Some Properties of Infinite-dimensional Systems Capable of Asymptotically Tracking Bounded Uniformly Continuous Signals

Abstract

We study several properties of infinite-dimensional systems capable of asymptotically tracking (i.e., regulating) a given bounded uniformly continuous (BUC) reference signal y _ref using a feedforward controller. We show that the regulability of the signal can be characterized by the solvability of the regulator equations. Using this information we construct the so called regulable space $\mathcal{R}_{y_{\rm ref}}$ associated to the reference signal y _ref. This space contains other reference signals which can also be regulated if we know that y _ref can be regulated. Using methods of harmonic analysis we obtain a complete description of the elements of $\mathcal{R}_{y_{\rm ref}}$ whenever y _ref is almost periodic, and we show that ergodic vectors and isolated spectral points of y _ref yield nontrivial information about $\mathcal{R}_{y_{\rm ref}}$ even if y _ref is not almost periodic. We also show that certain spectral points of the regulable reference signals—and their generators—cannot be zeros of the feedforward control system. These results generalize the well known results from finite-dimensional control theory; a crucial feature in our approach is the infinite-dimensionality of the exogenous signal generator. We conclude the article by demonstrating how infinite-dimensional exosystems can be used in practice.