Vincent Laurain

Refined instrumental variable methods for identification of LPV Box-Jenkins models

The identification of linear parameter-varying systems in an input-output setting is investigated, focusing on the case when the noise part of the data generating system is an additive colored noise. In the Box-Jenkins and output-error cases, it is shown that the currently available linear regression and instrumental variable methods from the literature are far from being optimal in terms of bias and variance of the estimates. To overcome the underlying problems, a refined instrumental variable method is introduced. The proposed approach is compared to the existing methods via a representative simulation example.

Instrumental variable scheme for closed-loop LPV model identification

Identification of real-world systems is often applied in closed loop due to stability, performance or safety constraints. However, when considering Linear Parameter-Varying (LPV) systems, closed-loop identification is not well-established despite the recent advances in prediction error approaches. Building on the available results, the paper proposes the closed-loop generalization of a recently introduced instrumental variable scheme for the identification of LPV-IO models with a Box-Jenkins type of noise model structures. Estimation under closed-loop conditions with the proposed approach is analyzed from the stochastic point of view and the performance of the method is demonstrated through a representative simulation example.

Refined instrumental variable methods for identification of Hammerstein continuous-time Box-Jenkins models

This article presents instrumental variable methods for direct continuous-time estimation of a Hammerstein model. The non-linear function is a sum of known basis functions and the linear part is a Box-Jenkins model. Although the presented algorithm is not statistically optimal, this paper further shows the performance of the presented algorithms and the advantages of continuous-time estimation on relevant simulations.

Model structure learning: A support vector machine approach for LPV linear-regression models

Accurate parametric identification of Linear Parameter-Varying (LPV) systems requires an optimal prior selection of a set of functional dependencies for the parametrization of the model coefficients. Inaccurate selection leads to structural bias while over-parametrization results in a variance increase of the estimates. This corresponds to the classical bias-variance trade-off, but with a significantly larger degree of freedom and sensitivity in the LPV case. Hence, it is attractive to estimate the underlying model structure of LPV systems based on measured data, i.e., to learn the underlying dependencies of the model coefficients together with model orders etc. In this paper a Least-Squares Support Vector Machine (LS-SVM) approach is introduced which is capable of reconstructing the dependency structure for linear regression based LPV models even in case of rational dynamic dependency. The properties of the approach are analyzed in the prediction error setting and its performance is evaluated on representative examples.

The CONTSID toolbox for Matlab: extensions and latest developments

This paper describes the latest developments for the CONtinuous-Time System IDentification (CONTSID) toolbox to be run with Matlab which includes time-domain identification methods for estimating continuous-time models directly from sampled data. The main additions to the new version aim at extending the available methods to handle wider practical situations in order to enhance the application field of the CONTSID toolbox. The toolbox now includes routines to solve errors-in-variables and closed-loop identification problems, as well as non-linear continuous-time model identification techniques.

An instrumental least squares support vector machine for nonlinear system identification

Least-Squares Support Vector Machines (LS-SVMs), originating from Stochastic Learning theory, represent a promising approach to identify nonlinear systems via nonparametric es- timation of nonlinearities in a computationally and stochastically attractive way. However, application of LS-SVMs in the identification context is formulated as a linear regression aim- ing at the minimization of the l2 loss in terms of the prediction error. This formulation corresponds to a prejudice of an auto-regressive noise structure, which, especially in the non- linear context, is often found to be too restrictive in practical applications. In [1], a novel Instrumental Variable (IV) based estimation is integrated into the LS-SVM approach provid- ing, under minor conditions, a consistent identification of nonlinear systems in case of a noise modeling error. It is shown how the cost function of the LS-SVM is modified to achieve an IV-based solution. In this technical report, a detailed derivation of the results presented in Section 5.2 of [1] is given as a supplement material for interested readers.

LPV system identification under noise corrupted scheduling and output signal observations

Most of the approaches available in the literature for the identification of Linear Parameter-Varying (LPV) systems rely on the assumption that only the measurements of the output signal are corrupted by the noise, while the observations of the scheduling variable are considered to be noise free. However, in practice, this turns out to be an unrealistic assumption in most of the cases, as the scheduling variable is often related to a measured signal and, thus, it is inherently affected by a measurement noise. In this paper, it is shown that neglecting the noise on the scheduling signal, which corresponds to an error-in-variables problem, can lead to a significant bias on the estimated parameters. Consequently, in order to overcome this corruptive phenomenon affecting practical use of data-driven LPV modeling, we present an identification scheme to compute a consistent estimate of LPV Input/Output (IO) models from noisy output and scheduling signal observations. A simulation example is provided to prove the effectiveness of the proposed methodology.

Nonparametric identification of LPV models under general noise conditions : an LS-SVM based approach

Abstract Parametric identification approaches in the Linear Parameter-Varying (LPV) setting require optimal prior selection of a set of functional dependencies, used in the parametrization of the model coefficients, to provide accurate model estimates of the underlying system. Consequently, data-driven estimation of these functional dependencies has a paramount importance, especially when very limited a priori knowledge is available. Existing overparametrization and nonparametric methods dedicated to nonlinear estimation offer interesting starting points for this problem, but need reformulation to be applied in the LPV setting. Moreover, most of these approaches are developed under quite restrictive auto-regressive noise assumptions. In this paper, a nonparametric Least-Squares Support Vector Machine (LS-SVM) approach is extended for the identification of LPV polynomial models. The efficiency of the approach in the considered noise setting is shown, but the drawback of the auto-regressive noise assumption is also demonstrated by a challenging LPV identification example. To preserve the attractive properties of the approach, but to overcome the drawbacks in the estimation of polynomial LPV models in a general noise setting, a recently developed Instrumental Variable (IV)-based extension of the LS-SVM method is applied. The performance of the introduced IV and the original LS-SVM approaches are compared in an identification study of an LPV system with unknown noise dynamics.

Refined instrumental variable methods for identifying hammerstein models operating in closed loop

This article presents an instrumental variable method dedicated to non-linear Hammerstein systems operating in closed loop. The linear process is a Box-Jenkins model and the non-linear part is a sum of known basis functions. The performance of the proposed algorithm is illustrated by a numerical example.

Introducing instrumental variables in the LS-SVM based identification framework

Least-Squares Support Vector Machines (LS-SVM) represent a promising approach to identify nonlinear systems via nonparametric estimation of the nonlinearities in a computationally and stochastically attractive way. All the methods dedicated to the solution of this problem rely on the minimization of a squared-error criterion. In the identification literature, an instrumental variable based optimization criterion was introduced in order to cope with estimation bias in case of a noise modeling error. This principle has never been used in the LS-SVM context so far. Consequently, an instrumental variable scheme is introduced into the LS-SVM regression structure, which not only preserves the computationally attractive feature of the original approach, but also provides unbiased estimates under general noise model structures. The effectiveness of the proposed scheme is demonstrated by a representative example.