论文标题
关于有限运算符的Zames-Falb乘数的必要性和充分性
On the Necessity and Sufficiency of the Zames-Falb Multipliers for Bounded Operators
论文作者
论文摘要
本文使用Zames-Falb乘数分析了单输入单输出稳定线性时间流动（LTI）系统的稳健反馈稳定性。贡献是四倍。首先，我们提出了一个广义的S-保护无损定理，该定理涉及大量无限数量的二次形式。其次，我们确定了一类不确定的系统，在这些系统上，鲁棒反馈稳定性意味着基于广义的S-Procedure无损定理的适当Zames-Falb乘法器的存在。同时，我们表明，这种Zames-Falb乘数的存在足以在较小类别的不确定系统上具有强大的反馈稳定性。第三，当仅限于静态（又称无内存）时，第二类系统与倾斜限制的单调非线性的类别相吻合，以及使用Zames-Falb乘数来确保反馈稳定性的经典结果。最后，当限制为LTI时，第二类被证明是第三类的子集，并且Zames-Falb乘数的存在被证明是足够的，但对于强大的反馈稳定性而言并不是必需的。
This paper analyzes the robust feedback stability of a single-input-single-output stable linear time-invariant (LTI) system against four different classes of nonlinear systems using the Zames-Falb multipliers. The contribution is fourfold. Firstly, we present a generalised S-procedure lossless theorem that involves a countably infinite number of quadratic forms. Secondly, we identify a class of uncertain systems over which the robust feedback stability implies the existence of an appropriate Zames-Falb multiplier based on the generalised S-procedure lossless theorem. Meanwhile, we show that the existence of such a Zames-Falb multiplier is sufficient for the robust feedback stability over a smaller class of uncertain systems. Thirdly, when restricted to be static (a.k.a. memoryless), the second 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. Lastly, when restricted to be LTI, the second class is demonstrated to be a subset of the third, and the existence of a Zames-Falb multiplier is shown to be sufficient but not necessary for the robust feedback stability.
