多自由度非线性系统动态参数化模型建模方法研究

针对多自由度非线性系统的动态模型辨识问题,基于NARX(Non-linear Autoregressive with Exogenous inputs)模型的建模方法,考虑系统的物理设计参数,建立非线性系统动态参数化模型.首先,根据系统输入、输出数据建立系统不同参数下的NARX模型,并通过EFOR(Extended Forward Orthogonal Regression)算法对不同参数下NARX模型进行修正,以统一辨识得到的系统模型结构.随后,建立NARX模型系数与物理设计参数间的函数关系,得到多自由度非线性系统的动态参数化模型.以单输入、单输出两自由度非线性系统为例,根据数值仿真结果,对系统的动态参数化模型建模过程进行说明.最后,以带非线性涂层阻尼的悬臂梁作为试验对象,建立其动态参数化模型以反映其动力学特性.试验结果表明,非线性系统动态参数化模型能准确预测多自由度非线性系统的输出响应,为非线性系统的分析与优化设计提供了理论基础.     

Order and delay selection for neural network modelling by identification of linearized models

An approach to selecting the order and delay for neural network modelling of nonlinear dynamic systems is proposed by identifying local linear models at points spanning the system operating range. The method is based on relationships between linear and nonlinear models and is developed for three popular nonlinear model structures; nonlinear autoregressive with exogenous inputs (NARX), NARX with a linear noise model and nonlinear autoregressive moving-average with exogenous inputs (NARMAX). Simulation results illustrate the application of the method, and the suitability of the orders and delays selected are demonstrated by nonlinear system identification using radial basis function neural networks. The method is also shown to indicate the suitability of a particular nonlinear model structure for representing a system.     

Real-time design of an adaptive nonlinear predictive controller

Based on real-time identification and using the concept of NARX (Nonlinear AutoRegressive with exogenous inputs) models, a new adaptive nonlinear predictive controller (ANPC) design is proposed. NARX models represent a natural way to describe the input-output relationship of severely nonlinear systems. From an initial batch of input-output data, a parsimonious NARX model is obtained using the Modified Gram-Schmidt (MGS) orthogonalization algorithm. Following this initial off-line identification and model reduction procedure, the control loop is closed. The ANPC directly uses the obtained structure and initial parameter estimates, which are updated each time step using recursive identification. The controller is designed similar to a typical linear predictive controller based on solving a nonlinear programming (NLP) problem. This paper shows how to solve this NLP problem on-line without the knowledge of the NARX model structure. The design is given for the multi-input multi-output (MIMO) case.     

Chlorophyll-a Predicting Model Based on Dynamic Neural Network

Algal blooms are one of the most prevalent global problems. Studying the Chlorophyll-a (Chl-a) predicting model helps to control algal blooms. Predicting the behavior of algae is difficult because of the complex physical, chemical, and biological processes involved. Artificial neural network (ANN) models have been determined to be useful and efficient, especially for such problems for which the characteristics of the processes are difficult to describe using numerical models. An indoor simulated environment is designed for algal cultivation to analyze the temporal change in the algae biomass of Taihu Lake during summer. A Chl-a prediction model based on a nonlinear autoregressive neural network with exogenous inputs (NARX) that can detect and consider within the time dependency is proposed. The NARX model is compared to a static neural network and a dynamic neural network: feedforward neural network (FNN) and Elman recurrent neural network (ERNN). The performance of the proposed NARX model was examined with experimental data collected over 3 months in 2010. The results showed that the NARX model outperformed the other ANN models and significantly enhance the accuracy of Chl-a prediction.     

Neural model input selection for a MIMO chemical process

In modelling non-linear systems using neural networks (NN), a commonly used method for the selection of network inputs, or to determine system order and time-delay, is to try different combinations of the system input–output data and choose the best one, giving minimum prediction error. The method is increasingly difficult to apply to industrial systems, due to their multivariable nature and complexity. A systematic method for the selection of model order and time-delay is developed in this paper, and applied to the neural modelling of a multivariable chemical process rig. The method is much simpler compared to the structure identification of the Non-linear Auto-Regressive with eXogenous inputs model (NARX), since the latter also needs to determine the significant terms from a linear-in-parameters polynomial. The orders and delays for system input and output are determined by identifying linearised models of the system. The method can also be applied to other approximations of a MIMO non-linear system, such as fuzzy logic models, etc. The application example demonstrates the selection procedure. Finally, the process rig is modelled using NNs according to the chosen structure, and the modelling error is compared with that of models with different structures to show the effectiveness of the method.     

