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A criterion for exact output tracking, for some SISO systems
Abstract
In control theory we often consider equations of the form $x'(t) = Ax(t) + Bu(t)$ and try to choose the command $u(t)$ so as $x(t)$ reaches some target, e.g. $x(T) = x_T$ for some given $x_T$ and fixed time $T$. In some cases it is actually better suited to achieve another control objective: the output tracking. For this we define an output signal $y(t) = Cx(t) + Du(t)$ and we aim at selecting $u(t)$ so as $y(t)$ matches some reference signal, e.g. $y(t) = y_{ref}(t)$ for all times $t$. With this in mind, one of the most basic question one may ask is: what are these output signals $y$ that the system may generate? Surprisingly there is no satisfactory answer in the literature, even in the basic case where the system has a single input and a single output (SISO), meaning that both $u(t)$ and $y(t)$ are numbers. In finite dimension the situation is fully understood thanks to the theory of Volterra integral equations. In infinite dimension the latter does not help anymore and we rely on a new result for exterior multipliers on Hardy spaces, providing a partial answer to the previous question.
