Subspace State Space System Identification Research Articles

Identification of Bilinear Systems With White Noise Inputs: An Iterative Deterministic-Stochastic Subspace Approach

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> In this technical brief, a new subspace state space system identification algorithm for multi-input multi-output bilinear systems driven by white noise inputs is introduced. The new algorithm is based on a uniformly convergent Picard sequence of linear deterministic-stochastic state space subsystems which are easily identifiable by any linear deterministic-stochastic subspace algorithm such as MOESP, N4SID, CVA, or CCA. The key to the proposed algorithm is the fact that the bilinear term is a second-order white noise process. Using a standard linear Kalman filter model, the bilinear term can be estimated and combined with the system inputs at each iteration, thus leading to a linear system with extended inputs of dimension <formula formulatype="inline"><tex Notation="TeX">$m(n+1)$</tex></formula>, where <formula formulatype="inline"> <tex Notation="TeX">$n$</tex></formula> is the system order and <formula formulatype="inline"> <tex Notation="TeX">$m$</tex></formula> is the dimension of the inputs. It is also shown that the model parameters obtained with the new algorithm converge to those of the true bilinear model. Moreover, the proposed algorithm has the same consistency conditions as the linear subspace identification algorithms when <formula formulatype="inline"><tex Notation="TeX">$i\rightarrow\infty$</tex> </formula>, where <formula formulatype="inline"><tex Notation="TeX">$i$</tex> </formula> is the number of block rows in the past/future block Hankel data matrices. Typical bilinear subspace identification algorithms available in the literature cannot handle large values of <formula formulatype="inline"> <tex Notation="TeX">$i$</tex></formula>, thus leading to biased parameter estimates. Unlike existing bilinear subspace identification algorithms whose row dimensions in the data matrices grow exponentially, and hence suffer from the “curse of dimensionality,” in the proposed algorithm the dimensions of the data matrices are comparable to those of a linear subspace identification algorithm. A case study is presented with data from a heat exchanger experiment. </para>

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.

Rank reduction and volume minimization approach to state-space subspace system identification

In this paper we consider the reduced rank regression problem minrank L=n,L3 det(Yα - LPβ - L3Uα)(Yα - LPβ - L3Uα)T solved by maximum-likelihood-inspired state-space subspace system identification ...

Initial Results in Power System Identification From Injected Probing Signals Using a Subspace Method

In this paper, the authors use the Numerical algorithm for Subspace State Space System IDentification (N4SID) to extract dynamic parameters from phasor measurements collected on the western North American Power Grid. The data were obtained during tests on June 7, 2000, and they represent wide area response to several kinds of probing signals, including low-level pseudo-random noise (LLPRN) and single-mode square wave (SMSW) injected at the Celilo terminal of the Pacific HVDC Intertie (PDCI). An identified model is validated using a cross validation method. Also, the obtained electromechanical modes are compared with the results from Prony analysis of a ringdown and with signal analysis of ambient data measured under similar operating conditions. The consistent results show that methods in this class can be highly effective, even when the probing signal is small

Identification, Model Updating, and Response Prediction of an Instrumented 15-Story Steel-Frame Building

Identification of the modal properties of the UCLA Factor Building, a 15-story steel moment-resisting frame, is performed using low-amplitude earthquake and ambient vibration data. The numerical algorithm for subspace state-space system identification is employed to identify the structural frequencies, damping ratios, and mode shapes corresponding to the first nine modes. The frequencies and mode shapes identified based on the data recorded during the 2004 Parkfield earthquake ( Mw=6.0) are used to update a three-dimensional finite element model of the building to improve correlation between analytical and identified modal properties and responses. A linear dynamic analysis of the updated model excited by the 1994 Northridge earthquake is performed to assess the likelihood of structural damage.

Rank reduction and volume minimization approach to state-space subspace system identification

In this paper we consider the reduced rank regression problem min rank L ¯ = n , L 3 det ( Y α - L ¯ P β - L 3 U α ) ( Y α - L ¯ P β - L 3 U α ) T solved by maximum-likelihood-inspired state-space subspace system identification algorithms. We conclude that the determinant criterion is, due to potential rank-deficiencies, not general enough to handle all problem instances. The main part of the paper analyzes the structure of the reduced rank minimization problem and identifies signal properties in terms of geometrical concepts. A more general minimization criterion is considered, rank reduction followed by volume minimization. A numerically sound algorithm for minimizing this criterion is presented and validated on both simulated and experimental data.

System Order Determination by Using LDL&lt;sup&gt;T&lt;/sup&gt; Decomposition and Information Criteria

In this paper, we present a new order determination method of applying the LDLT decomposition and information criteria. A statistical test is available for the system order determination, and is based on the relation between the system order and the rank of the correlation matrix of the Hankel matrix consists of input-output data. We apply LDLT decomposition and Akaike's Information Criterion (AIC) [3] to determine the rank, so that we can determine the system order by the minimum AIC estimate (MAICE). AIC requires the heavy computation because the maximum likelihood estimates should he obtained by solving the nonlinear optimization problem. Thus we apply a subspace method, e.g., N4SID (Numerical algorithms for Subspace State Space System IDentification) [4], to simplify the computation of the log-likelihood functions. But in subspace methods, stability of the obtained model cannot he guaranteed. When the model obtained is unstable, the maximum likelihood estimates cannot be obtained. So we apply the method [6] that can guarantee the stability of the obtained model to MAICE.

