A new identification method for fuzzy linear models of nonlinear dynamic systems

The most promising methods for identifying a fuzzy model are data clustering, cluster merging and subsequent projection of the clusters on the input variable space. This article proposes to modify this procedure by adding a cluster rotation step, and a method for the direct calculation of the consequence parameters of the fuzzy linear model. These two additional steps make the model identification procedure more accurate and limits the loss of information during the identification procedure. The proposed method has been tested on a nonlinear first order model and a nonlinear model of a bioreactor and results are very promising.     

Robust inference in generalized partially linear models

In many situations, data follow a generalized partly linear model in which the mean of the responses is modeled, through a link function, linearly on some covariates and nonparametrically on the remaining ones. A new class of robust estimates for the smooth function η, associated to the nonparametric component, and for the parameter , related to the linear one, is defined. The robust estimators are based on a three-step procedure, where large values of the deviance or Pearson residuals are bounded through a score function. These estimators allow us to make easier inferences on the regression parameter and also improve computationally those based on a robust profile likelihood approach. The resulting estimates of turn out to be root-n consistent and asymptotically normally distributed. Besides, the empirical influence function allows us to study the sensitivity of the estimators to anomalous observations. A robust Wald test for the regression parameter is also provided. Through a Monte Carlo study, the performance of the robust estimators and the robust Wald test is compared with that of the classical ones.     

基于Hammerstein模型描述的非线性系统辨识新方法

Hammerstein模型常用来描述pH值或具有幂函数、死区、开关等特性的过程,本文提出了一种辨识此类对象模型结构和参数的新方法,首先将非线性静态部分和线性动态部分分别用非线性基和Laguerre级数表示,然后通过最小二乘法、矩阵特征值分解和矩阵扩维,辨识出两部分参数．并证明了该方法在输出端存在白噪声情况下误差的收敛性．此方法仅需假设输入为持续激励,适用范围广,计算简单,辨识精度高．最后通过pH中和滴定实验验证了以上结论．     

Kernel based nonlinear fuzzy regression model

Recent years have seen a surge of interest in extending statistical regression to fuzzy data. Most of the recent fuzzy regression models have undesirable performance when functional relationships are nonlinear. In this study, we propose a novel version of fuzzy regression model, called kernel based nonlinear fuzzy regression model, which deals with crisp inputs and fuzzy output, by introducing the strategy of kernel into fuzzy regression. The kernel based nonlinear fuzzy regression model is identified using fuzzy Expectation Maximization (EM) algorithm based maximum likelihood estimation strategy. Some experiments are designed to show its performance. The experimental results suggest that the proposed model is capable of dealing with the nonlinearity and has high prediction accuracy. Finally, the proposed model is used to monitor unmeasured parameter level of coal powder filling in ball mill in power plant. Driven by running data and expertise, a strategy is first proposed to construct fuzzy outputs, reflecting the possible values taken by the unmeasured parameter. With the engineering application, we then demonstrate the powerful performance of our model.     

Absolute penalty and shrinkage estimation in partially linear models

In the context of a partially linear regression model, shrinkage semiparametric estimation is considered based on the Stein-rule. In this framework, the coefficient vector is partitioned into two sub-vectors: the first sub-vector gives the coefficients of interest, i.e., main effects (for example, treatment effects), and the second sub-vector is for variables that may or may not need to be controlled. When estimating the first sub-vector, the best estimate may be obtained using either the full model that includes both sub-vectors, or the reduced model which leaves out the second sub-vector. It is demonstrated that shrinkage estimators which combine two semiparametric estimators computed for the full model and the reduced model outperform the semiparametric estimator for the full model. Using the semiparametric estimate for the reduced model is best when the second sub-vector is the null vector, but this estimator suffers seriously from bias otherwise. The relative dominance picture of suggested estimators is investigated. In particular, suitability of estimating the nonparametric component based on the B-spline basis function is explored. Further, the performance of the proposed estimators is compared with an absolute penalty estimator through Monte Carlo simulation. Lasso and adaptive lasso were implemented for simultaneous model selection and parameter estimation. A real data example is given to compare the proposed estimators with lasso and adaptive lasso estimators.     

On linear models for nonlinear systems

Best linear time-invariant (LTI) approximations are analysed for several interesting classes of discrete nonlinear time-invariant systems. These include nonlinear finite impulse response systems and a class of nonsmooth systems called bi-gain systems. The Fréchet derivative of a smooth nonlinear system is studied as a potential good LTI model candidate. The Fréchet derivative is determined for nonlinear finite memory systems and for a class of Wiener systems. Most of the concrete results are derived in an ?_∞ signal setting. Applications to linear controller design, to identification of linear models and to estimation of the size of the unmodelled dynamics are discussed.     

