On state-space characterisation for output negative imaginary systems with possible poles at the origin and their internal stability result

This paper derives a new and unifying state-space characterisation for the entire class of real, rational, proper Output Negative Imaginary (ONI) systems, allowing poles on the imaginary axis even at the origin. The proposed result captures the existing versions of the NI state-space characterisations, particularly the ones that apply to the NI systems with poles at the origin. A necessary and sufficient LMI condition has been derived to test the strict/non-strict ONI properties of an LTI system with a given minimal state-space realisation. The LMI-based characterisation offers easy and convenient execution due to the easily accessible SDP solver packages. Finally, a necessary and sufficient internal stability theorem is also derived for a positive feedback ONI systems interconnection containing pole(s) at the origin. The proposed stability result specialises to the earlier versions when the earlier assumptions are imposed. Numerical examples are given to show the usefulness of the proposed theoretical results.