Frequency Response Function Models Research Articles

Assessment of hybrid machine learning models for non-linear system identification of fatigue test rigs

The prediction of system responses for a given fatigue test bench drive signal is a challenging problem, since highly dynamic loads from measurement campaigns must be reproduced accurately. Linear frequency response function models are commonly used for this system identification task, but energy intensive experimental iterations are required to account for system non-linearities. Two novel hybrid modeling strategies are suggested, which augment existing approaches using non-linear Long Short-Term Memory networks. These are trained and deployed on short subsequences of measurement data and recombined using a windowing technique, which enables their application to measurement data with high sampling rates. In addition to fatigue test bench commissioning, these hybrid models can also be employed in the field of virtual sensing. The approach is tested using non-linear experimental data from a servo-hydraulic test rig and this dataset is made publicly available. A variety of metrics in time and frequency domains, as well as fatigue strength under variable amplitudes, are employed in the evaluation. It is shown, that hybrid models can successfully use frequency response function models as a linear baseline estimate, which is further improved by Long Short-Term Memory networks to enable non-linear predictions.

On the hierarchical Bayesian modelling of frequency response functions

Structural health monitoring (SHM) strategies seek to evaluate, predict, and maintain structural integrity, to improve the safety and design service life of structures in operation. Many of these strategies involve monitoring changes in structural dynamics, as damage can affect modal properties and present as changes in the characteristics of the resonance peaks of the frequency response function (FRF). While recent advances have improved the safety and reliability of structures, a number of challenges remain, impeding the practical implementation and generalisation of these systems. Like damage, benign variations, such as those caused by changes in temperature or other environmental fluctuations, can affect dynamic properties, making it difficult to distinguish between damage and normal operating conditions. In addition, newly-deployed structures can have insufficient data to describe the normal operating conditions (i.e., data scarcity), which can impair the development of data-based prediction models. Another common challenge is data loss (i.e., data sparsity), which may result from transmission issues, sensor failure, a sample-rate mismatch between sensors, and other causes. Missing data in the time domain will result in decreased resolution in the frequency domain, which can impair dynamic characterisation.For situations that may benefit from information sharing among datasets, e.g., population-based SHM of similar structures, the hierarchical Bayesian approach provides a useful modelling structure. Hierarchical Bayesian models learn statistical distributions at the population (or parent) and the domain levels simultaneously, to bolster statistical strength among the parameters. As a result, variance is reduced among the parameter estimates, particularly when data are limited. In this paper, a combined probabilistic FRF model is developed for a small population of nominally-identical helicopter blades, using a hierarchical Bayesian structure, to support information transfer in the context of sparse data. The modelling approach is also demonstrated in a traditional SHM context, for a single helicopter blade exposed to varying temperatures, to show how the inclusion of physics-based knowledge can improve generalisation beyond the training data, in the context of scarce data. These models address critical challenges in SHM, by accommodating benign variations that present as differences in the underlying dynamics, while also considering (and utilising), the similarities among the domains.

A developed model updating method based on extended frequency response functions and its application study of offshore structures

Accurate finite element (FE) models are extremely vital to the analysis of the dynamic characteristics of engineering structures. However, it is challenging for the frequency response function (FRF)-based model updating method to obtain reliable numerical results as the limited number of FRFs at the selected peak positions usually leads to an unacceptable or failure of model updating. Moreover, the increased number of updating coefficients in complex engineering problems generates nonlinear optimization models with the local solutions by traditional optimization techniques. To tackle these two issues, a new model updating method is proposed to make full use of FRFs extracted from measured field data of structures and the improved particle swarm optimization (PSO) technique for accurately estimating physical parameters of structures. The novelties of this research include: (1) The signature assurance criterion (SAC)-based FRF is evaluated to eliminate the influence of limited frequency points on the accuracy of the updated model using the normalized acceleration component; (2) The enhanced (PSO) is developed to realize the adaptive selection of inertia factors for the better diversity by introducing an average value of the fitness function, and then accurate predictions of updating coefficients within a smaller number of iterations are achieved using the developed constraint factor. The effectiveness of the proposed method is verified by mathematical model of a jacket platform. Results show that the proposed method can accurately obtain the updating coefficients under spatial incomplete condition, and the maximum error of natural frequencies is 0.779% using the accelerations containing 5% noise. To prove the robustness of the proposed method, experimental studies of monopile offshore wind turbine are also conducted and the maximum error of natural frequencies is 1.887% under the consideration of spatial incompleteness represented by the structural stiffness degradation. Finally, the feasibility of the proposed method is evaluated by a test of a complex jacket platform, whose variation is simulated by weakening the connections of some elements. The maximum error of natural frequencies predicted by the updated model is only 4.831% as compared with experimental results. Throughout these examples, the extended FRF model updating method provides engineers and designers with a useful insight into the development of reliable techniques to accurately predict dynamic responses of offshore structures.

