A Cyclic Small Phase Theorem for Nonlinear Systems

A new definition of phase, called the γ-segmental phase, is proposed for stable nonlinear systems from an input-output perspective. The definition is graphically established upon a disk segment covering the normalized numerical range of a nonlinear system plotted in the unit disk. The Hilbert transform is employed which acts as an enabling instrument for complexifying real-valued input-output trajectories of a system. A small phase stability criterion is developed for a cyclic feedback system consisting of multiple subsystems, namely, \begin{equation*}\Psi_{\gamma_{1}}\left(\boldsymbol{P}_{1}\right)+\Psi_{\gamma_{2}}\left(\boldsymbol{P}_{2}\right)+\cdots+\Psi_{\gamma_{m}}\left(\boldsymbol{P}_{m}\right) \subset(-\pi, \pi)\end{equation*} where the interval-valued $\Psi_{\gamma}(\cdot)$ denotes the γ-segmental phase of a system. The proposed criterion can be regarded as a phase counterpart to the monumental small gain theorem.