离散时间LTI系统的可跟踪性：简要回顾和新见解
Trackability for Discrete-Time LTI Systems: A Brief Review and New Insights
摘要：
“跟踪性”，系统遵循任意参考命令的能力，在这项工作中进行了研究。可控性在解释系统输出的跟踪行为方面没有用，这是一个经常被忽视的差距。可跟踪性通过表征系统动态是否允许跟踪某些参考命令来解决这个差距，而不管所使用的控制器/控制动作。虽然早期的文献讨论了密切相关的想法，缺乏一致性的术语和重点的文献中有必要定义跟踪性在一个简单而严格的方式。我们提出了基于初等线性代数的可跟踪性的定义，并提出了一个秩检验确定系统的可跟踪性，而跟踪控制作为一个存在性问题。我们讨论了一些详细的秩检验与文献中的其他结果的关系。然后，这些结果用于生成关于可跟踪系统的见解，更重要的是，用于获得对不可跟踪系统的跟踪行为的见解。我们还定义了三个指标，表明预期的跟踪系统的行为。此外，我们还提出了一个维恩图，详细解释了跟踪性和其他基本系统属性，如可控性，可观测性和输出可控性之间的联系，同时讨论了几个事实，阐述这些连接。具体来说，我们表明，跟踪性是从根本上不同的可控性和输出可控性。所提出的工作，预计将作为一个基础，更深入的调查跟踪控制的主题，并提供了一个框架，接收新的理论见解跟踪。
英文摘要：
`Trackability', the ability of systems to follow arbitrary reference commands, is investigated in this work. Controllability is not useful in explaining the tracking behavior of system outputs, a gap that is often overlooked. Trackability addresses this gap by characterizing whether system dynamics permits tracking of certain reference commands, irrespective of the controller/control action used. While earlier literature has discussed closely-related ideas, lack of consistency in terminology and focus in the literature has necessitated defining trackability in a simple but rigorous manner. We present definitions for trackability based on elementary linear algebra and propose a rank test for determining trackability of systems, while posing tracking control as an existence question. We discuss in some detail the relationship of this rank test with other results in the literature. These results are then used to generate insights about trackable systems and more importantly to gain insights in to tracking behavior of systems that are not trackable. We also define three indices that indicate the expected tracking behavior of the system. Furthermore, we also present a Venn diagram explaining in detail the connections between trackability and other fundamental system properties like controllability, observability and output controllability, while discussing several facts that elaborate these connections. Specifically, we show that trackability is fundamentally different from controllability and output controllability. The presented work is expected to serve as a foundation for a deeper investigation into the topic of tracking control and provide a framework for receiving new theoretical insights on trackability.
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