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Some (probably the most basic) version of balanced truncation is already implemented in ControlSystems.jl. You can refer to that implementation to get started.
https://arxiv.org/abs/1907.12084
https://web.stanford.edu/group/frg/course_work/CME345/CA-CME345-Ch6.pdf
http://web.mit.edu/6.242/www/images/lec5_6242_2004.pdf
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