Nonlinear distortions and multisine signals. I. Measuring the bestlinear approximation

This paper examines the effects of nonlinearities on frequency response function measurements using periodic multifrequency signals. A class of broadband pilot test signals is proposed, termed sparse odd multisines, which can be used to establish the system bandwidth and detect nonlinearities. Signals are then defined within this class which allow the measurement of the best linear approximation of a nonlinear system. A comparison is made with related work in this area     

Design of test signals for identification of linear systems withnonlinear distortions

The design of nonbinary multifrequency excitation signals (multisines) for identifying linear systems in the presence of nonlinear distortions is examined. The use of such signals in parametric and nonparametric frequency-domain estimation is addressed, and a comparison with the maximum length binary sequence is made. The performance of a multisine composed of only prime harmonics is assessed for robustness to nonlinear effects. Practical results are presented, and guidelines for test signal design are outlined     

Nonlinear disturbance errors in system identification usingmultisine test signals

The errors introduced into linear system identification by a class of nonlinear distortions are examined. A theoretical framework is presented for the distortion generated by odd-power-nonlinearities when using multisine test signals for frequency-domain identification. It is shown that the distortion is a function of the number of test harmonics, their harmonic values, and their phases. This leads to the definition of a novel class of signals, termed no interharmonic distortion (NID) multisines, with interesting properties. An explanation of previously published practical results is then given. Application of multisines to system testing with a method of compensating for nonlinearities is illustrated with practical results     

Practical choices in the FRF measurement in presence of nonlineardistortions

The frequency response function (FRF) of the best linear approximation to a nonlinear system is usually measured by averaging system responses to a normally distributed, filtered, and clipped random excitation (Gaussian noise). This signal is compared to the multisine signal with random phases. It is shown that a random phase multisine signal defined over an even-odd frequency grid is superior to the Gaussian signal in terms of the variance and the bias of the measured FRF of the approximated nonlinear system     

Schlieren systems with coherent illumination for quantitative measurements

Schlieren systems with a coherent light source were investigated by the Fourier optics technique. The imaging properties of the systems with various cutoff filters were studied. Systems with a graded piecewise linear filter and a Gaussian step function convolution (graded) filter are considered, demonstrating that the image can be approximated by the geometrical-optics theory of conventional schlieren systems. A nonlinear phase contribution was estimated, allowing for the measurement of strong phase objects. Within the framework of linear approximation the results are described by the phase derivative point-spread function, introduced in this paper as the schlieren point-spread function. In addition, modification of the Lopez cutoff filter is proposed, demonstrating its superiority over the piecewise linear and the Gaussian step convolution filters. Simulations of coherent schlieren imaging as well as phase derivative measurements were performed. Finally, the imaging properties of the schlieren systems with the different filters are compared.     

Estimating parameterized scalable models from the best linear approximation of nonlinear systems for accurate high-level simulations

System designers of communication systems need to compare the simulated behavior of a system with the linear and nonlinear specifications. They need high-level models to perform these simulations fast. The existing high-level models for nonlinear components do not scale smoothly with external parameters like the input power. To overcome this problem, a modeling technique based on the best linear approximation is developed. The parameterized models describe trajectories of the poles and zeros as a function of the input power. The resulting models accurately describe both the linear and nonlinear behavior of the system components. They can easily be implemented in modern simulators.     

Path integral, functional method, and stochastic dynamical systems

The path-integral approach to dynamical behavior of systems subject to Gaussian white noise is presented in a straightforward manner. Starting from the Chapman-Kolmogorov equation, the transition probability density, and therefore moments and other statistics of the random response are ultimately expressed in terms of functional integrals over the sample-path space. Accordingly, various characteristic functions are replaced by a single generating functional from which moments of all orders are simply calculated through functional differentiation. This generating functional is proven to satisfy a closed system of functional differential equations. These equations are solved in the case of linear systems, their generating functional being obtained in explicit form. Also given in this paper is an integral equation satisfied by the probability densities. Three kinds of approximation method, namely perturbation expansion, Feynman's variational method, and the WKB method, are developed based on the path-integral formalism. They can be used to study the transient as well as stationary behavior of nonlinear systems.     

