New results on PWARX model identification based on clustering approach

This paper deals with the problem of piecewise auto regressive systems with exogenous input(PWARX) model identification based on clustering solution. This problem involves both the estimation of the parameters of the affine sub-models and the hyper planes defining the partitions of the state-input regression. The existing identification methods present three main drawbacks which limit its effectiveness. First, most of them may converge to local minima in the case of poor initializations because they are based on the optimization using nonlinear criteria. Second, they use simple and ineffective techniques to remove outliers. Third, most of them assume that the number of sub-models is known a priori. To overcome these drawbacks, we suggest the use of the density-based spatial clustering of applications with noise(DBSCAN) algorithm. The results presented in this paper illustrate the performance of our methods in comparison with the existing approach. An application of the developed approach to an olive oil esterification reactor is also proposed in order to validate the simulation results.     

Parameter identification of PWARX models using fuzzy distance weighted least squares method

PieceWise AutoRegressive eXogenous (PWARX) models represent one of the broad classes of the hybrid dynamical systems (HDS). Among many classes of HDS, PWARX model used as an attractive modeling structure due to its equivalence to other classes. This paper presents a novel fuzzy distance weight matrix based parameter identification method for PWARX model. In the first phase of the proposed method estimation for the number of affine submodels present in the HDS is proposed using fuzzy clustering validation based algorithm. For the given set of input–output data points generated by predefined PWARX model fuzzy c-means (FCM) clustering procedure is used to classify the data set according to its affine submodels. The fuzzy distance weight matrix based weighted least squares (WLS) algorithm is proposed to identify the parameters for each PWARX submodel, which minimizes the effect of noise and classification error. In the final phase, fuzzy validity function based model selection method is applied to validate the identified PWARX model. The effectiveness of the proposed method is demonstrated using three benchmark examples. Simulation experiments show validation of the proposed method.     

Weighted least squares based recursive parametric identification for the submodels of a PWARX system

A piecewise affine autoregressive system with exogenous inputs (PWARX) is composed of a finite number of ARX subsystems, each of which corresponds to a polyhedral partition of the regression space. In this work a weighted least squares (WLS) estimator is suggested to recursively estimate the parameters of the ARX submodels, in which a sequence of kernel functions are introduced. Conditions on the input signal and the PWARX system are imposed to guarantee the almost sure convergence of the WLS estimates. Some numerical examples are included to illustrate performances of the algorithm.     

基于模糊聚类的PWA系统的模型辨识

针对一类分段仿射结构的离散时间混杂系统，其模型辨识可等价成对系统数据的分类、分类边界的优化及分类数据的线性回归问题.利用改进的G-K 模糊聚类算法，克服聚类迭代过程出现的非数值解问题；以综合性能指标最优确定最佳的子模型个数，从而获得最佳的分类数据; 以隶属度为权值，采用加权最小二乘算法提高子模型辨识精度；通过聚类中心最短法则确定两两相邻的子数据集，利用支持向量机思想，构造出一个标准的二次规划问题，得到凸多面体的方程系数. 仿真结果验证了该方法的有效性和实用性.     

A bounded-error approach to piecewise affine system identification

This paper proposes a three-stage procedure for parametric identification of piecewise affine autoregressive exogenous (PWARX) models. The first stage simultaneously classifies the data points and estimates the number of submodels and the corresponding parameters by solving the partition into a minimum number of feasible subsystems (MIN PFS) problem for a suitable set of linear complementary inequalities derived from data. Second, a refinement procedure reduces misclassifications and improves parameter estimates. The third stage determines a polyhedral partition of the regressor set via two-class or multiclass linear separation techniques. As a main feature, the algorithm imposes that the identification error is bounded by a quantity /spl delta/. Such a bound is a useful tuning parameter to trade off between quality of fit and model complexity. The performance of the proposed PWA system identification procedure is demonstrated via numerical examples and on experimental data from an electronic component placement process in a pick-and-place machine.     

A PWA model identification method based on optimal operating region partition with the output-error minimization for nonlinear systems

When piecewise affine (PWA) model-based control methods are applied to nonlinear systems, the first question is how to get sub-models and corresponding operating regions. Motivated by the fact that the operating region of each sub-model is an important component of a PWA model and the parameters of a sub-model are strongly coupled with the operating region, a new PWA model identification method based on optimal operating region partition with the output-error minimization for nonlinear systems is initiated. Firstly, construct local data sets from input-output data and get local models by using the least square (LS) method. Secondly, cluster local models according to the feature vectors and identify the parameter vectors of sub-models by weighted least squares (WLS) method. Thirdly, get the initial operating region partition by using a normalized exponential function, which is to partition the operating space completely. Finally, simultaneously determine the optimal parameter vectors of sub-models and the optimal operating region partition underlying the output-error minimization, which is executed by particle swarm optimization (PSO) algorithm. Simulation results demonstrate that the proposed method can improve model accuracy compared with two existing methods.     

Identification of hybrid and linear parameter‐varying models via piecewise affine regression using mixed integer programming

This article presents a two‐stage algorithm for piecewise affine (PWA) regression. In the first stage, a moving horizon strategy is employed to simultaneously estimate the model parameters and to classify the training data by solving a small‐size mixed‐integer quadratic programming problem. In the second stage, linear multicategory separation methods are used to partition the regressor space. The framework of PWA regression is adapted to the identification of PWA AutoRegressive with eXogenous input (PWARX) models as well as linear parameter‐varying (LPV) models. The performance of the proposed algorithm is demonstrated on an academic example and on two benchmark experimental case studies. The first experimental example concerns modeling the placement process in a pick‐and‐place machine, while the second one consists in the identification of an LPV model describing the input‐output relationship of an electronic bandpass filter with time‐varying resonant frequency.     

A clustering technique for the identification of piecewise affine systems

We propose a new technique for the identification of discrete-time hybrid systems in the piecewise affine (PWA) form. This problem can be formulated as the reconstruction of a possibly discontinuous PWA map with a multi-dimensional domain. In order to achieve our goal, we provide an algorithm that exploits the combined use of clustering, linear identification, and pattern recognition techniques. This allows to identify both the affine submodels and the polyhedral partition of the domain on which each submodel is valid avoiding gridding procedures. Moreover, the clustering step (used for classifying the datapoints) is performed in a suitably defined feature space which allows also to reconstruct different submodels that share the same coefficients but are defined on different regions. Measures of confidence on the samples are introduced and exploited in order to improve the performance of both the clustering and the final linear regression procedure.     

基于多K最近邻回归算法的软测量模型

针对单一模型预测精度较低的问题,提出多K最近邻回归算法(MKNN)的软测量建模方法.该方法采用高斯过程选择软测量模型的辅助变量,通过自适应仿射传播聚类方法将输入样本数据分成多组数据,对每组数据用K最近邻回归(KNN)算法建立子模型,各个子模型的预测输出通过主元回归(PCR)方法连接.用该方法建立粗汽油干点软测量模型,仿真研究表明,该算法的预测精度和泛化能力优于单KNN模型.     

Identification of piecewise affine models in noisy environment

This paper addresses the identification of non-linear systems. A wide class of these systems can be described using non-linear time-invariant regression models, that can be approximated by means of piecewise affine prototypes with an arbitrary degree of accuracy. This work concerns the identification of piecewise affine model parameters through input-output data affected by additive noise. In order to show the effectiveness of the developed technique, the results obtained in the identification of both a simple simulated system and a real dynamic process are reported.