Characterization of Input–Output Negative Imaginary Systems in a Dissipative Framework

In this article, we define the notion of stable input–output negative imaginary (IONI) systems. This new class captures and unifies all the existing stable subclasses of negative imaginary (NI) systems and is capable of distinguishing between the strict subclasses (e.g., strongly strictly negative imaginary, output strictly negative imaginary (OSNI), input strictly negative imaginary, etc.) in the literature. In addition to a frequency-domain definition, the proposed IONI class has been characterized in a time-domain dissipative framework in terms of a new quadratic supply rate
$w(u,\bar{u},\dot{\bar{y}})$
. This supply rate consists of the system’s input (
$u$
), an auxiliary input (
$\bar{u}$
) that is a filtered version of the system’s input, and the time-derivative of an auxiliary output of the system (
$\dot{\bar{y}}$
). This supply rate corrects earlier supply rate attempts in the literature, which were only expressed in terms of the input (
$u$
) and the time-derivative of the system’s output (
$\dot{y}$
). In this article, IONI systems are proved to be a class of dissipative systems with respect to the proposed supply rate
$w(u,\bar{u},\dot{\bar{y}})$
. Subsequently, an equivalent frequency-dependent
$(Q(\omega), S(\omega), R(\omega))$
dissipative supply rate is also proposed for IONI systems. These findings reveal the connections between the NI property and classical dissipativity in both the time domain and frequency domain. We also provide linear matrix inequality (LMI) tests on the state-space matrices to check whether a system belongs to the IONI class or any of its important subclasses. Finally, the derived results are specialized for OSNI systems since such systems exhibit interesting closed-loop stability properties when connected, in a positive feedback loop, to NI systems without poles at the origin. Several illustrative numerical examples are provided to make the results intuitive and useful.