Iterated derivatives of the output of a nonlinear dynamic system and Faà di Bruno formula

The Faà di Bruno formula enables us to compute the derivatives of a function of several variables. We show here, in spite of the noncommutativity of causal computations, that the Faà di Bruno grammar makes it possible to compute the derivatives of any causal functional. Using this property and some computations on derivatives of causal functionals, this paper presents the first step towards solving the problem of “Exact Algebraic Identification”. This problem consists in computing the coefficients of a noncommutative generating series when only the Taylor expansion of some inputs (at t = 0), and the Taylor expansion (at t = 0) of associated outputs are known.