Balanced Truncation of $k$-Positive Systems
Christian Grußler, Tobias Damm, Rodolphe J. Sepulchre · IEEE Transactions on Automatic Control · 2021
This article considers balanced truncation of discrete-time Hankel$k$-positive systems, characterized by Hankel matrices whose minors up to order$k$are nonnegative. Our main result shows that if the truncated system has order$k$or less, then it is Hankel totally positive ($\infty$-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.