基于NARX神经网络航空发动机参数动态辨识模型

针对航空发动机参数非线性动态特性,提出一种基于外部输入非线性自回归（NARX）神经网络的发动机参数动态辨识模型。主要思路是根据NARX网络的非线性时序预测特性,结合发动机参数的稳态和动态参数,提出一种基于偏稳态差值预测的NARX参数动态模型结构。设计了SP-P辨识结构,整定了模型内部结构参数并建立N1（低压转子转速）、N2（高压转子转速）、EGT（涡轮后排气温度）参数非线性差分预测模型。最后依据某发动机试车样本,对推杆加减速时N1、N2、EGT动态辨模型进行仿真。仿真结果表明,N2相对误差小于0.2%,N1相对误差小于0.3%,EGT相对误差小于[1℃],满足发动机试车仿真需要。最后,将所建模型应用于某A320机务维修训练器的发动机仿真系统。     

基于非线性多参数模型的软件老化检测

提出了一种软件系统的非线性有源自回归(Nonlinear AutoRegressive models with eXogenous Inputs,NARX)网络模型的老化检测方法。解决了目前软件老化方法未考虑多变量间关联性及历史数据的延迟影响的问题。该方法首先通过对实验采集的HelixServer-VOD服务器性能数据进行主成分分析,确定网络的输入维数,根据AIC准则确定最佳模型阶数,最终选取合理的网络模型结构;使用已知的未老化状态样本对NARX网络进行训练,建立系统的辨识模型;然后运用序贯概率比检验(Sequential Probability Ratio Test,SPRT)对NARX辨识模型的残差进行假设检验,判断系统的老化状态。实验分析表明,基于NARX网络模型的故障检测方法能够有效地应用于软件老化的检测。     

Nonlinear identification and control of a high-purity distillation column: a case study

Identification and control of ill-conditioned, interactive and highly nonlinear processes pose a challenging problem to the process industry. In the absence of a reasonably accurate model, these processes are fairly difficult to control. Using a high-purity distillation column as an example, model identification and control issues are addressed in this paper. The structure of the identified models is that of the polynomial type nonlinear autoregressive models with exogenous inputs (NARX). While most of the work in this area has concentrated on linear models (one-time scale and two-time scale models), this work is aimed at identifying the inherent nonlinearities. Comparisons are drawn between the identified models based on statistical criteria (AIC etc.) and other validation tests. Simulation results are provided to demonstrate the closed-loop performance of the nonlinear ARX models in the control of the distillation column. The controller employed is based on a nonlinear model predictive scheme with state and parameter estimation.     

一种基于小波分解的非线性系统辨识的新方法

提出了一种结合小波理论和NARX模型的新辨识算法．该算法利用小波(多维小波)函数有效的逼近能力避免了通常确定NARX模型结构时的复杂过程，构成了一个相当通用且不依赖于系统先验信息的辨识框架．应用递推最小二乘算法估计模型参数时，该算法可实现系统的在线辨识．两仿真算例说明了这种算法的有效性．     

Wavelet based non-parametric NARX models for nonlinear input–output system identification

Wavelet based non-parametric additive NARX models are proposed for nonlinear input–output system identification. By expanding each functional component of the non-parametric NARX model into wavelet multiresolution expansions, the non-parametric estimation problem becomes a linear-in-the-parameters problem, and least-squares-based methods such as the orthogonal forward regression (OFR) approach, coupled with model size determination criteria, can be used to select the model terms and estimate the parameters. Wavelet based additive models, combined with model order determination and variable selection approaches, are capable of handling problems of high dimensionality.     

Forecasting peak air pollution levels using NARX models

Air pollution has a negative impact on human health. For this reason, it is important to correctly forecast over-threshold events to give timely warnings to the population. Nonlinear models of the nonlinear autoregressive with exogenous variable (NARX) class have been extensively used to forecast air pollution time series, mainly using artificial neural networks (NNs) to model the nonlinearities. This work discusses the possible advantages of using polynomial NARX instead, in combination with suitable model structure selection methods. Furthermore, a suitably weighted mean square error (MSE) (one-step-ahead prediction) cost function is used in the identification/learning process to enhance the model performance in peak estimation, which is the final purpose of this application. The proposed approach is applied to ground-level ozone concentration time series. An extended simulation analysis is provided to compare the two classes of models on a selected case study (Milan metropolitan area) and to investigate the effect of different weighting functions in the identification performance index. Results show that polynomial NARX are able to correctly reconstruct ozone concentrations, with performances similar to NN-based NARX models, but providing additional information, as, e.g., the best set of regressors to describe the studied phenomena. The simulation analysis also demonstrates the potential benefits of using the weighted cost function, especially in increasing the reliability in peak estimation.     

