Many engineering applications involve systems whose dynamics vary with time. Consider, for example, the dynamics of the wings of a plane that is airborne. The resonance modes of the wings, and the associated damping and stiffness are functions of the flight speed and height. Therefore, the dynamics vary as the plane accelerates and/or its altitude increases. This paper provides a methodology for tracking the evolving dynamics of linear, slowly time-varying systems. The responses of the systems to multisine excitations are explained and shown to provide an insight into the timevarying behavior of the system. The acquired knowledge of the responses is used to estimate non-parametrically the instantaneous frequency response function from one single experiment. A simplified case, where the variations are linear with time, is discussed in this paper. A glance of a generalization procedure to a polynomial variation is given. Results are illustrated on simulations of a lumped time-varying system.

We consider the least squares minimization problem in which both complex- and real-valued parameters are simultaneously present. A prominent example of this is the estimation of the frequency response function (FRF) in the presence of missing output data. In this case, the FRF parameters are complex-valued, while the missing output samples are real-valued. In the extended local polynomial method (ELPM), the missing samples are treated as global variables and are estimated simultaneously with the FRF parameters under the least squares criterion. This returns a mixed-valued least squares problem. We provide a formal setting, in which we show mixed-valued least squares problems are well-defined and have a unique solution. We introduce an iterative method for solving the resulting system of equations, which splits the complex-valued and real-valued least squares designs. We show that the resulting method is algebraically equivalent to applying a 2-cyclic matrix splitting to the mixed-valued normal equations, which ensures its convergence and provides an additional adaptive scheme to improve the convergence speed. Finally, we conduct a detailed case-study on the ELPM and compare our method to the original ad-hoc method. We present impressive improvements to the computation time by exploiting the separate physical structures within both regression matrices.

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.

Even-though data concatenation is a well-known technique for identifying Linear Time-Invariant models from multiple records, the study of the asymptotic properties of the estimator continues to be limited. Therefore, we investigated consistency and asymptotic normality as the number or records tend to infinity, with focus on the identification of discrete-time parametric models for single-input single-output systems operating in open loop. This paper presents the results of a consistency and asymptotic normality study based on the analysis of the prediction error cost function and Monte Carlo simulations. We show that for persistently exciting input signals (filtered white noise), model structures such as Output-Error, AR and ARX are consistently estimated, and the estimated parameters are asymptotically normally distributed. On the other hand, ARMA, ARMAX and Box–Jenkins present a bias on the estimated parameters. However, this bias asymptotically disappears for longer records

Nonparametric kernel-based modelling of dynamical systems offers important advantages over other nonparametric techniques; the estimate is a continuous function, the model complexity is continuously tuneable, and stability, causality and smoothness are imposed on the impulse response estimate. However, for lightly damped systems, most of the existing kernel-based approaches for estimating the impulse- or frequency response function fail because classical kernels are not appropriate for describing lowly damped resonances. Smoothness is imposed on the entire frequency axis with the diagonal correlated or stable spline kernel, with as a result that resonances cannot be captured well. By introducing the superposition of different kernels, carrying prior knowledge about the resonant poles of the system, we make the kernel-based modelling of lightly-damped systems possible with high-accuracy. We use a frequency domain local rational modelling technique as preprocessing step to determine the most dominant poles, and include these as prior knowledge in the kernels. The performance of the new kernel is demonstrated on a highly resonating simulated system and compared to the state of the art nonparametric frequency domain approaches.

This paper studies the linear dynamic errors-in-variables (EIV) problem in a fairly general condition where the input–output disturbing noises are colored and the input is quasi-stationary. A novel formulation of the extended frequency domain maximum likelihood (ML) estimator is developed which reduces the number of nonlinear normal equations to be solved. Sufficient conditions are provided to achieve local identifiability of the EIV model for specified noise cases of interest. The parameter estimates are calculated via a numerically stable Gauss–Newton minimization scheme started by an initial value generation strategy. Also, both the consistency and accuracy of the extended ML estimate are analyzed in detail. The performance of the proposed method is finally demonstrated on simulated dynamic systems.

Electrical Impedance Tomography (EIT) is mainly used to display information about the respiration of a patient. However, also cardiac-related signals are present, and, although they have small amplitude, they can be distinguished by their frequencies. In this contribution, we report a method based on harmonic analysis to separate respiration and perfusion. These are described by a summation of amplitude-modulated signals at different frequencies. We report the mathematical background of the method and its application on the global impedance and, finally, show how it is possible to create frequency-related images highlighting either the respiration or the perfusion inside an EIT video.

Nonlinear systems are appearing in all engineering applications. Deriving models for these systems is important for instance for prediction and control. The goal of this paper is to estimate models of a class of nonlinear systems, from experimental data. When considering slowly varying setpoints, nonlinear systems can be approximated by linear time-varying models. That is, the nonlinear system is linearised around a trajectory of setpoints. The approach followed in this paper formulates the identification problem of a nonlinear system as an exploration through the relevant range of setpoints, which are identifiable by using tools for linear time-varying systems. This approach is demonstrated on an idealised simulation example, and on a real-life robotic application.

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.

Kernel-based modelling of dynamical systems offers important advantages such as imposing stability, causality and smoothness on the estimate of the model. Here, we improve the existing frequency domain kernel-based approach for estimating the transfer function of a linear time-invariant system from noisy data. This is done by introducing prior knowledge in the kernel. We use a local rational modelling technique to determine the most significant poles, and include these poles as prior knowledge in the kernel. This results in accurate models for the identification of lightly-damped systems.

