On the performance of antipode algorithms for the multivariable output feedback Hopf algebra

The feedback interconnection of two systems written in terms of Chen-Fliess series can be described explicitly in terms of the antipode of the output feedback Hopf algebra. At present, there are four known computational approaches to calculating this antipode. The main goal of this paper is to compare the computational performances of the three latest methods, two coproduct recursion methods and the derivation method, using a Mathematica implementation. The main conclusion is that the derivation method yields the best performance.