Nonparametric Frequency Response Function Research Articles

Estimating frequency response functions by using transient responses of systems to simple excitations

This study develops an efficient method using input–output data on estimating non-parametric and then parametric frequency response functions (FRFs) associated with a dynamic system. Contrary to most existing FRF estimation methods that have been theoretically based on the steady-state or stationary responses, the proposed method uses the transient responses that are induced by simple, such as sinusoidal or two-exponential, excitations instead. Its numerical procedure for obtaining a non-parametric FRF is nothing more than solving a set of linear equations in which the unknown variables are individual components of the sought FRF. Furthermore, from the non-parametric FRF, its corresponding poles and residues are extracted through a procedure that involves a usage of the inverse discrete Fourier transform and the Prony-SS method; afterwards, a partial fraction FRF in terms of poles and residues can be achieved. Three numerical examples, including computer simulations and lab experiment, are provided to demonstrate the superior performance of the developed method.

Frequency Response Function-Based Learning Control: Analysis and Design for Finite-Time Convergence

Iterative learning control (ILC) enables substantial performance improvement by using past operational data in combination with approximate plant models. The aim of this article is to develop an ILC framework based on nonparametric frequency response function (FRF) models that requires very limited modeling effort. These FRF models describe the behavior of a system in periodic steady state, yet are employed for the control of arbitrary finite-length tasks. A detailed analysis and design framework is developed to construct noncausal learning filters directly from uncertain FRF models, that achieve ILC convergence for arbitrary tasks. The resulting framework provides a unification between ILC and iterative inversion-based control, where the latter is a learning method specifically developed for periodic tasks.

A Wavelet-Based Approach to FRF Identification From Incomplete Data

Frequency Response Function (FRF) estimation from measured data is an essential step in the design, the control, and the analysis of complex dynamical systems, including thermal and motion systems. Especially for systems that require long measurement time, missing samples in the data record, e.g., due to measurement interruptions, often occur. The aim of this paper is to achieve accurate identification of non-parametric FRF models of periodically excited systems from noisy output measurements with missing samples. An identification framework is established that exploits a wavelet-based transform to separate the effect of the missing samples in the time domain from the system characteristics in the frequency domain. The framework encompasses both a time-invariant and a time-varying wavelet-based estimator, which provide different mechanisms to address the missing samples. Experimental results from a thermo-dynamical system confirm that the estimators enable accurate identification.

A frequency domain approach for local module identification in dynamic networks

In classical approaches of dynamic network identification, in order to identify a system (module) embedded in a dynamic network, one has to formulate a Multi-input-Single-output (MISO) identification problem that requires identification of a parametric model for all the modules constituting the MISO setup including (possibly) the noise model, and determine their model order. This requirement leads to model order selection steps for modules that are of no interest to the experimenter which increases the computational complexity for large-sized networks. Also, identification using a parametric noise model (like BJ method) can suffer from local minima, however neglecting the noise model has its impact on the variance of the estimates. In this paper, we provide a two-step identification approach to avoid these issues such that the complexity of the problem is independent of the complexity of the network. The first step involves performing a non-parametric indirect approach for a MISO identification problem to get the non-parametric frequency response function (FRF) estimates and its variance as a function of frequency. In the second step, the estimated non-parametric FRF of the target module is smoothed using a parametric frequency domain estimator with the estimated variance from the previous step as the non-parametric noise model. The developed approach is practical with weak assumptions on noise, uses the available toolbox, requires a parametric model only for the target module of interest, and uses a non-parametric noise model to reduce the variance of the estimates. Numerical simulations illustrate the potentials of the introduced method in comparison with the classical identification methods.

First Results on Modelling of a Plate Heat Exchanger of a District Heating System

This paper describes the first results of the modelling of a plate heat exchanger which is installed in the VUB district heating network. A properly controlled energy exchange through the heat exchanger improves the economy of the entire system, minimizes the pollutant emissions and fossil fuel consumption, in accordance with the main goals of the European energy plan till 2030. To achieve these goals, an accurate model of the plate heat exchanger is required. In this article two approaches are considered to accurately model the heat exchanger: a physics-based white-box and a data-based black-box modelling technique. The white-box modelling approach uses the unsteady energy balance and heat transfer equations to predict the temperature evolutions in the plate heat exchanger. For the (data-driven) black-box modelling approach, there are three interrelated steps. Step 1 is the (semi-automatic) preprocessing of the measurements and the building of a nonparametric frequency response function (FRF) model. Step 2 is the building of a linear parametric (state-space) model, based on the FRF estimate. Step 3 is building a polynomial nonlinear state-space model that relies on the original linear state-space model. The proposed black-box method outperforms the white-box techniques in terms of prediction capabilities and computational needs.