A nonlinear model predictive control system based on Wiener piecewise linear models

In this paper a nonlinear model predictive control (NMPC) based on a Wiener model with a piecewise linear gain is presented. This approach retains all the interested properties of the classical linear model predictive control (MPC) and keeps computations easy to solve due to the canonical structure of the nonlinear gain. Some guidelines for the identification of the nominal model as well as the uncertainty bounds are discussed, and two examples that show the possibility of application of this control scheme to real life problems are presented.     

System identification of nonlinear state-space models

This paper is concerned with the parameter estimation of a general class of nonlinear dynamic systems in state-space form. More specifically, a Maximum Likelihood (ML) framework is employed and an Expectation Maximisation (EM) algorithm is derived to compute these ML estimates. The Expectation (E) step involves solving a nonlinear state estimation problem, where the smoothed estimates of the states are required. This problem lends itself perfectly to the particle smoother, which provides arbitrarily good estimates. The maximisation (M) step is solved using standard techniques from numerical optimisation theory. Simulation examples demonstrate the efficacy of our proposed solution.     

On identification of parallel block-cascade nonlinear models

The class of nonlinear systems studied in this paper is assumed to be modelled by parallel block-cascades. Such models are composed of parallel branches where each branch has a linear block in cascade with a zero-memory nonlinear block followed by another linear block. These types of models are extensively used to represent nonlinear dynamic systems and are known in the literature as Wiener-Hammerstein models. Using a zero-mean stationary white gaussian sequence as an input to such models, a structure identification criterion is developed, utilizing the bispectrum estimate of the output sequence only. The application of this criterion is shown by several simulation examples. Also, impulse response estimation of an example of such a model is considered to show the effectiveness of the proposed identification technique.     

Non-linear system modelling based on non-parametric identification and linear wavelet estimation of SDP models

This paper describes a data-based approach to the identification and estimation of non-linear dynamic systems which exploits the concept of a state dependent parameter (SDP) model structure. The major attractive features of the proposed approach are: (1) the initial non-parametric identification of the non-linear system structure using an SDP algorithm based on recursive fixed interval smoothing; (2) a compact parameterization of this initially identified model structure via a linear wavelet functional approximation; and (3) final optimized model structure selection using the predicted residual sums of squares (PRESS) statistic, prior to final parametric optimization using this optimized, parsimonious structure. Two simulation examples are used to demonstrate the proposed approach.     

Robust likelihood inference for regression parameters in partially linear models

A robust likelihood approach is proposed for inference about regression parameters in partially-linear models. More specifically, normality is adopted as the working model and is properly corrected to accomplish the objective. Knowledge about the true underlying random mechanism is not required for the proposed method. Simulations and illustrative examples demonstrate the usefulness of the proposed robust likelihood method, even in irregular situations caused by the components of the nonparametric smooth function in partially-linear models.     

Stationary linear and nonlinear system identification and predictor set completeness

A general consistency theorem for stationary nonlinear prediction error estimators is presented. Since this theorem does not require the existence of a parameterized system generating the observations, it applies to the practical problem of modeling complex systems with simple parameterized models. In order to measure the quality of fit between a set of observed processes and a given candidate set of predictors, the notion of predictor set completeness is introduced. Several examples are given to illustrate this idea; in particular, a negative result concerning the completehess of certain sets of linear predictors is presented. The relationship of Ljung's definitions of identifiability to various notions of predictor set completeness is examined, and the strong consistency of maximum likelihood estimators for Gaussian autoregressive moving average systems is obtained via an application of our techniques. Finally, problems for future research are described.     

偏最小二乘线性模型及其非线性动态扩展模型综述

偏最小二乘(Partial least square,PLS)是一种基于数据驱动可以处理多个因变量对多个自变量的回归建模方法,因其具有提取质量相关信息的特性,在质量相关复杂工业过程监控中得到广泛的应用,成为近几十年复杂工业过程故障检测和诊断领域的研究热点.对此,介绍线性、非线性、动态PLS模型及其故障检测技术.首先,介绍标准PLS模型,在此基础上对传统PLS模型进行细化分并指出其优缺点,针对标准PLS存在的两个问题以及工业过程数据的两种极端情况,从数据预处理类、多空间类和分块类三方面梳理线性PLS模型的发展和改进历程;其次,将非线性PLS模型扩展方法分为两类,重点介绍核函数非线性PLS模型的研究现状;再次,指出动态扩展方法的两种基本思路,对PLS动态模型进行分类,阐明动态特性的成因,从本质上揭示两种动态扩展方法的原理,按照分类综述动态PLS模型的发展现状;最后,指出该领域亟需解决的问题和未来研究方向.     

