Nonparametric frequency response function estimators based on nonlinear averaging techniques

Engineers and scientists want mathematical models of the observed system for understanding, design, and control. Many mechanical and civil structures are nonlinear. This paper illustrates a combined nonparametric and parametric system identification framework for modelling a nonlinear vibrating structure. First step of the process is the analysis: measurements are (semiautomatically) preprocessed, and a nonparametric best linear approximation (BLA) method is applied. The outcome of the BLA analysis results in nonparametric frequency response function, noise and nonlinear distortion estimates. Second, based on the information obtained from the BLA process, a linear parametric (state-space) model is built. Third, the parametric model is used to initialize a complex 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 significantly 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 F16 aircraft.KeywordsMIMO systemsNonlinearityDecouplingSystem identificationGround vibration testing

Accurate Estimation of Multivariable Frequency Response Functions

Non-parametric frequency response estimation using a local rational model

Frequency-Domain Gray-Box Identification of Industrial Robots

Traditional Frequency Response Function (FRF) estimation techniques used for analysis of Human Controller (HC) dynamics in tracking tasks assume HC dynamics to be linear, but generally do not quantify or compensate for the effects of human nonlinearities. The robust and fast Best Linear Approximation (BLA) techniques for estimating an FRF do provide such quantification of nonlinear distortions caused by Period-In-Same-Period-Out (PISPO) nonlinearities and can reduce the effect of PISPO nonlinear operations on the FRF estimate. This paper investigates the application of these BLA techniques to both measured and simulated HC data. For the simulated data, a linear HC model was deliberately extended with a symmetric PISPO deadzone nonlinear operator and a realistic level of HC noise. Overall, both the measured and the simulated data indicate that due to the high levels of remnant noise inherent to HC data, no consistent estimate of PISPO nonlinear contributions could be made. This also means that the improvement of using BLA techniques and averaging over multiple forcing function realizations does not result in a substantial improvement over the current practice of estimating HC FRFs from repeated measurements of a single forcing function.

Frequency response function (FRF) estimation from measured data is an essential step in the design, control, and 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 article is to achieve accurate identification of nonparametric 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 tre frequency domain. The framework encompasses both a time-invariant and a time-varying wavelet-based estimator, which provides different mechanisms to address the missing samples. Experimental results from a thermodynamical system confirm that the estimators enable accurate identification.

In experimental modal analysis, the use of characterized excitation and the choice of window functions are fundamental to the estimation of system frequency response function (FRF) data. It greatly influences the quality of the resulting FRF measurements. Discussed are combination methods of several commonly-used excitation signals with window function, which is relative to lightly damped systems. Study shows that burst random signal with Rectangular works well for lightly damped structures, and pure random with Hanning will bring about higher error. A modal testing of white car body was done. Experimental parameters analysis are given to demonstrate the study result presented above. So it poses a practicable method for improving the accuracy of frequency response function (FRF) measurements of lightly damped structure by using proper excitation signal with appropriate window function.

Frequency response function estimation techniques and the corresponding coherence functions: A review and update

When controlling an aircraft the pilot perceives the motion kinematics in different ways. For instance, he sees the motion due to changes in the outside visual scene or a display and he feels the motion with his vestibular organ in the inner ear or from sensors in his skin. In this paper the pilot’s use of these different cues in a manual aircraft control task is studied using both single-loop and multi-loop identification methods. The task of the pilot was to control the roll-axis of an aircraft with double integrator dynamics. Two kinds of control tasks were analyzed: the disturbance task, where the pilot had to compensate for a disturbance working on the aircraft, and the target-following task, where the pilot had to minimize the difference of the roll-angle with a reference. The cues presented to the pilot were the error at a central display and the actual roll-angle on a peripheral display. In the single-loop experiments the total, lumped, frequency response function is identified. In the multi-loop method two frequency response functions are identified separately, representing the feedback of the central visual and the peripheral visual cues. The experiment showed that due to the addition of peripheral visual the response to the central visual decreases. The central visual also generates more phase in case of the target task. The estimation of the peripheral frequency response function with the multiloop method was difficult, which could be due to the choice of the forcing functions. Pilot behavior did not change when applying the multi-loop identification method, a result that paves the way for employing this method in other multi-sensory environments.

