  This paper considers a string of vehicles where the local control law uses
the states of the vehicle's immediate predecessor and follower. The coupling
towards the preceding vehicle can be chosen different to the coupling towards
the following vehicle, which is often referred to as an asymmetric
bidirectional string. Further, the asymmetry for the velocity coupling can be
chosen differently to the asymmetry in the position coupling. It is
investigated how the effect of disturbance on the control errors in the string
depends on the string length. It is shown, that in case of symmetric position
coupling and asymmetric velocity coupling, linear scaling can be achieved. For
symmetric interaction in both states, i.e., in symmetric bidirectional strings,
the errors scale quadratically in the number of vehicles. When the coupling in
position is asymmetric, exponential scaling may occur or the system might even
become unstable. The paper thus gives a comprehensive overview of the
achievable performance in linear, asymmetric, bidirectional platoons. The
results reveal that symmetry in the position coupling and asymmetry in velocity
coupling qualitatively improves the performance of the string. Extensive
numerical results illustrate the theoretical findings.
