R. Pintelon

Equilibria evolving over time, also called trends, are often present in ongoing measurements of real-life systems. These trends are considered as disturbances when computing the best linear time-varying approximation (BLTVA) of the system. Current techniques for dealing with trends in BLTVA measurements consist of modeling the trend with a finite number of basis functions. However, in measurements with dominant trends, the trend cannot always be captured well enough by this set of basis functions, and hence, the uncertainty on the BLTVA increases. As a consequence, one loses low-frequency information. In this article, the state-of-the-art method for estimating the BLTVA is extended by removing the trend with a differencing operator. It is shown that with this novel technique, low-frequency information becomes more visible. Moreover, the novel method decreases the variance on the BLTVA and allows to measure fewer periods. Hence, the novel technique improves the route for treating arbitrary out-of-equilibrium or also called operando, measurements. As an illustration, it is applied to operando time-varying impedance measurements of three electrochemical processes: the charging of a Li-ion battery cell, the electrorefining of copper, and the anodizing of aluminum.

This article presents a method for estimating a linear time-varying approximation of a general class of nonlinear time-varying (NLTV) systems. It starts from noisy measurements of the response of the NLTV system to a special class of periodic excitation signals. These measurements are subject to measurement noise, process noise, and a trend. The proposed method is a two-step procedure. First, the disturbing noise variance is quantified. Next, using this knowledge, the linear time-varying dynamics are estimated together with the NLTV distortions. The latter are split into even and odd contributions. As a result, the signal-to-nonlinear-distortion ratio is quantified. It allows one to decide whether or not a linear approximation is justifiable for the application at hand. The two-step algorithm is fully automatic in the sense that the user only has to choose upper bounds on the number of basis functions used for modeling the response signal. The obtained linear time-varying approximation is the best in the sense that the difference between the actual nonlinear response and the response predicted by the linear approximation is uncorrelated with the input. Therefore, it is called the best linear time-varying approximation (BLTVA). Finally, the theory is validated on a simulation example and illustrated on two measurement examples: the crystallographic pitting corrosion of aluminum and copper electrorefining.

The frequency response function (FRF) is a fundamental property of a linear time-invariant system that describes the system's frequency-dependent behavior. Accurately measuring the FRF of multivariable lightly damped systems in a time-efficient manner is difficult due to spectral leakage and long-lasting transient phenomena. In the past, local rational modeling techniques have been introduced that mitigate these difficulties. This article presents an overview of the existing techniques for multivariable systems and proposes a new local rational modeling technique that is both consistent and stochastically efficient in measuring the nonparametric FRF. The proposed technique is successfully validated on simulation examples of a generic multivariable lightly damped system. Its practical applicability is illustrated by characterizing the FRF of an aluminum plate.

In many engineering applications, the level of nonlinear distortions in frequency response function (FRF) measurements is quantified using specially designed periodic excitation signals called random phase multisines and periodic noise. The technique is based on the concept of the best linear approximation (BLA), and it allows one to check the validity of the linear framework with a simple experiment. Although the classical BLA theory can handle measurement noise only, in most applications, the noise generated by the system-called process noise-is the dominant noise source. Therefore, there is a need to extend the existing BLA theory to the process noise case. In this article, we study in detail the impact of the process noise on the BLA of nonlinear continuous-time systems operating in a closed loop. It is shown that the existing nonparametric estimation methods for detecting and quantifying the level of nonlinear distortions in FRF measurements are still applicable in the presence of process noise. All results are also valid for discrete-time systems and systems operating in an open loop.

A finite measurement time is at the origin of transient (sometimes called leakage) errors in nonparametric frequency response function (FRF) estimation. If the FRF varies significantly over the frequency resolution of the experiment (= reciprocal of the measurement time), then these transients (leakage) errors cause important bias and variance errors in the FRF estimate. To decrease these errors, several local modeling techniques have been proposed in the literature. This article presents an overview of the existing methods and gives an in-depth bias and variance analysis of the FRF and disturbing noise variance estimates. In addition, a new local modeling approach is described, which combines the small bias error of the local rational approximation with the low noise sensitivity of the local polynomial approximation. It is based on an automatic local model-order selection procedure applied to a specific subclass of rational functions.

The class of nonlinear time-varying (NLTV) systems includes all possible systems and, hence, is difficult to identify. Still, when the nonlinearities are not too strong then, depending on the application, a linear model might be sufficient for approximating the true response. To quantify the approximation error of the linear model, detecting and quantifying the nonlinear behavior is of key importance. In this paper we propose a fully automated procedure for detecting, classifying and quantifying the nonlinear distortions in the response, possibly subject to a trend, of a specific class of NLTV systems to odd random phase multisine excitations. The result is a measurement of the time-varying frequency response function together with uncertainty bounds due to noise and nonlinear distortions. The user only has to specify four integer numbers: an upper bound on (i) the degree on the time-domain polynomial modelling of the trend, (ii) the degree of the frequency-domain polynomial basis function and (iii) the number of frequency-domain hyperbolic-like basis functions, all used for modeling the output spectrum; and (iv) a quality measure – called degrees-of-freedom – of the noise variance estimate. Guidelines are provided for obtaining reasonable values for these upper bounds.

In many engineering applications the level of nonlinear distortions in frequency response function (FRF) measurements is quantified using specially designed periodic excitation signals called random phase multisines and periodic noise. The technique is based on the concept of the best linear approximation (BLA) and it allows one to check the validity of the linear framework with a simple experiment. Although the classical BLA theory can handle measurement noise only, in most applications the noise generated by the system -- called process noise -- is the dominant noise source. Therefore, there is a need to extend the existing BLA theory to the process noise case. In this paper we study in detail the impact of the process noise on the BLA of nonlinear continuous-time systems operating in a closed loop. It is shown that the existing nonparametric estimation methods for detecting and quantifying the level of nonlinear distortions in FRF measurements are still applicable in the presence of process noise. All results are also valid for discrete-time systems and systems operating in open loop.