Robust Kalman filter and smoother for errors‐in‐variables state space models with observation outliers based on the minimum‐covariance determinant estimator

In this paper, we propose a robust Kalman filter and smoother for the errors‐in‐variables (EIV) state space models subject to observation noise with outliers. We introduce the EIV problem with outliers and then present the minimum covariance determinant (MCD) estimator which is a highly robust estimator in terms of protecting the estimate from the outliers. Then, we propose the randomized algorithm to find the MCD estimate. However, the uniform sampling method has a high computational cost and may lead to biased estimates, therefore we apply the sub‐sampling method. A Monte Carlo simulation result shows the efficiency of the proposed algorithm. Copyright © 2011 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

A new subspace identification approach based on principal component analysis

Principal component analysis (PCA) has been widely used for monitoring complex industrial processes with multiple variables and diagnosing process and sensor faults. The objective of this paper is to develop a new subspace identification algorithm that gives consistent model estimates under the errors-in-variables (EIV) situation. In this paper, we propose a new subspace identification approach using principal component analysis. PCA naturally falls into the category of EIV formulation, which resembles total least squares and allows for errors in both process input and output. We propose to use PCA to determine the system observability subspace, the A, B, C, and D matrices and the system order for an EIV formulation. Standard PCA is modified with instrumental variables in order to achieve consistent estimates of the system matrices. The proposed subspace identification method is demonstrated using a simulated process and a real industrial process for model identification and order determination. For comparison the MOESP algorithm and N4SID algorithm are used as benchmarks to demonstrate the advantages of the proposed PCA based subspace model identification (SMI) algorithm.     

Robust identification for nonlinear errors-in-variables systems using the EM algorithm

This article presents a robust identification approach for nonlinear errors-in-variables (EIV) systems contaminated with outliers. In this work, the measurement noise is modelled using the t-distribution, instead of the traditional Gaussian distribution, to mitigate the effect of the outliers. The heavier tails of the t-distribution, through the adjustable degrees of freedom, is used to account for noise and outliers concomitantly. Further, to avoid the intricacies related to the direct nonlinear identification, we propose to approximate the nonlinear EIV dynamics using multiple local ARX models and aggregating them using an exponential weighting strategy. The parameters of the local models and weighting parameters are estimated using the expectation maximization (EM) algorithm, under the framework of the maximum likelihood estimation (MLE). The studies with simulated numerical examples and an experiment on a multi-tank system demonstrate the superiority of the proposed method.     

基于子空间辨识的状态空间模型预测控制

针对无法从工业过程中获得准确状态空间模型的问题,提出一种基于子空间辨识的状态空间模型预测控制方法。利用子空间辨识方法得到的状态空间模型作为系统模型,给出约束条件下的预测控制算法。以CD播放器机械臂系统为例,通过状态空间模型预测控制方法实现对系统输出的跟踪控制,仿真结果表明,该方法控制效果良好。     

Frequency domain subspace identification of commensurate fractional order input time delay systems

A frequency domain subspace identification of fractional order systems with input timedelay is studied in this paper. A new identification method, which combines the merits of differential evolution (DE) algorithm and subspace identification algorithm in frequency domain, is presented. For the optimal search of fractional commensurate differential order and time delay parameters, the DE algorithm is applied. For fixed fractional commensurate differential order and time delay, subspace method is performed to obtain the state space model. Simulation results validate the proposed fractional order system identification method.     

Robust identification of MISO neuro‐fractional‐order Hammerstein systems

This paper introduces a multiple‐input–single‐output (MISO) neuro‐fractional‐order Hammerstein (NFH) model with a Lyapunov‐based identification method, which is robust in the presence of outliers. The proposed model is composed of a multiple‐input–multiple‐output radial basis function neural network in series with a MISO linear fractional‐order system. The state‐space matrices of the NFH are identified in the time domain via the Lyapunov stability theory using input‐output data acquired from the system. In this regard, the need for the system state variables is eliminated by introducing the auxiliary input‐output filtered signals into the identification laws. Moreover, since practical measurement data may contain outliers, which degrade performance of the identification methods (eg, least‐square–based methods), a Gaussian Lyapunov function is proposed, which is rather insensitive to outliers compared with commonly used quadratic Lyapunov function. In addition, stability and convergence analysis of the presented method is provided. Comparative example verifies superior performance of the proposed method as compared with the algorithm based on the quadratic Lyapunov function and a recently developed input‐output regression‐based robust identification algorithm.     

