Volterra高阶核的新算法在航空涡轮转速控制上的应用

针对航空发动机的转速控制这个难题,提出了对非线性动态系统做建模研究的Volterra泛函方法的任意高阶核估计方法;该方法在核(kernel method)理论基础上,构造线性空间,将求解Volterra泛函各阶核的问题转化为求输出观测向量在希尔伯特空间(Hilbert space)子空间上的分量,利用线性空间中向量内积的求解而间接辨识出复杂的非线性动态系统;相对于其它在时域或频域估计Volterra核的理论,该方法数学基础牢固、计算量不随辨识精度增高而大量增加、理论上能够对任意高阶核进行估计,可对强非线性动态系统进行辨识。     

一种泛函级数建模的新方法及其应用

针对非线性动态系统较难做任意精度逼近的这一问题,提出了使用Volterra级数高阶核估算的全新估计方法。该方法在核函数理论基础上,构造特殊线性空间,将求解Volterra级数的各阶核的问题转化为求用输出观测向量在希尔伯特空间中某一子空间上的投影的问题,使原本复杂、难于计算的非线性系统的Volterra级数的逼近问题在所构建的线性空间中巧妙地以向量内积的方式解决。给出了具体计算方法。相比于其他时域或频域估计Volterra核的方法,该算法的优点在于理论体系严密、计算量不会随着阶数增高而成几何级数增加,辨识精度高,理论上能够辨识出任意阶的核,改善了现有的估计Volterra核的方法难以估计超过4阶或更高阶核的缺点,特别能够应用在对动态系统和强非线性系统的建模上。通过对电厂汽轮机轴系统的辨识和仿真,证明了该方法的有效性。     

基于双阶跃信号输入的Volterra模型辨识

Volterra模型作为非线性领域的一种非线性模型,由于其对工业过程可以以任意精度逼近,使得该模型有很广泛的应用研究意义。在将该模型运用到实际控制系统中之前,模型的高精度辨识显得尤为重要。在以往针对Volterra模型的辨识算法中,基本上主要是采用通用辨识算法识别模型参数,比如最小二乘法及各种改进的最小二乘法。这些通用的辨识算法在辨识Volterra模型时,不能充分考虑其非线性特点,同样不能在辨识过程中充分利用该特点。本文在充分考虑Volterra模型非线性的前提下,提出了一种基于双阶跃信号输入的Volterra模型辨识算法,该算法辨识原理简单,计算量较小,论文最后将该辨识算法应用到典型非线性CSTR系统的的辨识中,辨识结果证明了算法的有效性。     

基于线性空间投影的计算Volterra级数高阶核的方法

研究了对非线性动态系统作任意精度逼近的Volterra级数高阶核的全新估计方法。该方法在核函数理论基础上构造特殊线性空间,将求解Volterra级数的各阶核的问题转换为求用输出观测向量在希尔伯特空间中某一子空间上的投影问题,使原本复杂、难以计算的非线性系统的Volterra级数的逼近问题在所构建的线性空间中巧妙地以向量内积的方式解决,并给出了具体算法。相比于其他时域或频域估计Volterra核的方法,该算法的优点在于理论体系严密、计算量不会随着阶数增高而呈几何级数增加,辨识精度高,理论上能够辨识出任意阶的核,弥补了迄今现有的各种估计Volterra核的方法难以估计超过四阶或更高阶核的缺点,特别能够应用在对动态系统和强非线性系统的建模上。仿真研究的结果证明了该方法的有效性。     

非线性系统广义脉冲响应函数的盲辨识

探讨减少非线性系统广义脉冲响应函数（GIRF）盲辨识所需计算量问题。 基于线性MIMO模型，应用多项式矩阵理论和子空间盲辨识技术，研究使用部分噪声向量对非线性Volterra系统的GIRF盲辨识方法。该方法的优点是能有效减少GIRF盲辨识所需的计算量。这对GIRF盲辩识方法的在线应用是有利的。仿真结果说明了这一方法的有效性。     

基于支持向量回归的非线性系统辨识

本文将支持向量回归方法应用于非线性系统辨识问题．基于高斯支持向量回归及ε不敏感损失函数的基本思想，本文提出一个非线性系统辨识的新算法，并将其与用于系统辨识的径向基函数神经网络进行了比较．模拟实验表明，支持向量回归方法可以成为非线性系统辨识的有力工具．     

应用改进粒子群算法辨识Hammerstein模型

研究非线性系统辨识问题.针对非线性系统中单输入单输出Hammerstein模型,由于传统辨识方法对Hammerstein模型中非线性部分具有不易辨识的缺陷,造成辨识精度低、辨识效果差等问题.为此,在基本粒子群算法的基础上,提出了一种带有收缩因子的改进的粒子群算法对非线性系统进行辨识的方法,可将参数辨识问题转换为参数空间上的函数优化问题,然后利用粒子群算法的并行搜索能力进行参数寻优.通过MATLAB软件进行仿真,并与基本粒子群算法进行比较,结果表明,利用改进算法不仅提高了辨识精度而且获得了良好的辨识效果,从而验证了算法的有效性和可行性.     

