Frequency Response Function Identification Research Articles

Beyond Nyquist in Frequency Response Function Identification: Applied to Slow-Sampled Systems

Fast-sampled models are essential for control design, e.g., to address intersample behavior. The aim of this paper is to develop a non-parametric identification technique for fast-sampled models of systems that have relevant dynamics and actuation above the Nyquist frequency of the sensor, such as vision-in-the-loop systems. The developed method assumes smoothness of the frequency response function, which allows to disentangle aliased components through local models over multiple frequency bands. The method identifies fast-sampled models of slowly-sampled systems accurately in a single identification experiment. Finally, an experimental example demonstrates the effectiveness of the technique.

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.

Operational Impact Excitation Method for Milling Robot End Frequency Response Function Identification Under Movement State and Pose-Dependent Dynamic Compliance Analysis

Abstract Robot milling has become an important means of machining large structural parts, and the dynamic compliance of the robot end is the key factor affecting machining quality and efficiency. The dynamic characteristics of the milling robot end are different in movement state and static state and have significant pose dependence. In order to effectively evaluate the dynamic compliance of the robot end in the workspace under the movement state, the operational impact excitation method for robot joint relative frequency response function (FRF) identification based on operational modal analysis (OMA) used in machine tool is established, the inertial force generated by the joint acceleration and deceleration movements is used as the excitation force, the robot end relative dynamic compliance index (RERDCI) is proposed to evaluate the dynamic compliance of robot end in different poses, and verified by cutting experiments. Based on RERDCI, the effect of the robot pose on end dynamic compliance is analyzed, and some theoretical guidance for improvement of dynamic performance to resist vibration in milling is given.

Frequency response function identification using fused filament fabrication-3D-printed embedded ArUco markers

The assessment of modal components is a fundamental step in structural dynamics. While experimental investigations are generally performed through full-contact techniques, using accelerometers or modal hammers, this research proposes a non-contact Frequency Response Function identification measurement technique based on ArUco markers displacement detection. The proposed method is presented using a harmonically excited FFF 3D-printed flexible structure, equipped with multiple embedded-printed markers, whose displacement is measured with an industrial camera. Comparison with numerical simulation and an established experimental approach is finally provided for the results validation.

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.

Analysis of the Creeping Phenomenon of Linear Feed System Based on the Rigid-Flexible Coupling Model

A mathematical model of the creeping phenomenon based on the mechanical model of the linear feed system was established. The dynamic characteristic parameters of each fixed joint were obtained by Yoshimura’s integral. Using the method, only the dynamic characteristic parameters of the joint surface per unit area with simple structure need to be studied, and then, the dynamic characteristic parameters of the whole joint surface can be obtained by integration. Based on the principle of the half-power bandwidth method and the frequency response function identification, the dynamic parameters of each moving joint were solved by the method of experimental modal analysis. Through the parameters of the fixed and moving joints, a rigid body model of the feed system and a flexible body model including the power transmission parts (ball screw pair) and the motion guide parts (guide slide pair and rolling bearing) were, respectively, established. And then, a rigid-flexible coupling dynamic model of the feed system was obtained through the constraint relationships between joints. The influence of both the external load and the feed rate on the fluctuation of motion speed of the system was analyzed from this model. The difference between the experimental results and the simulation results on a feed system platform is not greater than 10%, which verifies the creeping phenomenon. This conclusion can provide a basis for the optimization of the dynamic performance of the ball screw linear-feeding workbench.

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 identification for multivariable motion control: Optimal experiment design with element-wise constraints

Frequency Response Functions (FRFs) are essential in mechatronic systems and its application ranges from system design and validation to controller design and diagnostics. The aim of this paper is to optimally design experiments for FRF identification of multivariable motion systems subject to element-wise power constraints. A multivariable excitation design framework is established that explicitly addresses the frequency-wise directionality of the system to be identified. The design problem involves solving a rank-constrained optimization problem, which is non-convex and NP-hard in most cases. Two algorithms to solving this problem approximately are presented that rely on a convex (semi-definite) relaxation of the original problem. Additionally, exact solutions for several special cases are presented. The two algorithms are shown to overcome the limitations of traditional excitation design. This is confirmed by experimental results from a 7 × 8 wafer stage setup, which show a significant improvement of the FRF quality using the proposed techniques over traditional design approaches.

Frequency response function identification of periodically scheduled linear parameter-varying systems

For Linear Time-Invariant (LTI) systems, Frequency Response Functions (FRFs) facilitate dynamics analysis, controller design, and parametric modeling, while many practically relevant systems are in fact more accurately described by Linear Parameter-Varying (LPV) models. The aim of this paper is to develop an FRF modeling framework for periodically scheduled Single-Input Single-Output (SISO) LPV systems, that enables the identification of LPV FRF models from global experiments. This is achieved by developing an appropriate definition of the harmonic FRF for input–output LPV systems and by developing a method to compute a suitable harmonic FRF estimator. The developed approach generalizes the Empirical Transfer Function Estimate (ETFE) to the class of periodically-scheduled LPV systems, and the classical ETFE is recovered for LTI systems as a special case. The developed method is successfully used to estimate a SISO LPV FRF of an experimental motion system, thereby confirming the potential of the developed framework.