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.

Frequency response function method for dynamic gas flow modeling and its application in pipeline system leakage diagnosis

Timely diagnosing leakages is significant for the safety and reliability of gas pipeline systems, where an efficient and accurate modeling and solution method is critical to identify the leakage rate and location. Here, a frequency response function-based modeling method for dynamic gas flow in pipelines is proposed. Based on the frequency-domain flow resistance of pipelines and Fourier Transform, original dynamic flow model is transformed into the frequency response function model consisting of linear lumped transmission constraints, nonlinear frequency response function, and nonlinear pipeline resistance characteristics, which are efficiently solved according to their different mathematical properties. Dynamic simulation cases show the proposed modeling and solution method greatly improves the computational efficiency keeping the accuracy. Besides, taking the difference between measurements and simulation results under leakage condition as objective, an iterative bilayer optimization algorithm is proposed based on the mathematical categorization of system constraints to efficiently diagnose leakages. Four leakage diagnosis cases in single pipeline and pipeline network with measurement noise and transient boundary conditions are studied to validate the leakage diagnosis method. Absolute location errors in the results are all within 1%, demonstrating the accuracy and robustness of the proposed method for single leakages and multiple leakages occurring successively or simultaneously.

Run-Time Cutting Force Estimation Based on Learned Nonlinear Frequency Response Function

Abstract The frequency response function (FRF) provides an input–output model that describes the system dynamics. Learning the FRF of a mechanical system can facilitate system identification, adaptive control, and condition-based health monitoring. Traditionally, FRFs can be measured by off-line experimental testing, such as impulse response measurements via impact hammer testing. In this paper, we investigate learning FRFs from operational data with a nonlinear regression approach. A regression model with a learned nonlinear basis is proposed for FRF learning for run-time systems under dynamic steady state. Compared with a classic FRF, the data-driven model accounts for both transient and steady-state responses. With a nonlinear function basis, the FRF model naturally handles nonlinear frequency response analysis. The proposed method is tested and validated for dynamic cutting force estimation of machining spindles under various operating conditions. As shown in the results, instead of being a constant linear ratio, the learned FRF can represent different mapping relationships under different spindle speeds and force levels, which accounts for the nonlinear behavior of the systems. It is shown that the proposed method can predict dynamic cutting forces with high accuracy using measured vibration signals. We also demonstrate that the learned data-driven FRF can be easily applied with the few-shot learning scheme to machine tool spindles with different frequency responses when limited training samples are available.

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.

A data-driven approach for approximating non-linear dynamic systems using LSTM networks

The analysis of sensor data in nonlinear dynamical systems plays a fundamental role in a variety of modern engineering problems like Fatigue Analysis and Predictive Maintenance. In many applications, the accurate approximation of sensor signals supports and complements experimental efforts in order to accelerate the development process and conserve resources.Virtual sensing techniques aim to replace physical sensors in a system by using the data from available sensors to estimate additional unknown quantities of interest. Forward prediction estimates the system response for a given drive signal during the commission of component test rigs. Data-driven approaches can be convenient tools for forward prediction and virtual sensing, as they only require a sufficiently large dataset of the desired input and output quantities for the purpose of model parametrization. The presented approach explores two hybrid modeling strategies, which combine Frequency Response Function models with Long Short-Term Memory networks to approximate the behavior of non-linear dynamic systems with multiple input and output channels.The proposed method utilizes short subsequences of signals to carry out model training and prediction. Long sequence estimations are generated by combining the individual subsequence predictions using a windowing technique.Our approach is tested on a large, non-linear experimental dataset, obtained from a servo hydraulic fatigue test bench. To enable a comprehensive evaluation of the model quality, both hybrid modeling approaches are compared on virtual sensing and forward prediction tasks using multiple error metrics relevant to fatigue analysis.

