String instability analysis of heterogeneous coupled oscillator systems

In this paper we consider the disturbance response for a heterogeneous string of harmonic oscillators. We assume the individual controllers can communicate with a limited range of neighbors and be tuned independently. Motivated by the speed-dependent spacing policy in vehicle platooning, we develop an extension of the concept of time headway for oscillator systems. Using fundamental limitations theory, we derive a lower bound on the peak magnitude response of the transfer function that describes how a disturbance applied to the lead oscillator propagates along the string to produce a response in the last oscillator. We then give sufficient conditions for string instability in terms of the separation policy, communication ranges, certain time domain performance specifications, and bandwidth limitations.