This article considers balanced truncation of discrete-time Hankel k-positive systems, characterized by Hankel matrices whose minors up to order kare nonnegative. Our main result shows that if the truncated system has order kor less, then it is Hankel totally positive (positive), meaning that it is a sum of first-order lags. This result can be understood as a bridge between two known results: the property that the first-order truncation of a positive system is positive (k=1), and the property that balanced truncation preserves state-space symmetry. It provides a broad class of systems where balanced truncation is guaranteed to result in a minimal internally positive system.
