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 $(Q_{\sigma} (\omega),S_{\sigma} (\omega),R_{\sigma} (\omega)) -$dissipativity”, as an alternative to the conventional $(Q(\omega),S(\omega),R(\omega)) -$dissipativity, to capture the TD-ONI systems, particularly the ones having imaginary-axis poles. The shifted $(Q_{\sigma} (\omega),S_{\sigma} (\omega),R_{\sigma} (\omega)) -$dissipativity is defined w.r.t. a shifted imaginary axis $(\sigma + \quad j\omega, \sigma \gt 0)$ 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