String instability in coupled harmonic oscillator systems

In this paper we consider the problem of disturbance response for a simple system of coupled harmonic oscillators. We suppose that the oscillators are connected in a string in which each oscillator tries to track its predecessor. Motivated by terminology from the problem of vehicle platooning, we say that the system is string unstable if the effect of a disturbance to the lead oscillator is amplified as it propagates along the string. By using a new Bode-like integral relation that must be satisfied by the complementary sensitivity function, we provide sufficient conditions for string instability.