Frequency Response Function Identification from Incomplete Data: A Wavelet-based Approach

Frequency Response Function (FRF) identification plays a crucial role in the design, the control, and the analysis of complex dynamical systems, including thermal and motion systems. Especially for applications that require long measurements, missing data samples, e.g., due to interruptions in the data transmission or sensor failure, often occur. The aim of this paper is to accurately identify nonparametric FRF models of periodically excited systems from noisy output measurements with missing samples. The presented method employs a wavelet-based transformation to address the identification problem in the time-frequency plane. A simulation example confirms that the developed techniques produce accurate estimates, even when many samples are missing.

An empirical study on decoupling PNLSS models illustrated on an airplane

Abstract This paper illustrates a combined nonparametric and parametric system identification framework for modeling nonlinear vibrating structures. First step is the analysis: multiple-input multiple-output measurements are (semi-automatically) preprocessed, and a nonparametric Best Linear Approximation (BLA) method is performed. The outcome of the BLA analysis results in nonparametric frequency response function, noise and nonlinear distortion estimates. Based on this information, a linear parametric (state-space) model is built. This model is used to initialize a high complexity Polynomial Nonlinear State-Space PNLSS model. The nonlinear part of a PNLSS model is manifested as a combination of high-dimensional multivariate polynomials. The last step in the proposed approach is the decoupling: transforming multivariate polynomials into a simplified, alternative basis, thereby dramatically reducing the number of parameters. In this work a novel filtered canonical polyadic decomposition (CPD) is used. The proposed methodology is illustrated on, but of course not limited to, a ground vibration testing measurement of an air fighter.

A Frequency Domain Approach to Model Reference Control

Abstract 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 Cramer-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.

Frequency Response Function Measurements of Multivariable Systems via Local Rational Modeling

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.

Multivariable nonparametric learning: A robust iterative inversion‐based control approach

SummaryLearning control enables significant performance improvement for systems by utilizing past data. Typical design methods aim to achieve fast convergence by using prior system knowledge in the form of a parametric model. To ensure that the learning process converges in the presence of model uncertainties, it is essential that robustness is appropriately introduced, which is particularly challenging for multivariable systems. The aim of the present article is to develop an optimization‐based design framework for fast and robust learning control for multivariable systems. This is achieved by connecting robust control and nonparametric frequency response function identification, which results in a design approach that enables the synthesis of learning and robustness parameters on a frequency‐by‐frequency basis. Application to a multivariable benchmark motion system confirms the potential of the developed framework.

Frequency Response Function Measurements via Local Rational Modeling, Revisited

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.

Finite-Time Learning Control Using Frequency Response Data With Application to a Nanopositioning Stage

Learning control enables significant performance improvement for systems that perform repeating tasks. Achieving high tracking performance by utilizing past error data typically requires noncausal learning that is based on a parametric model of the process. Such model-based approaches impose significant requirements on modeling and filter design. The aim of this paper is to reduce these requirements by developing a learning control framework that enables performance improvement through noncausal learning without relying on a parametric model. This is achieved by explicitly using the discrete Fourier transform to enable learning by using a nonparametric frequency response function model of the process. The effectiveness of the developed method is illustrated by application to a nanopositioning stage.

Accurate estimation of the non-parametric FRF of lightly-damped mechanical systems using arbitrary excitations

Abstract Lightly-damped mechanical systems exhibit strong resonant behaviour which could potentially result in life-threatening situations. To prevent these situations from happening, frequency response function measurements are essential to accurately characterise the resonant modes of the mechanical system. Unfortunately, these measurements are distorted by leakage and (long) transient phenomena. Local modelling techniques have been introduced in the past to resolve these complications but either they do not use the correct model structure or they introduce a bias. This paper proposes a local rational modelling technique which completely removes the bias from the estimation procedure and is applicable to large-scale multiple-input, multiple-output systems. The developed technique uses the bootstrapped total least squares estimator which provides unbiased estimates for the local rational model and generates accurate uncertainty bounds for the obtained non-parametric frequency response function. The proposed technique is successfully verified using a simulation example of a large-scale system which contains 100 resonances and has 100 inputs and 100 outputs. Its practical applicability is illustrated by characterising the resonant behaviour of the tailplane of a glider.

Impact of the Missing Data Pattern, the Oversampling, the Noise Level, and the Excitation on Nonparametric Frequency Response Function Estimates

Nonparametric frequency response function estimation (FRF) is a first important step towards successful parametric modelling of the dynamics. In some applications such as, for example, low-cost wireless sensor networks, sensors are subject to failure (clipping, outliers) and the transmission errors of the wireless communication can be as high as 30%. Hence, nonparametric estimation of the FRF in the presence of missing data is an important issue. In this paper we study the impact of the missing data pattern, the missing data fraction, the oversampling (w.r.t. the bandwidth of the system), the signal-to-noise ratio and the type of excitation on the bias and variance of the FRF estimates.

