Nonparametric Frequency Response Function Research Articles

Non-parametric frequency response estimation using a local rational model

A review of the relationship between the frequency response function of linear system and the DFT of the input and output signals show that the output DFT is a sum of two terms. The first term contain the FRF multiplied with the input DFT and the second term capture the effect when the system is not operating in a periodic fashion. The utilization of these two terms when performing non-parametric frequency response function estimation has led to the previously developed Local Polynomial Method. This paper acknowledge that the two terms can better be approximated by local rational functions with a common denominator polynomial and a new method called Local Rational Method has been developed. Numerical simulations illustrate the performance of the new rational method in comparison with the polynomial one. The results suggest that the new rational method gives better performance when the system has a resonant behavior.

Improved (non-)parametric identification of dynamic systems excited by periodic signals

The steady state response of a system to a periodic input is still subject to noise transients. For lightly damped systems these noise transients can significantly increase the variance of the estimated frequency response function (FRF). This paper presents a method that suppresses the influence of the noise transients (leakage errors) in nonparametric FRF and noise (co-)variance estimates of dynamic systems excited by periodic signals. The method is based on a local polynomial approximation of the noise leakage errors on the FRF. Compared with the classical approaches, the proposed procedure is more robust and needs less measurement time (two signal periods are sufficient). The theory is supported by simulation and real measurement examples.

Nonlinear gray-box identification using local models applied to industrial robots

In this paper, we study the problem of estimating unknown parameters in nonlinear gray-box models that may be multivariable, nonlinear, unstable, and resonant at the same time. A straightforward use of time-domain predication-error methods for this type of problem easily ends up in a large and numerically stiff optimization problem. We therefore propose an identification procedure that uses intermediate local models that allow for data compression and a less complex optimization problem. The procedure is based on the estimation of the nonparametric frequency response function (FRF) in a number of operating points. The nonlinear gray-box model is linearized in the same operating points, resulting in parametric FRFs. The optimal parameters are finally obtained by minimizing the discrepancy between the nonparametric and parametric FRFs. The procedure is illustrated by estimating elasticity parameters in a six-axis industrial robot. Different parameter estimators are compared and experimental results show the usefulness of the proposed identification procedure. The weighted logarithmic least squares estimator achieves the best result and the identified model gives a good global description of the dynamics in the frequency range of interest for robot control.

Study of the Variance of Parametric Estimates of the Best Linear Approximation of Nonlinear Systems

Abstract-This paper analyzes the variance of the estimated parametric best linear approximation (BLA) Ĝ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">BLA</sub> (q,θ) of a nonlinear system that is driven by random excitations. The estimated model Ĝ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">BLA</sub> (q; θ) varies not only due to the disturbing measurement and process noise but also over different realizations of the random excitation because the nonlinear distortions depend on the input realization. For the nonparametric frequency response function (FRF) estimate, it has been shown that the variance expression is still valid in the presence of nonlinear distortions, and the same formulas can be used in the linear as in the nonlinear case. This result does not hold for the variance σ(Ĝ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">BLA</sub> (q,θ)) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> on the parametric estimate Ĝ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">BLA</sub> (q,θ). It is shown in this paper that it is still possible to upperbound the variance σ(Ĝ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">BLA</sub> (q,θ)) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> using linear expression by introducing an additional scaling factor that depends upon the maximal degree of nonlinearity.

ANALYSIS OF THE NONLINEAR INDUCED VARIANCE IN LINEAR SYSTEM IDENTIFICATION

- This paper analyses the variance of linear models that are identified in the presence of nonlinear distortions. The best linear approximation GBLA to a nonlinear system, driven by a Gaussian distributed random input, is identified in least squares sense. Even in the absence of disturbing noise, the estimated model ĜBLA will vary from one realization of the excitation signal to the other. In this paper it will be shown that the variance of the nonparametric frequency response function (FRF) ĜBLA(k) can still be obtained from the experimental data using the classical variance formulas. However, for parametric estimates ĜBLA(q, θ), an increased variance is observed. The contributing terms to this additional variance are studied in this paper.

