REAL-TIME MODAL PARAMETER ESTIMATION USING SUBSPACE METHODS: THEORY

This article describes the underlying theory of a newly developed algorithm for online modal parameter identification. These online subspace estimation methods use eigenanalysis for data filtering, and are derived from a recent multi-input, multi-output batch algorithm. One method is obtained by deriving a new efficient data update expression combined with a recently developed modified singular value decomposition known as the URV method. The second method combines an existing data update expression with the URV method. The URV method enables recursive update of the signal subspace. The close relationship of a modified form of the batch estimation approach to the Eigensystem Realization Algorithm (ERA) is also shown through the introduction of an extended ERA method.     

Recursive parameter estimation for Hammerstein-Wiener systems using modified EKF algorithm

This paper focuses on the recursive parameter estimation for the single input single output Hammerstein-Wiener system model, and the study is then extended to a rarely mentioned multiple input single output Hammerstein-Wiener system. Inspired by the extended Kalman filter algorithm, two basic recursive algorithms are derived from the first and the second order Taylor approximation. Based on the form of the first order approximation algorithm, a modified algorithm with larger parameter convergence domain is proposed to cope with the problem of small parameter convergence domain of the first order one and the application limit of the second order one. The validity of the modification on the expansion of convergence domain is shown from the convergence analysis and is demonstrated with two simulation cases.     

THE EXTENDED KALMAN FILTER IN THE FREQUENCY DOMAIN FOR THE IDENTIFICATION OF MECHANICAL STRUCTURES EXCITED BY SINUSOIDAL MULTIPLE INPUTS

A modal parameter identification method applied to mechanical structures excited by correlated sinusoidal multiple inputs was developed. The algorithm is based on the same formulation of the extended Kalman filter, applied as a system parameter identifier in the frequency domain to mechanical structures subject to excitations characterised by an inherently high degree of correlation. The algorithm was validated by using simulated data on a multi-degree-of-freedom system. The tests demonstrate that the proposed technique is of practical application value. The method is devoted to the identification of the modal parameters of supporting structures of rotating machinery, using data obtained during the normal operation of the machines. Nonetheless, it can have an interest for more general applications in the field of the experimental modal analysis. This study was undertaken in the framework of the BRITE EURAM III project MODIAROT (MOdel based DIAgnosis of ROTors in power plants).     

基于时变多变量Prony法的时变振动系统模态参数辨识

在经典的Prony法理论的基础上,提出了可以同时处理多维非平稳信号的时变多变量Prony法,并将其应用于时变多自由度振动系统的模态参数辨识。对传统的递推最小二乘算法加以改进,解决了时变多变量参数模型中时变参数矩阵估计的难题。对时变平面两杆操作臂系统进行仿真和分析,得出了较满意的结果。证明该算法在时变结构模态参数辨识方面,具有有效、准确的计算能力和较强的过程跟踪能力。     

用于线性时变系统辨识的固定长度平移窗投影估计递推子空间方法

给出了一个新的用于线性时变参数结构系统模态参数识别的基于固定长度平移窗投影估计的递推子空间方法。首先推得基于平移窗投影估计的子空间跟踪算法,以代替奇异值分解,再推得系统数据矩阵的一阶修正形式,从而得到新的基于平移窗投影估计的递推子空间方法。该方法可有效地降低算法的计算量。最后通过刚度随时间变化的3自由度系统和一具有移动质量的机械臂系统的时变模态频率辨识仿真表明该方法可有效地辨识线性时变系统的伪模态参数。     

Recursive subspace identification for on-line tracking of structural modal parameter

The objective of this paper is to develop an on-line tracking of system parameter estimation and damage detection techniques using response measurements. To avoid the singular-value-decomposition in data Hankel matrix, a new subspace identification algorithm was developed. Seismic response data of a 3-story steel frame with abrupt change of inter-story stiffness from the shaking table test was used to verify the proposed recursive subspace identification (RSI) method by using both input and output measurements. With the implementation of forgetting factor in RSI method the ability of on-line damage detection of the abrupt change of structural stiffness can be enhanced. Then, the recursive stochastic subspace identification (RSSI) algorithm is also developed for continuous structural health monitor of structure by using the output-only measurements. Verification of the proposed RSSI method by using the white noise response data of a 2-story reinforced concrete frame from its low level white noise excitation was used. Discussion of the subspace identification model parameters is also investigated.     

