Scaling down the plant condition number scales up the size of uncertainty

It is shown for a special case that design limitations due to an ill-conditioned plant are invariant under scaling even though the size of the condition number is not. The argument is as follows. The standard design problem utilized to study ill-conditioned plants is a robust performance problem. One cannot study robust performance without descriptions of the modeling uncertainty and of the performance goals. These descriptions must necessarily be stated in the same set of units as those used to describe the plant. Changing the plant description by changing units also changes the description of the uncertainty and performance goals. A careful analysis reveals that the robustness difficulty therefore remains the same