Feedback and optimal sensitivity: Model reference transformations, multiplicative seminorms, and approximate inverses
Feedback and optimal sensitivity: Model reference transformations, multiplicative seminorms, and approximate inverses
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DOI：
10.1109/tac.1981.1102603
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发表时间：
1981-04
影响因子：
6.8
通讯作者：
G. Zames
中科院分区：
文献类型：
--
作者：
G. Zames
在本文中，反馈减少的灵敏度问题被表述为优化问题，并与稳定性问题分开。从给定设备可获得的稳定反馈方案被参数化。导出了灵敏度降低方案的显着特性，并表明设备不确定性降低了反馈降低灵敏度的能力。该理论是在 Banach 代数的一般设置下针对输入输出系统开发的，然后专门用于一类多变量、时不变系统，其特征是 H^{\infty} 频率响应函数的 n × n 矩阵，右半平面有或没有零。该方法基于使用算子代数的加权半范数来测量灵敏度，以及近似逆的概念。植物的近似可逆性被证明是敏感性降低的充分必要条件。引入了近似可逆性的指标，称为奇异性度量。线性时不变对象的奇异性度量由其右半平面零点的位置确定。在没有工厂不确定性的情况下，对输出扰动的敏感性可以降低到接近奇点测量的最佳值。特别是，如果不存在右半平面零点，则灵敏度可以任意小。灵敏度优化中使用的反馈方案类似于经典控制设计的超前滞后网络。介绍了它们的一些属性以及在特殊情况下构造它们的方法。
In this paper, the problem of sensitivity, reduction by feedback is formulated as an optimization problem and separated from the problem of stabilization. Stable feedback schemes obtainable from a given plant are parameterized. Salient properties of sensitivity reducing schemes are derived, and it is shown that plant uncertainty reduces the ability, of feedback to reduce sensitivity. The theory is developed for input-output systems in a general setting of Banach algebras, and then specialized to a class of multivariable, time-invariant systems characterized by n \times n matrices of H^{\infty} frequency response functions, either with or without zeros in the right half-plane. The approach is based on the use of a weighted seminorm on the algebra of operators to measure sensitivity, and on the concept of an approximate inverse. Approximate invertibility, of the plant is shown to be a necessary and sufficient condition for sensitivity reduction. An indicator of approximate invertibility, called a measure of singularity, is introduced. The measure of singularity of a linear time-invariant plant is shown to be determined by the location of its right half-plane zeros. In the absence of plant uncertainty, the sensitivity, to output disturbances can be reduced to an optimal value approaching the singularity, measure. In particular, if there are no right half-plane zeros, sensitivity can be made arbitrarily small. The feedback schemes used in the optimization of sensitivity resemble the lead-lag networks of classical control design. Some of their properties, and methods of constructing them in special cases are presented.