This paper analyzes the robust feedback stability of a single-input-single-output stable linear time-invariant (LTI) system against three different classes of nonlinear systems using the Zames–Falb multipliers. The contribution is threefold. Firstly, we identify a class of uncertain systems over which the robust feedback stability is equivalent to the existence of an appropriate Zames–Falb multiplier. Secondly, when restricted to be static (a.k.a. memoryless), such a class of systems coincides with the class of sloped-restricted monotone nonlinearities, and the classical result of using the Zames–Falb multipliers to ensure feedback stability is recovered. Thirdly, when restricted to be LTI, the first class is demonstrated to be a subset of the second, and the existence of a Zames–Falb multiplier is shown to be sufficient but not necessary for the robust feedback stability.

本文使用 Zames-Falb 乘法器分析了单输入单输出稳定线性时不变 (LTI) 系统对三类不同非线性系统的鲁棒反馈稳定性。贡献是三方面的。首先，我们确定了一类不确定系统，其稳健反馈稳定性相当于存在适当的 Zames-Falb 乘子。其次，当限制为静态（又名无记忆）时，此类系统与倾斜限制单调类一致非线性，以及使用 Zames-Falb 乘法器确保反馈稳定性的经典结果得到恢复。第三，当限制为 LTI 时，第一类被证明是第二类的子集，并且 Zames-Falb 乘子的存在被证明是足够的，但对于稳健的反馈稳定性不是必需的。