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Converse negative imaginary theorems

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.

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