Convergence aspects on internal model principle and tracking control in linear dynamical systems: classification and properties

This paper talks about asymptotic convergence aspects about the internal model principle (IMP) and trajectory tracking control (TTC) in linear dynamical systems. First, IMP and asymptotic TTC are re-formulated and explicated. Second, TTC under internal models of trajectory and disturbance is classified in terms of tracking error bias into bounded; nonzero steady-state; zero steady-state; unbounded (tracking failure). Third, TTC attainability and tracking error biases are examined via asymptotic convergence analysis. In particular, bounded TTC is attainable if trajectory and disturbance are essentially bounded; zero/nonzero-steady-state TTC is attainable if trajectory and disturbance $ t\to \infty $ t→∞ limits are well-defined so that the Laplace ultimate value theorem holds; zero-steady-state TTC is realisable if internal models exactly reflect all unstable modes of trajectory and disturbance. Fourth, when internal models are subject to mismatches and sampled-data approximates, several interesting convergence facts are revealed. More specifically, for addressing zero-steady-state TTC to periodic trajectory and disturbance, stabilisation may be insufficient; for addressing TTC under sampled-data internal models, unknown convergence features are claimed. Finally, TTC controller parametrisation via pole assignment stabilisation is scrutinised under static output and state feedback. Numerical examples are included to illustrate the main results.