In this study, we extend the theory of negative imaginary (NI) systems to a nonlinear framework using a frequency-domain approach. The extended notion is completely characterized via a finite-frequency integration over a “kernel function” on energy-bounded input and output signal pairs. The notion is closely related to and carefully contrasted with the well-studied extension of negative imaginariness — the theory of counterclockwise dynamics. Conditions for feedback stability of the proposed nonlinear NI systems are then developed based on the technique of integral quadratic constraints. Examples and simulations on feedback interconnections of typical nonlinear systems are provided to demonstrate the effectiveness.

在这项研究中，我们使用频域方法将负虚 (NI) 系统的理论扩展到非线性框架。扩展概念完全通过有限频率积分在能量有界输入和输出信号对上的“核函数”上进行表征。这个概念与经过充分研究的负想象扩展——逆时针动力学理论密切相关，并与之形成鲜明对比。然后基于积分二次约束技术开发了所提出的非线性NI系统的反馈稳定性条件。提供了典型非线性系统反馈互连的示例和仿真来证明其有效性。