Mixed model estimation methods for the Rasch model

Mixed models take the dependency between observations based on the same person into account by introducing one or more random effects. After introducing the mixed model framework, it is explained, by taking the Rasch model as a generic example, how item response models can be conceptualized as generalized linear and nonlinear mixed models. Common estimation methods for generalized linear and nonlinear models are discussed. In a simulation study, the performance of four estimation methods is assessed for the Rasch model under different conditions regarding the number of items and persons, and the degree of interindividual differences. The estimation methods included in the study are: an approximation of the integral over the random effect by means of Gaussian quadrature; direct maximization with a sixth-order Laplace approximation to the integrand; a linearized approximation of the nonlinear model employing PQL2; and finally a Bayesian MCMC method. It is concluded that the estimation methods perform almost equally well, except for a slightly worse recovery of the variance parameter for PQL2 and MCMC.     

Capacitated dynamic production and remanufacturing planning under demand and return uncertainty

This paper considers a stochastic dynamic multi-product capacitated lot sizing problem with remanufacturing. Finished goods come from two sources: a standard production resource using virgin material and a remanufacturing resource that processes recoverable returns. Both the period demands and the inflow of returns are random. For this integrated stochastic production and remanufacturing problem, we propose a nonlinear model formulation that is approximated by sample averages and a piecewise linear approximation model. In the first approach, the expected values of random variables are replaced by sample averages. The idea of the piecewise linear approximation model is to replace the nonlinear functions with piecewise linear functions. The resulting mixed-integer linear programs are solved to create robust (re)manufacturing plans.     

Tail-equivalent linearization method for nonlinear random vibration

A new, non-parametric linearization method for nonlinear random vibration analysis is developed. The method employs a discrete representation of the stochastic excitation and concepts from the first-order reliability method, FORM. For a specified response threshold of the nonlinear system, the equivalent linear system is defined by matching the “design points” of the linear and nonlinear responses in the space of the standard normal random variables obtained from the discretization of the excitation. Due to this definition, the tail probability of the linear system is equal to the first-order approximation of the tail probability of the nonlinear system, this property motivating the name Tail-Equivalent Linearization Method (TELM). It is shown that the equivalent linear system is uniquely determined in terms of its impulse response function in a non-parametric form from the knowledge of the design point. The paper examines the influences of various parameters on the tail-equivalent linear system, presents an algorithm for finding the needed sequence of design points, and describes methods for determining various statistics of the nonlinear response, such as the probability distribution, the mean level-crossing rate and the first-passage probability. Applications to single- and multi-degree-of-freedom, non-degrading hysteretic systems illustrate various features of the method, and comparisons with results obtained by Monte Carlo simulations and by the conventional equivalent linearization method (ELM) demonstrate the superior accuracy of TELM over ELM, particularly for high response thresholds.     

Measurement of multivariable frequency response functions in the presence of nonlinear distortions-some practical aspects

The aim of this paper is the design of optimal multi-input periodic excitations (i.e., sets of multisines) that allow accurate calculation of a nonparametric multivariable frequency response function (MFRF) of a multiple input multiple output (MIMO) system in the presence of nonlinear distortions. Due to the presence of nonlinear distortions, it is essential that the different experiments necessary to determine the MFRF are performed at the same operating point/region. It will be shown that, in order to reduce the noise on the measured MFRF, one has to consider the class of excitation signals for which the matrix formed by the input Fourier vectors used in the different experiments has a condition number close to one for all the considered frequencies (modified E-optimality). Within this selected class of excitations with condition number close to one, the uncertainty on the measured MFRFs can further be minimized by maximizing the determinant of the Fisher information matrix. This is achieved by using crest factor optimized excitation signals. The theoretical results are experimentally verified in the field of electric machinery using a synchronous machine. Finally, a precompensation technique is proposed that further improves the signal-to-noise ratio (SNR) of the measurements.     

