  In this paper, we propose a definition of phase for a class of stable
nonlinear systems called semi-sectorial systems, from an input-output
perspective. The definition involves the Hilbert transform as a critical
instrument to complexify real-valued signals since the notion of phase arises
most naturally in the complex domain. The proposed nonlinear system phase,
serving as a counterpart of $\mathcal{L}_2$-gain, quantifies the passivity and
is highly related to the dissipativity. It also possesses a nice physical
interpretation which quantifies the tradeoff between the real energy and
reactive energy. A nonlinear small phase theorem is then established for
feedback stability analysis of semi-sectorial systems. Additionally, its
generalized version is proposed via the use of multipliers. These nonlinear
small phase theorems generalize a version of the classical passivity theorem
and a recently appeared linear time-invariant small phase theorem.
