Phase of Nonlinear Systems

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.