Laurent Vanbeylen

Blind Maximum-Likelihood Identification of Wiener Systems

This paper is about the identification of discrete-time Wiener systems from output measurements only (blind identification). Assuming that the unobserved input is white Gaussian noise, that the static nonlinearity is invertible, and that the output is observed without errors, a Gaussian maximum-likelihood estimator is constructed. Its asymptotic properties are analyzed and the Cramer-Rao lower bound is calculated. A two-step procedure for generating high-quality initial estimates is presented as well. The paper includes the illustration of the method on a simulation example.

Nonlinear LFR Block-Oriented Model: Potential Benefits and Improved, User-Friendly Identification Method

Nowadays, there is a high need for accurate, parsimonious nonlinear dynamic models. Block-oriented nonlinear model structures are known to be excellent candidates for this task. The nonlinear linear fractional representation model, composed of a static nonlinearity (SNL) and a multiple-input-multiple-output (MIMO) linear time-invariant (LTI) part, is highly flexible since it creates an arbitrary MIMO-LTI interconnection between the models input and output and the SNLs input and output. First of all, it can cope with the nonlinear feedback (which is very important in oscillators and mechanical applications). Secondly, it incorporates certain classical block-oriented models as special cases. Finally, it does not postulate the SNLs location prior to the identification. Starting from two classical frequency response measurements of the system, the method generates the best possible MIMO-LTI configuration and estimates the SNL in an automated, user-friendly, and efficient (noniterative) way. The method will be illustrated on simulation examples and experimental data.

Identification of a block-structured model with several sources of nonlinearity

This paper focuses on a state-space based approach for the identification of a rather general nonlinear block-structured model. The model has several Single-Input Single-Output (SISO) static polynomial nonlinearities connected to a Multiple-Input Multiple-Output (MIMO) dynamic part. The presented method is an extension and improvement of prior work, where at most two nonlinearities could be identified. The location of the nonlinearities or their relation to other parts of the model does not have to be known beforehand: the method is a black-box approach, in which no states, internal signals or structural properties need to be measured or known. The first step is to estimate a partly structured polynomial (nonlinear) state-space model from input-output measurements. Secondly, an algebraic approach is used to split the dynamics and the nonlinearities by decomposing the multivariate polynomial coefficients.

Application of Blind Identification to Nonlinear Calibration

This paper handles the identification of nonlinear discrete-time Wiener systems from output measurements only (blind identification). Assuming that the unobserved input is white Gaussian noise, that the static nonlinearity is invertible, and that the output is observed without errors, a Gaussian maximum likelihood estimator is constructed. A two-step procedure for generating high quality starting values is presented as well. Finally, the proposed scheme is applied to both a simulation example and a laboratory experiment, illustrating the potential usefulness of the method for nonlinear calibration applications.

Identification of nonlinear LFR systems with two nonlinearities

When identifying a system (e.g. mechanical, electrical or chemical) based on inand output measurements and without physical knowledge, an engineer faces many choices. First of all, there exist standard linear models, but when those do not sufficiently well describe the data, nonlinear models should be considered. There are many kinds of nonlinear models and it is often hard to choose among them. Most likely, the engineer will prefer a simple model (containing as few parameters as possible), which is yet flexible enough to describe the data. This paper presents an identification method that results in a block-structured model. The block-structure consists of a linear dynamic part and two (single-input single-output) static nonlinearities. Because of this structure, the model complexity remains reasonable, whereas the structure is flexible enough to describe any system with two static nonlinearities (including Hammerstein-Wiener, Wiener-Hammerstein, nonlinear feedback etc.).

Identification of a Block-Structured Model with Localised Nonlinearity

Abstract This paper considers the identification of a rather general nonlinear time-invariant system, consisting of a Multiple-Input Multiple-Output (MIMO) linear dynamic part and one static nonlinear part. It is sometimes referred to as Linear Fractional Transformation (LFT) or Linear Fractional Representation (LFR). The structure will be called nonlinear LFR and includes many standard block-structured models, such as Wiener, Hammerstein, Wiener-Hammerstein and nonlinear feedback. The identification does assume neither the states, nor the internal signals over the static nonlinearity to be measured. The static nonlinearity (SNL) is assumed to be polynomial. After estimation of a nonlinear state-space model with certain structural properties, the SNL can be separated from the MIMO linear part. Next, the linear system is represented by a combination of four linear dynamic blocks, yielding extra insight. The method is illustrated via an experimental-data example.

Nonlinear Induced Variance of Frequency Response Function Measurements

This paper analyzes the variance of the estimated frequency response function (FRF) Ĝ_BLA of the best linear approximation Ĝ_BLA to a nonlinear system that is driven by random excitations. Ĝ_BLA varies not only due to the disturbing measurement and process noise but also over different realizations of the random excitation because the nonlinear distortions depend on the input realization. It will be shown that the variance expression σ_Ĝ_BLA² that is obtained in the linear framework can also be used to calculate the variance that is induced by the nonlinear distortions. This validates the common engineering practice, where the linear FRF methodology is often used under nonlinear conditions.

Comparison of some initialisation methods for the identification of nonlinear state-space models

In many measurement applications, highly accurate models are needed. Linear models can fail to represent the measured phenomena in a satisfactory way. In the class of nonlinear dynamic models, a nonlinear state-space model is an attractive option due to its good modelling capabilities. In order to obtain optimal results with this model, good initial estimates for the nonconvex optimisation problem are crucial. We show that the classical approach (via the best linear approximation) can get trapped in local minima and we present some alternative, recently developed, initialisation methods (nonlinear models). A simulation example is used to compare their performance, and the results (including time-efficiency and flexibility) are discussed.

Improved, user-friendly initialization for the identification of the nonlinear LFR block-oriented model

Nowadays, there is a high need for accurate, parsimonious nonlinear dynamic models. Block-oriented nonlinear model structures are known to be excellent candidates for this task. The nonlinear LFR (Linear Fractional Representation) model, composed of a static nonlinearity (SNL) and a multiple-input-multiple-output (MIMO) linear time-invariant (LTI) part, is highly flexible since it creates an arbitrary MIMO-LTI interconnection between the models inand output and the SNLs in- and output. It can create nonlinear feedback (which is very important in oscillators and mechanical applications), incorporates e.g. the Wiener-Hammerstein model as a special case and does not postulate the SNLs location prior to the identification. Starting from 2 classical frequency response measurements of the system, the method generates the best possible MIMO-LTI configuration and estimates the SNL in an automated, user-friendly, and efficient way. The resulting model parameters are fine-tuned via a subsequent optimization. The method will be illustrated via simulation experiments.

Measuring the stability of nonlinear feedback systems

This paper is concerned with measuring the stability or instability of nonlinear feedback systems. For such kinds of systems, it may happen that the stable or unstable behaviour depends on the nature of the (stochastic) input signal. E.g. the system can show a stable behaviour at low amplitudes, and an unstable behaviour at high amplitudes. In between there can be a transition zone between low and high probabilities of instability. Although the measurement of instability phenomena is relevant for the instrumentation and measurement society, it has not been handled inside this community. This paper partly fills this gap, by introducing a stability definition, and proposing a method that makes a statistical statement about the stable or unstable operation condition of the nonlinear feedback system, starting from input-output measurements.