Nonparametric frequency response function estimates for switching piecewise linear systems

This paper proposes two algorithms (ATIRMM and ALPM) for estimating the time-varying behavior of single input single output (SISO) switching piecewise linear systems. Walsh basis functions are used to capture the non-smooth fast varying dynamics. The piecewise time-varying frequency response function (TV-FRF) is approximated by the sum of a series of LTI FRFs multiplied with a set of Walsh functions. The best linear time invariant approximation (BLTIA) of the TV-FRF is estimated with small uncertainty. Besides the BLTIA, the two methods are capable of estimating the noise power spectrum and the TV-FRF. The error analysis shows that ATIRMM delivers more accurate TV-FRF and BLTIA estimations, while ALPM has better performance in noise power spectrum estimation. The conclusions are illustrated by simulations.

Maximum Likelihood Estimation of the Non-Parametric FRF for Pulse-Like Excitations

This technical note introduces the closed form maximum likelihood estimator for estimating the coefficients of the non-parametric frequency response function from system identification experiments. It is assumed that the experiments consist of repeated pulse excitations and that both the excitation and system response are measured which leads to an error-in-variables setting. Monte Carlo simulations indicate that the estimator achieves efficiency at low signal-to-noise ratios with only few measurements. Comparison with the least-squares estimator shows that better, unbiased results are obtained.

Frequency response function measurement and parametric SISO system modelling of a gyro-stabilized infrared electro optic gimbal system

In this research study, a single axis (inner azimuth gimbal) of a four-axis gyro-stabilized electro optic gimbal system is modelled through experimental investigation in the frequency domain, and the results of this investigation are presented. The dynamic behaviour of the mechanical system is obtained from input and output signals, strictly speaking, nonparametric measurements. Detecting and measuring the nonlinear distortions allows a better understanding and gives an intuitive insight into the error sources on frequency response function measurements. In this study, a nonparametric frequency response function and its uncertainty (noise) are measured; nonlinear distortions are quantified on a real-time system. The dynamic system is modelled by its parametric transfer function with various different estimation techniques and their efficiencies, convergence properties, and bias errors are compared and discussed. It turns out that, the nonparametric noise model allows the estimator to weight the cost function and reach statistically better results. As an original contribution, a common problem encountered with gimbals having small rotational movement capacity (in this case study, ±5° for the inner gimbals) is formulated as an optimization problem. An optimization procedure is studied to achieve a better signal-to-noise ratio in the frequency band of interest, while satisfying device-specific constraints.

Non-parametric frequency response function tissue modeling in bipolar electrosurgery.

High-frequency radio energy is applied to tissue therapeutically in a number of different medical applications. The ability to model the effects of RF energy on the collagen, elastin, and liquid content of the target tissue would allow for the refinement of the control of the energy in order to improve outcomes and reduce negative side-effects. In this paper, we study the time-varying impedance spectra of the circuit. It is expected that the collagen/elastin ratio does not change over time such that the time-varying impedance is a function of the liquid content. We apply a non-parametric model in which we characterize the measured impedance spectra by its frequency response function. The measurements indicate that the changing impedance as a function of time exhibit a polynomial shift which we characterize by a polynomial regression. Finally, we quantify the uncertainty to obtain prediction intervals for the estimated polynomial describing the time variation of the impedance spectra.

Transient suppression in FRF measurement: Comparison and analysis of four state-of-the-art methods

This paper compares four well selected methods for computing the non-parametric Frequency Response Function (FRF) of a periodically excited linear time invariant system. The suppression of the transient is mandatory when its influence in the data is large. Better suppression of the transient leads to a better non-parametric FRF estimate. A good non-parametric FRF estimate can be used to validate the parametric transfer function model in a second step. The suppression of the transient will be highlighted using the mean squared error of the non-parametric FRF estimate. Temperature transients caused by heat diffusion are used as example. The selected methods consist of two standard windowing methods and two methods based on the Local Polynomial Method (LPM). LPM was designed to find a non-parametric FRF estimate in the presence of nonlinearities. This paper will modify LPM to find a non-parametric FRF estimate for linear systems using a single experiment. The mean squared error of the four non-parametric FRF estimates will be compared and analyzed, based on a simulation and a measurement example.

Extension of Local Polynomial Method for Periodic Excitations

This paper extends the Local Polynomial Method (LPM) for linear and time invariant systems excited by periodic signals. LPM is a robust and fast method for finding a non-parametric Frequency Response Function (FRF) estimate. A good FRF estimate is important in designing a good controller. Since both the system FRF and the transient behave smooth as a function of the frequency, LPM assumes that these functions can be approximated locally by a low degree polynomial. However, if the FRF varies strongly as a function of the frequency this assumption results in bias errors due to under-modeling. That is why this paper presents a transient LPM. This transient LPM suppresses the transients as well as the original LPM but does not introduce bias errors due to under-modeling. The variance of the FRF estimate via the transient LPM will be slightly larger than the variance of the FRF estimate via LPM. However, when these non-parametric FRF estimates are used to find a parametric estimate, this variance difference will not affect the result. Thus, the reduced bias of the FRF estimate via the transient LPM will lead to a better parametric FRF estimate. A disadvantage is that the transient LPM cannot estimate the level of the nonlinear distortions..