  We provide a comprehensive impossibility result towards achieving string
stability, i.e. keeping local relative errors in check with local controllers
independently of the size of a chain of subsystems. We significantly extend
existing results, from the LTI setting to any homogeneous controllers that can
be nonlinear, time-varying, and locally communicating. We prove this
impossibility for a set of definitions with various norm choices, including the
$L_2$-type which is more standard in the literature and a BIBO type criterion.
All results hold for a general discrete-time controller which should cover most
applications.