Improvement of identification of blast furnace ironmaking process by outlier detection and missing value imputation

The control of blast furnace ironmaking process requires model of process dynamics accurate enough to facilitate the control strategies. However, data sets collected from blast furnace contain considerable number of missing values and outliers. These values can significantly affect subsequent statistical analysis and thus the identification of the whole process, so it becomes much important to deal with these values. This paper considers a data processing procedure including missing value imputation and outlier detection, and examines the impact of processing to the identification of blast furnace ironmaking process. Missing values are imputed based on the decision tree algorithm and outliers are identified and discarded through a set of multivariate outlier detection methods. The data sets before and after processing are then used for identification. Two classic identification methods, N4SID (numerical algorithms for state space subspace system identification) and PEM (prediction error method) are considered and a comparative study is presented.     

Identification of discrete-time state affine state space models using cumulants

This paper is concerned with the identification of discrete-time, time invariant, state affine state space models driven by an independent identically distributed (IID) random input, and in the presence of process and measurement noise. The identification problem is treated using a cumulant based approach. It is shown that the input-output and input-state crosscumulant equations in the time domain have the form of a linear autonomous system. An algorithmic procedure is then developed, for the computation of the unknown system matrices, based on a standard deterministic linear subspace identification algorithm, provided the input signal has some persistent excitation properties. The special case of Gaussian IID input is also examined. The proposed method is computationally very efficient and its accuracy is illustrated by simulations.     

N4SID and MOESP algorithms to highlight the ill-conditioning into subspace identification

In this paper,an analysis for ill conditioning problem in subspace identifcation method is provided.The subspace identifcation technique presents a satisfactory robustness in the parameter estimation of process model which performs control.As a frst step,the main geometric and mathematical tools used in subspace identifcation are briefly presented.In the second step,the problem of analyzing ill-conditioning matrices in the subspace identifcation method is considered.To illustrate this situation,a simulation study of an example is introduced to show the ill-conditioning in subspace identifcation.Algorithms numerical subspace state space system identifcation(N4SID)and multivariable output error state space model identifcation(MOESP)are considered to study,the parameters estimation while using the induction motor model,in simulation(Matlab environment).Finally,we show the inadequacy of the oblique projection and validate the efectiveness of the orthogonal projection approach which is needed in ill-conditioning;a real application dealing with induction motor parameters estimation has been experimented.The obtained results proved that the algorithm based on orthogonal projection MOESP,overcomes the situation of ill-conditioning in the Hankel s block,and thereby improving the estimation of parameters.     

Subspace identification by data orthogonalization and model decoupling

It has been observed that identification of state-space models with inputs may lead to unreliable results in certain experimental conditions even when the input signal excites well within the bandwidth of the system. This may be due to ill-conditioning of the identification problem, which occurs when the state space and the future input space are nearly parallel.We have in particular shown in the companion papers (Automatica 40(4) (2004) 575; Automatica 40(4) (2004) 677) that, under these circumstances, subspace methods operating on input-output data may be ill-conditioned, quite independently of the particular algorithm which is used. In this paper, we indicate that the cause of ill-conditioning can sometimes be cured by using orthogonalized data and by recasting the model into a certain natural block-decoupled form consisting of a “deterministic” and a “stochastic” subsystem. The natural subspace algorithm for the identification of the deterministic subsystem is then a weighted version of the PI-MOESP method of Verhaegen and Dewilde (Int. J. Control 56 (1993) 1187-1211). The analysis shows that, under certain conditions, methods based on the block-decoupled parametrization and orthogonal decomposition of the input-output data, perform better than traditional joint-model-based methods in the circumstance of nearly parallel regressors.     

