Magnitude bounds of generalized frequency response functions for nonlinear Volterra systems described by NARX model

In order to reveal the relationship between system time domain model parameters and system frequency response functions, new magnitude bounds of frequency response functions for nonlinear Volterra systems described by NARX model are established. The magnitude bound of the nth-order generalized frequency response function (GFRF) can be expressed as a simple n-degree polynomial function of the magnitude of the first order GFRF, whose coefficients are functions of the model parameters and frequency variables. Thus the system output spectrum can also be bounded by a polynomial function of the magnitude of the first order GFRF. These results demonstrate explicitly the analytical relationship between model parameters and system frequency response functions, and provide a significant insight into the magnitude based analysis and synthesis of nonlinear systems in the frequency domain.     

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.     

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.     

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.     

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.     

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.     

A parametric frequency response method for non-linear time-varying systems

A new parametric frequency response algorithm is introduced to investigate linear and non-linear dynamic systems with time-varying parameters. In the new algorithm the time-varying parameters are regarded as additional inputs of the systems and the non-linear generalised frequency response functions for multi-input-single-output systems are then employed to obtain Zadeh's system functions from a differential equation representation. The parametric frequency response method reveals how the time-varying parameters affect the behaviour of the systems through a time-varying term. The new method can be applied to both linear and non-linear time-varying systems.     

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.     

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.     

多频简谐激励下裂纹梁的非线性振动响应

主要对含裂纹梁在振动与超声波联合激励下所出现的非线性动力响应的机理和特性进行研究.将疲劳裂纹在外加激励下的状态简化为周期性张开-闭合的非线性过程,基于圣维南原理,采用有限元方法建立了含非对称疲劳裂纹梁的非线性数值分析模型.利用非线性输出频率响应函数(NOFRFs)概念,对裂纹梁在高-低频简谐激励下所出现的非线性动力响应特性的机理进行了解释.具体以悬臂梁为例,仿真分析了裂纹深度和裂纹位置等参数的变化对系统非线性动力响应特性的影响规律.     

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.     

基于NARMAX模型和NOFRF结构损伤检测的实验研究

非线性输出频率响应函数是由Volterra级数发展而来的频域概念,可方便在频域对非线性系统进行分析,它是频率的一维函数.本文主要介绍了利用NARMAX模型以及NOFRF对结构进行损伤检测的方法,并利用实验研究证实了该损伤检测方法的可行性.另外,由于系统非线性特性可用来做结构损伤检测,且具有对系统状态比较敏感的优点,而基于NOFRF的损伤检测方法是利用非线性方法来分析系统的状态,该方法提取出的特征属于非线性特征,所以该损伤检测方法可以用来做结构损伤检测,且具有对系统状态比较敏感的优点.     

Non‐Parametric Identification Method of Volterra Kernels for Nonlinear Systems Excited by Multitone Signal

To solve the problem of Volterra frequency‐domain kernels (VFKs) of nonlinear systems, which can be difficult to identify, we propose a novel non‐parametric identification method based on multitone excitation. First, we have studied the output properties of VFKs of nonlinear systems excited by the multitone signal, and derived a formula for identifying VFKs. Second, to improve the efficiency of the non‐parametric identification method, we suggest an increase in the number of tones for multitone excitation to simultaneously identify multi‐point VFKs with one excitation. We also propose an algorithm for searching the frequency base of multitone excitation. Finally, we use the interpolation method to separate every order output of VFK and extract its output frequency components, then use the derived formula to calculate the VFKs. The theoretical analysis and simulation results indicate that the non‐parametric method has a high precision and convenience of operation, improving the conventional methods, which have the defects of being unable to precisely identify VFKs and identification results are limited to three‐order VFK.     

Lyapunov characterization of forced oscillations

This paper develops a Lyapunov approach to the analysis of input-output characteristics for systems under the excitation of a class of oscillatory inputs. Apart from sinusoidal signals, the class of oscillatory inputs include multi-tone signals and periodic signals which can be described as the output of an autonomous system. The Lyapunov approach is developed for linear systems, homogeneous systems (differential inclusions) and nonlinear systems (differential inclusions), respectively. In particular, it is established that the steady-state gain can be arbitrarily closely characterized with Lyapunov functions if the output response converges exponentially to the steady-state. Other output measures that will be characterized include the peak of the transient response and the convergence rate. Tools based on linear matrix inequalities (LMIs) are developed for the numerical analysis of linear differential inclusions (LDIs). This paper's results can be readily applied to the evaluation of frequency responses of general nonlinear and uncertain systems by restricting the inputs to sinusoidal signals. Guided by the numerical result for a second order LDI, an interesting phenomenon is observed that the peak of the frequency response can be strictly larger than the L₂ gain.     

Energy transfer properties of non-linear systems in the frequency domain

In this paper, an analysis of the energy transfer properties of non-linear systems in the frequency domain is studied based on a new concept known as non-linear output frequency response functions (NOFRFs). The new concept allows the analysis to be implemented in a manner similar to the analysis of linear systems in the frequency domain, and provides great insight into the mechanisms which dominate the non-linear behaviour. The new analysis is also helpful for the design of non-linear systems in the frequency domain.     

Efficient computation of higher order frequency response functions for nonlinear systems with,and without,a constant term

The behaviour of a nonlinear system can be profoundly affected by the presence of a constant or dc term in the system governing equation. These changes are reflected in the nonlinear frequency response characteristics of the system which provide a powerful insight into the system's dynamics. In this article, a new and efficient algorithm is presented for computing the higher order Volterra frequency response functions from nonlinear time-domain models that may contain a constant term. A comparison with previous methods is included to demonstrate the significant gains in computational efficiency that are achieved using the new method. The algorithm is applicable to systems modelled by nonlinear differential, or difference, equations and is easily automated. Several examples are used to illustrate the method, and to highlight the importance of dc terms in nonlinear system analysis.     

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。