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In this paper, the Polynomial NonLinear State Space (PNLSS) approach is applied to model a nonlinear system with a Wiener-Hammerstein structure. To obtain good initial estimates, the best linear approximation of the system under test is first identified. Next, this linear model is extended to a polynomial nonlinear state space model to capture also the system's nonlinear behavior. The identification procedure is applied to measurement data

This paper describes the identification of Wiener-Hammerstein models and two recently suggested algorithms are applied to the SYSID'09 benchmark data. The most difficult step in the identification process of such block-oriented models is to generate good initial values for the linear dynamic blocks so that local minima are avoided. Both of the considered algorithms obtain good initial estimates by using the best linear approximation (BLA) which can easily be estimated from data. Given the BLA, the two algorithms differ in the way the dynamics are separated into two linear parts. The first algorithm simply considers all possible splits of the dynamics. Each of the splits is used to initialize one Wiener-Hammerstein model using linear least-squares and the best performing model is selected. In the second algorithm, both linear blocks are initialized with the entire BLA model using basis function expansions of the poles and zeros of the BLA. This gives over-parameterized linear blocks and their order is decreased in a model reduction step. Both algorithms are explained and their properties are discussed. They both give good, comparable models on the benchmark data.

In this paper, we focus on the nonlinear modelling of Wiener-Hammerstein structures. The most difficult step in the identification process of such block-oriented models is to generate good initial values for the linear dynamic blocks. Here, good initial estimates are obtained by using the Best Linear Approximation of the system under test, which can easily be extracted from the data. The idea is to write the linear dynamic blocks as a linear combination of basis functions containing the poles and the zeros of the Best Linear Approximation. Next, these initial values are further tuned via a nonlinear optimization procedure. The proposed identification method is applied to real measurement data.

Dynamical systems describing a physical process with a dominant diffusion phenomenon require a large dimensional model due to their long memory. Without prior knowledge, it is however not straightforward to know if/whether one deals with a fractional order system or long memory effects. Since the parametric modeling of a fractional system is very involved, we tackle the question whether fractional insight can be gathered in a non-parametric way. In this paper we show that the classical Fourier basis leading to the Frequency Response Function (FRF) lacks fractional insight. Therefore, we introduce a TaylorFourier basis to obtain non-parametric insight in the fractional system. This analysis proposes a novel type of spectrum to visualize the spectral content of a fractional system: the Taylor-Fourier spectrum.

Engineers find interpreting plots of a measured physical variable more straightforward than doing a formal statistical analysis. The default choice to display the data behavior is the histogram. The histogram’s performance has proved to be sufficient. However, histograms have a number of limitations including sensitivity to the binwidth and a non-physical roughness. Over the past years, statisticians have developed different techniques to address these problems. These techniques provide a much clearer visualization of the probability density and a more accurate estimation of the statistical properties of the measured data. Despite their increasing use in other fields, these techniques are rarely used in the measurement community. For instance, most measurement instruments provide histograms only. This review article revisits these techniques from an engineer viewpoint to encourage its use. Different examples that include known and unknown densities result in practical guidelines that help the measurement engineer to visualize the probability content.

When modeling a linear system in a parametric way, one needs to deal with (i) model structure selection, (ii) model order selection as well as (iii) an accurate fit of the model. The most popular model structure for linear systems has a rational form which reveals crucial physical information and insight due to the accessibility of poles and zeros. In the model order selection step, one needs to specify the number of poles and zeros in the model. Automated model order selectors like Akaike@?s Information Criterion (AIC) and the Minimum Description Length (MDL) are popular choices. A large model order in combination with poles and zeros lying closer to each other in frequency than the frequency resolution indicates that the modeled system exhibits some fractional behavior. Classical integer order techniques cannot handle this fractional behavior due to the fact that the poles and zeros are lying to close to each other to be resolvable and not enough data is available for the classical integer order identification procedure. In this paper, we study the use of fractional order poles and zeros and introduce a fully automated algorithm which (i) estimates a large integer order model, (ii) detects the fractional behavior, and (iii) identifies a fractional order system.

In measurement science mathematical models are often used as an indirect measurement of physical properties which are mapped to measurands through the mathematical model. Dynamical systems describing a physical process with a dominant diffusion or dispersion phenomenon requires a large dimensional model due to its long memory. Ignoring a dominant difussion or dispersion component acts as a confounder which may introduce a bias in the estimated quantities of interest. For linear systems it has been observed that fractional order models outperform classical rational forms in terms of the number of parameters for the same fitting error. However it is not straightforward to deal with a fractional order system or long memory effects without prior knowledge. Since the parametric modeling of a fractional system is very involved, we put forward the question whether fractional insight can be gathered in a non-parametric way. In this paper we show that classical Fourier basis leading to the frequency response function lacks fractional insight. To circumvent this problem, we introduce a fractional Taylor–Fourier basis to obtain non-parametric insight in the fractional system. This analysis proposes a novel type of spectrum to visualize the spectral content of a fractional system: Taylor–Fourier spectrum. This spectrum is fully measurement driven which can be used as a first to explore the fractional dynamics of a measured diffusion or dispersion system.

In this paper, the Polynomial NonLinear State Space (PNLSS) approach is applied to model a nonlinear system with a Wiener–Hammerstein structure. To obtain good initial estimates, the best linear approximation of the system under test is first identified. Next, this linear model is extended to a polynomial nonlinear state space model to capture also the system's nonlinear behavior. The identification procedure is applied to measurement data.

In this paper, a method is proposed to initialize the linear dynamic blocks of a Wiener-Hammerstein model. The idea is to build these blocks from the poles and the zeros of the best linear approximation of the system under test, which can easily be extracted from the data. This approach results in an easy to solve problem (linear-in-the-parameters) from which initial estimates for the linear dynamics can be obtained. The proposed method is applied to simulation data from a Wiener-Hammerstein system.

Analyzing functional MRI data is often a hard task due to the fact that these periodic signals are strongly disturbed with noise. In many cases, the signals are buried under the noise and not visible, such that detection is quite impossible. However, it is well known that the amplitude measurements of such disturbed signals follow a Rice distribution which is characterized by two parameters. In this paper, an alternative Bayesian approach is proposed to tackle this two-parameter estimation problem. By incorporating prior knowledge into a mathematical framework, the drawbacks of the existing methods (i.e. the maximum likelihood approach and the method of moments) can be overcome. The performance of the proposed Bayesian estimator is analyzed theoretically and illustrated through simulations. Finally, the developed approach is successfully applied to measurement data for the analysis of functional MRI.

In this paper, we propose a method to model nonlinear systems using polynomial nonlinear state space equations. Obtaining good initial estimates is a major problem in nonlinear modelling. It is solved here by identifying first the best linear approximation of the system under test. The proposed identification procedure is successfully applied to measurements of two physical systems.