Further Results about the Internal Model Principle and Trajectory Tracking Control in Linear Dynamical Systems

The paper talks about the internal model principle and its application to trajectory tracking control in linear timeinvariant dynamical systems subject to disturbances. Firstly, standard LTI state-space internal models and closed-loop tracking control configurations are examined. Secondly, trajectory tracking control is re-formulated and classified according to tracking error into one with ultimate bounded bias (bounded), or one with non-zero steady-state bias (biased), or one with zero steady-state bias (unbiased), or one with unbounded bias (unbounded biased). Thirdly, with regard to static state feedback, disturbances and trajectories, trajectory tracking attainability and bias-types are claimed and proved via stabilization by time- and/or complex-domain methods. More specifically, it is revealed that bounded trajectory tracking is attainable if disturbance and trajectory are bounded; biased trajectory tracking is achievable if the limits of disturbance and trajectory as time goes to infinity are well-defined in the sense that the ultimate value theorem of the Laplace transform holds true; unbiased trajectory tracking can be realized if the internal model in terms of pole position and multiplicity exactly reflects unstable modes in disturbance and trajectory. Finally, several numerical examples are included to illustrate the main results.