Modeling nonlinear behavior of Buckling-Restrained Braces via different artificial intelligence methods

Five AI models are presented to model the dynamic nonlinear behavior of Buckling-Restrained Braces (BRBs). The AI techniques utilized in the models are: Time-Delayed Neural Networks (TDNN), Nonlinear Auto-Regressive eXogenous (NARX) neural networks, Gaussian-Mixture Models Regression (GMMR), Adaptive Neuro-Fuzzy Inference Systems (ANFIS) and Polynomial Classifier Regression (PCR). The models are developed using time-delayed brace displacements inputs and brace force outputs to predict updated brace forces during load reversals. The training and testing of the AI models are performed using experimental data from BRB specimens tested at the Pacific Earthquake Engineering Research (PEER) Center. The training stage for every method makes use of the experimental data from one specimen. In order to assess the models’ learning and generalization capabilities, three sets of experimental data for different specimens are used. To arrive at an optimized architecture that best models the phenomenon, the model performance with different parameters is evaluated. The brace force predicted by the proposed model shows excellent resemblance to the experimental results for the training sample, for all techniques. The predicted behavior of the testing samples shows noticeable accuracy and further demonstrates the generalization and prediction capability of the proposed modeling techniques. The various techniques are compared on the basis of selected performance criteria. It is found that the performance of two AI techniques standout among the others: the NARX and the PCR. Although the NARX demonstrates a slight advantage in the prediction accuracy over the PCR, the latter is far more superior in terms of computational efficiency. Thus, the PCR would be recommended for scenarios where online training is needed. The BRB design and performance investigation processes can be facilitated by the developed modeling techniques thus minimizing the need for, and extent of, experimental testing.     

On the dynamical modeling with neural fuzzy networks.

In the literature, researchers have introduced delay feedback (or recurrent) networks and claimed that those networks could accurately model dynamical systems without knowing their system orders. In this paper, we have studied those delay feedback networks and also proposed a better version of delay feedback neural-fuzzy networks, called additive delay feedback neural-fuzzy networks (ADFNFN). From our simulations for various examples, it is clearly evident that ADFNFN can have the best modeling accuracy among those existing delay feedback networks. Nevertheless, we also showed by examples that those delay feedback networks can only reach the accuracy of nonlinear autoregressive with exogenous inputs (NARX) models with order two, and that the number of delays in delay feedback networks plays the same role as the order in NARX models.     

A unified wavelet-based modelling framework for non-linear system identification: the WANARX model structure

A new unified modelling framework based on the superposition of additive submodels, functional components, and wavelet decompositions is proposed for non-linear system identification. A non-linear model, which is often represented using a multivariate non-linear function, is initially decomposed into a number of functional components via the well-known analysis of variance (ANOVA) expression, which can be viewed as a special form of the NARX (non-linear autoregressive with exogenous inputs) model for representing dynamic input–output systems. By expanding each functional component using wavelet decompositions including the regular lattice frame decomposition, wavelet series and multiresolution wavelet decompositions, the multivariate non-linear model can then be converted into a linear-in-the-parameters problem, which can be solved using least-squares type methods. An efficient model structure determination approach based upon a forward orthogonal least squares (OLS) algorithm, which involves a stepwise orthogonalization of the regressors and a forward selection of the relevant model terms based on the error reduction ratio (ERR), is employed to solve the linear-in-the-parameters problem in the present study. The new modelling structure is referred to as a wavelet-based ANOVA decomposition of the NARX model or simply WANARX model, and can be applied to represent high-order and high dimensional non-linear systems.     

A randomised approach for NARX model identification based on a multivariate Bernoulli distribution

The identification of polynomial NARX models is typically performed by incremental model building techniques. These methods assess the importance of each regressor based on the evaluation of partial individual models, which may ultimately lead to erroneous model selections. A more robust assessment of the significance of a specific model term can be obtained by considering ensembles of models, as done by the RaMSS algorithm. In that context, the identification task is formulated in a probabilistic fashion and a Bernoulli distribution is employed to represent the probability that a regressor belongs to the target model. Then, samples of the model distribution are collected to gather reliable information to update it, until convergence to a specific model. The basic RaMSS algorithm employs multiple independent univariate Bernoulli distributions associated to the different candidate model terms, thus overlooking the correlations between different terms, which are typically important in the selection process. Here, a multivariate Bernoulli distribution is employed, in which the sampling of a given term is conditioned by the sampling of the others. The added complexity inherent in considering the regressor correlation properties is more than compensated by the achievable improvements in terms of accuracy of the model selection process.     

