Proof of relation between singular directions and external flows in distillation

It has been observed that inputs which affect the external flows of a distillation column excite the high-gain direction of the product compositions, while inputs that do not affect these flows excite the low-gain direction of the compositions. This conjecture is proved in the paper. The proof is based on bounds, derived in the paper, on the input singular directions of 2×2 systems with relative gains outside the interval (0,1), and general properties of distillation columns that can be derived from first principles. Simple low-order distillation models that depend on the proved property have been suggested in the literature. The result is also useful for control-relevant identification of distillation columns