COUPLING OF NON-LINEAR SUBSTRUCTURES USING VARIABLE MODAL PARAMETERS

This paper presents a general methodology for the coupling analysis of systems with relatively weak non-linearities by assuming that the response remains harmonic under harmonic excitation. Standard coupling methods and their current shortcomings were discussed first. Two ways of obtaining non-linear modal parameter variations, namely profile constructing and parameter extracting, were presented next. The profile constructing method uses the system's spatial data directly, while the parameter extracting method is based on a non-linear modal analysis of measured response data. Through numerical test cases, it was shown that both methods yielded virtually identical results. An iterative algorithm for the coupling of non-linear subsystems was presented in a form compatible with profile building. A six-degree-of-freedom system with cubic stiffness non-linearity was chosen for a detailed numerical study. Two subsystems, one linear and the other non-linear, were coupled to obtain the modal parameter variations of the coupled system. Using the non-linear modal parameters, the response of the coupled system was predicted at various force levels and the findings were checked via direct simulations using the harmonic balance method. Finally, the methodology was validated by coupling experimentally derived non-linear modal models of two substructures. As for the numerical study, the response of the coupled non-linear structure was predicted at various force levels and the findings were checked against direct measurements. Very good agreement was obtained in all cases studied.     

Recursive generalized extended least squares and RML algorithms for identification of bilinear systems with ARMA noise

Bilinear systems are considered as a particular class of nonlinear systems including the state variables which are typically used for online identification. By using a recursive identification method and the maximum likelihood principle, this paper presents two recursive-based algorithms to identify the parameters of bilinear in parameter systems with ARMA noise. In this regard, recursive generalized extended least squares (RGELS) and recursive Maximum Likelihood (RML) algorithms have been proposed for identification of bilinear systems. These algorithms can be used as an alternative choice in system identification with acceptable performance. The proposed algorithms estimate the correlated noise parameters with high accuracy by making full use of the measurement data. Simulation results indicate that the proposed algorithms are effective for online identification of bilinear in parameter systems with high convergence speed.     

Structural identification based on optimally weighted modal residuals

The structural parameter estimation problem based on measured modal data is often formulated as a weighted least-squares problem in which modal residuals measuring the fit between experimental and model predicted modal properties are build up into a single weighted residuals metric using weighting factors. Standard optimisation techniques are then used to find the optimal values of the structural parameters that minimise the weighted residuals metric. Due to model error and measurement noise, the results of the optimisation are affected by the values assumed for the weighting factors. In this work, the parameter estimation problem is first formulated as a multi-objective identification problem for which all Pareto optimal structural parameter values are obtained, corresponding to all possible values of the weights. A Bayesian statistical framework is then used to rationally select the optimal values of the weights based on the measured modal data. It is shown that the optimal weight value for a group of modal properties is asymptotically, for large number of measured data, inversely proportional to the optimal value of the residuals of the modal group. A computationally efficient algorithm is proposed for simultaneously obtaining the optimal weight values and the corresponding optimal values of the structural parameters. The proposed framework is illustrated using simulated data from a multi-dof spring–mass chain structure. In particular, compared to conventional parameter estimation techniques that are based on pre-selected values of the weights, it is demonstrated that the optimal parameter values estimated by the proposed methodology are insensitive to large model errors or bad measured modal data.     

Recursive least squares based sliding mode approach for position control of DC motors with self-tuning rule

Journal of Mechanical Science and Technology - In this paper, a self-tuning rule-based position control algorithm is proposed for DC motors with system parameter estimation using the recursive...     

