Frequency-domain dissipativity analysis for output negative imaginary systems allowing imaginary-axis poles

This brief addresses the frequency-domain dissipativity problem of a broader class of Output Negative Imaginary systems, termed as the time-domain ONI (or TDONI) systems, which have been defined in the time domain w.r.t. an abstract energy supply rate function. This definition encompasses the existing strict/non-strict NI subsets, including those having imaginary-axis poles. This paper introduces the idea of a “shifted (Qσ(ω); Sσ(ω);Rσ(ω))-dissipativity”, as an alternative to the conventional (Q(ω); S(ω);R(ω))-dissipativity, to capture the TD-ONI systems, particularly the ones having imaginaryaxis poles. The shifted (Qσ(ω); Sσ(ω);Rσ(ω))-dissipativity is defined w.r.t. a shifted imaginary axis (σ + jω; σ > 0) and thereby, it overcomes the limitation of earlier frequency-domain dissipative frameworks to capture systems with imaginary-axis poles. The paper has also established the relationship between the time-domain and frequency-domain dissipativity of TD-ONI systems. Finally, a closed-loop stability theorem is also given for a positive feedback interconnection of two TD-ONI systems.