Balanced truncation of $k$-positive systems

This papers shows that balanced truncation of single-input-single-output systems with $k$-positive Hankel operators is totally positive for any approximation of order up to $k$. The results generalize the fact that balanced truncation of a relaxation system are known to be totally positive for any order ($k=\infty$) and extends our preliminary work on first order approximation of internally positive systems to externally (input-output) positive systems ($k=1$).