Decoupling the linear and nonlinear parts in Hammerstein model identification

The main result of this paper is to show that the linear part can be made decoupled from the nonlinear part in Hammerstein model identification. Therefore, identification of the linear part for a Hammerstein model becomes a linear problem and accordingly enjoys the same convergence and consistency results as if the unknown nonlinearity is absent.     

Fuzzy functions based ARX model and new fuzzy basis function models for nonlinear system identification

In this study, auto regressive with exogenous input (ARX) modeling is improved with fuzzy functions concept (FF-ARX). Fuzzy function with least squares estimation (FF-LSE) method has been recently developed and widely used with a small improvement with respect to least squares estimation method (LSE). FF-LSE is structured with only inputs and their membership values. This proposed model aims to increase the capability of the FF-LSE by widening the regression matrix with lagged input–output values. In addition, by using same idea, we proposed also two new fuzzy basis function models. In the first, basis of the fuzzy system and lagged input–output values are structured together in the regression matrix and named as “L-FBF”. Secondly, instead of using basis function, the membership values of the lagged input–output values are used in the regression matrix by using Gaussian membership functions, called “M-FBF”. Therefore, the power of the fuzzy basis function is also enhanced. For the corresponding models, antecedent part parameters for the input vectors are determined with fuzzy c-means (FCM) clustering algorithm. The consequent parameters of the all models are estimated with the LSE. The proposed models are utilized and compared for the identification of nonlinear benchmark problems.     

Kernel based approaches to local nonlinear non-parametric variable selection

In this paper, we consider the variable selection problem for a nonlinear non-parametric system. Two approaches are proposed, one top-down approach and one bottom-up approach. The top-down algorithm selects a variable by detecting if the corresponding partial derivative is zero or not at the point of interest. The algorithm is shown to have not only the parameter but also the set convergence. This is critical because the variable selection problem is binary, a variable is either selected or not selected. The bottom-up approach is based on the forward/backward stepwise selection which is designed to work if the data length is limited. Both approaches determine the most important variables locally and allow the unknown non-parametric nonlinear system to have different local dimensions at different points of interest. Further, two potential applications along with numerical simulations are provided to illustrate the usefulness of the proposed algorithms.     

Robust nonlinear system identification using neural-network models.

We study the problem of identification for nonlinear systems in the presence of unknown driving noise, using both feedforward multilayer neural network and radial basis function network models. Our objective is to resolve the difficulty associated with the persistency of excitation condition inherent to the standard schemes in the neural identification literature. This difficulty is circumvented here by a novel formulation and by using a new class of identification algorithms recently obtained by Didinsky et al. (1995). We present a class of identifiers which secure a good approximant for the system nonlinearity provided that some global optimization technique is used. Subsequently, we address the same problem under a third, worst case L(infinity) criterion for an RBF modeling. We present a neural-network version of an H(infinity)-based identification algorithm from Didinsky et al., and show how it leads to satisfaction of a relevant persistency of excitation condition, and thereby to robust identification of the nonlinearity.     

Robust recursive identification of multidimensional linear regression models

Stochastic adaptive estimation and control algorithms involving recursive prediction estimates have guaranteed convergence rates when the noise is not ‘too’ coloured, as when a positive-real condition on the noise mode is satisfied. Moreover, the whiter the noise environment the more robust are the algorithms. This paper shows that for linear regression signal models, the suitable introduction of while noise into the estimation algorithm can make it more robust without compromising on convergence rates. Indeed, there are guaranteed attractive convergence rates independent of the process noise colour. No positive-real condition is imposed on the noise model.     

Profile inference on partially linear varying-coefficient errors-in-variables models under restricted condition

In this paper, we investigate the estimation and testing problems of partially linear varying-coefficient errors-in-variables (EV) models under additional restricted condition. The restricted estimators of parametric and nonparametric components are established based on modified profile least-squares method, and their asymptotic properties are also studied under some regularity conditions. Moreover, the modified profile Lagrange multiplier test statistic is constructed under additional restricted condition. It is shown that the modified profile Lagrange multiplier test statistic is asymptotically distribution-free and follows a Chi-squared distribution under the null hypothesis. Some simulation studies are carried out to assess the performance of the proposed methods. A real dataset is analyzed for illustration.