Robustness of controllers for SISO-plants and signals generated by an infinite-dimensional exosystem

Robust regulation of signals generated by infinite-dimensional exosystems have received some attention lately. Robustness have been understood in the sense that strong stability and solvability of a Sylvester equation should imply asymptotic tracking. It is not known what perturbations preserve strong stability or solvability of the Sylvester equation, so it is not entirely clear how robust a robustly regulating controller actually is. The purpose of this article is to give a detailed analysis of the robustness properties of a controller. We give a simple necessary and sufficient condition that guarantees the solvability of the Sylvester equation for single input single output plants. The solvability condition relates smoothness of the reference signals to the solvability of the Sylvester equation and helps us to understand robustness of controllers.