基于子空间方法的非均匀周期刷新和采样系统辨识

针对非均匀周期刷新和采样系统的建模问题, 对于含有提升变量的状态空间模型, 提出基于子空间技术的辨识方法. 首先, 通过系统的采样数据建立由Hankel 矩阵组成的扩展状态空间方程; 然后, 利用斜交投影的原理、方法和奇异值分解, 通过子空间辨识算法确定增广观测矩阵和状态向量, 通过最小二乘方法确定模型的参数矩阵; 最后, 通过仿真实例表明了所提出算法的有效性.

     

Subspace identification of continuous time models for process fault detection and isolation

This paper proposes a novel subspace approach towards identification of optimal residual models for process fault detection and isolation (PFDI) in a multivariate continuous-time system. We formulate the problem in terms of the state space model of the continuous-time system. The motivation for such a formulation is that the fault gain matrix, which links the process faults to the state variables of the system under consideration, is always available no matter how the faults vary with time. However, in the discrete-time state space model, the fault gain matrix is only available when the faults follow some known function of time within each sampling interval. To isolate faults, the fault gain matrix is essential. We develop subspace algorithms in the continuous-time domain to directly identify the residual models from sampled noisy data without separate identification of the system matrices. Furthermore, the proposed approach can also be extended towards the identification of the system matrices if they are needed. The newly proposed approach is applied to a simulated four-tank system, where a small leak from any tank is successfully detected and isolated. To make a comparison, we also apply the discrete time residual models to the tank system for detection and isolation of leaks. It is demonstrated that the continuous-time PFDI approach is practical and has better performance than the discrete-time PFDI approach.     

分数阶时滞系统的频域子空间辨识

研究了利用频率响应数据辨识分数阶时滞系统子空间模型的问题,给出了一种差分进化算法与频域子空间方法相结合的辨识算法.利用差分进化算法搜索最优分数微分阶次和时滞参数,而对于固定的分数微分阶次和时滞,则采用分数阶频域子空间辨识方法得到状态空间模型.通过仿真算例验证了该算法的有效性.     

A CLOSED‐LOOP SUBSPACE IDENTIFICATION METHOD FOR CONTINUOUS‐TIME SYSTEMS BASED ON δ‐OPERATOR MODEL

This paper is concerned with a subspace identification of a continuous‐time plant operating in closed‐loop in the framework of the joint input‐output approach. The main procedure consists of two steps. Firstly, the dual‐Youla parametrization of the plant is used for obtaining an equivalent open‐loop problem to the original closed‐loop identification problem. Then, a δ‐operator based IV‐MOESP type subspace identification algorithm is developed to estimate the state space model for the joint input‐output process, whereby a higher‐order state space model of the plant is obtained by an algebraic operation. Subsequently, a model reduction procedure is employed to derive a lower‐order plant model removing irrelevant modes from the higher order model. Simulation results by using numerical and chemical plant models demonstrate the feasibility of the proposed method.     

UD分解与偏差补偿结合用于变量带误差模型辨识

本文提出一种基于UD(upper-diagonal)分解与偏差补偿结合的辨识方法,用于变量带误差(errors-in-variables,EIV)模型辨识.考虑单输入单输出(single input and single output,SISO)线性动态系统,当输入和输出含有零均值、方差未知的高斯测量白噪声时,该类系统的模型参数估计是一种典型的EIV模型辨识问题.为了获得这种EIV模型参数的无偏估计,本文先推导出最小二乘模型参数估计偏差量与输入输出噪声方差以及最小二乘损失函数与输入输出噪声方差的关系,然后采用UD分解方法递推获得模型参数估计值,再利用输入输出噪声方差估计值补偿模型参数估计偏差,以此获得模型参数的无偏估计.本文还讨论了算法实现过程中遇到的一些问题及修补方法,并通过仿真例验证了所提辨识方法的有效性.     

