New bound characteristics of NARX model in the frequency domain

New results about the bound characteristics of both the generalized frequency response functions (GFRFs) and the output frequency response for the NARX (Non-linear AutoRegressive model with eXogenous input) model are established. It is shown that the magnitudes of the GFRFs and the system output spectrum can all be bounded by a polynomial function of the magnitude bound of the first order GFRF, and the coefficients of the polynomial are functions of the NARX model parameters. These new bound characteristics of the NARX model provide an important insight into the relationship between the model parameters and the magnitudes of the system frequency response functions, reveal the effect of the model parameters on the stability of the NARX model to a certain extent, and provide a useful technique for the magnitude based analysis of nonlinear systems in the frequency domain, for example, evaluation of the truncation error in a volterra series expression of non-linear systems and the highest order needed in the volterra series approximation. A numerical example is given to demonstrate the effectiveness of the theoretical results.     

Mapping from parametric characteristics to generalized frequency response functions of non-linear systems

Based on the parametric characteristic of the nth-order generalized frequency response function (GFRF) for non-linear systems described by a non-linear differential equation (NDE) model, a mapping function from the parametric characteristics to the GFRFs is established, by which the nth-order GFRF can be directly written into a more straightforward and meaningful form in terms of the first order GFRF, i.e., an n-degree polynomial function of the first order GFRF. The new expression has no recursive relationship between different order GFRFs, and demonstrates some new properties of the GFRFs which can explicitly unveil the linear and non-linear factors included in the GFRFs, and reveal clearly the relationship between the nth-order GFRF and its parametric characteristic, as well as the relationship between the nth-order GFRF and the first order GFRF. The new results provide a novel and useful insight into the frequency domain analysis and design of non-linear systems based on the GFRFs. Several examples are given to illustrate the theoretical results.     

Uniquely connecting frequency domain representations of given order polynomial Wiener–Hammerstein systems

The notion of frequency response functions has been generalized to nonlinear systems in several ways. However, a relation between different approaches has not yet been established. In this paper, frequency domain representations for nonlinear systems are uniquely connected for a class of nonlinear systems. Specifically, by means of novel analytical results, the generalized frequency response function (GFRF) and the higher order sinusoidal input describing function (HOSIDF) for polynomial Wiener–Hammerstein systems are explicitly related, assuming the linear dynamics are known. Necessary and sufficient conditions for this relation to exist and results on the uniqueness and equivalence of the HOSIDF and GFRF are provided. Finally, this yields an efficient computational procedure for computing the GFRF from the HOSIDF and vice versa.     

Output frequency response function of nonlinear Volterra systems

An expression for the output frequency response function (OFRF), which defines the explicit analytical relationship between the output spectrum and the system parameters, is derived for nonlinear systems which can be described by a polynomial form differential equation model. An effective algorithm is developed to determine the OFRF directly from system simulation or experimental data. Simulation studies demonstrate the significance of the OFRF concept, and verify the effectiveness of the algorithm which evaluates the OFRF numerically. These new results provide an important basis for the analytical study and design of a wide class of nonlinear systems in the frequency domain.     

基于GFRF的二次调节流量耦联系统的频域非线性H∞控制

二次调节流量耦联系统为非线性系统, 在Volterra 级数描述该系统的基础上, 通过SISO多项式类非线性系统的GFRF 递推算式获得二次调节流量耦联系统的广义频率响应函数GFRF(generalized frequency response function). 基于系统的GFRF, 在频域内应用非线性控制理论为系统设计了镇定控制器和非线性H_∞控制器, 不仅使系统稳定, 而且能达到无超调、无稳态误差. 此外, 在白噪声条件下证明该控制器比线性控制器抗干扰性强.     

Frequency domain analysis for non-linear Volterra systems with a general non-linear output function

For non-linear Volterra systems which include a non-linear state equation and a general non-linear output function, the system frequency response functions and some related frequency response characteristics are developed and discussed in this study. These new results establish the frequency response functions for this general form of non-linear systems by extending some existing theory, and provide an analytical insight into the relationship between model parameters and frequency response functions, and the relationship between model parameters and the magnitude bound of frequency response functions. Several examples are given to illustrate the new results.     