基于Volterra泛函级数的涡轮建模研究

给出了对非线性动态系统做任意精度逼近的Volterra级数高阶核的全新估算方法并将其应用在涡喷发动机的转速控制上。该方法在核函数理论基础上,构造线性空间,将求解Volterra级数各阶核的问题转化为求输出观测向量在希尔伯特空间中某一子空间上的投影的问题,使原本复杂的非线性系统的Volterra级数的逼近问题在线性空间中以向量内积的方式得到解决。与其他时域或频域估算Volterra核的方法相比较,该算法的优点在于理论体系严密、计算量不随阶数增高而成几何级数增加、辨识精度高。该方法理论上能够估算任意阶核,弥补了现有方法难以估算四阶以上核的缺点,可应用于动态系统和强非线性系统的建模。将发动机动态过程描述为四阶的Volterra级数模型。     

Volterra模型辨识及其在蒸馏塔预测控制中的应用

本文研究一类单输入单输出非线性系统的预测函数控制问题，这类系统能用有限阶离散Volterra级数模型表示，采用最小二乘法进行参数辨识，并通过求解高次方程得到控制律。针对化工过程蒸馏塔控制系统，通过仿真计算验证了该方法的有效性。     

基于支持向量机的非线性系统辨识

支持向量机（SVM）是一种新的通用学习机器，它从结构风险最小化的角度，分析了学习过程的一致性，收敛速度等。SVM能以任意精度逼近一类函数，而与输入的维数无关，克服了传统神经网络用于系统辨识的维数灾问题及结构难以确定等,在于这一辨识的维数灾问题及结构难以确定等特点，基于这一特性研究了对非线性动态系统的辨识问题，仿真结果表明SVM用于系统辨识有良好的辨识效果，并指出了今后研究的方向。     

二阶Volterra变记忆长度LMP算法

针对实际应用中非线性系统记忆长度未知致使Volterra自适应滤波器可能无法达到最优性能的问题,提出一种二阶Volterra变记忆长度LMP算法。利用Volterra滤波器二阶权系数矩阵的对称性和对称矩阵可对角化分解性质,推导得到了一阶权系数与二阶权系数个数相同的信号矢量与权系数矢量内积的二阶Volterra滤波器输出信号表达式;提出了基于DCT的二阶Volterra自适应滤波器（CSVF）及其LMP算法（CSVLMP）;采用FIR抽头长度的自适应调整思想,提出了基于DCT的二阶Volterra变记忆长度LMP算法（CSVMLMP）。记忆长度未知的非线性系统辨识的仿真结果表明,在[α]稳定分布噪声背景下,该算法在收敛速度、稳态性能和计算复杂度之间达到了较好的折中。     

一种基于Volterra模型的非线性系统具有全解耦结构的递阶式自适应辨识方法

研究了在输入输出观测数据均含有噪声时如何对基于Volterra级数描述的非线性系统进行解耦自适应辨识的问题. 按照Volterra级数模型的伪线性组合结构, 采用总体最小二乘辨识技术的原理, 导出了一种总体全解耦辨识的思想. 从而建立了一种具有全解耦结构的递阶式自适应辨识算法, 给出了该算法的结构图. 相比于部分解耦辨识算法, 该算法的优点在于它能够在全噪声数据环境下得到更高的收敛速度和精度. 仿真研究的结果证明了本文方法的有效性.     

Approximation-theoretic methods for nonlinear deconvolution and inversion

Nonlinear deconvolution and nonlinear inversion are cast as inverse problems in generalized Fock spaces. Generalized Fock spaces, introduced by de Figueiredo and Dwyer in [1], are reproducing kernel Hilbert spaces (RKHSs) of input-output maps represented by Volterra series equipped with an appropriately weighted inner product, the choice of the weights in the inner product depending on the particular problem under consideration. The solution to the nonlinear deconvolution problem presented here is the same as the one obtained previously for the nonlinear system identification problem [1–4]. However, the present solution to the nonlinear inversion problem consists of a new approach, whereby the unknown samples of the input are obtained from the given samples of the output by means of an efficient sequential algorithm. The algorithm is based on a framework which interpolates the input samples by an appropriate spline, and its sequential nature is elicited by the use of a truncated function basis to represent the spline.     

