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

We consider the distributed control of a network of heterogeneous agents with double integrator dynamics to maintain a rigid formation in 1D. The control signal at a vehicle is allowed to use relative position and velocity with its immediate neighbors. 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 eigenvalue. By using a PDE approximation, we show that heterogeneity has little effect while asymmetry has a significant effect on the stability margin. When control is symmetric, in which information from front and back neighbors are weighted equally, the stability margin decays to 0 as O(1/N²), 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). With equal amount of asymmetry in both velocity and position feedback gains, the closed loop is stable for arbitrary N. Numerical computations of the eigenvalues are provided that corroborate the PDE-based analysis.