  Converse negative imaginary theorems for linear time-invariant systems are
derived. In particular, we provide necessary and sufficient conditions for a
feedback system to be robustly stable against various types of negative
imaginary (NI) uncertainty. Uncertainty classes of marginally stable NI systems
and stable strictly NI systems with restrictions on their static or
instantaneous gains are considered. It is shown that robust stability against
the former class entails the strictly NI property, whereas the latter class
entails the NI property. We also establish a non-existence result that no
stable system can robustly stabilise all marginally stable NI uncertainty,
thereby showing that the uncertainty class of NI systems is too large as far as
robust feedback stability is concerned, thus justifying the consideration of
subclasses of NI systems with constrained static or instantaneous gains.