On Convergence of Volterra Series Expansion of a Class of Nonlinear Systems

A fundamental issue in conducting the analysis and design of a nonlinear system via Volterra series theory is how to ensure the excitation magnitude and/or model parameters will be in the appropriate range such that the nonlinear system has a convergent Volterra series expansion. To this aim, parametric convergence bounds of Volterra series expansion of nonlinear systems described by a NARX model, which can reveal under what excitation magnitude or within what parameter range a given NARX system is able to have a convergent Volterra series expansion subject to any given input signal, are investigated systematically in this paper. The existing bound results often are given as a function of the maximum input magnitude, which could be suitable for single‐tone harmonic inputs but very conservative for complicated inputs (e.g. multi‐tone or arbitrary inputs). In this study, the output response of nonlinear systems is expressed in a closed form, which is not only determined by the input magnitude but also related to the input energy or waveform. These new techniques result in more accurate bound criteria, which are not only functions of model parameters and the maximum input magnitude but also consider a factor reflecting the overall input energy or wave form. This is significant to practical applications, since the same nonlinear system could exhibit chaotic behavior subject to a simple single‐tone input but might not with respect to other different input signals (e.g. multi‐tone inputs) of the same input magnitude. The results provide useful guidance for the application of Volterra series‐based theory and methods from an engineering point of view. The Duffing equation is used as a benchmark example to show the effectiveness of the results.     

Applications of nonlinear Volterra equations of the first kind to the control problem for heat exchange dynamics

We consider a class of nonlinear integral Volterra equations of the first kind related to the automatic control problem for a nonlinear dynamical system (object) which is a black box with vector input and no output feedback. The characteristic features of the algorithms are illustrated on the example of mathematical modeling for nonlinear heat exchange processes.     

基于Volterra泛函级数的非线性系统的鲁棒辨识

针对弱非线性系统的鲁棒建模问题, 基于Volterra泛函级数, 结合集员辨识理论, 提出了广义频率响应函数的鲁棒辨识方法, 形成了一套较完整的弱非线性系统的鲁棒建模方法, 仿真结果表明该方法是行之有效的.     

Stochastic functional fourier series, Volterra series, and nonlinear systems analysis

A functional Fourier series is developed with emphasis on applications to the nonlinear systems analysis. In analogy to Fourier coefficients, Fourier kernels are introduced and can be determined through a cross correlation between the output and the orthogonal basis function of the stochastic input. This applies for the class of strict-sense stationary white inputs, except for a singularity problem incurred with inputs distributed at quantized levels. The input may be correlated if it is zero-mean Gaussian. The Wiener expansion is treated as an example corresponding to the white Gaussian input and this modifies the Lee-Schetzen algorithm for Wiener kernel estimation conceptually and computationally. The Poisson-distributed white input is dealt with as another example. Possible links between the Fourier and Volterra series expansions are investigated. A mutual relationship between the Wiener and Volterra kernels is presented for a subclass of analytic nonlinear systems. Connections to the Cameron-Martin expansion are examined as well The analysis suggests precautions in the interpretation of Wiener kernel data from white-noise identification experiments.     

Volterra-series-based equivalent nonlinear system method for subharmonic vibration systems

Subharmonics generation in the nonlinear system, directly using the traditional finite Volterra series, cannot generally represent the aimed system. In this paper, a new approach is presented, which is an extension of single finite Volterra series for representation and analysis of the subharmonic vibration system based on equivalent nonlinear system. The equivalent nonlinear system, which is constructed by pre-compensating the subharmonic vibration system with the super-harmonic nonlinear model, yields the input–output relation between the virtual source and the response of the aimed nonlinear system. Orthogonal least square method is employed to identify the truncated order of Volterra series and predominant Volterra kernels of the equivalent nonlinear system. The MGFRFs (modified generalised frequency response functions) of the equivalent nonlinear system is obtained from the data of the virtual source and response, and verified by comparing the response estimated by the MGFRFs with its true value. Therefore, the aimed subharmonic vibration system can be analysed by taking advantage of a truncated Volterra series based on the equivalent nonlinear system. Numerical simulations were carried out, whose results have shown that the proposed method is valid and feasible, and suitable to apply on representation and analysis of subharmonic vibration systems.     

On the role of prefiltering in nonlinear system identification

Data prefiltering is often used in linear system identification to increase model accuracy in a specified frequency band, as prefiltering is equivalent to a frequency weighting on the prediction error function. However, this interpretation applies only to a strictly linear setting of the identification problem. In this note, the role of data and error prefiltering in nonlinear system identification is analyzed and a frequency domain interpretation is provided, based on the Volterra series representation of nonlinear systems. Simulation results illustrate the conclusions of the analysis.