Nonlinear Modeling of a Plate Heat Exchanger of a District Heating System

The integration of renewable and residual energy sources requires a detailed level of knowledge of district heating components such as thermal pipes, pipes, pumps, and heat exchanger of a substation. The thermal energy stored in these components can be utilized to match the heat demand and heat generation in time for next generation district heating networks. The goal of this paper is to provide accurate substation models that can be utilized for controlled energy exchange between primary and secondary networks. The proposed method introduces an efficient, user-friendly, combined parametric and nonparametric nonlinear data-driven modeling technique illustrated on a plate heat exchanger which is installed in the district heating network of the Vrije Universiteit Brussel. A fair comparison between a physics-based white-box and a data-driven black-box modeling techniques is provided. The white-box model is developed on the first principle modeling approach of the heat balance equations, which allows calculating the temperature evolution and heat transfer in the plate heat exchanger. As for the data-driven modeling approach, there are four interrelated consecutive steps. First, the measurement is processed to obtain a frequency response function model. Second, a linear parametric (state-space) model is estimated. Third, a polynomial nonlinear model is built. Last, a decoupled model structure is obtained based on the previously obtained PNLSS model. The proposed black-box method outperforms the white-box techniques in terms of prediction capabilities and computational needs.

A Virtual Sensing approach for approximating nonlinear dynamical systems using LSTM networks

AbstractIn this contribution, we introduce a hybrid model for virtual sensing applications which combines a frequency response function model with a Long Short‐Term Memory network. It estimates the behavior of non‐linear dynamic systems with multiple input and output channels by generating predictions on short subsequences of signals and recombining them using a windowing technique. The approach is tested on an experimental dataset composed of measurements from a 3‐component servo hydraulic fatigue test bench. The model is parameterized using noise data, while fatigue serviceloads with variable amplitudes are used for validation and testing.

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.

Application of FRF with SISO and MISO model for accelerometer-based in-cylinder pressure reconstruction on a 9-L diesel engine

Engine control with feedback from engine combustion process diagnostics can help improve fuel efficiency and assist in meeting stricter emission regulations. The standard is to use in-cylinder pressure measurements with analysis including rate of heat release. The measurement is usually obtained with intrusive sensors that require a special mounting process and engine structure modification. The potential of the low-cost non-intrusive accelerometer as an alternative means to reconstruct the in-cylinder pressure has been demonstrated by previous investigations. In this work, start of injection (SOI) sweep test conditions at varied speed spanning both low load and high load were conducted on an inline 6-cylinder, 9 L diesel engine. The relationship between the in-cylinder pressure and the accelerometer signal was quantified with frequency response function (FRF). The robustness of the obtained FRF was evaluated by applying the single-test-based FRF to reconstruct the in-cylinder pressures for other test conditions. Two models, single-input single-output (SISO) and multiple-input single-output (MISO), were investigated and compared where the accelerometer signal was taken as the input and in-cylinder pressure as the output. The optimal channel used to acquire the input signal in the SISO model was selected on the basis of coherence analysis. Results show that the MISO model assisted by principal component analysis (PCA) and offset-compensation processes can result in better in-cylinder pressure estimation than the SISO model for conditions with 2200 rpm engine speed. With the purpose of minimizing the cost for accelerometer employment, the minimum number of inputs used to reconstruct the in-cylinder pressure in the MISO model was pursued. Thresholds were set based on three estimated in-cylinder pressure parameters to select the qualified input channels and two input channels were finally determined. Results showed that the two-input single-output FRF model coupled with the PCA and offset-compensation processes improves the FRF’s robustness for the in-cylinder pressure estimation in comparison to the SISO FRF model based on all the tests conducted in this paper.

