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We 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.

This paper presents the first step towards solving the problem of the exact algebraic identification of causal functionals. 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.

The Faa 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 Faa 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.

We provide the drawing of the output of dynamical system (Σ), particularly when the output is rough or near instability points. (Σ) being analytical in a neighborhood of the initial state q(0) and described by its state equations, its output y(t) in a neighborhood of t=0 is obtained by "evaluating" its generating series. Our algorithm consists in juxtaposing local approximating outputs on successive time intervals [ti,ti+1]0≤i≤n−1, to draw y(t) everywhere as far as possible. At every point ti+1 we calculate at order k an approximated value of each component qr of the state; on every interval [ti,ti+1]0≤i≤n−1 we calculate an approximated output. These computings are obtained from the symbolic expressions of the generating series of qr and y, truncated at order k, specified for t=ti and "evaluated". A Maple package is built, providing a suitable result for oscillating outputs or near instability points when a Runge-Kutta method is wrong.

A dynamical system being described by its state equations and its initial state, we develop a method for drawing its output: It is based on the juxtaposition of local approximating outputs on successive time intervals $[t_i, t_(i+1)]0

Recently, many papers have proposed automatic techniques for • extraction of knowledge from numerical data [18], [19], [20], [1]. • minimization of the number of rules [2], [3], [4]. But few works have been developed for design of experiments and datum plane covering. Most of optimization methods make the assumption that datum plane is sufficiently covered. If this assumption no longer holds, we will see that these methods may not work, since it implies that, before optimization, the fuzzy system gives acceptable results. We present in this paper, an algorithm for selection of fuzzy rules based on datum plane covering. We apply this method to two applications : • the problem of treatment of diabetics. Taking the insulin infusion rate as the input and the blood glucose rate as the output, we consider the patient as a black box [23], [22], whose model has to be obtained from the available measures of inputs-outputs. We dispose of a glycaemia file automatically produced for every person, and an insulin file shared by several persons. • the problem of detecting fraud in electronic payment systems. Along with the development of credit cards, fraudulent activities become a major problem in electronic payment systems [5], [6], [7]. Our model is based on specific and usual customer behavior, and deviation from such patterns is suspect.

In previous papers, we provided a modeling of the behavior "insulin delivery/glycaemia" of the diabetic patient under continuous insulin infusion, continuous glucose monitoring and we provided a method of regulation of his glycaemia. This behavioral model is bilinear, predicting the behavior on an interval of 15 minutes, with an average error of 15%. And consequently, the model is adjusted for every 15 minutes. The aim of this paper is to study the Bounded-Input-Bounded-Output (BIBO) stability of the bilinear model in order to point out that the patient is entering in a period of stable/unstable equilibrium. In case of stable equilibrium, the prediction will be valid for a longer time interval, when in case of unstable equilibrium, it will leads one to reduce the time intervals and to pilot closely the variations of insulin delivery. The BIBO stability is studied by computing the generating series G of the model. This series, generalization of the transfer function, presents an useful tool for analyzing the stability of bilinear systems. It is a rational power series in noncommutative variables and by evaluating it, a formal expression of the output in form of iterated integrals is provided. Three cases arise: firstly, the output can be explicitly computed; secondly, the output can be bounded/unbounded if the input is bounded; thirdly, no conclusion seems available about the BIBO stability by using G. In this case, we propose a stabilizing constant input eta by studying the univariate series Geta.

We present and study, in this paper, a Tagaki-Sugeno(TS) fuzzy model that consists in a family of linear models mixed together with nonlinear membership functions. We apply this method to the problem of treatment of diabetics. Taking the insulin infusion rate as the input and the blood glucose rate as the output, we consider the patient as a black box [12], [11], whose model has to be obtained from the available measures of inputs-outputs. We dispose of a glycaemia file automatically produced for every person, and an insulin file shared by several persons. We investigate the model's quality on two criteria: a convergence measure and the impact of datum plane covering on the outcome of a fuzzy inference system. We emphasize on the importance of datum plane covering. Many papers propose fuzzy algorithms for extraction of knowledge from numerical data [7], [8], [9]. But few works have been developed for design of experiments and datum plane covering. We propose a measure used to pre-validate a fuzzy model. This pre-validation takes place after design of the inference system. So, when the model is not pre-validated, we do not have to carry out the next steps, optimisation and validation.

We present an algorithm for the identification of any unknown dynamical system and a partial validation of the produced model. The method consists in computing a dynamical system (B k ) whose behaviour approximates the exact system one’s, “up to order k”, when knowing only a finite set of input/output data. The approximated dynamical system provided is a bilinear system depending on some parameters. In order to preserve the generic feature of the computation, the algorithm produces a pair (bilinear system B, constraint C on parameters), where B is produced under constraint C. A computational tool is provided, in the form of Maple package DYNAMICMODEL. The model validation is a central problem in system identification [2]. It consists generally in a test that falsifies or not falsifies the model, using a validation data set. For this continuous and symbolic model, we propose an overestimation of the mistake resulting from the systems (B k ) at order k and we show the convergence towards the exact output system. We propose also a majoration, for a certain class (DP), of the gap between two consecutive outputs.

Etant donnee une serie formelle en variables non commutatives sur un alphabet fini X, a coefficients dans un corps K, on propose une construction de serie rationnelle approximante au sens de la valuation X-adique, et qui soit de rang minimal

Given an analytic system, we compute a bilinear system of minimal dimension which approximates it up to order k (i.e. the outputs of these two systems have the same Taylor expansion up to order k).