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

Trackability is a known system property and has been studied extensively in the literature both from a theoretical standpoint and from the standpoint of constructing inversion-based tracking controllers. There is also a significant body of literature on closely related ideas focusing on existence of tracking control inputs but using varied terminology and tools. In this paper, we start with the existence question, given an linear time-invariant (LTI) discrete-time system, when does there exists control inputs to enable the system to track arbitrary reference commands. We adopt an algebraic approach that then allows us to define the sets of all trackable output reference commands and untrackable output reference commands, and subsequently permitting us to determine conditions for existence of a right inverse of the system based on these sets and a simple rank test. These results permit us to then gain insights into tracking behavior of systems that are not trackable. We discuss in some detail the relationship of these rank tests with other results in the literature. We also define three indices that indicate the expected tracking behavior of any given system (whether trackable or not). 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. The presented work is expected to provide a framework for receiving new theoretical insights on the topic of tracking control and trackability.