On the approximation properties of TP model forms

TLDR: It is shown that the set of functions that can be approximated arbitrarily well by TP forms with bounded number of components lies no-where dense in the setof continuous functions.

Abstract: The tensor product (TP) based models have been applied widely in approximation theory and approximation techniques. Recently, a controller design framework working on dynamic systems has also been established based on TP model transformation combined with linear matrix inequalities (LMI) within parallel distributed compensation (PDC) framework. The effectiveness of the control design framework strongly depends on the approximation property of the TP model used. Therefore, the primary aim of this paper is to investigate the approximation capabilities of dynamic TP model. It is shown that the set of functions that can be approximated arbitrarily well by TP forms with bounded number of components lies no-where dense in the set of continuous functions. This drawback necessitates the application of trade-off techniques between accuracy and complexity of TP form. Such requirements are very difficult to consider in the analytical framework, but TP model transformation offers an easy way to deal with them.