I. Introduction
String stability roughly requires that a chain of subsystems (e.g., vehicles), controlled by local feedback from relative position measurements and subject to bounded local perturbations, keeps local deformations bounded independently of the size of the chain. In the absence of further variables intervening in the system, local deformations are defined as the deviations of the distances between consecutive vehicles from a constant target value. The topic has gathered attention from the observation that, for second-order integrator subsystems, by using any linear time-invariant (LTI) controller reacting to their predecessor's distance, the amplification of some disturbance components along the chain is unavoidable [1], [2]. The historical application is a chain of acceleration-controlled vehicles, but other distributed systems like mechanical structures [3], [4], or more fundamental models, should also benefit from string stability insight. This has initiated a rich line of results such as the following.
Establishing various impossibility results and scalings for LTI controllers [1], [5], [6], mostly with subsystems reacting to their immediate predecessor and immediate follower. Often PD controllers are used as a proxy for bandwidth limitation, but problems associated with integral control have also been characterized [7].
Designing a passive LTI controller, with symmetric coupling to the predecessor and follower, to keep in check, at least in some sense, the effect of disturbances acting on the leading subsystem [3].
Adding dependence on absolute velocity into the pure double integrator, to overcome the string instability issue [8]–[10]. This can take the form of strong enough drag, which, e.g., for terrestrial vehicle applications are easy to ensure but would be in tradeoff with fuel efficiency; in other applications, ensuring strong drag may not always be trivial. Alternatively, the “time-headway” spacing policy makes the target intervehicle distance dependent on absolute velocity. For terrestrial vehicles, again, this appears easy to achieve, but questionable in terms of chain performance (possibly big spacing at high velocities, instead of moving efficiently like a big rigid body at all speeds); for other applications, the necessity to measure absolute velocity of each subsystem with respect to a common reference may even pose sensing questions.
Adding local communication between subsystems, as in adaptive cruise control [11], [12]. Remarkably, the existing studies with this approach all incorporate a dependence on absolute velocity, as mentioned in the previous item. In a recent paper [13], a variation on time headway is proposed for a generic controlled system, and, in the nominal case, the controller uses both absolute state information and communication with the leader.