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

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Document pages: 47 pages

Abstract: `Trackability , the ability of systems to follow arbitrary referencecommands, is investigated in this work. Controllability is not useful inexplaining the tracking behavior of system outputs, a gap that is oftenoverlooked. Trackability addresses this gap by characterizing whether systemdynamics permits tracking of certain reference commands, irrespective of thecontroller control action used. While earlier literature has discussedclosely-related ideas, lack of consistency in terminology and focus in theliterature has necessitated defining trackability in a simple but rigorousmanner. We present definitions for trackability based on elementary linearalgebra and propose a rank test for determining trackability of systems, whileposing tracking control as an existence question. We discuss in some detail therelationship of this rank test with other results in the literature. Theseresults are then used to generate insights about trackable systems and moreimportantly to gain insights in to tracking behavior of systems that are nottrackable. We also define three indices that indicate the expected trackingbehavior of the system. Furthermore, we also present a Venn diagram explainingin detail the connections between trackability and other fundamental systemproperties like controllability, observability and output controllability,while discussing several facts that elaborate these connections. Specifically,we show that trackability is fundamentally different from controllability andoutput controllability. The presented work is expected to serve as a foundationfor a deeper investigation into the topic of tracking control and provide aframework for receiving new theoretical insights on trackability.

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