MIMO线性系统输入输出信号统计特征的双谱评定方法

基于高阶统计量具备处理随机信号的特性,提出了一种利用三阶谱(双谱)评定MIMO线性系统时域输入输出信号统计特征的新方法。通过建立线性系统双谱数学模型,根据系统响应、所测得的频响函数以及离散信号的双谱数值估计算法,经逆运算获得系统的双谱驱动信号,随后利用高阶谱对高斯随机信号的盲性判定其输入信号的高斯性。将上述方法与采用传统相位随机化法(对功率谱添加随机相位)所获得的驱动信号分别应用于一悬臂梁模拟控制系统中,通过对输入信号的分析及控制结果的比较,发现基于双谱所生成的时域随机驱动信号呈现出较强的非高斯性且收敛速度更快。对于输出信号统计特征的评定,提出从输入信号与系统频带接近的程度入手,再次利用高阶统计量对高斯随机信号的盲性进行定性判定,对于无法判别满足何种非高斯统计分布特征的,不管是对于输入信号还是输出信号,一律采用绘制信号的概率分布特征曲线进行定量评定。     

A survey of quantitative and qualitative methods of sensitivity analysis for stochastic dynamic systems

In this paper a brief literature survey concerning sensitivity analysis of stochastic dynamic systems described by Ito equations is presented. In the first part of the paper we review the quantitative methods in the time and frequency domain. We quote the definitions of output and moment sensitivity measures and we present applications for linear systems and nonlinear oscillators with stochastic coefficients under stochastic excitations. The cases of white and coloured noise are considered. This is followed by a discussion of the application of the output sensitivity process to response approximation of nonlinear oscillators. We discuss also the application of sensitivity methods to the approximation of characteristics of the solution of stochastic differential equations. We present another approach to the study of response sensitivity of stochastic systems in the frequency-domain with applications to secondary structural systems. In the second part of the paper we review the qualitative methods. The concept of exponential and practical insensitivity for linear and a class of nonlinear stochastic dynamic systems is presented. We quote the definitions and criteria of the exponential and practical insensitivity which are very close to the definitions of stability concerning selected response variables.     

Probabilistic analysis of concrete cracking using neural networks and random fields

In this paper an algorithm for the probabilistic analysis of concrete structures is proposed which considers material uncertainties and failure due to cracking. The fluctuations of the material parameters are modeled by means of random fields and the cracking process is represented by a discrete approach using a coupled meshless and finite element discretization. In order to analyze the complex behavior of these nonlinear systems with low numerical costs a neural network approximation of the performance functions is realized. As neural network input parameters the important random variables of the random field in the uncorrelated Gaussian space are used and the output values are the interesting response quantities such as deformation and load capacities. The neural network approximation is based on a stochastic training which uses wide spanned Latin hypercube sampling to generate the training samples. This ensures a high quality approximation over the whole domain investigated, even in regions with very small probability.     

Application of the proper orthogonal decomposition for linear and nonlinear structures under transient excitations

Model reduction has become very important in order to save calculation time. In particular, in structural dynamics, computations become very time-consuming when the critical time step of explicit integrators becomes very small. The main focus of this paper is on the application of the Proper Orthogonal Decomposition (POD) method to a structure subjected to transient earthquake loading. It is shown that based on the information of only a small portion of the transient excitation and the structure (“snapshots”), it is possible to assemble a reduced-order model, which yields a very accurate and time-saving approximation of the response to the entire earthquake. The POD reduction method is applied not only to linear, but also to nonlinear structures under earthquake loading. In the linear case, the POD results can be compared to those obtained by the classical method of modal truncation. In the nonlinear case, base isolation systems (friction pendulum systems) are integrated in the structure. Error estimations are applied in order to assess the solution of the POD-reduced system of the linear and the nonlinear systems. The POD can be applied successfully if the snapshots within the chosen time interval describe the main behavior of the system well. In both the linear and nonlinear cases, the approximation of the system as reduced by the POD is very accurate even if only a few POD modes are used. The advantage over the method of Modal Truncation is not only the optimality of the POD modes concerning their associated energy, but also its applicability to nonlinear systems.     

