An Overview of Generalized KYP Lemma Based Methods for FiniteFrequency Analysis and Design
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摘要: 频域方法是控制理论与工程领域的一种基本研究手段，许多控制问题都可归结为有限频域性能指标的分析与综合问题.广义Kalman-Yakubovich-Popov（KYP）引理建立了频域方法（传递函数）与时域方法（状态空间）之间的一座桥梁，成为近年来系统与控制理论领域的研究热点之一.本文首先从信号和系统两个角度阐明有限频域分析与设计的背景和意义，并依次讨论三种主要研究方法（经典控制理论方法、频率加权法和广义性能指标法）各自的优缺点.然后简单介绍广义KYP引理的主体内容，并详细总结当前基于广义KYP引理的有限频域分析与设计的主要方向及研究进展.最后给出在使用广义KYP引理时很重要但容易忽视的几点注记，同时指明该领域目前存在并值得未来进一步研究的关键问题.
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关键词:
- 有限频域 /
- 广义Kalman-Yakubovich-Popov (KYP)引理 /
- 控制器设计 /
- 滤波 /
- 模型降阶
Abstract: Frequency-domain methods are a fundamental research approach in control theory and engineering. Many control problems can be viewed as analysis and design issues of finite frequency specifications. The generalized Kalman-Yakubovich-Popov (KYP) lemma, which bridges frequency-domain methods (transfer functions) and time-domain methods (state-space models), has been one of the hotspots in systems and control theory in recent years. In this paper, the background and significance of finite frequency analysis and design are first introduced from signal and system perspectives, respectively. Three main research methods (classical control theory, frequency-weighting strategy and generalized system specification based methodology) are discussed with respect to their individual advantages and disadvantages. The body of the generalized KYP lemma is then introduced briefly, which is followed by a detailed summary of main directions and recent progresses in finite frequency analysis and design based on the generalized KYP lemma. Finally, a few notes are presented, which are important but commonly overlooked in applying the generalized KYP lemma, and a few critical problems in the field are also pointed out, which are worth future investigation. -
表 1 集合$\Omega $与$\Lambda $以及矩阵$\Phi $和$\Psi $的取值
Table 1 The values of sets$\Omega $and$\Lambda $and matrices$\Phi $and$\Psi $
$\Lambda $ $\Phi $ $\Omega $ $\Psi $ 连续系统 $\left[ \begin{matrix} 0 & 1 \\ 1 & 0 \\ \end{matrix} \right]$ 低频 $\left[ \begin{matrix} -1 & 0 \\ 0 & \omega _{l}^{2} \\ \end{matrix} \right]$ 中频 $\left[ \begin{matrix} -1 & \text{j}{{\omega }_{c}} \\ -\text{j}{{\omega }_{c}} & -{{\omega }_{1}}{{\omega }_{2}} \\ \end{matrix} \right]$ 高频 $\left[ \begin{matrix} 1 & 0 \\ 0 & -\omega _{h}^{2} \\ \end{matrix} \right]$ 离散系统 $\left[ \begin{matrix} 1 & 0 \\ 0 & -1 \\ \end{matrix} \right]$ 低频 $\left[ \begin{matrix} 0 & 1 \\ 1 & -2\cos {{\omega }_{r}} \\ \end{matrix} \right]$ 中频 $\left[ \begin{matrix} 0 & {{\text{e}}^{\text{j}\omega c}} \\ {{\text{e}}^{-\text{j}\omega \text{c}}} & -2\cos {{\omega }_{r}} \\ \end{matrix} \right]$ 高频 $\left[ \begin{matrix} 0 & -1 \\ -1 & 2\cos {{\omega }_{h}} \\ \end{matrix} \right]$ 
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