Regressor and structure selection in NARX models using a structured ANOVA approach

Regressor selection can be viewed as the first step in the system identification process. The benefits of finding good regressors before estimating complex models are especially clear for nonlinear systems, where the class of possible models is huge. In this article, a structured way of using the tool analysis of variance (ANOVA) is presented and used for NARX model (nonlinear autoregressive model with exogenous input) identification with many candidate regressors.     

Least squares support vector machines with tuning based on chaotic differential evolution approach applied to the identification of a thermal process

In the past decade, support vector machines (SVMs) have gained the attention of many researchers. SVMs are non-parametric supervised learning schemes that rely on statistical learning theory which enables learning machines to generalize well to unseen data. SVMs refer to kernel-based methods that have been introduced as a robust approach to classification and regression problems, lately has handled nonlinear identification problems, the so called support vector regression. In SVMs designs for nonlinear identification, a nonlinear model is represented by an expansion in terms of nonlinear mappings of the model input. The nonlinear mappings define a feature space, which may have infinite dimension. In this context, a relevant identification approach is the least squares support vector machines (LS-SVMs). Compared to the other identification method, LS-SVMs possess prominent advantages: its generalization performance (i.e. error rates on test sets) either matches or is significantly better than that of the competing methods, and more importantly, the performance does not depend on the dimensionality of the input data. Consider a constrained optimization problem of quadratic programing with a regularized cost function, the training process of LS-SVM involves the selection of kernel parameters and the regularization parameter of the objective function. A good choice of these parameters is crucial for the performance of the estimator. In this paper, the LS-SVMs design proposed is the combination of LS-SVM and a new chaotic differential evolution optimization approach based on Ikeda map (CDEK). The CDEK is adopted in tuning of regularization parameter and the radial basis function bandwith. Simulations using LS-SVMs on NARX (Nonlinear AutoRegressive with eXogenous inputs) for the identification of a thermal process show the effectiveness and practicality of the proposed CDEK algorithm when compared with the classical DE approach.     

A Note on the Equivalence of NARX and RNN

This paper presents several aspects with regards the application of the NARX model and Recurrent Neural Network (RNN) model in system identification and control. We show that every RNN can be transformed to a first order NARX model, and vice versa, under the condition that the neuron transfer function is similar to the NARX transfer function. If the neuron transfer function is piecewise linear, that is f(x):=x if uxu , 1 and f(x):=sign(x) otherwise, we further show that every NARX model of order larger than one can be transformed into a RNN. According to these equivalence results, there are three advantages from which we can benefit: (i) if the output dimension of a NARX model is larger than the number of its hidden unit, training an equivalent RNN will be faster, i.e. the equivalent RNN is trained instead of the NARX model. Once the training is finished, the RNN is transformed back to an equivalent NARX model. On the other hand, (ii) if the output dimension of a RNN model is less than the number of its hidden units, the training of a RNN can be speeded up by using a similar method; (iii) the RNN pruning can be accomplished in a much simpler way, i.e. the equivalent NARX model is pruned instead of the RNN. After pruning, the NARX model is transformed back to the equivalent RNN.     

Learning long-term dependencies in NARX recurrent neural networks

It has previously been shown that gradient-descent learning algorithms for recurrent neural networks can perform poorly on tasks that involve long-term dependencies, i.e. those problems for which the desired output depends on inputs presented at times far in the past. We show that the long-term dependencies problem is lessened for a class of architectures called nonlinear autoregressive models with exogenous (NARX) recurrent neural networks, which have powerful representational capabilities. We have previously reported that gradient descent learning can be more effective in NARX networks than in recurrent neural network architectures that have "hidden states" on problems including grammatical inference and nonlinear system identification. Typically, the network converges much faster and generalizes better than other networks. The results in this paper are consistent with this phenomenon. We present some experimental results which show that NARX networks can often retain information for two to three times as long as conventional recurrent neural networks. We show that although NARX networks do not circumvent the problem of long-term dependencies, they can greatly improve performance on long-term dependency problems. We also describe in detail some of the assumptions regarding what it means to latch information robustly and suggest possible ways to loosen these assumptions.     

Identification of the rainfall–runoff relationship in urban drainage networks

The calibration of conceptual models for the design of urban drainage networks is an important and well-known problem in hydraulic engineering. In this paper the problem is analysed and the use of black-box identification methods is proposed and applied to experimental data. Both linear (ARX and state space) and nonlinear (polynomial and neural NARX) models are considered and their performance in the simulation and prediction of the network flow from rainfall measurements is evaluated.