基于广义频率响应函数的非线性控制系统的闭环稳定性研究

本文基于广义频率响应函数，讨论了一类非线性控制系统的闭环性问题，给出了输入输出频域稳定性判据条件，最后，利用仿真例子对结论进行了验证。     

结合非线性频谱与核主元分析的复杂系统故障诊断方法

传统非线性频谱分析方法对复杂系统进行故障诊断时,求解出的非线性频谱数据量庞大,不便于直接用于故障检测与分类识别.本文提出了一种非线性频谱特征与核主元分析(KPCA)结合的故障诊断方法,首先通过最小二乘算法估计出前3阶Volterra时域核,由多维傅立叶变换求取出广义频率响应函数,然后利用KPCA方法对谱数据进行压缩与提取谱特征,最后利用多分类最小二乘支持向量机进行多故障检测与识别.考虑到频谱数据具有非线性的特点,KPCA中的核函数选用由多项式函数与径向基函数构成的混合核函数,兼顾了局部特性与全局特性.论文基于非线性频谱数据,给出了核主元模型建立与在线故障诊断的具体算法.对非线性模拟电路和数控机床伺服传动系统进行了仿真实验,结果表明本文方法能够大幅度降低频谱数据维数,故障识别率高,是一种实用的故障诊断方法.     

Estimation of Multi‐Order Spectra for Nonlinear Closed‐Loop Systems

In this paper, a multi‐order spectra estimation method is proposed for a nonlinear closed‐loop system based on the Volterra series. Owing to the correlation between the noise and the input, the estimation accuracy is poor when the nonlinear spectra of the plant is obtained using the traditional estimation method. In order to overcome this problem, a two‐step scheme is used to estimate the multi‐order spectrum of a nonlinear system operating in closed‐loop. Firstly, the generalized frequency response functions (GFRFs) from the reference signal to the input of the plant are estimated, and they are used to simulate the noise‐free input spectra of the plant. Secondly, the GFRFs of the controlled plant are estimated using the noise‐free input spectra and the output spectra. Because the GFRFs are multidimensional functions, the required amount of calculation for the estimation is very large. To reduce computational complexity, a simplified GFRF model is adopted to estimate the multi‐order nonlinear spectrum of the plant. In this model, the GFRF is transformed to a one‐dimensional function. Two simulation experiments are provided to illustrate the proposed approach.     

Component mode synthesis and polynomial chaos expansions for stochastic frequency functions of large linear FE models

This paper presents a methodological approach for the numerical investigation of frequency transfer functions for large FE systems with linear and nonlinear stochastic parameters. The component mode synthesis methods are used to reduce the size of the model and are extended to stochastic structural vibrations. The statistical first two moments of frequency transfer functions are obtained by an adaptive polynomial chaos expansion. Free and fixed interface methods with and without reduction of interface dof are used. The coupling with the first and second order polynomial chaos expansion is elaborated for beams and assembled plates with linear and nonlinear stochastic parameters.     

Identification and frequency domain analysis of non-stationary and nonlinear systems using time-varying NARMAX models

This paper introduces a new approach for nonlinear and non-stationary (time-varying) system identification based on time-varying nonlinear autoregressive moving average with exogenous variable (TV-NARMAX) models. The challenging model structure selection and parameter tracking problems are solved by combining a multiwavelet basis function expansion of the time-varying parameters with an orthogonal least squares algorithm. Numerical examples demonstrate that the proposed approach can track rapid time-varying effects in nonlinear systems more accurately than the standard recursive algorithms. Based on the identified time domain model, a new frequency domain analysis approach is introduced based on a time-varying generalised frequency response function (TV-GFRF) concept, which enables the analysis of nonlinear, non-stationary systems in the frequency domain. Features in the TV-GFRFs which depend on the TV-NARMAX model structure and time-varying parameters are investigated. It is shown that the high-dimensional frequency features can be visualised in a low-dimensional time–frequency space.     

