# A Method for the Rapid Propagation of Emergency Event Notifications in a Long Vehicle Convoy
### String Stability  
The main challenge facing vehicle convoys is a phenomenon known as string stability. Typically, a convoy of vehicles tries to maintain a certain headway distance or time between each other using sensors such as radar. If a vehicle experiences a disturbance in its velocity (such as applying the brakes), then the vehicles behind it apply actuations according to their control law to regain the nominal headway. If the disturbance is large enough or fast enough, then it may send the system into an unstable region, depending on various factors. A familiar example of string instability is in \"phantom traffic jams\", where small decreases in the speed of a leader can cause follower vehicles to overcorrect. This, in turn, causes even greater overcorrection as the \"wave\" propagates rearward to the point that some vehicles may come to a complete stop on a highway. In short, this is a problem of delays. Even if the system could instantaneously recognize and decide upon an appropriate course of action, there are still unavoidable delays in vehicle actuation. The follower vehicle's sensors only notice a change after the leader vehicle's actuation delays have concluded. Thus, when considering its own actuation delays, the follower has less time to react and so may need to apply a higher magnitude control input compared to the leader. If the follower could be informed of the leader's actuation intent at the time that the decision is made, then the effect of actuation delays can be largely negated. At this point, the problem becomes one of communication delay. While this work does not address how communication delays affect a string-stable controller design, there is a wealth of prior research on this type of problem. [PERSON] et al. discussed the general effects of string stability in a vehicle platoon that is controlled using a fixed inter-vehicle spacing [6]. Their findings showed that disturbance sensitivity increases for rearward vehicles when only information on the immediate neighbor is known, such as only using sensors or only having communication with the vehicle's immediate leader. Only upon incorporating information on the behavior of the ultimate leader did the sensitivity decrease further down the platoon. [PERSON] et al. created a more real-world style of experiment by directly addressing communication delays, among other things [7]. In particular, they utilized anticipatory information broadcast (what a vehicle plans to do in the near future), a fully non-overlapping transmission schedule, and synchronized vehicle actuation to maintain the stability of the platoon. The most relevant aspect here is in how heavily the platoon leader's information is weighted in the follower's controller. When the leader's information has no effect outside its immediate follower, the platoon displays instability. The platoon becomes stable when the leader's information is almost exclusively used. [PERSON] et al. discussed how a vehicle platoon's internal stability is affected by communication topologies and the vehicles' inertial delays [8]. While their work did not specifically deal with string stability, it did show how delays in information flow dramatically change the characteristics of the vehicle platoon. In their results, the topologies that do not incorporate a direct link to the leader result in much greater spacing errors further rearward in the platoon when a disturbance occurs, as compared to topologies that do incorporate a link to the leader. [PERSON] and [PERSON] worked toward finding string stability even when communication delay is randomly distributed [9]. What they showed is a method for determining a minimum headway time that guarantees string stability given a certain packet delivery probability. Specifically, their method allows for a certain percentage of message delays to be greater than a threshold value while still maintaining string stability. This work is pertinent here in that it shows how smaller inter-vehicle spacings become feasible as the delays become shorter and more consistent.