Frequency-Domain Gray-Box Identification of Industrial Robots

This paper considers identification of unknown parameters in elastic dynamic models of industrial robots. Identifying such models is a challenging task since an industrial robot is a multivariable, nonlinear, resonant, and unstable system. Unknown parameters (mainly spring-damper pairs) in a physically parameterized nonlinear dynamic model are identified in the frequency domain, using estimates of the nonparametric frequency response function (FRF) in different robot configurations/positions. The nonlinear parametric robot model is linearized in the same positions and the optimal parameters are obtained by minimizing the discrepancy between the nonparametric FRFs and the parametric FRFs (the FRFs of the linearized parametric robot model). In order to accurately estimate the nonparametric FRFs, the experiments must be carefully designed. The selection of optimal robot configurations for the experiments is also part of the design. Different parameter estimators are compared and experimental results show the usefulness of the proposed identification procedure. The weighted logarithmic least squares estimator achieves the best result and the identified model gives a good global description of the dynamics in the frequency range of interest.

Optimized Excitation Signals for MIMO Frequency Response Function Measurements

Many different excitation signals can be chosen for the nonparametric frequency response function measurements of a linear multiple-input-multiple-output (MIMO) system. In the presence of output noise they will result in different signal-to-noise ratios (SNRs) of the frequency response function measurements. In this paper, these effects are analyzed and compared for three classes of multisine excitations: random multisines, approximately orthogonal random multisines, and orthogonal random multisines, taking also into account the crest-factor minimization. It is shown that the orthogonalization of the inputs yields an essential increase in the measured SNR

MEASURING LINEAR APPROXIMATION TO WEAKLY NONLINEAR MIMO SYSTEMS

The choice of the input signals has impact on the nonparametric frequency response function (FRF) measurements of a nonlinear MIMO system. It is shown that Gaussian noise, periodic noise, and random multisines are equivalent, yielding in the limit the same linear approximation to a nonlinear MIMO system. Even in the noiseless case, variability of the FRF is observed due to the presence of the nonlinearity and the randomness of the excitation. Based upon these effects a new class of equivalent input signals is proposed, decreasing essentially the variance of the nonlinear FRF measurements, while the same linear approximation is retrieved.

Frequency domain control-relevant identification of MIMO AMB rigid rotor

This paper describes an application of frequency domain control-relevant identification and controller design based on an LQ criterion, which has moderately enhanced the performance of a MIMO active magnetic bearing (AMB) rigid rotor system. Unlike existing schemes using LQG criteria, the controller gain has not been tuned using frequency weighting calculated by the differences between measured and predicted input–output signals during iteration. Instead, the performance enhancement provided by frequency domain control-relevant identification of an open-loop plant is emphasized. The non-parametric frequency response functions of a MIMO open-loop plant including all additional dynamics such as sensor, amplifier, and discretization are estimated from measured input and output signals of the closed-loop plant. An approximate MIMO model of the matrix fractional description form is obtained with control-relevant weightings and the LQG/LTR controller is designed iteratively based on the identified model. The performance enhancements through the frequency domain control-relevant identification of an open-loop plant may not be as dramatic as those obtained with the existing schemes, since we limit the order of the controller by eliminating the frequency weighting in the controller design stage. However, the performance of a MIMO AMB rigid rotor system has been moderately enhanced with the resulting limited-order controller.

Improved frequency response function measurements for random noise excitations

This paper presents a method to get improved nonparametric frequency response function (FRF) measurements for random noise excitations. The basic idea is to remove the effect of leakage on the FRF by iterative improvement.

Non-Parametric Estimation of the Frequency Response Functions of the Linear Blocks of a Wienerhammerstein Model.

This paper proposes a method to estimate the non-parametric frequency response functions of the linear parts of a Wiener-Hammerstein model using special designed periodic excitation signals. No assumptions are made concerning the static non-linearity. The estimator is developed within a stochastic errors-in-variables framework. The applicability of the method is illustrated using simulations.

Nonparametric frequency response function estimators based on nonlinear averaging techniques

Nonparametric frequency response function (FRF) estimators based on nonlinear averaging techniques are proposed. The suitability of such estimators for FFT based signal analyzers is studied. Under mild assumptions, it is shown that their bias is a function of the fourth-order moments of the perturbing errors on the input-output Fourier coefficients while the bias of the classical H/sub 1/ estimator is a function of the second-order moments. The performances of these estimators are analyzed in different situations: uncorrelated input-output perturbing errors (i.e., open-loop measurements), mutually correlated input-output disturbances (i.e., closed-loop measurements) and in presence of outliers. Analytical expressions are given for the bias when the input-output errors are zero-mean normal distributed (and mutually correlated). >