基于多维泰勒网的非线性时间序列预测方法及其应用

针对非线性时间序列, 提出一种基于多维泰勒网的时间序列预测方法. 其特点在于利用非线性时间序列的观测数据, 通过多维泰勒网得到?? 元一阶多项式差分方程组, 在无需待预测系统的任何先验知识和机理的情况下获得动力学特性描述, 实现对非线性时间序列的预测. 最后分别采用Lorenz 混沌时间序列, 以及某大型建筑在顶升施工安全性监测中的结构振动响应数据进行实证研究, 所得结果表明了该方法的有效性.

     

动态系统"类等效"模型测辩及多可调参数的Padé模型降阶

本文提出了一种时域和频域测辨相结合的线性定常系统测辨新方法.当高阶系统传递函 数已知时,它就是具有多可调参数的Padé逼近.此法由测辨高阶系统前n阶时间矩和测辨 描述系统主要动态性能的一些典型频率响应数据,确定动态系统"类等效"简化模型的参数. 此外,"外推函数"的使用有效地提高了时矩测辨的精度.实例表明,本方法简易、精确和灵活, 适于工程建模,便于工程设计.     

An investigation into the characteristics of non-linear frequency response functions. Part 1: Understanding the higher dimensional frequency spaces

The characteristics of generalized frequency response functions (GFRFs) of non-linear systems in higher dimensional space are investigated using a combination of graphical and symbolic decomposition techniques. It is shown how a systematic analysis can be achieved for a wide class of non-linear systems in the frequency domain using the proposed methods. The paper is divided into two parts. In Part 1, the concepts of input and output frequency subdomains are introduced to give insight into the relationship between one dimensional and multi-dimensional frequency spaces. The visualization of both magnitude and phase responses of third order generalized frequency response functions is presented for the first time. In Part 2 symbolic expansion techniques are introduced and new methods are developed to analyse the properties of generalized frequency response functions of non-linear systems described by the NARMAX class of models. Case studies are included in Part 2 to illustrate the application of the new methods.     

Analysis of FLC with changing fuzzy variables in frequency domain

This paper discusses a simple method for analyzing FLC in frequency domain based on describing function. Since nonlinear characteristics of FLC make it difficult FLC analysis, it usually requires a big deal of trial-and-error procedures based on computer simulation. The proposed method is simple and easy to understand, because it is based on the Nyquist stability criterion used to analyze absolute and relative stability, phase and gain margin of a linear system. To linearize in frequency domain, a describing function for FLC is derived by using a piecewise linearization of the FLC response plot. This describing function is represented as a function of magnitude of input sinusoid and nonlinear parameters x ₁ and x ₂ which change consequence fuzzy variables and nonlinearity of FLC. The describing function is redefined without the magnitude of sinusoid input because maximum values of the describing function can explain the stability of the system. This redefined describing function is used to get minimum stability characteristic, an absolute stability, phase margin and gain margin, of FLC. Using this function, we can explicitly figure out various characteristic of FLC according to x ₁ and x ₂ in frequency domain. In this work, we suggest a minimum phase margin (MPM) and a minimum gain margin (MGM) for FLC which can be used to determine whether the system is stable or not and how stable it is. For simplicity, we use one-input FLC with three rules. For various nonlinear response of FLC, changing fuzzy variables of a consequence membership function is used. Simulation results show that these parameters are effective in analyzing FLC.     

Identification of linear systems using polynomial kernels in the frequency domain

In prior work we presented an identification algorithm using polynomials in the time domain. In this article, we extend this algorithm to include polynomials in the frequency domain. A polynomial is used to represent the imaginary part of the Fourier transform of the impulse response. The Hilbert transform relationship is used to compute the real part of the frequency response and hence the complete process model. The polynomial parameters are computed based on the computationally efficient linear least square method. The order of the polynomial is estimated based on residue decrement. Simulated and experimental results show the effectiveness of this method, particularly for short input/output data sequence with high signal to noise ratio. The frequency domain polynomial model complements the time domain methods since it can provide a good estimate of the time to steady state for time domain FIR (finite impulse response) models. Confidence limits in time or frequency domain can be computed using this approach. Noise rejection properties of the algorithm are illustrated using data from both simulated and real processes.     

