A combinatorial Hopf algebra for nonlinear output feedback control   systems
  In this work a combinatorial description is provided of a Faa di Bruno type Hopf algebra which naturally appears in the context of Fliess operators in nonlinear feedback control theory. It is a connected graded commutative and non-cocommutative Hopf algebra defined on rooted circle trees. A cancellation free forest formula for its antipode is given. 
Luis A. Duffaut Espinosa, Kurusch Ebrahimi-Fard, W. Steven Gray
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