Trackability for Discrete-Time LTI Systems: A Brief Review and New Insights

TL;DR

This paper reviews the concept of trackability in discrete-time LTI systems, introduces a rank-based test for assessing it, and explores its relationship with other system properties, providing new theoretical insights.

Contribution

It defines a rigorous, simple framework for evaluating trackability, including a rank test and indices, and clarifies its distinction from controllability and output controllability.

Findings

Proposes a rank test for system trackability.

Defines indices to predict tracking behavior.

Elucidates the relationship between trackability and other properties.

Abstract

`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.