Construction of bode envelopes using REP based range finding algorithms

The frequency domain analysis of systems is an important topic in control theory. Powerful graphical tools exist in classic control, such as the Nyquist plot, Bode plots, and Nichols chart. These methods have been widely used to evaluate the frequency domain behavior of system. A literature survey shows that various approaches are available for the computation of the frequency response of control systems under different types of parametric dependencies, such as affine, multi-linear, polynomial, etc. However, there is a lack of tools in the literature to construct the Bode envelopes for the general nonlinear type of parametric dependencies. In this paper, we address the problem of computation of the envelope of Bode frequency response of a non-rational transfer function with nonlinear parametric uncertainties varying over a box. We propose two techniques to compute the Bode envelopes:first, based on the natural interval extensions (NIE) combined with uniform subdivision and second, based on the existing Taylor model combined with subdivision strategy. We also propose the algorithms to further speed up both methods through extrapolation techniques.     

The vibration transmissibility of a single degree of freedom oscillator with nonlinear fractional order damping

Motivated by the theoretical analysis of the effects of nonlinear viscous damping on vibration isolation using the output frequency response function approach, the output frequency response function approach is employed to investigate the effects of the nonlinear fractional order damping on vibration isolation based on Volterra series in the frequency domain. First, the recursive algorithm which is proposed by Billings et al. is extended to deal with the system with fractional order terms. Then, the analytical relationships are established among the force transmissibility, nonlinear characteristic coefficients and fractional order parameters for the single degree of freedom oscillator. Consequently, the effects of the nonlinear system parameters on the force transmissibility are discussed in detail. The theoretical analysis reveals that the force transmissibility of the oscillator is suppressed due to the existence of the fractional order damping, but presents different effects on suppressing the force transmissibility of the oscillator over the frequency region by varying the fractional order parameters. Moreover, the fractional order parameters, which affect the force transmissibility, the bandwidth of the frequency region and the resonance frequency, can be used as designing parameters for vibration isolation systems. At last, numerical studies are presented to illustrate the theoretical results.     

Modeling Multibody Systems with Uncertainties. Part I: Theoretical and Computational Aspects

This study explores the use of generalized polynomial chaos theory for modeling complex nonlinear multibody dynamic systems in the presence of parametric and external uncertainty. The polynomial chaos framework has been chosen because it offers an efficient computational approach for the large, nonlinear multibody models of engineering systems of interest, where the number of uncertain parameters is relatively small, while the magnitude of uncertainties can be very large (e.g., vehicle-soil interaction). The proposed methodology allows the quantification of uncertainty distributions in both time and frequency domains, and enables the simulations of multibody systems to produce results with “error bars”. The first part of this study presents the theoretical and computational aspects of the polynomial chaos methodology. Both unconstrained and constrained formulations of multibody dynamics are considered. Direct stochastic collocation is proposed as less expensive alternative to the traditional Galerkin approach. It is established that stochastic collocation is equivalent to a stochastic response surface approach. We show that multi-dimensional basis functions are constructed as tensor products of one-dimensional basis functions and discuss the treatment of polynomial and trigonometric nonlinearities. Parametric uncertainties are modeled by finite-support probability densities. Stochastic forcings are discretized using truncated Karhunen-Loeve expansions. The companion paper “Modeling Multibody Dynamic Systems With Uncertainties. Part II: Numerical Applications” illustrates the use of the proposed methodology on a selected set of test problems. The overall conclusion is that despite its limitations, polynomial chaos is a powerful approach for the simulation of multibody systems with uncertainties.     

A bound for the magnitude characteristics of nonlinear output frequency response functions Part 2: Practical computation of the bound for systems described by the nonlinear autoregressive model with exogenous input

In Part 1 of this paper the concept of a bound for the output frequency response magnitude characteristics of nonlinear systems was proposed, and general calculation and analysis procedures were developed. In this, Part 2 of the paper, a new recursive algorithm for the computation of the gain bounds for the generalized frequency response functions of the polynomial nonlinear autoregressive model with exogenous input is proposed, and effective procedures for the practical computation of the new bound are developed. Simulated examples are included to verify the effectiveness of the proposed procedures.