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Balanced Truncation of k-Positive Systems

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This paper 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 (-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.



External positivity, internal positivity, k-positivity, model order reduction, nonnegative matrix factorization, positive systems, total positivity

Journal Title

IEEE Transactions on Automatic Control

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Institute of Electrical and Electronics Engineers (IEEE)


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European Research Council (670645)
T We would like to thank the anonymous reviewers for their outstanding support. The research leading to these results was completed while the first author was a postdoctoral research associate at the University of Cambridge. The research has received funding from the European Research Council under the Advanced ERC Grant Agreement Switchlet n.670