Most Exigent Eigenvalue Problem for the Stability Analysis of the Vehicle Platoon With Delay

In vehicle platoons, elucidating the essential relationship between the closed-loop stability and the communication topology is critical for expediting the determination of the exact delay margin (EDM) and enhancing controller synthesis. To unveil this essential relationship, this study investigates the most exigent eigenvalue (MEE) problem of the third-order vehicle platoon with delay under generic communication topologies. Compared to existing studies on the MEE problem, the vehicle model employed more accurately reflects real-world traffic, and the limitation of undirected communication topologies is overcome. In this paper, an analytical method is proposed to search for the MEE, particularly for vehicle platoons equipped with adaptive cruise control (ACC) and cooperative adaptive cruise control (CACC) systems. This method derives analytical results regarding the MEE by examining the monotonicity relationship between the EDM and the eigenvalues of the Laplacian matrix. In particular, we examine the coupled effects of the moduli and phases of the eigenvalues on searching for the MEE. Finally, simulation results demonstrate that the proposed method reduces the computation time required to calculate the EDM by three orders of magnitude compared to the traditional method.