Flower Pollination Heuristics for Nonlinear Active Noise Control Systems

Abstract In this paper, a novel design of the flower pollination algorithm is presented for model identification problems in nonlinear active noise control systems. The recently introduced flower pollination based heuristics is implemented to minimize the mean squared error based merit/cost function representing the scenarios of active noise control system with linear/nonlinear and primary/secondary paths based on the sinusoidal signal, random and complex random signals as noise interferences. The flower pollination heuristics based active noise controllers are formulated through exploitation of nonlinear filtering with Volterra series. The comparative study on statistical observations in terms of accuracy, convergence and complexity measures demonstrates that the proposed meta-heuristic of flower pollination algorithm is reliable, accurate, stable as well as robust for active noise control system. The accuracy of the proposed nature inspired computing of flower pollination is in good agreement with the state of the art counterpart solvers based on variants of genetic algorithms, particle swarm optimization, backtracking search optimization algorithm, fireworks optimization algorithm along with their memetic combination with local search methodologies. Moreover, the central tendency and variation based statistical indices further validate the consistency and reliability of the proposed scheme mimic the mathematical model for the process of flower pollination systems.     

Explicit solutions for the response probability density function of nonlinear transformations of static random inputs

This work is the second paper of two companion ones. Both of them show the use of a new version of the Probabilistic Transformation Method (PTM) for finding the probability density function (pdf) of a limited number of response quantities in the transformations of static random inputs. This is made without performing multi-dimensional integrals of the response total joint pdf for saturating the non-interested variables. While in the first paper the linear transformations have been considered, in the present one some nonlinear systems are taken into account. In particular, first the case when the loads on a linear structural system are a nonlinear combination of static random inputs is studied. Then the attention is placed on the case of nonlinear structural systems, for which the new version of the PTM allows to determine approximated, but accurate, results.     

Stochastic Perturbation Model for Optimal Position Transfer Characteristic

An analytic model describing the transfer properties of a discrete photosensitive element position detection system is developed. Improvement of the inherently nonlinear transfer characteristic is shown to be possible when the target is subject to perturbation in the form of Gaussian noise. Applying a spline function approximation approach, relationship between the element threshold, noise variance, and element displacement are obtained which achieve the best (Chebyschev) approximation to an ideal (linear) transfer characteristic. Effective smoothing is shown to be obtainable for relatively small noise variance.     

一种新的估计多项式相位信号瞬时频率的参数化时频分析方法

通过多项式非线性核函数取代线性调频小波变换中的线性核函数,提出一种新的参数化时频分析方法：非线性调频小波变换。对瞬时频率是时间任意连续函数的信号而言,选择合适的多项式核特征参数,非线性调频小波变换的时频分布有良好的时频聚集性。应用非线性调频小波变换分析任意阶次多项式相位信号。由于非线性调频小波变换的性能取决于多项式核特征参数,本文还给出非线性调频小波变换的核特征参数估计算法,进一步可实现多项式相位信号的瞬时频率和参量估计。仿真信号验证算法的有效性。     

Nonstationary equivalent linearization of nonlinear continuous systems

This paper presents a technique for obtaining the nonstationary stochastic response of a nonlinear continuous system. The method of equivalent linearization is generalized to continuous systems subjected to nonstationary random excitation. This technique allows replacement of the original nonlinear system with a time-varying linear continuous system. A numerical implementation is also described. In this procedure, the linear replacement system is discretized by the finite element method. Application to systems satisfying the one-dimensional wave equation with a constitutive nonlinearity is discussed. Results are presented for nonlinear stress-strain laws of a strain-hardening type.