Asymmetric Decentralized Flocks

This paper analyzes the transient dynamics of one dimensional flocks, platoons, i.e., a finite collection of identical vehicles moving on the line, with a single leader with independent motion. We show for a class of platoon control laws that if the information flow is asymmetric then a motion change of the leader will cause system transients with amplitudes that grow at an exponential rate as the length of the platoon increases. With suitable choice of the control parameters the system is asymptotically stable and in steady state all vehicles move at the same velocity as the leader and at the required separation. When the leader changes velocity, over very long time scales the vehicles in the platoon tend to the steady state dictated by the leader's new velocity. The transient dynamics in the intermediary regime can however appear quite unstable, where the trailing vehicle can undergo oscillatory motion with amplitudes that grow exponentially large with the number of cars N in the platoon, or may be irresponsive over an exponentially long time to the change in the motion of the leader. In this paper we prove that if the control law is asymmetric then such transient errors, measured in terms of displacement between the leader and the trailing car, grow at an exponential rate in N, the length of the platoon. This contrasts sharply with the symmetric (bidirectional) case when such transient errors grow only linearly in the length of the platoon, the theoretical minimum for decentralized linear time-invariant platoon control systems with a constant vehicle spacing policy. These results suggest that symmetry of the information flow is an important design parameter for safe control laws for platoons.