  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.

Adaptation and Self-Organizing Systems