Multivariable Experiment Design with Application to a Wafer Stage: a Sequential Relaxation Approach for Dealing with Element-Wise Constraints

Optimal Experiment Design (OED) is an essential aspect in accurate Frequency Response Function (FRF) identification of complex systems. The aim of this paper is to optimally design experiments for FRF identification of multivariable motion systems subject to element-wise power constraints. This design problem involves solving a non-convex and generally NP-hard optimization problem. An algorithm to solving this problem approximately is presented based on sequential semi-definite relaxations. Experimental results on a wafer stage show an improvement of the FRF quality using the proposed techniques over traditional excitation design methods.

Local polynomial method for frequency response function identification

Here we propose the local polynomial method to solve the problem of estimating the frequency response function in the linear system. Compared with other nonparametric identification methods based on the windowing strategies, this new identification method can be remarkably efficient in reducing the effect caused by the leakage error when the discrete Fourier transform is used under a non-periodic input excited signal. Considering the constraints between the coefficients of the polynomials at neighbour frequencies, we modify the proposed local polynomial method to get one constrained local polynomial method. The modified local polynomial method reduces the mean square error of the frequency response function and the estimation of the frequency response function is identified by one multi-objective least squares criterion. Finally the simulation example results confirm the identification theoretical results.

Experimental study of the variations in the electromechanical properties of piezoelectric energy harvesters and their impact on the frequency response function

Abstract This work presents a comprehensive study of the uncertainties involved in the Frequency Response Function (FRF) of Piezoelectric Energy Harvesters (PEHs) based on experimental results. A proper experimental setup is designed facilitating the FRF identification for different PEHs. The test protocol consists in three test that quantify the aleatoric and epistemic uncertainties by the study of repeated measurements, variation in the mounting process, and variations in the electromechanical properties of the materials employed in the harvester. The experimental setup and the test protocol are tested employing two different bimorph-type harvesters identified as Model A and B. 20 harvesters corresponding to Model A and 10 to Model B are tested multiples times under the test protocol proposed. Results clearly indicate that the most important source of uncertainties deals with the variability of the electromechanical properties of the PEH, which is particularly greater than: (i) the noise on the measurements, (ii) the variations introduced by the clamping condition, and (iii) the dimensional variations in the geometry. The results are also compared with predictions performed by adopting a recent numerical framework to propagate uncertainties in PEHs. The present work is particularly important since it presents the experimental evidence of the variations in the electromechanical properties of PEH and their influence on the FRF. Additionally, the results not only validate the uncertainty propagation procedures previously proposed but also suggest that their use is mandatory to estimate the FRF. This work presents, to the best of the authors’ knowledge, the first effort to experimentally quantify the uncertainties in PEHs.

Frequency Response Function Identification of LPV Systems: a Global Approach with Application to Mechanical Systems

Frequency Response Function (FRF) modeling of Linear Parameter Varying (LPV) systems facilitates analysis, controller design and parametric modeling of a large class systems, including position-dependent mechanical systems. The aim of this paper is to enable FRF identification of LPV systems using global experiments. This is achieved by developing an appropriate definition of the FRF for input-output LPV systems and by developing a method to compute a statistically optimal estimator of the FRF, which reduces to the Empirical Transfer Function Estimate (ETFE) for frozen scheduling. The developed method is successfully used to estimate a position-dependent FRF of a wide-format printer, confirming the potential of the approach.

Non-parametric identification of multivariable systems: A local rational modeling approach with application to a vibration isolation benchmark

Abstract Frequency response function (FRF) identification is often used as a basis for control systems design and as a starting point for subsequent parametric system identification. The aim of this paper is to develop a multiple-input multiple-output (MIMO) local parametric modeling approach for FRF identification of lightly damped mechanical systems with improved speed and accuracy. The proposed method is based on local rational models, which can efficiently handle the lightly-damped resonant dynamics. A key aspect herein is the freedom in the multivariable rational model parametrizations. Several choices for such multivariable rational model parametrizations are proposed and investigated. For systems with many inputs and outputs the required number of model parameters can rapidly increase, adversely affecting the performance of the local modeling approach. Therefore, low-order model structures are investigated. The structure of these low-order parametrizations leads to an undesired directionality in the identification problem. To address this, an iterative local rational modeling algorithm is proposed. As a special case recently developed SISO algorithms are recovered. The proposed approach is successfully demonstrated on simulations and on an active vibration isolation system benchmark, confirming good performance of the method using significantly less parameters compared with alternative approaches.

