What does one do if output regulation is not possible?

Abstract

We studied so far output regulation whether it be exact or almost type under a variety of transient, robustness and other performance constraints. Also, we studied output regulation when actuators are subject to amplitude and rate saturation. As discussed throughout the book, output regulation is not always possible. It requires certain conditions. One of the important conditions that needs to be satisfied is the solvability of a couple of algebraic equations which together are generally known as regulator equation. One of the questions that arises is this; what can a designer do if either exact or almost output regulation is not possible? To answer this question, let us first interpret the error signal e whenever output regulation is possible. Obviously, in that case e is an energy signal. On the other hand, whenever output regulation is not possible, under some mild conditions, e can be seen to be a power signal. In this case, since e does not asymptotically go to zero, one could minimize in the asymptotic sense the power of the signal e. In other words, in the classical output regulation we seek to render e asymptotically zero, where as whenever it is not possible to do so we could seek to render the power of e as small as possible.