Lilianne Denis-Vidal

Some effective approaches to check the identifiability of uncontrolled nonlinear systems

The problem of identifiability of parameters has hardly ever been considered in the case of uncontrolled systems whereas many efficient methods have been developed for controlled systems. In this paper, we are pointing out two procedures to get global identifiability results of uncontrolled nonlinear systems. The first one derives from an algorithm proposed by Ljung and Glad. It is based on differential algebra and its complexity, due to the system size, does not increase as fast as the complexity of their algorithm. The second one is a heuristic approach. It builds a new model from various input datasets which expresses an experimental reality. Therefore, we will analyze the identifiability of this new model. Indeed, this procedure has been tested on an intricate system for which the other methods failed and it has given global identifiability results.

Equivalence and identifiability analysis of uncontrolled nonlinear dynamical systems

The problem of parameter identifiability has been considered from different points of view in the case of nonlinear dynamical systems. For analytic systems the standard approach for uncontrolled systems is the Taylor series approach (Pohjanpalo, Math. Biosciences 41 (1978) 21), or the approaches based on differential algebra for polynomial and rational systems. The similarity transformation approach, based on the local state isomorphism theorem, gives a sufficient and necessary condition for global identifiability of nonlinear controlled systems. But it leads only to a necessary condition for identifiability in the case of some uncontrolled systems. Our contribution consists in using the equivalence of systems, based on the straightening out theorem, to analyse the identifiability of uncontrolled systems. From this theory, we state the necessary or sufficient identifiability conditions, some of them depending on the state variable dimension.

Technical communique: An optimal input design procedure

This technical communique presents a method which aims at improving input design searching in some dynamical systems. The original idea of the proposed method is the combination of a dynamic programming method working on square wave inputs followed by a Quasi-Newton algorithm working on infinite differential switching mode inputs. These infinite differential functions approximate square wave inputs in order to enlarge the set of admissible inputs while ensuring a reasonable computation requirement which is not the case of classical methods based on dynamic programming only. Moreover, they correspond to practical inputs. The precise description of the approach is followed by an application in aerospace sciences.

Technical Communique: Some Remarks about an Identifiability Result of Nonlinear Systems

The aim of this paper is to give an example that shows the bad use of Corollary 2 (Vajda et al., 1989, p. 233) in various applications (Chappell et al., 1990; Chappell and Godfrey, 1992) in order to prove identifiability results in uncontrolled systems.

System Identifiability (Symbolic Computation) and Parameter Estimation (Numerical Computation)

A system identification based on physical laws often involves a parameter estimation. Before performing an estimation problem, it is necessary to investigate its identifiability. This investigation leads often to painful calculations. Generally, the numerical computation of the parameters does not use these calculus. In this contribution we propose least-squares methods to link identifiability approaches with numerical parameter estimation.

An Algorithm to Test Identifiability of Non-Linear Systems

Abstract This paper considers an approach for analyzing the identifiability of nonlinear controlled or uncontrolled dynamical systems. The method is based on the computation of the ideal containing the differential algebraic relations between the input and the output of the model, a such ideal is called input-output ideal. Our contribution consists in developing the corresponding algorithm in a symbolic computing language. This algorithm is based on differential algebra.

Brief paper: Identifiability of discrete-time nonlinear systems: The local state isomorphism approach

A new theorem is provided to test the identifiability of discrete-time systems with polynomial nonlinearities. That extends to discrete-time systems the local state isomorphism approach for continuous-time systems. Two examples are provided to illustrate the approach.

A new method for estimating derivatives based on a distribution approach

In many applications, the estimation of derivatives has to be done from noisy measured signal. In this paper, an original method based on a distribution approach is presented. Its interest is to report the derivatives on infinitely differentiable functions. Thus, the estimation of the derivatives is done only from the signal. Besides, this method gives some explicit formulae leading to fast calculus. For all these reasons, it is an efficient method in the case of noisy signals as it will be confirmed in several examples.

Aircraft Parameter Estimation: Successive Steps

Abstract System identification based on physical laws often involves parameter estimation. Before performing estimation problem, it is necessary to investigate its identifiability. A signifiant increase in accuracy of the parameter estimation may be obtained by a suitable choice of experimental conditions. This paper considers a model describing aircraft dynamics. Our contribution consists in solving the problem of optimal design. Then two successive steps are used to solve this problem. The first one is based on dynamic programming and the second one solves an optimization problem by a gradient algorithm using the results of the previous method.

Identifiability and estimation of aircraft parameters and delays by using optimal input design

This paper considers test of identifiability, optimal input design and estimation problem given by aerospace domain describing aircraft nonlinear dynamics with time delays. The original idea is to use an approximation well in line with the given system and an algebraic approach to analyze identifiability. Then the approximate model is considered to obtain the optimal input and to estimate the parameters and delays of the original model. Numerical results are given and compared.