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In this work a combinatorial description is provided of a Faa di Bruno typeHopf algebra which naturally appears in the context of Fliess operators innonlinear feedback control theory. It is a connected graded commutative andnon-cocommutative Hopf algebra defined on rooted circle trees. A cancellationfree forest formula for its antipode is given.

A combinatorial Hopf algebra for nonlinear output feedback control systems