A Simple Robust Controller for the Approximate Regulation of Almost Periodic Signals for Linear Systems

By using a delay line as a signal generator, we design a simple robust controller for the approximate regulation (i.e., approximate asymptotic tracking and rejection) of almost periodic signals for linear systems. As opposed to the related existing results, our controller design does not require online parameter tuning. Besides the usual assumptions about stabilizability, detectability, and nonexistence of transmission zeros for the plant, the controller design algorithm of this paper only requires knowledge of the desired tracking accuracy $\epsilon>0$. Based on this information alone, we can only design a robust controller achieving, in the presence of unknown trigonometric polynomial disturbances and additive perturbations to the plant and the controller, $\lim\sup_{t\rightarrow\infty}\Vert y(t)-y_{ref}(t)\Vert≪\epsilon\Vert y_{ref}\Vert$. Here, $y$ is the output of the plant and $y_{ref}$ is an arbitrary almost periodic reference signal taken from an infinite-dimensional generalized Sobolev space of almost periodic functions. In this paper, we also study robust output regulation in the limiting case where $\epsilon\rightarrow 0$. It turns out that, due to the general loss of exponential closed loop stability, asymptotic tracking/rejection can also be lost as $\epsilon\rightarrow 0$, unless the exogenous signals are smooth enough. This last result extends some recent theorems on (nonrobust) open loop output regulation for infinite-dimensional exosystems. The results of this paper are new and potentially useful for finite-dimensional plants, but the main results are also true for infinite-dimensional systems with bounded control and observation operators.