The use of block overlapping and time windowing in the estimation of frequency response functions of linear systems is considered. The systems are subjected to stationary white random excitation and there is no external noise. The sources of errors for the bias and variance of the frequency response are identified to be due to the Fourier transforms of the transient terms or end effects that arise due to non-periodic signals. Using a pure time delay as an example for the system, it is shown that there is a limit to the improvement achievable in reducing the variance as the block overlap is increased.

A ‘Sub-PolyMAX’ method is proposed in this paper not only for estimating modal parameters, but also for identifying structural flexibility by processing the impact test data in narrow frequency bands. The traditional PolyMAX method obtains denominator polynomial coefficients by minimizing the least square (LS) errors of frequency response function (FRF) estimates over the whole frequency range, but FRF peaks in different structural modes may have different levels of magnitude, which leads to the modal parameters identified for the modes with small FRF peaks being inaccurate. In contrast, the proposed Sub-PolyMAX method implements the LS solver in each subspace of the whole frequency range separately; thus the results identified from a narrow frequency band are not affected by FRF data in other frequency bands. In performing structural identification in narrow frequency bands, not in the whole frequency space, the proposed method has the following merits: (1) it produces accurate modal parameters, even for the modes with very small FRF peaks; (2) it significantly reduces computation cost by reducing the number of frequency lines and the model order in each LS implementation; (3) it accurately identifies structural flexibility from impact test data, from which structural deflection under any static load can be predicted. Numerical and laboratory examples are investigated to verify the effectiveness of the proposed method.

Non-parametric hemodynamic response function (HRF) estimation in noisy functional Magnetic Resonance Imaging (fMRI) plays an important role when investigating the temporal dynamic of a brain region response during activations. Assuming the drift Lipschitz continuous; a new algorithm for non-parametric HRF estimation is derived in this paper. The proposed algorithm estimates the HRF by applying a first order differencing to the fMRI time series samples. It is shown that the proposed HRF estimator is √(N) consistent. Its performance is assessed using both simulated and a real fMRI data sets obtained from an event-related fMRI experiment. The application results reveal that the proposed HRF estimation method is efficient both computationally and in term of accuracy.

Flexible multibody system simulation allows for fast and adequate investigation of the dynamics of mechanical systems. But in case of a system response with large deformations the time response does not uncover the causes, i. e. the resonances of the system. The identification of the systems eigenfrequencies gives more insight in resonance phenomena, but in case of periodic time-variant systems the often used snap-shot-eigenfrequencies do not reveal the real system dynamics, which has to be described by more than only one frequency response function. Based on the formulation of a flexible multibody system and the theory of ordinary linear periodic differential equations, partial frequency response functions, describing the real characteristics of a periodic system, are calculated and compared to the snapshot-frequency response functions.

A Frequency Domain Approach to Model Reference Control

As an alternative to operational modal analysis and classical experimental modal analysis (EMA), a novel method was introduced previously, namely impact-synchronous modal analysis (ISMA). The effectiveness ISMA on rotor and structural dynamic systems has been proven in previous literature. More recently, an automated impact device (AID) was introduced which utilized tachometer pulse as initiation signal and its effectiveness on ISMA was proven. An attempt to further enhance this device in term of equipment and cost is then proposed by replacing the tachometer with the in-use tri-axial accelerometer through utilizing the filtered response of cyclic load component as an initiation signal to control the impact device, which is also the primary aim for this study. Prior to modal testing, accuracy of this device is illustrated at desired phase angles of 0 deg, 90 deg, 180 deg, and 270 deg. Subsequently, frequency response function (FRF) estimations obtained for ISMA using enhanced AID has demonstrated the suppression capabilities of this device on disturbances, i.e., reduction of 93.58% at 30 Hz and 57.78% at 60 Hz, resulting in a high correlation for signature assurance criterion (SAC) and cross signature assurance criterion (CSAC). Modal parameters extracted from the EMA and ISMA using impact device are presented and compared, for the first three natural modes of the test rig. It is found that natural frequencies are deviating by less than 6%, whereas modal assurance criterion (MAC) values between the mode shapes of the two tests are found to be above 0.9.