On algebraic identification of causal functionals

AbstractWe present here a second step in solving the Algebraic Identification Problem for the causal analytic functionals in the sense of Fliess. These functionals are symbolically represented by noncommutative formal power series G=∑w∈Z★〈G|w〉w, where w is a word on a finite-encoding alphabet Z. The problem consists in computing the coefficients 〈G|w〉 from the choice of a finite set of informations on the input/output behaviour of the functional. In a previous work, we already presented a first step: we showed that one can compute the contributions of G relative to a family of noncommutative polynomials gμ with integer coefficients, indexed by the set of partitions. Hence it remains to inverse these relations by computing the words w as linear combinations of the gμ. An answer could be found in two ways: firstly by providing an identification computation tool, secondly by solving the ‘Identifiability Problem’: is the previous identification effectively computable at any order? A computational tool is here presented, in the form of a concise Maple package IDENTALG that computes the polynomials gμ by a block recursive matrix implementation, and allows then to test the identification (when possible) at any order by matrix inversion. It requires a combinatorial study of the differential monomials on the inputs. The computation of a test set covering the identification of 2048 words is presented. This package is given in the widely significant case of functionals depending on ‘a single input with drift part’. It can be used without change in case of ‘two inputs without drift’. It could be extended very easily to the case of ‘several inputs with drift part’. Finally, we discuss the Identifiability Problem: we summarize the current state of our results, and we conclude with a conjecture in a weak form and in a strong form