Subspace System Identification Algorithm Research Articles

Observer Kalman Filter Identification for Output-Only Systems Using Interactive Structural Modal Identification Toolsuite

AbstractSeveral modal identification techniques have been developed in the past few decades, and their use is rapidly expanding due to new focus on the instrumentation of major structures. This paper focuses on the expansion of the eigenvalue realization algorithm (ERA)–observer Kalman filter identification (OKID) to identify modal parameters of output-only systems (OO) by splitting the state-space model into deterministic and stochastic subsystems (ERA-OKID-OO). The performance is then compared with other output-only identification methods in terms of the level of accuracy and efficiency. A newly developed software package [Structural Modal Identification Toolsuite (SMIT)] is used to provide a uniform and convenient way of utilizing several system identification (SID) methods, including variations of ERA, auto-regressive with exogenous terms (ARX) models, system realization using information matrix (SRIM), and numerical algorithms for subspace state space system identification (N4SID). The main purpose o...

Diagnostics of data outliers using subspace identification

We propose a diagnostics technique for the state space model fitting formed by deleting observations from the data and measuring the change in the estimates of the parameters. A method is proposed for distinguishing an observational outlier from an innovational one. The presented subspace system identification algorithm is robust and less sensitive to outliers. We give a numerical result to show effectiveness of the proposed method.

Power System Critical Eigenvalue Estimation Using Flexibility of Subspace System Identification

In this research, a new algorithm and proposition are suggested to estimate critical eigenvalues of power systems. The algorithm is based on the characteristics of five renowned deterministic subspace system identification algorithms: the multivariable output error state space, past output multivariable output error state space, numerical algorithm for subspace identification, canonical correlation analysis, and orthogonal decomposition technique. Suggesting a novel proposition, the mentioned subspace system identification algorithms are reformulated in a general realization framework in order to clarify their similarities and differences. Numerous implementation characteristics of the algorithms are explored in order to automate system order selection and state matrix estimation. Performance of the proposed algorithm, when applied for eigenvalue estimation, is investigated in the concern of estimation accuracy and computational loads. Different aspects of the proposition and the algorithm are explored through conducting several simulations. The simulations may be carried out using a simple test power system (two-area four-machine system), an average system (New England 10-machine system) and a complicated power system benchmark (IEEE 50-machine system). In accordance with the test results, it is claimed that among the investigated subspace system identification algorithms, the past output multivariable output error state space has very good performance for power system eigenvalue, damping ratio, and frequency estimation when concerning estimation accuracy and CPU processing time.

Subspace algorithms for identifying separable-in-denominator two-dimensional systems with deterministic inputs

The class of subspace system identification algorithms is used here to derive new identification algorithms for 2-D causal, recursive, and separable-in-denominator (CRSD) state space systems in the Roesser form. The algorithms take a known deterministic input-output pair of 2-D signals and compute the system order (n) and system parameter matrices {A,B,C,D}. Since the CRSD model can be treated as two 1-D systems, the proposed algorithms first separate the vertical component from the state and output equations and then formulate a set of 1-D horizontal subspace equations. The solution to the horizontal subproblem contains all the information necessary to compute (n) and {A,B,C,D}. Four algorithms are presented for the identification of CRSD models directly from input-output data: an intersection algorithm, (N4SID), (MOESP), and (CCA). The intersection algorithm is distinguished from the rest in that it computes the state sequences, as well as the system parameters, whereas N4SID, MOESP, and CCA differ primarily in the way they compute the system parameter matrices {A1,C1}. The advantage of the intersection algorithm is that the identified model is in balanced coordinates, thus ideally suited for 2-D model reduction. However, it is computationally more expensive than the other algorithms. A comparison of all algorithms is presented.

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.