RCSA-Based Method for Tool Frequency Response Function Identification Under Operational Conditions Without Using Noncontact Sensor

The stability lobe diagrams predicted using the tool frequency response function (FRF) at the idle state usually have discrepancies compared with the actual stability cutting boundary. These discrepancies can be attributed to the effect of spindle rotating on the tool FRFs which are difficult to measure at the rotating state. This paper proposes a new tool FRF identification method without using noncontact sensor for the rotating state of the spindle. In this method, the FRFs with impact applied on smooth rotating tool and vibration response tested on spindle head are measured for two tools of different lengths clamped in spindle–holder assembly. Based on those FRFs, an inverse receptance coupling substructure analysis (RCSA) algorithm is developed to identify the FRFs of spindle–holder–partial tool assembly. A finite-element modeling (FEM) simulation is performed to verify the validity of inverse RCSA algorithm. The tool point FRFs at the spindle rotating state are obtained by coupling the FRFs of the spindle–holder–partial tool and the other partial tool. The effects of spindle rotational speed on tool point FRFs are investigated. The cutting experiment demonstrates that this method can accurately identify the tool point FRFs and predict cutting stability region under spindle rotating state.

A control oriented multivariable identification procedure for turbocharged diesel engines

In this paper, multisine frequency response function identification is used for non-parametric modelling of the air path of a turbocharged diesel engine around a fixed operating point. The variable geometry turbine (VGT) and exhaust gas recirculation (EGR) valve are used as inputs. The nitrogen oxide emissions, air-fuel equivalence ratio, and pressure difference between the intake and exhaust manifold are the considered outputs. A time-efficient and accurate identification procedure for this input-output system is devised, analysed, and executed experimentally for a range of operating points. The analysis quantifies the individual effects of noise and nonlinearities and shows that the identified linear models capture the actual local behaviour with over 97.2% accuracy.

Improved RCSA technique for efficient tool-tip dynamics prediction

Chatter prediction requires Frequency Response Function (FRF) at the tool-tip, which can be reliably identified by modal testing. However this procedure is time-consuming and unsuitable for industrial application since measurements must be performed for each tool attached to the machine. Receptance coupling technique can be adopted to speed up FRF identification. This technique computes tool-tip response combining experimental spindle-holder dynamics with numerical/analytical model of the tool. The main drawback is to identify rotational degree of freedom responses that are generally obtained by performing additional calibrations tests. This work presents a new method overcoming this limitation, obtaining rotational responses by a novel inverse formulation that reduces the number of required experiments. The proposed method is based on receptance coupling theory and requires FRF measurements of a single machine setup, thus increasing the efficiency of receptance coupling approach for tool-tip FRF prediction. The proposed technique is proven by experimental validation and the same accuracy of state of the art methods is obtained.

In-process tool point FRF identification under operational conditions using inverse stability solution

Self-excited vibrations of machine tools during cutting result in process instability, poor surface finish and reduced material removal rate. In order to obtain stability lobe diagrams to avoid chatter vibrations, tool point frequency response function (FRF) must be determined. In classical machine tool studies, tool point FRF is obtained experimentally or analytically for the idle state of the machine. However, during cutting operations, discrepancies are frequently observed between the stability diagrams predicted by using the FRFs measured at the idle state and the actual stability of the process. These deviations can be attributed to the changes in machine tool dynamics under cutting conditions which are difficult to measure. In this study, a new identification method is proposed for the identification of in-process tool point FRFs. In this method, experimentally determined chatter frequency and corresponding axial depth of cut are used in order to identify tool point FRF. The proposed method is applied to a real machining center and by using chatter tests it is demonstrated that the tool point FRF can be accurately identified under operational conditions.

Power Spectra Density Replication Control Strategy for 2-Axis Hydraulic Shaker Based on HV Estimator

Hydraulic shaker, equipment of simulating laboratory vibration environment, can accurately replicate the given power spectral density (PSD) and time history with an appropriate control algorithm. By studying method Hv estimator of frequency response function (FRF) estimation, a FRF identification strategy based on the Hv estimator is designed to increase the convergence rapidity and improve the system response function specialty. The system amplitude-frequency characteristics in some frequency points or frequency bands have large fluctuation. To solve this issue, a step-varying and frequency-sectioning iterative correction control algorithm is proposed for the control of 2-axial exciter PSD replication tests and the results show that the algorithm has a good effect on the control of hydraulic shaker, and can achieve reliable and high-precision PSD replication.

Direct synthesis of pseudo-random ternary perturbation signals with harmonic multiples of two and three suppressed

This paper proposes an analytical method to directly synthesise pseudo-random ternary perturbation signals for the identification of frequency response functions, through the multiplication of two components signals which satisfy a prescribed set of properties. The ternary signals generated have harmonic multiples of two and three suppressed; this specification is useful for reducing the effects of nonlinear distortion on the linear estimate. The signals have uniform magnitude in all, except two, of the nonzero harmonics. The method is significant in overcoming the existing problem of sparsity in the available periods when analytical methods are used, as well as the relatively long computational time required in approaches based on exhaustive search or computer optimisation. The proposed technique presents a breakthrough as it eliminates the sequence-to-signal conversion stage required in the existing conventional methods. A direct consequence is the increased signal power within amplitude constraints, which now equals the theoretical limit for the specification considered. If it is necessary to further increase the number of available periods, the mathematical derivation can be extended to a class of suboptimal direct synthesis signals; however, these possess reduced signal power compared with direct synthesis signals.