Root locus plots are one of the basic design tools in classical control. They help the designer tune control gains which appear linearly in the coefficients of the closed loop characteristic polynomial. And they give considerable intuition to the designer, based on the simple rules that root loci must follow. When designing a control system, one wants to know where the zeros are, but when designing a digital control system new issues appear. The original zero locations when mapped to discrete time are functions of the new parameter, the sample time T (as well as the pole locations). In addition, new zeros are usually introduced by the discretization process. The purpose of this paper is to give a general understanding of the nature of root loci of discrete time transfer function zeros as a function of this parameter T. We consider the complete range of values from T equal zero to infinity to understand the full plot. Reasonable sample rates will only use part of the plots. The characteristic polynomial coefficients are nonlinear functions of T so the usual root locus rules do not apply. One can be amazed at how the usual root locus rules are repeatedly violated, and what new kinds of unexpected behavior can be observed.

中文翻译：

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离散时间系统零点的根轨迹与采样率的关系

根轨迹图是经典控制中的基本设计工具之一。它们帮助设计人员调整控制增益，该增益线性出现在闭环特性多项式的系数中。根据根基因座必须遵循的简单规则，它们为设计人员提供了相当直观的认识。在设计控制系统时，人们想知道零在哪里，但是在设计数字控制系统时，就会出现新的问题。映射到离散时间的原始零位置是新参数（采样时间T）的函数（以及极点位置）。此外，离散化过程通常会引入新的零。本文的目的是对作为参数T的函数的离散时间传递函数零的根基因座的性质进行一般性的理解。我们认为从T等于零到无穷大的值的完整范围可以理解整个图。合理的采样率将仅使用部分图。特征多项式系数是T的非线性函数，因此通常的根轨迹规则不适用。令人惊讶的是，如何屡次违反常规的根源规则，以及可以观察到哪些新的意外行为。