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arXiv · 2012.00692

Phase of Nonlinear Systems

Chao Chen·Di Zhao·Wei Chen·Sei Zhen Khong·Li Qiu
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Abstract

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.

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Chao Chen, Di Zhao, Wei Chen, Sei Zhen Khong, Li Qiu. 2020-11-30. Phase of Nonlinear Systems. https://arxiv.org/abs/2012.00692

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