有界实引理和结构奇异值与对角标度：自由非交换设置
Bounded Real Lemma and structured singular value versus diagonal scaling: the free noncommutative setting
摘要：
结构奇异值$μ$是由Doyle和Safanov独立提出的，用于分析系统参数存在结构不确定性时系统稳定性和性能的鲁棒性。虽然结构化奇异值提供了关于结构化不确定球的鲁棒性的必要且充分的标准，但众所周知，实际计算是困难的。另一方面，对角（或简称为“D”）缩放的方法为结构化奇异值提供了一个易于计算的上限（我们称之为$\hat μ$），但仅在特殊情况下提供了对$μ$的精确评估（甚至是对$μ$有用的上限）。然而，在20世纪90年代发现，不确定性结构的某种增强（即，让不确定性参数是无限维可分希尔伯特空间上的自由非交换线性算子）导致$D$ -标度过程导致$μ_{\text{enhanced}}$（$μ_{\text{enhanced}} = \hat μ$）的精确评估，至少对于完全详细分析的易处理的特殊情况。一方面，这种增强的不确定性从物理的角度来看有一定的吸引力：可以允许在工厂参数的不确定性是随时间变化的，或者更一般地说，可以捕捉到的不确定性所造成的设计师的决定不建模的更复杂的（例如非线性）动态的真实工厂。另一方面，这种增强的不确定性结构的精确数学表述与自由非对易参数中的解析函数和自由非对易不确定性中的相关形式幂级数的增长理论的发展相联系。在本文中，我们得到了一个更令人满意的一般设置的$\widetilde μ= \hat μ$定理。
英文摘要：
The structured singular value $μ$ was introduced independently by Doyle and Safanov as a tool for analyzing robustness of system stability and performance in the presence of structured uncertainty in the system parameters. While the structured singular value provides a necessary and sufficient criterion for robustness with respect to a structured ball of uncertainty, it is notoriously difficult to actually compute. The method of diagonal (or simply "D") scaling, on the other hand, provides an easily computable upper bound (which we call $\hat μ$) for the structured singular value, but provides an exact evaluation of $μ$ (or even a useful upper bound for $μ$) only in special cases. However it was discovered in the 1990s that a certain enhancement of the uncertainty structure (i.e., letting the uncertainty parameters be freely noncommuting linear operators on an infinite-dimensional separable Hilbert space) resulted in the $D$-scaling procedure leading to an exact evaluation of $μ_{\text{enhanced}}$ ($μ_{\text{enhanced}} = \hat μ$), at least for the tractable special cases which were analyzed in complete detail. On the one hand this enhanced uncertainty has some appeal from the physical point of view: one can allow the uncertainty in the plant parameters to be time-varying, or more generally, one can catch the uncertainty caused by the designer's decision not to model the more complex (e.g. nonlinear) dynamics of the true plant. On the other hand, the precise mathematical formulation of this enhanced uncertainty structure makes contact with developments in the growing theory of analytic functions in freely noncommuting arguments and associated formal power series in freely noncommuting indeterminates. In this article we obtain the $\widetilde μ= \hat μ$ theorem for a more satisfactory general setting.
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