一种新的基于线性EIV模型的鲁棒估计算法

提出了一种新的基于线性EIV模型的鲁棒估计算法——鲁棒扩充算法．该算法从结构化数据区域出发，逐渐扩充模型数据集，并不断更新模型参数的估计，直至找到所有模型数据．在每次迭代中，使用C-Step方法对集合进行调整，从而保证了算法的鲁棒性．同时，提出了关于粗差数据和结构化数据分布的结构化密度假设，结合Mean Shift算法，完成对算法的初始位置选取．仿真结果表明，该算法可以有效地处理含有多个结构和大量离群样本的混杂数据，与现有算法相比，具有更强的鲁棒性和更高的精度．     

An outlier mining algorithm based on constrained concept lattice

Traditional outlier mining methods identify outliers from a global point of view. These methods are inefficient to find locally biased data points (outliers) in low dimensional subspaces. Constrained concept lattices can be used as an effective formal tool for data analysis because constrained concept lattices have the characteristics of high constructing efficiency, practicability and pertinency. In this paper, we propose an outlier mining algorithm that treats the intent of any constrained concept lattice node as a subspace. We introduce sparsity and density coefficients to measure outliers in low dimensional subspaces. The intent of any constrained concept lattice node is regarded as a subspace, and sparsity subspaces are searched by traversing the constrained concept lattice according to a sparsity coefficient threshold. If the intent of any father node of the sparsity subspace is a density subspace according to a density coefficient threshold, then objects contained in the extent of the sparsity subspace node are considered as bias data points or outliers. Our experimental results show that the proposed algorithm performs very well for high red-shift spectral data sets.     

An extended EM algorithm for subspace clustering

Clustering high dimensional data has become a challenge in data mining due to the curse of dimensionality. To solve this problem, subspace clustering has been defined as an extension of traditional clustering that seeks to find clusters in subspaces spanned by different combinations of dimensions within a dataset. This paper presents a new subspace clustering algorithm that calculates the local feature weights automatically in an EM-based clustering process. In the algorithm, the features are locally weighted by using a new unsupervised weighting method, as a means to minimize a proposed clustering criterion that takes into account both the average intra-clusters compactness and the average inter-clusters separation for subspace clustering. For the purposes of capturing accurate subspace information, an additional outlier detection process is presented to identify the possible local outliers of subspace clusters, and is embedded between the E-step and M-step of the algorithm. The method has been evaluated in clustering real-world gene expression data and high dimensional artificial data with outliers, and the experimental results have shown its effectiveness.     

Closed-loop subspace identification using the parity space

It is known that many subspace algorithms give biased estimates for closed-loop data due to the existence of feedback. In this paper we present a new subspace identification method using the parity space employed in fault detection in the past. The basic algorithm, known as subspace identification method via principal component analysis (SIMPCA), gives consistent estimation of the deterministic part and stochastic part of the system under closed loop. Column weighting for SIMPCA is introduced which shows improved efficiency/accuracy. A simulation example is given to illustrate the performance of the proposed algorithm in closed-loop identification and the effect of column weighting.     

多输入多输出变量带误差模型的最坏情况频域辨识

本文将单输入单输出(SISO)变量带误差(EIV)模型的频域最坏情况辨识方法推广应用于多输入多输出 (MIMO)情况. 类似于SISO情况, 多输入多输出变量带误差(MIMO EIV)模型的辨识模型集合由估计的系统名义模型及 其最坏情况误差界描述. 所估计的系统名义模型表征为正规右图符号, 其最坏情况误差界具有可能的更少保守性, 可利 用EIV 模型的先验信息和后验信息由v-gap度量量化得到. 因此, 这种模型集合非常适合于后期利用Vinnicombe提出 的H1回路成形法设计鲁棒控制器. 最后, 利用一数值仿真实例验证所提出辨识方法的有效性.