Integrated stepped-sine system for modal analysis

The inherent qualities of the stepped-sine excitation technique for frequency response measurements are generally accepted, but several factors have made its practical implementation as a valid alternative to today's broadband excitation methods very difficult. The dedicated hardware for harmonic testing and the incompatibility of the obtained frequency response functions with current commonly used parameter estimation methods inhibited the integration of both methods into a single modal analysis system. In order to overcome these limitations, a versatile measurement front-end, ensuring the compatibility of these procedures, as well as new multiple input parameter estimation methods have been developed. This paper discusses the integration of stepped-sine excitation into a standard modal testing system and illustrates its application with a complete dual input test example.     

Grey forecasting run-to-run control system in copper chemical mechanical polishing

In this paper, an online grey forecasting run-to-run control system was proposed with the integration of run-to-run control system, recursive least-squares (RLS) algorithm, and grey forecasting model (GFM). One of the objectives of this study is to explore the possibility and feasibility of applying GFM to run-to-run control system in copper chemical mechanical polishing. Under the condition of limited experiment data, GFM is excellent at estimating and forecasting error of the next batch online. To keep the process under control, the controllers are then employed to adjust the process parameters in order to compensate the error. In addition, the RLS algorithm is used to construct dynamically a system estimation matrix for the purpose of stating precisely the relationship between process quality and process parameters, and to consequently improve processing performances. From the computer simulation and the experiment results, the proposed new method developed in this study was able not only to confine the processing performances’ error within the range of 5% but also to supplement, when the process parameters are saturated, the control capability through adjusting other unsaturated process parameters, thus maintaining good processing performances.     

ARMAX modal parameter identification in the presence of unmeasured excitation—II: Numerical and experimental verification

The performance of an autoregressive moving average with exogenous excitation (ARMAX) time-domain parameter estimation algorithm is assessed using data simulating the response of a two degree-of-freedom system and also experimental data obtained from a cantilever aluminium beam. The algorithm is aimed at identifying modal parameters in the presence of significant measurement noise and unmeasured sources of periodic and random excitation. The simulated system exhibits lightly damped modes and is excited with measured random excitation as well as unmeasured random and periodic excitation. A second case is considered where the modal frequencies of the simulated system are separated by 0.345 Hz and the frequency of an unmeasured periodic excitation is 0.257 Hz below the first modal frequency. Electrodynamic shakers as well as piezoceramic plates are used to apply multiple independent excitation forces, including measured random noise and also unmeasured random and periodic signals. Successful identification of modal parameters is achieved in the presence of up to 10% random measurement noise, as well as over 200% unmeasured periodic and random excitations. Results showed that the ARMAX algorithm achieved better results than ARX models estimated from experimental data corrupted by the effects of unmeasured excitations. The use of piezoceramic plates for structural excitation was demonstrated in experiments and the ability to excite modes was observed to be dependent on the deflection of a particular mode in the contact area of the piezoceramic plates.     

A fast iterative recursive least squares algorithm for Wiener model identification of highly nonlinear systems

In this paper, an online identification algorithm is presented for nonlinear systems in the presence of output colored noise. The proposed method is based on extended recursive least squares (ERLS) algorithm, where the identified system is in polynomial Wiener form. To this end, an unknown intermediate signal is estimated by using an inner iterative algorithm. The iterative recursive algorithm adaptively modifies the vector of parameters of the presented Wiener model when the system parameters vary. In addition, to increase the robustness of the proposed method against variations, a robust RLS algorithm is applied to the model. Simulation results are provided to show the effectiveness of the proposed approach. Results confirm that the proposed method has fast convergence rate with robust characteristics, which increases the efficiency of the proposed model and identification approach. For instance, the FIT criterion will be achieved 92% in CSTR process where about 400 data is used.     

