Phase of Multivariable Systems: A Revisit via $\mathcal{H}_{2}^{T}$ -Dissipativity

A new notion of phase of multi-input multi-output (MIMO) systems was recently defined and studied, leading to new understandings in various fronts including a formulation of small phase theorem, a performance criterion named
$\mathcal{H}_{\infty}$
phase sector, and a sectored real lemma, etc. In this paper, we define a new notion of
$\mathcal{H}_{2}^{T}$
-dissipativity and show the connection between the phase of a multivariable linear time-invariant (LTI) system and the
$\mathcal{H}_{2}^{T}$
-dissipativity. The
$\mathcal{H}_{2}^{T}$
-dissipativity, roughly speaking, is dissipativity restricted to the time-domain
$\mathcal{H}_{2}$
space which consists of
$\mathcal{L}_{2}$
signals with only positive frequency components. In addition, by exploiting the newly defined
$\mathcal{H}_{2}^{T}{-}$
dissipativity, we also study the phase of a feedback system and provide a physical interpretation of the sectored real lemma.