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.     

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 state space system identification for industrial processes

We give a general overview of the state-of-the-art in subspace system identification methods. We have restricted ourselves to the most important ideas and developments since the methods appeared in the late eighties. First, the basics of linear subspace identification are summarized. Different algorithms one finds in literature (such as N4SID, IV-4SID, MOESP, CVA) are discussed and put into a unifying framework. Further, a comparison between subspace identification and prediction error methods is made on the basis of computational complexity and precision of the methods by applying them on 10 industrial data sets.     

Numerical conditioning and asymptotic variance of subspace estimates

New formulas for the asymptotic variance of the parameter estimates in subspace identification, show that the accuracy of the parameter estimates depends on certain indices of ‘near collinearity’ of the state and future input subspaces of the system to be identified. This complements the numerical conditioning analysis of subspace methods presented in the companion paper (On the ill-conditioning of subspace identification with inputs, Automatica, doi:10.1016/j.automatica.2003.11.009).     

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.     

Identification of the Tennessee Eastman challenge process with subspace methods

The Tennessee Eastman challenge process is a realistic simulation of a chemical process that has been widely used in process control studies. In this case study, several identification methods are examined and used to develop MIMO models that contain seven inputs and ten outputs. ARX and finite impulse response models are identified using reduced-rank regression techniques (PLS and CCR) and state-space models identified with prediction error methods and subspace algorithms. For a variety of reasons, the only successful models are the state-space models produced by two popular subspace algorithms, N4SID and canonical variate analysis (CVA). The CVA model is the most accurate. Important issues for identifying the Tennessee Eastman challenge process and comparisons between the subspace algorithms are also discussed.     

Subspace identification of Hammerstein systems using least squares support vector machines

This paper presents a method for the identification of multiple-input-multiple-output (MIMO) Hammerstein systems for the goal of prediction. The method extends the numerical algorithms for subspace state space system identification (N4SID), mainly by rewriting the oblique projection in the N4SID algorithm as a set of componentwise least squares support vector machines (LS-SVMs) regression problems. The linear model and static nonlinearities follow from a low-rank approximation of a matrix obtained from this regression problem.     

Asymptotic variance of subspace methods by data orthogonalization and model decoupling: a comparative analysis

This is a companion of the paper Chiuso and Picci (2004d) where we do asymptotic error analysis of a weighted PI-MOESP type method and compare accuracy with respect to estimates obtained by customary “joint” subspace methods. The analysis shows that, under certain conditions, methods based on orthogonal decomposition of the input-output data and block-decoupled parametrization perform better than traditional joint-model based methods in the circumstance of nearly parallel regressors.     

工业过程的子空间模型辨识

子空间模型辨识方法(SMI)是一类新兴的直接估计线性状态空间模型的黑箱建模方法,近年来获得了广泛关注.和传统的线性建模方法相比,SMI的优势不仅在于算法本身的简单可靠,也在于它的状态空间表达.本文首先简要介绍了SMI的基本思想以及3种基本算法(N4SID,MOESP,CVA).然后将这类方法应用于一个实际的工业过程建模,同时对3种SMI基本算法和一种传统辨识算法—预测误差方法(PEM)进行了研究对比.     

Improved closed-loop subspace identification with prior information

It has been proven that combining open-loop subspace identification with prior information can promote the accuracy of obtaining state-space models. In this study, prior information is exploited to improve the accuracy of closed-loop subspace identification. The proposed approach initially removes the correlation between future input and past innovation, a significant obstacle in closed-loop subspace identification method. Then, each row of the extended subspace matrix equation is considered an optimal multi-step ahead predictor and prior information is expressed in the form of equality constraints. The constrained least squares method is used to obtain improved results, so that the accuracy of the closed-loop subspace can be enhanced. Simulation examples are provided to demonstrate the effectiveness of the proposed algorithm.     

A RECURSIVE 4SID FROM THE INPUT-OUTPUT POINT OF VIEW

In this paper, recursive algorithms of subspace state-space system identification (4SID) for multiple-input, multiple-output (MIMO), finite dimensional, linear time-invariant (FDLTI) systems are proposed. These algorithms are derived based on the Matrix Inversion Lemma. The investigation of our algorithms clarifies that a series of 4SID is the extension of the classical least square method to identification for multivariable systems, and also that our algorithms are the direct extension of the recursive least square algorithm to such ones. For PO-MOESP (the ordinary MOESP scheme with instrumental variables constructed from Past input and Output measurements), we show the mechanism of how the effect of the process and measurement noises is eliminated asymptotically by a projection related to the input and regressor matrices. “MOESP” is an abbreviation for “the MIMO output-error state-space model identification.”     

The role of vector autoregressive modeling in predictor-based subspace identification

Subspace identification for closed loop systems has been recently studied by several authors. A class of new and consistent closed-loop subspace algorithms is based on identification of a predictor model, in a way similar as prediction error methods (PEM) do. Experimental evidence suggests that these methods have a behavior which is very close to PEM in certain examples. The asymptotical statistical properties of one of these methods have been studied recently allowing to show (i) its relation with CCA and (ii) that Cramér-Rao lower bound is not reached in general. Very little, however, is known concerning their relative performance.In this paper we shall discuss the link between these “predictor-based” methods; to this purpose we exploit the role which Vector Auto Regressive with eXogenous input models play in all these algorithms. The results of this paper provide a unifying framework under which all these algorithms can be viewed; also the link with VARX modeling have important implications as to computational complexity is concerned, leading to very computationally attractive implementations.We also hope that this framework, and in particular the relation with VARX modeling followed by model reduction will turn out to be useful in future developments of subspace identification, such as the quest for efficient procedures and the statistical analysis with finite-data.     