一种基于卡尔曼滤波的非线性模型在线模糊辨识算法

文章提出一种用于非线性模型在线辨识的模糊算法。该算法将非线性输入输出系统用时变线性系统模型来拟和，并把此非线性系统模型表示成模糊模型的形式，用在线调节模糊模型的方法来辨识时变线性模型的相关参数。本文将递推模糊聚类方法与卡尔曼滤波法用于在线调整模糊模型参数。仿真算例表明了此算法的有效性。     

Parameter estimation for a dual-rate system with time delay

This paper investigates the parameter estimation problem of the dual-rate system with time delay. The slow-rate model of the dual-rate system with time delay is derived by using the discretization technique. The parameters and states of the system are simultaneously estimated. The states are estimated by using the Kalman filter, and the parameters are estimated based on the stochastic gradient algorithm or the recursive least squares algorithm. When concerning state estimate of the dual-rate system with time delay, the state augmentation method is employed with lower computational load than that of the conventional one. Simulation examples and an experimental study are given to illustrate the proposed algorithm.     

基于十字链叉的NURBS曲面插补算法及仿真分析

针对目前曲面插补算法存在的实时性不强,插补算法需要大量的额外空间,算法步骤繁琐以及对插补过程产生的大量数据点的数据处理问题,为了保证作为数控技术核心模块的曲面插补算法的高速、高精度性能,基于德布尔递推算法思想,重新推导出了适合于曲面插补的新的NURBS曲面表达式,在此基础上重新设计了曲面插补快速算法,并着重研究了新算法的实时性.另外,为验证新算法的正确性和有效性,提出并实现了十字链叉数据表达结构,对NURBS曲面插补点微细步长及插补质量进行了分析.建立了以VC6.0/MFC为平台的、集弓高误差、速度、加速度等功能分析于一体的算法仿真验证系统,并以汽轮机叶片为例进行了验证.研究结果表明,该算法可以满足曲面插补高速、高精度的性能要求.     

基于Hankel矩阵联合近似对角化的结构模态识别

针对基于二阶盲辨识(second order blind identification,简称SOBI)的模态参数识别方法存在的不足,提出了一种基于Hankel矩阵联合近似对角化(Hankel matrix joint approximate diagonalization,简称HJAD)技术的结构运行模态分析(operational modal analysis,简称OMA)的新方法。该方法通过对随机子空间类模态识别方法常用的Hankel矩阵进行联合近似对角化,以分离各阶模态响应,进行模态识别。与基于SOBI的模态识别方法相比,在具体实施过程中,仅需要在分析数据中添加与实测振动响应对应的时间延迟的数据,实现难度较小。数值算例和物理模型试验的分析结果表明,所提出的基于HJAD技术的结构运行模态分析方法,不仅具有鲁棒性强和计算效率高的优点,还可以克服传统的基于SOBI的模态识别方法的模态识别能力受测点数目限制的问题。     

基于模态参数识别的塔式起重机金属结构损伤检测有限元方法

以塔式起重机的一个标准节为研究对象,采用有限元软件对其金属结构进行了模态分析,分析了金属结构损伤位置和模态参数之间的关系,推导出了表征系统损伤位置的系统损伤矩阵表达式。并在Msc.Patran软件上进行了建模仿真,通过测量该模型系统的模态参数,利用损伤矩阵找出了结构的单一和多重损伤,验证了该方法的在该结构上的可行性。     

基于SVM的多传感器信息融合算法

支持向量机(Support Vector machine,简称SVM)是一种基于结构风险最小化原理,具有很高泛化性能的学习算法。针对工业多传感器测控系统中,被测系数与相关参数之间存在有较大的非线性和模糊关系,提出了一种基于支持SVM的多传感器信息融合模型及算法。为小样本、非线性、高维数一类多传感器信息融合问题的建模提供了一种有效的途径。通过对“纸张水份在线测量系统”应用表明,基于SVM的多传感器信息融合模型及算法在测量精度和推广性能上都具有一定的优越性。