Recursive 4SID algorithms using gradient type subspace tracking

Sometimes we obtain some prior information about a system to be identified, e.g., the order, model structure etc. In this paper, we consider the case where the order of a MIMO system to be identified is a priori known. Recursive subspace state-space system identification algorithms presented here are based on the gradient type subspace tracking method used in the array signal processing. The algorithms enable us to estimate directly the subspace spanned by the column vectors of the extended observability matrix of the system to be identified without performing the singular value decomposition. Also, a new convergence proof of the gradient type subspace tracking is given in this paper. Under the condition of a step size between 0 and 1, we prove the convergence property of the recursive equation of the gradient type subspace tracking. A numerical example illustrates that our algorithm is more robust with respect to the choice of the initial values than the corresponding PAST one.

Integration of multivariate SPM and FDD by parity space technique for a food pasteurization process

Multivariate statistical process monitoring (MSPM), contribution plots, and parity space fault diagnosis (FD) techniques are used to detect abnormal operation of dynamic processes and diagnose sensor and actuator faults. The methods are illustrated by monitoring the critical control points (CCP) and diagnosing causes of abnormal operation of a pilot pasteurization plant. An empirical model of the process is developed by using subspace state space system identification methods and normal process data. The process data collected under the influence of different magnitude and duration of faults in sensors and actuators are used to validate the MSPM and FD techniques. T 2 and squared prediction error (SPE N ) charts are used as MSPM charts. A parity space technique for dynamic stochastic systems and dynamic trends in contribution plots of T 2 and SPE N statistics are used for FD. The detection and FD by these techniques show significant improvements over univariate methods.

Modeling for Tandem Cold Mill Using Model Identification

In this paper, a new linear model of each stand in a five-stand TCM(tandem cold mill) is constructed by using a model identification method. The model identification for each stand is achieved by considering an interstand interference, natural dynamic characteristics of the TCM. In the model identification, first input and output data for model identification are generated by a nonlinear simulator of the TCM, and then the linear model of each stand from these data is identified by using least square method and N4SID(numerical algorithms for subspace state space system identification) method. Through the comparison of the identified model with the existing Taylor linearized model of the TCM, it is shown that the identified model also has the dynamic characteristics of the TCM.

On State Estimation of Unknown Systems Based on Subspace State-Space System Identification

In this paper, we consider a state estimation problem of a MIMO model with which a MIMO system is identified by using Subspace State-Space System Identification (4SID). This problem is very important because the knowledge of the present state of a model can be a clue to design an adaptive control law for a system. First, we review the recursive 4SID algorithm based on the matrix inversion lemma shown in our previous paper. Then, we propose a scheme for estimating the state of models obtained by the recursive 4SID algorithm. A major advantage of 4SID schemes is that we can select the order of an identified model as the number of dominant singular values of a certain matrix which consists of I/O data. However, it becomes a bottleneck on considering the recursive state estimation because the dimension of state space changes from one time to another during the identification proceeds. To circumvent this problem, we introduce a new kind of state (named"quasi-state") which is still meaningful over different update steps. Finally, we illustrate the usefulness of our idea by considering a tracking control problem.

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.

部分空間同定法の逐次化について

部分空間同定法の逐次化について

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.

Joint Design of Model-Subspace based State-Space Identification and Control

Recently, the importance of the joint design of identification and control has been recognized, and several controller design methods based on the iteration of identification and controller re-design have been proposed. In these methods, the frequency weighted identification plays an important role. On the other hand, as a powerful identification method, subspace state-space system identification (4SID) method has been proposed. However, it is difficult to use the frequency weight in conventional 4SID methods. In this paper, we propose a new frequency weighted subspace state-space system identification method for the joint design of identification and control. First, we give a new 4SID method using not only the input-output data of the system but also the nominal model data. Second, we show how to introduce the frequency weight to our identification method, which is relevant to the cost function for control. And third, we evaluate the effectiveness of the proposed method by a numerical example using an inverted pendulum system.

Subspace State Space System Identification using Prior Knowledge

In this paper, we propose a subspace state space system identification method using prior knowledge. In this method, we give an algorithm to identify the partial state which corresponds to unknown modes of the system. This scheme is effective when some system poles cannot be identified because of the measurement noise or a high gain feedback controller. The effectiveness of the proposed method is evaluated by numerical examples including the comparison with the conventional 4SID method

A linear regression approach to state-space subspace system identification

Recently, state-space subspace system identification (4SID) has been suggested as an alternative to the more traditional prediction error system identification. The aim of this paper is to analyze the connections between these two different approaches to system identification. The conclusion is that 4SID can be viewed as a linear regression multistep-ahead prediction error method with certain rank constraints. This allows us to describe 4SID methods within the standard framework of system identification and linear regression estimation. For example, this observation is used to compare different cost-functions which occur rather implicitly in the ordinary framework of 4SID. From the cost-functions, estimates of the extended observability matrix are derived and related to previous work. Based on the estimates of the observability matrix, the asymptotic properties of two pole estimators, namely the shift invariance method and a weighted subspace fitting method, are analyzed. Expressions for the asymptotic variances of the pole estimation error are given. From these expressions, difficulties in choosing userspecified parameters are pointed out. Furthermore, it is found that a row-weighting in the subspace estimation step does not affect the pole estimation error asymptotically.

N4SID: Subspace algorithms for the identification of combined deterministic-stochastic systems

Recently a great deal of attention has been given to numerical algorithms for subspace state space system identification (N4SID). In this paper, we derive two new N4SID algorithms to identify mixed deterministic-stochastic systems. Both algorithms determine state sequences through the projection of input and output data. These state sequences are shown to be outputs of non-steady state Kalman filter banks. From these it is easy to determine the state space system matrices. The N4SID algorithms are always convergent (non-iterative) and numerically stable since they only make use of QR and Singular Value Decompositions. Both N4SID algorithms are similar, but the second one trades off accuracy for simplicity. These new algorithms are compared with existing subspace algorithms in theory and in practice.