Design and Subspace System Identification of an Ex Vivo Vascular Perfusion System

Numerical algorithms for subspace system identification (N4SID) are a powerful tool for generating the state space (SS) representation of any system. The purpose of this work was to use N4SID to generate SS models of the flowrate and pressure generation within an ex vivo vascular perfusion system (EVPS). Accurate SS models were generated and converted to transfer functions (TFs) to be used for proportional integral and derivative (PID) controller design. By prescribing the pressure and flowrate inputs to the pumping components within the EVPS and measuring the resulting pressure and flowrate in the system,_four TFs were estimated;_two for a flowrate controller (H(RP,f) and H(RPP,f)) and two for a pressure controller (H(RP,p) and H(RPP,p)). In each controller,_one TF represents a roller pump (H(RP,f) and H(RP,p)),_and the other represents a roller pump and piston in series (H(RPP,f) and H(RPP,p)). Experiments to generate the four TFs were repeated five times (N=5) from which average TFs were calculated. The average model fits, computed as the percentage of the output variation (to_the_prescribed_inputs) reproduced by the model, were 94.93+/-1.05% for H(RP,p), 81.29+/-0.20% for H(RPP,p), 94.45+/-0.73% for H(RP,f), and 77.12+/-0.36% for H(RPP,f). The simulated step, impulse, and frequency responses indicate that the EVPS is a stable system and can respond to signals containing power of up to 70_Hz.

Identification of errors-in-variables state space models with observation outliers based on minimum covariance determinant

In this paper, a subspace system identification algorithm for the errors-in-variables (EIV) state space models subject to observation noise with outliers has been developed. By using the minimum covariance determinant (MCD) estimator, the outliers have been identified and deleted. Then the classical EIV subspace system identification algorithms have been applied to estimate the parameters of the state space models. In order to solve the MCD problem for the EIV state space models, a random search algorithm has been proposed. A Monte-Carlo simulation results demonstrate the effectiveness of the proposed algorithm.

Subspace identification of linear parameter-varying systems with innovation-type noise models driven by general inputs and a measurable white noise time-varying parameter vector

In this article, we introduce an iterative subspace system identification algorithm for MIMO linear parameter-varying systems with innovation-type noise models driven by general inputs and a measurable white noise time-varying parameter vector. The new algorithm is based on a convergent sequence of linear deterministic–stochastic state-space approximations, thus considered a Picard-based method. Such methods have proven to be convergent for the bilinear state-space system identification problem. Their greatest strength lies on the dimensions of the data matrices that are comparable to those of a linear subspace algorithm, thus avoiding the curse of dimensionality.

Identification of errors-in-variables models using the EM algorithm

This paper advocates a new subspace system identification algorithm for the errors-in-variables (EIV) state space model via the EM algorithm. To initialize the EM algorithm an initial estimate is obtained by the errors-in-variables subspace system identification method: EIV-MOESP (Chou et al. [1997]) and EIV-N4SID (Gustafsson [2001]). The EM algorithm is an algorithm to compute the maximum value for the likelihood function that is consists of two steps; namely the E- and M-steps. The E- and M-steps in the EM algorithm are calculated by computing the conditional expectation under the assumption that the input-output data is completely observed. Numerical example shows that the EM algorithm can monotonically improve the initial estimates obtained by subspace identification methods.

Identification of structural surfaces' positions on an F/A-18 using the subspace identification method from flight flutter tests

