On the algebraic identification problem: results and perspectives

As shown by M. Fliess, the input/output behaviour of the nonlinear analytical dynamical systems can be symbolically encoded in a non commutative formal power series. We get so, for the non linear dynamical systems, an equivalent of the Laplace transform. Here we are concerned by the converse process: starting from some input/output behaviour, is it possible to effectively compute its symbolic description, starting from some finite panels of input/output data?
More specifically, we present here several ways for effectively computing the terms of the generating series, taking as data the jets of some input at time $0$ and the jets at time $0$ of the corresponding output. And we restrict ourselves to the theoretic case of numerically exact data.
First we give a commutative approach, based on the representation of the output as an analytical expansion on a basis of the shuffle algebra.
Then we present a noncommutative combinatorial approach, based of the analysis of recurrence relations via Chen series and its successive derivatives. By this way, we have obtained a very compact and efficient identification software.
These results should have mainly two types of applications (not yet developed). We can use them as an help for real identification softwares. It would also allow to test real process, in order to decide if they reflect the behaviour of some unknown causal analytic functional.