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.     

Fuzzy modeling and stable model predictive tracking control of large-scale power plants

This paper develops a stable model predictive tracking controller (SMPTC) for coordinated control of a large-scale power plant. First, a Takagi–Sugeno (TS) fuzzy model is established to approximate the behavior of the boiler–turbine coordinated control system (CCS) using fuzzy clustering and subspace identification (SID). Then, an SMPTC is designed based on the fuzzy model to track the power and pressure set-points while guaranteeing the input-to-state stability and the input constraints of the system. An output-based objective function is adopted for the proposed SMPTC so that the controller could be directly applicable for the data-driven model. Moreover, the effect of modeling mismatches and unknown plant variations has been overcome by the use of a disturbance term and steady-state target calculator (SSTC). Simulation results for a 600 MW power plant show that an off-set free tracking performance can be achieved over a wide range load variation.     

基于正交分解的递推子空间闭环辨识方法

为实现闭环系统在线辨识,提出递推正交分解闭环子空间辨识方法(RORT)。首先,根据闭环系统状态空间模型和数据间投影关系,构建确定-随机模型,并利用GIVENS变换实现投影向量的递推QR分解;然后,引入带遗忘因子的辨识算法,构建广义能观测矩阵的递推更新形式,以减少子空间辨识算法中QR分解和SVD分解的计算量;最后,针对某型号陀螺仪闭环系统进行实验。实验结果表明, RORT法的辨识拟合度高于91%,能够对陀螺仪闭环系统模型参数进行在线监测。     

基于主元分析的子空间辨识算法

子空间辨识算法作为一种优良的多变量系统辨识算法,最近在国内发展很快.但是现在国内介绍的大多数子空间辨识算法在变量有误差(errors-in-variable)时和闭环辨识时辨识结果却是有偏的,这是因为大多数子空间辨识算法都假设输入变量是没有噪声及辨识算法中存在的一个投影过程.文中介绍了一种新的子空间辨识算法,这种算法利用主元分析(PCA)来获取系统矩阵,避免了其他算法中的投影过程,因此该算法在闭环辨识和变量有误差(errors-in-variable)的情况下,辨识结果也是无偏的.最后给出一个仿真例子说明这种辨识算法的辨识效果良好.     

Subspace identification methods for Hammerstein systems: rank constraint and dimension problem

This article studies the subspace identification methods (SIMs) for Hammerstein systems with major focus on a rank constraint and the related dimension problem. We analyse the effects of the rank constraint on the three steps of a unifying framework for SIMs: the rank constraint has no effect on the first two steps, but does so on the third step. If the rank constraint is ignored, as in the existing over-parametrised method (OPM) for Hammerstein system identification, the optimality of the resulting estimate can still be established. Even so, the OPM may suffer from the dimension problem resulting in a low numerical efficiency. To resolve the dimension problem, we propose a new subspace-based method, named as the least-parametrised method (LPM), for identification of Hammerstein systems with non-coupling input nonlinearities. Simulation results are provided to demonstrate the effectiveness of the LPM, and show the necessity of considering the rank constraint to improve the numerical efficiency.     

Consistency of noise covariance estimation in joint input–output closed-loop subspace identification with application in LQG benchmarking

The LQG trade-off curve has been used as a benchmark for control loop performance assessment. The subspace approach to estimating the LQG benchmark has been proposed in the literature which requires certain intermediate matrices in subspace identification as well as the covariance matrix of the noise. It is shown in this paper that many existing closed-loop identification methods do not give a consistent estimate of the noise covariance matrix. As a result, we propose an alternative subspace formulation for the joint input–output closed-loop identification for which the consistency of the required subspace matrices and noise covariance is guaranteed. Simulation studies and experimental results are provided to demonstrate the utility of the proposed method.     

Continuous-time System Identification with Nuclear Norm Minimization and GPMF-based Subspace Method

To improve the accuracy and effectiveness of continuous-time (CT) system identification, this paper introduces a novel method that incorporates the nuclear norm minimization (NNM) with the generalized Poisson moment functional (GPMF) based subspace method. The GPMF algorithm provides a simple linear mapping for subspace identification without the timederivatives of the input and output measurements to avoid amplification of measurement noise, and the NNM is a heuristic convex relaxation of the rank minimization. The Hankel matrix with minimized nuclear norm is used to determine the model order and to avoid the over-parameterization in subspace identification method (SIM). Furthermore, the algorithm to solve the NNM problem in CT case is also deduced with alternating direction methods of multipliers (ADMM). Lastly, two numerical examples are presented to evaluate the performance of the proposed method and to show the advantages of the proposed method over the existing methods.      

基于随机分布理论的递推子空间辨识

在实际生产过程中,采用传统子空间辨识法建立的离线模型并不能有效准确地跟踪系统的动态变化;奇异值分解等线性代数工具虽然增加算法的数值鲁棒性,但也相应增加了子空间辨识的在线递推困难.为解决上述问题,本文针对连续时间系统提出基于随机分布理论的递推子空间辨识方法.首先,通过随机分布理论构建系统的连续随机分布函数,并利用微分计算获得系统等价的输入输出矩阵方程.然后,采用将输入输出数据矩阵"R"规模固定的方法,达到数据压缩的目的.最后,通过最小二乘法和残差分析法递推更新模型的系统矩阵和噪声强度直至达到辨识要求.仿真结果验证了所提方法的有效性和精确性.