In the current paper, a linear state-space mathematical model, identified from flight flutter tests is presented, to simulate the aeroelastic deflections of specific structural parts of the NASA F/A-18 Active Aeroelastic Wing research aircraft. The flight flutter tests were performed in steady-level flight with Schroeder frequency excitation induced on the aircraft ailerons by an on-board excitation system activated by the pilot. The results of the flight flutter tests were used to generate an aeroelastic model in which the deflections of the specific aircraft surfaces are functions of the control inputs combined with the deflections of other aircraft surfaces. The F/A-18 linear model is conceived as nine third-order multiple input-single output (MISO) models. Each model has nine inputs and one output. The nine inputs are the differential ailerons deflection and the deflections of all the other parts of the aircraft. The output of each model is the structural deflection of a given aircraft structure. The model's parameters are estimated with the subspace system identification algorithm, an efficient non-iterative algorithm that computes the system matrices directly from the input and output data. The model's quality is evaluated by calculating the fit and correlation coefficients between the model's outputs and the outputs from flight flutter test data. Although the fit coefficient results are very good - between 89 and 99 per cent - the correlation coefficient method gave the best results (nearly 100 per cent). Finally, resampled inputs were used to validate the F/A-18 model's robustness. The model's aircraft structure was validated for flutter flight tests at different Mach numbers and altitudes. The estimated linear model fits the flight flutter test data very well. The subspace method is therefore very convernvenient for model identification from flight flutter tests.

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.

Finite Element Model Updating for Damage Localisation and Quantification Using Experimental Modal Data

In this contribution we present a validation of an identification procedure and a modeling method with regard to detection, localisation and quantification of damage in a structure. Vibration measurements of an excited experimental structure are used as input for a stochastic subspace system identification algorithm. The identified experimental modal data (eigenvalues and mode shapes) serve to update the underlying finite element model. The experimental setup consists of a cantilever beam and an additional equipment to damage the structure locally and progressively. In contrast to earlier contributions the evolution of damage is quantified in order to estimate the lifetime of the structure.

Enhance Accuracy in Pole Identification of System by Wavelet Transform De-Noising

Based on the multiscale character of wavelet transform, the method of wavelet transform de-noising (WTDN) is introduced to improve the signal-noise ratio (SNR) of sampled data and to enhance the accuracy of pole identification of system with N4SID (numerical algorithms for subspace state system identification) method. The WTDN method does not require extra limit of the frequency range for the processed signal, and does not need a prior estimate of impulse response for identified system. So it is especially suitable to de-noise the wide-band signal and the noisy impulse response of the blind system. The results of numerical simulation indicate that the WTDN method is reliable. The WTDN method is used to process the sampled data from actual preamplifier coupled to a gas detector. The experimental results show that the WTDN method improves effectively the SNR of sampled data, which can help to enhance the accuracy in pole identification of system with the N4SID method.

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.

COMPARISON STUDY OF SUBSPACE IDENTIFICATION METHODS APPLIED TO FLEXIBLE STRUCTURES

In the past few years, various time domain methods for identifying dynamic models of mechanical structures from modal experimental data have appeared. Much attention has been given recently to so-called subspace methods for identifying state space models. This paper presents a detailed comparison study of these subspace identification methods: the eigensystem realisation algorithm with observer/Kalman filter Markov parameters computed from input/output data (ERA/OM), the robust version of the numerical algorithm for subspace system identification (N4SID), and a refined version of the past outputs scheme of the multiple-output error state space (MOESP) family of algorithms. The comparison is performed by simulating experimental data using the five mode reduced model of the NASA Mini-Mast structure. The general conclusion is that for the case of white noise excitations as well as coloured noise excitations, the N4SID/MOESP algorithms perform equally well but give better results (improved transfer function estimates, improved estimates of the output) compared to the ERA/OM algorithm. The key computational step in the three algorithms is the approximation of the extended observability matrix of the system to be identified, for N4SID/MOESP, or of the observer for the system to be identified, for the ERA/OM. Furthermore, the three algorithms only require the specification of one dimensioning parameter.

A unifying theorem for three subspace system identification algorithms

The aim of this paper is to indicate and explore the similarities between three different subspace algorithms for the identification of combined deterministic-stochastic systems. The similarities between these algorithms have been obscured, due to different notations and backgrounds. It is shown that all three algorithms are special cases of one unifying theorem. The comparison also reveals that the three algorithms use exactly the same subspace to determine the order and the extended observability matrix, but that the weighting matrix, used to calculate a basis for the column space of the observability matrix is different in the three cases.

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.