Estimation of nonparametric noise and FRF models for multivariable systems—Part II: Extensions, applications

This is Part II of a series of two papers on the estimation of nonparametric noise and frequency response function models for multiple input, multiple output systems excited by arbitrary inputs. Part I (Pintelon et al. (2009)) [1] develops the methodology for linear dynamic multivariable systems with exactly known input and where the output is disturbed by stationary noise ( = output error problem ) . The contributions of Part II are the following. First, it is shown that the proposed method can be used for nonlinear systems and for parametric identification of the system dynamics in a generalized output error framework. Next, the methodology is generalized to handle noisy input–output data ( = errors -in-variables problem), and identification in feedback. Finally, the approach is illustrated on simulations and real measurements examples.

Estimation of nonparametric noise and FRF models for multivariable systems—Part I: Theory

This series of two papers presents a method for estimating nonparametric noise and frequency response function models of multivariable linear dynamic systems excited by arbitrary inputs. It extends the results of Schoukens et al. (2006) [1] and Schoukens and Pintelon (2009) [2] from single input, single output systems with known input and noisy output observations ( = output error problem ), to multiple input, multiple output systems where both the input and output are disturbed by noise ( = errors‐in‐variables problem ). In Part I, the theory is developed for linear dynamic multivariable output error problems. The results are supported by simulations. A detailed comparison with the classical spectral analysis based on correlation techniques shows that the proposed procedures are more robust. In Part II (Pintelon et al., 2009) [3], the method first is applied to nonlinear systems, and parametric identification within a generalized output error framework. Next, it is extended to handle errors-in-variables problems, and identification in feedback. Finally, it is illustrated on four real measurement examples.

An overview of MIMO-FRF excitation/averaging/processing techniques

The use of characterized excitation and choice of averaging techniques are fundamental to the estimation of multiple input, multiple output (MIMO) frequency response function (FRF) data. The characteristics of the excitation and averaging selected greatly influence the quality of the resulting MIMO-FRF measurements. Presented is an overview of the basic excitation methods, such as random, periodic random, pseudorandom, and burst random (random transient) as well as more advanced excitation methods, such as burst-cyclic random, slow random, MOOZ random, and periodic chirps. The application of these excitation and averaging methods is discussed relative to lightly or heavily damped systems, systems with small non-linearities, FRF models, and peak to RMS (crest factor) as well as signal-to-noise ratio (SNR) issues. Experimental examples are given to demonstrate the important issues.

FREQUENCY DOMAIN ARX MODEL AND MULTI-HARMONIC FRF ESTIMATORS FOR NON-LINEAR DYNAMIC SYSTEMS

Non-linear dynamic systems respond at frequencies other than the excitation frequency; however, standard frequency response function estimators for linear systems do not accommodate this harmonic distortion. A new multi-harmonic frequency response function estimator that utilizes discrete frequency models for non-linear systems is introduced here. The multi-harmonic estimator relates the frequency response at each frequency to the input and output spectra within a given frequency band in the same way that autoregressive exogenous input models relate inputs and outputs at particular samples in the time domain. Overdetermined, least-mean-squares calculations are used to minimize model error throughout a frequency band rather than at a single frequency as in the corresponding linear estimators. The resulting multi-harmonic frequency response function models are non-parametric (e.g., vary with amplitude) when linear functions are used and parametric when non-linear functions are used. A new sensitive indicator for experimentally characterizing non-linearity is introduced.

Near field acoustic source identification based on numerical models and operational pressure data

When pressure measurements are taken in the near field of possible sources, a global inversion of an input–output model of the acoustic medium can be accomplished to calculate the source velocities, and thus to identify the main noise radiating parts. The acoustic medium can be represented by a frequency response function (FRF) matrix which can either be derived analytically based on a point source assumption, or experimentally under laboratory conditions, as well as numerically using a finite or a boundary element model. The operational source velocities can be calculated by multiplying the inverted FRF matrix with the vector of operational pressure measurements. The inversion process, however, is usually an ill-conditioned problem. Regularization techniques based on a singular value decomposition can be used to solve for the ill-conditioned nature of the inverse acoustic problem. In this paper, the applicability of numerically derived FRF models using the boundary element modeling technique, is shown for the calculation of the velocities at the boundary of a source model, based on operational pressure measurements. The results are compared with near field acoustic holography calculations, and the usefulness of an L-curve criterion for the selection of the optimal regularization parameter is indicated based on simulations as well as experiments.