/N)$, but the sensitivity to disturbance becomes worse. Effect of asymmetry in position feedback gains alone and unequal amount of asymmetry in position and velocity feedback are open problems. Numerical computations are provided to corroborate the analysis.