On the ill-posedness of certain vehicular platoon control problems

We revisit the vehicular platoon control problems formulated by Levine & Athans (1966) and Melzer & Kuo (1971). We show that in each case, these formulations are effectively ill-posed. Specifically, we demonstrate that in the first formulation, the system´s stabilizability degrades as the size of the platoon increases, and that the system loses stabilizability in the limit of an infinite number of vehicles. We show that in the LQR formulation of Melzer & Kuo, the performance index is not detectable, leading to non-stabilizing optimal feedbacks. Effectively, these closed-loop systems do not have a uniform bound on the time constants of all vehicles. For the case of infinite platoons, these difficulties are easily exhibited using the theory of spatially invariant systems. We argue that the infinite case is a useful paradigm to understand large platoons. To this end, we illustrate numerically how stabilizability and detectability degrade as functions of a finite platoon size, implying that the infinite case is a reasonable approximation to the large, but finite case. Finally, we suggest a well-posed alternative formulation of the LQR problem based on penalizing absolute position errors in addition to relative ones in the performance objective.