Asymmetric control achieves size-independent stability margin in 1-D flocks

We consider the stability margin of a large 1-D flock of double-integrator agents with distributed control, in which the control at each agent depends on the relative information from its nearest neighbors. The stability margin is measured by the real part of the least stable eigenvalue of the closed-loop state matrix, which quantifies the rate of decay of initial errors. In [1], it was shown that with symmetric control, in which two neighbors put equal weight on information received from each other, the stability margin of the flock decays to 0 as O(1/N²), where N is the number of agents. Moreover, a perturbation analysis was used to show that with vanishingly small amount of asymmetry in the control gains, the stability margin can be improved to O(1/N). In this paper, we show that, in fact, with asymmetric control the stability margin of the closed-loop can be bounded away from zero uniformly in N. Asymmetry in control gains thus makes the control architecture highly scalable. We establish the results through distinct routes, using state-space analysis and also using a partial differential equation (PDE) approximation. Numerical verifications are also provided to corroborate our analysis.