Dynamic dissipative characterisation of time-domain input–output negative imaginary systems

This paper introduces the class of Time-Domain Input–Output Negative Imaginary (TD-IONI) systems. The new TD-IONI definition unifies the class of the existing Negative Imaginary (NI) systems, including those that have imaginary-axis poles. A new dynamic dissipative framework is proposed to define and characterise the TD-IONI systems. This framework does not impose any a priori conditions on the system, such as asymptotic stability, minimality, full normal rank constraint, etc., which are commonly used in the NI literature. Dynamic dissipativity of TD-IONI systems also leads to an LMI-based state-space characterisation, which can be conveniently used to classify the strict/non-strict TD-IONI properties of a given system. This paper also reveals the connections amongst the NI theory, dynamic dissipativity and classical dissipativity. Subsequently, a frequency-domain dissipative supply rate is also proposed to describe the whole class of TD-IONI systems, which is defined with respect to a shifted imaginary axis to capture particularly the systems having poles on the imaginary axis. This trick overcomes the limitation of earlier frequency-domain dissipative frameworks to capture systems with imaginary-axis poles. Finally, the derived results are specialised for the Time-Domain Output (Strictly) Negative Imaginary subclass since such systems exhibit useful closed-loop stability properties when connected in a positive feedback loop. Several illustrative numerical examples are provided to make the results intuitive and useful.