Abstract : System identification is an established field in the area of system analysis and control. It aims at determining mathematical models for dynamical systems based on measured data. Most of the techniques that have been developed assume the input signal to be perfectly known. However, there are cases when the input signal is measured, and thus also noise-corrupted. This situation, where the input signal and the output signal are both affected by noises -- named `errors-in-variables' (EIV) model identification -- is considered in the thesis.
In the introduction chapter, the use of EIV models is motivated. The problem considered is then formally stated, before underlining some of its inherent difficulties.
The second chapter deals with the identification of discrete-time EIV models, and is itself divided into two parts. The first part is about methods based on second-order statistics. In a first step, the main existing methods are recalled. A unified presentation of the various methods that aim at compensating the bias of the least squares estimate is then given. Afterwards, instrumental variable estimators are studied and a few estimators are proposed. The second part of the chapter deals with methods based on higher-order statistics. After a summary of the available methods, the least squares and iterative least squares methods are introduced, using the fact that the equation of the model is verified by the cumulants. The chapter ends with the computation of the expression of the asymptotic covariance matrix of the least squares estimate based on the third-order cumulants, that has been proposed earlier.
The third chapter is dedicated to the identification of continuous-time EIV models. While the identification of discrete-time errors-in-variables models has been extensively studied, continuous-time EIV model identification is still in its infancy. The interest of the direct identification of continuous-time models is first underlined. A review of the few available methods is given, and then estimators using the third- and fourth-order cumulants are proposed. In particular, since no structural hypothesis on the noises is required, they allow to handle the general case of coloured (and even mutually correlated) noises on input and output of the system.