Data-driven model reference control allows for the design of a controller from input and output data when a parametric model of the system is not available. In this work we propose to use advanced nonparametric frequency response function estimation methods to aid in the model reference control task. This allows for a convenient way to extend model reference control to continuous-time systems. Moreover, we also outline a procedure that implements frequency weighing to achieve the Cramér-Rao lower bound in the case that the ideal controller is realizable and in the case that the input is also perturbed by noise. The proposed methods are used to design an analog controller for a continuous-time system.

Additives are used in metal electroreduction processes to obtain a smooth and optimal metal deposit.However, the mechanism with which these additives influence the metal deposition is poorly understood.Operando odd random phase electrochemical impedance spectroscopy (ORP-EIS) experiments have been performed to study the effect of a chloride-thiourea additive mixture on the reduction of copper ions in a sulfuric acid solution.This additive mixture is crucial in the copper electrorefining process.An optimal experimental design and a wellfounded methodology is applied to study this complex additive mixture.The ORP-EIS data is studied to analyze non-stationarities and non-linearities and the time-varying impedance is resolved when necessary.Two equivalent circuits are proposed to model the impedance data and the model parameters are linked to physical processes in the system.The use of the time-varying impedance when studying thiourea in a plating process and the use of distributed circuit elements in the EIS model is discussed.

Indirect measurements of physical parameters of interest require a mathematical model in which these parameters are estimated from the gathered measurements. Within the least squares (LS) estimation, the parameters are estimated through a regression problem. The presence of dynamics, multiple sensors, and high sampling rates leads to high-dimensional regression matrices. This paper deals with solving such large-scale regression problems time efficiently. We revisit Renaut’s least squares multisplitting (LSMS) technique aimed at solving the ordinary LS problem in parallel. The LSMS decomposes the design matrix column-wise into several blocks. The global LS solution is subsequently replaced by an equivalent set of local LS problems that are to be solved in parallel. We study how the user should configure the partition of the multisplitting. We propose a partition design based on a clustering analysis and prove the consistency of this approach. The method is illustrated with dedicated numerical simulations for a highly scalable LS-based problem within engineering: frequency response function (FRF) estimation in the presence of missing output samples. Finally, its practical utility is shown with a laboratory measurement application.

A structured errors-in-variables (EIV) problem arising in metrology is studied. The observations of a sensor response are subject to perturbation. The input estimation from the transient response leads to a structured EIV problem. Total least squares (TLS) is a typical estimation method to solve EIV problems. The TLS estimator of an EIV problem is consistent, and can be computed efficiently when the perturbations have zero mean, and are independently and identically distributed (i.i.d). If the perturbation is additionally Gaussian, the TLS solution coincides with maximum-likelihood (ML). However, the computational complexity of structured TLS and total ML prevents their real-time implementation. The least-squares (LS) estimator offers a suboptimal but simple recursive solution to structured EIV problems with correlation, but the statistical properties of the LS estimator are unknown. To know the LS estimate uncertainty in EIV problems, either structured or not, to provide confidence bounds for the estimation uncertainty, and to find the difference from the optimal solutions, the bias and variance of the LS estimates should be quantified. Expressions to predict the bias and variance of LS estimators applied to unstructured and structured EIV problems are derived. The predicted bias and variance quantify the statistical properties of the LS estimate and give an approximation of the uncertainty and the mean squared error for comparison to the Cramér–Rao lower bound of the structured EIV problem.

Simultaneous fast and accurate measurement is still a challenging and active problem in metrology. A sensor is a dynamic system that produces a transient response. For fast measurements, the unknown input needs to be estimated using the sensor transient response. When a model of the sensor exists, standard compensation filter methods can be used to estimate the input. If a model is not available, either an adaptive filter is used or a sensor model is identified before the input estimation. Recently, a signal processing method was proposed to avoid the identification stage and estimate directly the value of a step input from the sensor response. This data-driven step input estimation method requires only the order of the sensor dynamics and the sensor static gain. To validate the data-driven step input estimation method, in this article, the uncertainty of the input estimate is studied and illustrated on simulation and real-life weighing measurements. It was found that the predicted mean-squared error of the input estimate is close to an approximate Cramer-Rao lower bound for biased estimators.

Linear time-varying systems are a class of systems, the dynamics of which evolve in time. This results in a time-varying frequency response function where each frequency has a time-varying gain. In classical identification techniques, basis functions are employed to fit these time-varying gains. In this paper a new method based on Gaussian process regression is presented. The advantage of the proposed method is a more convenient model structure and model order selection.

The Best Linear Approximation (BLA) framework has already proven to be a valuable tool to analyze nonlinear systems and to start the nonlinear modeling process. The existing BLA framework is limited to systems with additive (colored) noise at the output. Such a noise framework is a simplified representation of reality. Process noise can play an important role in many real-life applications. This paper generalizes the Best Linear Approximation framework to account also for the process noise, both for the open-loop and the closed-loop setting, and shows that the most important properties of the existing BLA framework remain valid. The impact of the process noise contributions on the robust BLA estimation method is also analyzed.

In this paper, we present a novel method for the identification of the local bending stiffness of a beam.We use shearography to capture measurements of vibrating beams, so the input data for the identification is the modal slope -the differential of the modal shape.The modal slope is fitted by two Fourier-series functions, one of which is derived from a thin-beam model.The local bending stiffness is identified as the one corresponding with the best match between the measured and the two fitted modal slopes.This identification method, which we call simultaneous Fourier-series fitting, is demonstrated on numerically-generated inputs, as well as on experimental measurements.We use a flat, concave and convex beam, as well as beams with locally varying bending stiffness mimicking local damage to verify the method.It is shown that the method gives accurate results and is robust to noise.Additionally, it has advantageous properties that make it useful and practical: using this method, it is possible to perform the identification from only a sub-region of a beam and even without specifying the boundary conditions.